Complexity Theoretic Mo del Theory and Algebra

Douglas Cenzer

Department of Mathematics

University of Florida Gainesville Fl

email cenzermathuedu

Jerey B Remmel

Department of Mathematics

University of California at San Diego

La Jolla CA

email jremmelucsdedu

November

Intro duction

In this pap er we will survey some recent results on complexity theoretic mo del

theory and algebra Essentially there are two ma jor themes in this work The

rst whichwe call complexity theoretic mo del theory deals with mo del ex

istence questions For example given a recursive mo del A is there there a

p olynomial time exp onential time p olynomial space etc mo del B whichis

isomorphic to A The second theme whichwe call complexity theoretic al

gebra xes a given p olynomial time structure and explores the prop erties of

that structure For example we can ask whether every p olynomial time ideal

of a given p olynomial time representation of the free Bo olean algebra can b e

extended to a maximal p olynomial time ideal In b oth cases one uses the rich

theory of recursive mo del theory and algebra as a reference but lo oks at resource

bounded versions of the results in those areas

It turns out that not only are there a number of contrasts b etween results

in recursive mo del theory and algebra and complexity theoretic mo del theory

and algebra but some new and interesting phenomena o ccur in the study of

complexity theoretic mo del theory and algebra That is there are results in

recursive mo del theory and algebra for which the natural complexity theoretic

analogue is true but requires a more delicate pro of which incorp orates the re

source b ounds There are also results in recursive mo del theory and algebra for

Dept of Commerce Agreement NANBH and NSF grant DMS

which the natural complexity theoretic analogue is false b ecause the pro of of the

recursive result uses the unb ounded resources allowed in recursive constructions

in a crucial wayHowever there are a numberofinteresting new phenomena

which arise due to the fact that not all innite p olynomial time sets are p oly

nomial time isomorphic or due to the fact that complexity theoretic results do

not relativize as is the case for most recursion theoretic results For example in

recursive mo del theory anytwo innite recursive sets are recursively isomorphic

so that one can restrict ones attention to mo dels whose universe is the set of

natural numb ers It is not the case that anytwo innite p olynomial time sets

are p olynomial time isomorphic so that the choice of a particular universe say

the tally representation of the natural numb ers versus the binary representation

of the natural numb ers makes a dierence

Also it is well known that the question of whether P NP is oracle dep en

dent That is Baker Gill and Solovay proved that there are recursive oracles

X X Y Y

X and Y such that P NP and P NP We shall see that some of the

natural complexity theoretic analogues of results in recursive algebra are oracle

dep endentaswell

There are several other areas of complexity theoretic mo del theory and al

gebra which will not b e co vered in this survey There is the work of Friedman

and Ko see for example and on p olynomial time analysis where

complexity theoretic versions of various theorems of analysis are studied Some

of these results are oracle dep endent and some are shown to b e equivalentto

P NP There is the work of Crossley Nero de and Remmel on ptime equiv

alence typ es and ptime isols as develop ed in and We will

present some results from on ptime equivalence typ es in section There is

the work of Khoussainov and Nero de on automatic or automata presentable

structures which is a further restriction of p olynomial time structures

We will start with a survey of complexity theoretic mo del theory In section

wewillprovide a general intro duction to complexity theoretic mo del theory

In section we shall give our basic complexity theoretic denitions and establish

notation In section we will give a series of lemmas which are useful for

building mo dels with standard universes such as the binary representation of the

natural numb ers Bin and the tally representation of the natural numbers

Tal In section weprovide a survey of the main existence theorems for

feasible mo dels In section wesurvey various feasible categoricity results

In section wegiveanintro duction to complexity theoretic algebra Then in

section we fo cus on the structure of the binary and tally representation of an

ver a p olynomial time eld In section we innite dimensional vector space o

lo ok at the semilattice of NP ideals of the binary and tally representation of

the free Bo olean algebra Finally in section wegive conclusions as well as

some directions for further work

Complexity Theoretic Mo del Theory

Complexity theoretic or feasible mo del theory is the study of resourceb ounded

structures and isomorphisms and their relation to computable structures and

computable isomorphisms The fo cus of complexity theoretic mo del theory in

this pap er is very dierent from classical complexity theory A primary fo cus

in classical complexity theory has b een to determine the complexity of certain

classes of nite mo dels enco ded as a decision problem That is one is interested

in classifying decision problems as b eing in P NP PSPACEetcAtypical

example is the graphcoloring problem where it is known that the family of

nite graphs which can b e colored is NP complete

Complexity theoretic mo del theory is more concerned with innite mo dels

whose universe functions and relations are in some well known complexity

class such as p olynomial time exp onential time p olynomial space etc Thus

if one studies graph colorings from this p oint of view one would study the

complexity of graph colorings in an innite p olynomial time graph as was done

by Cenzer and Remmel in However complexity theoretic mo del theory

has b een more concerned with the complexity of the mo del itself Thus one

can pickany complexity class and ask questions ab out what structures can b e

represented by mo dels in that complexity class By far the complexity class

that has received the most attention is p olynomial time The basic questions

that have b een consider are to classify which recursive mo dels are isomorphic

or recursively isomorphic to a p olynomial time mo del

To establish some notation let f g denote the set of natural

numbers Let denote the usual quadratictime pairing function m n

m nm n whichmaps onto Let denote the nary m

en



n

partial function on f g computed bytheeth Turing machine Then we

say that a structure

A A A

A A fR g ff g fc g

iS iT iU

i i i

where the universe A of A is a subset of f g isrecursive if A is a recursive

subset of f g S T andU are initial segments of the set of relations

A

fR g is uniformly recursive in the sense that there is a recursive function

iS

i

A

G such that for all i S Gin e where R is an n ary relation and

i i i

i

A A

computes the characteristic function of R the set of functions ff g

e n iT

i i

i i

is uniformly recursive in the sense that there is a recursive function F suchthat

A

for all i T F i n e where f is an n ary function and restricted

i i i e n

i i

i

n A

i

and there is a recursive function interpreting the constant to A computes f

i

symb ols in the sense that there is a recursive function H such that for all i U

A

H ic Note that if A is a recursive structure then the atomic diagram of

i

A is recursive

A A A

Wesay that a recursive structure A A fR g ff g fc g is

iS iT iU

i i i

polynomial time if A is a p olynomial time subset of f g and the set of re

A A

g are uniformly p olynomial g and the set of functions ff lations fR

iT iS

i i

time in the sense that in addition to the functions G and F dened ab ove

there are recursive functions G and F such that for i S G im where

i

n m

i i

for all x x inf g it takes at most maxf jx jjx jg

n n

i i

steps to compute x x and for all i T F iq where for all

e n n i

i i i

n q

i i

x x inf g ittakes at most maxf jx jjx jg steps to

n n

i i

compute x x Note that if A is a p olynomial time structure with

e n n

i i i

innitely many relation symb ols or with innitely many function symb ols then

our denition of a p olynomial time structure do es not ensure that the atomic

diagram of A is p olynomial time Thus wesay A is uniformly polynomial time

if the atomic diagram of A is p olynomial time Note that the fact that A is uni

formly p olynomial time implies among other things that the sequence of run

m q

i i

times fx i S g and fx i T g are b ounded by some xed p olynomial

Of course if A is a structure over a nite language then A is a p olynomial time

structure if and only if A is a uniformly p olynomial time structure Similar

denitions may b e given for other resourceb ounded classes

There are two basic typ es of questions whichhav e b een studied in p olyno

mial time mo del theory First as discussed ab ove there is the basic existence

problem ie whether a given innite recursive structure A isomorphic or re

cursively isomorphic to a p olynomial time mo del For example the authors

showed in p that every recursive relational structure is recursively

isomorphic to a p olynomial time mo del and that the standard mo del of arith

x

metic with addition subtraction multiplication order and the

place exp onential function is isomorphic to a p olynomial time mo del The fun

damental eective completeness theorem says that any decidable theory has a

decidable mo del It follows that any decidable relational theory has a p olyno

mial time mo del However one is naturally led to ask more rened existence

questions in complexity theoretic algebra than one asks in recursive algebra

That is since all innite recursive sets are recursively isomorphic it is easy to

see that any innite recursive structure is recursively isomorphic to a recursive

structure whose universe is It is certainly not the case that anytwo innite

p olynomialtime sets are p olynomialtime isomorphic For example the tally

representation of the natural numb ers is not p olynomial time isomorphic to the

binary representation of the natural numb ers Hence it no longer the case that

any innite p olynomialtime structure can b e identied with a p olynomialtime

structure whose universe is f g Thus a more rened existence questions is to

bers or take a xed universe such as the tally representation of the natural num

the binary representation of the natural numb ers and ask if a recursive mo del

is isomorphic or recursively isomorphic to a p olynomial time mo del with that

given universe

Here are two examples which illustrate b oth the negative and p ositiveout

comes to the simplest existence typ e question ie whether a given recursive

mo del is isomorphic to a p olynomial time mo del

Example Let A A SR where A fg that is the set of natural

n

numbers in unary representation S is the successor function that is S

n

and R is a unary relation that is a subset of fg Now if A is

B B B

isomorphic to a polynomialtime structure B B S R thenwecan

n B n B

test for membership in R as fol lows Given compute S y and

n

B B

then test whether y is in R Now if we assume that we can compute S x

n

i n

k n B k B k 

in jxj steps for jxj then it takes at most j j j j steps to

i

B r

Next we may assume that testing whether x R takes jxj steps compute y

n

n

B r k  n

if jxj so that it takes at most j j steps to test whether is in

R This means that R is a doublyexponentialtime set Thus if we start with

any recursive structure A A SR whereR isa recursive set but is not

doubly exponentialtime then A is not even isomorphic much less recursively

isomorphic to a polynomialtime structure

Despite this example there are lots of recursive structures which are recur

sively isomorphic to p olynomialtime structures

Example Let A A f where A fg and f is a unary function

m n

We say that and are in the same forbit if for some k either

k m n k n m

f or f If f is lengthincreasing then it is clear that

each forbit is isomorphic to AS Now let f and g be any two recursive length

increasing functions from fg into fg Then the structures Af and Ag

arerecursively isomorphic if and only if they have the same number of orbits

n an

Thus for example we can let f where a is Ackermanns function

and stil l be guaranteed that Af is recursively isomorphic to a polynomialtime

structure

Next consider the more restricted kind of existence question ie whether a

given recursive mo del is isomorphic or recursively isomorphic to a p olynomial

time mo del which has a standard universe such as the binary representation of

the natural numb ers Bin or the tally representation of the natural numbers

n

Tal f n g Grigorie proved that every recursive linear order

ing is isomorphic to a linear time linear ordering which has universe Bin

However Grigorie s result can not b e improved to the result that every recur

sive linear ordering is recursively isomorphic to a linear time linear ordering

over Bin That is Cenzer and Remmel p showed that for any

innite p olynomial time set A f g there exists a recursive copyofthe

linear ordering which is not recursively isomorphic to any p olynomial

time linear ordering whichhasuniverse AHere is the ordering obtained

by taking a copyof f g under the usual ordering followed byacopy

of the negativeintegers under the usual ordering

The general problem of determining which recursive mo dels are isomorphic

or recursively isomorphic to feasible mo dels has b een studied by the authors in

and For example it was shown in pp that any

recursive torsion Ab elian group G is isomorphic to a p olynomial time group A

and that if the orders of the elements of G are b ounded then A maybetaken

to have a standard universe ie either Bin orTal It was also shown

in p that there exists a recursive torsion Ab elian group whichisnot

even isomorphic to any p olynomial time or any primitive recursive group with

a standard universe Feasible linear orderings were studied by Grigorie

by Cenzer and Remmel and by Remmel Feasible vector spaces

were studied by Nero de and Remmel in and Feasible Bo olean algebras

were studied by Cenzer and Remmel in andby Nero de and Remmel in

Feasible p ermutation structures and feasible Ab elian groups were studied by

Cenzer and Remmel in and By a permutation structure A A f

we mean a set A together with a unary function f which maps A onetoone

A

and onto A Similarly an equivalence structure A A R consists of a set A

together with an equivalence relation

The second basic typ e of problem studied in p olynomial time mo del theory

Here wesay that a recursive mo del A is the problem of feasible categoricity

is recursively categorical if any other recursive mo del isomorphic to A is in fact

recursively isomorphic to A The notion of recursive categoricitywas rst de

ned by Malcev and is referred to in the Russian literature as autostability

Recursively categorical structures have b een widely studied in the literature of

recursive algebra and recursive mo del theory

The recursively categorical structures for various theories have b een classi

ed including Bo olean algebras indep endently by Goncharovand

LaRo che Ab elian groups by Smith and linear orderings indep endently

byDzgoev and Remmel For example Remmel showed in that a

recursive linear ordering L D is recursively categorical if and only if L

has only nitely many successivities where a pair abis a successivity if there

is no c with acb

Dening a natural analogue of feasible categoricity is complicated bythe

fact that unlike the case of innite recursive mo dels where anytwo innite

recursiveuniverses are recursively isomorphic it is not the case that anytwo

p olynomial time universes are p olynomial time isomorphic It turns out to b e

more natural to dene p olynomial categorical structures with resp ect to a xed

universe Th us wesay that a ptime structure A with universe D f g is

ptime categorical with respect to D if every ptime structure B with universe D

which is isomorphic to A is necessarily ptime isomorphic to A ie there exist

p olynomial time functions f g such that f restricted to D is an isomorphism

from A onto B and g restricted to D is an isomorphism from B onto A

Remmel showed in that there are no ptime categorical linear orderings

with resp ect to the standard universes Bin and Tal There are twoparts

to this strongly negative statement For any ptime linear ordering L with

universe B either Bin orTal there is a ptime linear ordering L with

universe B which is not primitive recursively isomorphic to LFurthermore if

L is not recursively categorical then L is not even recursively isomorphic to

L Similar results will b e shown for other structures The problem of feasible

categoricity for p ermutation structures and torsion Ab elian groups was studied

by Cenzer and Remmel in Here there are some limited p ositive results

In particular a p ermutation structure A f such that all orbits of f havethe

same nite size is ptime categorical over Tal There are also structures A

which are not ptime categorical over B but suchthatany ptime structure D

with universe B which is isomorphic to A must b e exp onential time isomorphic

or double exp onential time isomorphic to A More generallywecandenea

larger notion of feasibility qtime or iterated exp onential time computability

and show that there are many natural structures which are qtime categorical

over Bin and Tal

General semantic conditions for when a decidable mo del is recursively cate

gorical were given by Nurtazin and Goncharov and similar results were

found by Ash and Nero de for mo dels in which one can eectively decide

all formulas These metho ds are based on the existence of a socalled Scott

family of formulas We discuss in section various notions from of a feasible

Scott family of formulas for a feasible mo del and showthatanytwo families

which p ossess a common Scott family and have the same universe B are feasibly

isomorphic Structures considered in include linear orderings p ermutation

structures Ab elian groups and equivalence structures

Preliminari es

In this section we will give the basic denitions from complexity theory which

will b e needed for the rest of the pap er

Let b e a nite alphab et Then denotes the set of nite strings of

letters from and denotes the set of innite strings of letters from where

f g is the set of natural numbers For any natural number n

n

tal n is the tally representation of n and binni i i f g

 e

e

is the reverse binary representation of n if n i i i

 e

In general the kary representation b n i i i if n and i

k  e e

e

i i k i k and i Welettal bin b Then

 e e k

we let Tal ftal nn g Bin fbinnn g and for each

k B fb nn g Occasionallywe will wanttosaythat

k k

B Bin and that B Tal



For a string n j j denotes the length n of The

empty string has length and will b e denoted by A constant string of length

n

n will b e denoted by k For mj j dm is the string m

is an initial segment of written if dm for some m The

concatenation or sometimes just is dened by

m n

where j j m and j j n in particular we write a for aanda for

a

Our basic computation mo del is the standard multitap e Turing machine of

Hop croft and Ullman Note that there are dierent heads on each tap e and

that the heads are allowed to move indep endently This implies that a string

can b e copied in linear time An oracle machine is a multitap e Turing machine

M with a distinguished work tap e a query tap e and three distinguished states

QUERY YES and NO At some step of a computation on an input string

M may transfer into the state QUERY In state QUERY M transfers into the

state YES if the string currently app earing on the query tap e is in an oracle set

A Otherwise M transfers into the state NO In either case the query tap e is

instantly erased The set of strings accepted by M relative to the oracle set A

is LM Af j there is an accepting computation of M on input when the

oracle set is AgIfA we write LM instead of LM

Let tn b e a function on natural numb ers A Turing machine M is said

to b e tn time bounded if each computation of M on inputs of length n where

n requires at most tn steps A function f x on strings is said to b e in

DT I M E t if there is a tntime b ounded deterministic Turing machine M

which computes f x For a function f of several variables we let the length

of x x bejx j jx j A set of strings or a relation on strings is in

n n

DT I M E t if its characteristic function is in DT I M E t Welet

S

R fDT I M E n cc g

c

S

LI N fDT I M E cnc g

c

S

i

P fDT I M E n i g

i

S

cn

DE X T fDT I M E gand

c

S

cn



DOU B E X T fDT I M E g

c

S

c

n

EXPTIME g and in general fDT I M E

c

S

tn

DE X S DT I M E g

tnS

A function f x on strings is said to b e in NT IMEt if there is a tntime

b ounded nondeterministic Turing machine M which computes f x A set of

strings or a relation on strings is in NT IMEtifitscharacteristic function is

NT IMEt Welet in

S

i

fNT IMEn i g NP

i

S

cn

fNT IME g NEXT

c

S

c

n

NEXPT IME fNT IME g

c

S

cn



DOUBNEXT fNT IME g and in general

c

S

tn

NEXS NT IME g

tnS

of the p olynomial time b ounded Wexenumerations fP g and fN g

N i iN i i

deterministic oracle Turing machines and the p olynomial time b ounded non

deterministic oracle Turing machines resp ectivelyWemay assume that p n

i

i

maxn is a strict upp er b ound on the length of any computation by P or N

i i

X X

with any oracle X on inputs of length n P and N denote the oracle Turing

i i

machine using oracle X and in an abuse of notation we shall denote LP X

i

X X X X

by simply P and LN Xby N This given P fP i N g and

i

i i i

X X

NP fN i N g

i

P

For A B we shall write A B if there is a p olynomialtime function

m

P

f such that for all x x A i f x B We shall write A B if A

T

is p olynomial time Turing reducible to B For r equal to m or T we write

P P P P P

A B if A B and B A and we write A j B if not A B and not

r r r r r

P

B A

r

We dene the standard notions of feasibility as follows Wesay that a

function f xisquasirealtime if f x R This is slightly more general

than the usual notion of realtime as computable byaTuring machine which

simply reads the input one symb ol at a time from left to right or righttoleft

and simultaneously leaves the output in its place on the tap e In particular a

realtime function is always in DT I M E n The function f xislinear time

P nondeterministic polynomial time if if f x L polynomial time if f x

f x NP exponential time if f x DE X T nondeterministic exponential

time if f x NEXT andisdouble exponential time if f x DOU B E X T

Wesay that f x is exp onentially feasible if f x DE X T for a notion T of

feasibility In particular if f x DE X DOU B E X T then f x is said to b e

triple exponential time

The smallest class including P and closed under DE X can b e dened by

iterating DE X That is let

S

 n n

P P P DE X P for each n and Q P

n

A function f x Q is said to b e iterated exponential time or qtimeThe

iterated exp onential functions E x can b e dened recursively by E x x

n 

E x r

n

and E x for all n and x It is easy to see that x E xforall

n r

S

m

r from which it follows that Q DE X DT I M E n

m

We observe that the classes R LI N P NP andQ are all closed under com

p osition whereas the other classes dened ab ove are not In addition Tal

and Bin are qtime isomorphic

Observe that for a function f x x of several variables the ab ove

k

denitions are equivalent if we declare the size of the input x x tobe

k

the maximum of the sizes jx jjx j since we allowmultiple tap es This

k

o ccasionally simplies the computation of the complexityofvarious functions

We refer the reader to Odifreddi for the basic denitions of recursion

theoryLet b e the partial recursive function of n variables computed by

in

th

the i Turing machine M Ifn we will write instead of Given a

i i i

s

string f g we write es an output in s or fewer steps if M giv

i

i

s

when started on input string Thus the function is uniformly p olynomial

i

s

time Wewrite if s and if not

e e e

e

The notion of a ptime structure was dened in section We need a few

renements of that denition

Denition i A ptime function f is honest ptime if thereisapoly

nomial function q such that for al l x x

n

y f x x i njx j q jy j

n i

ii A ptime structure A is honest ptime if al l of its functions are honest

ptime

iii A structure A has honest witnesses if for any quantierfree

formula y x x thereisapolynomial q such that for any a a

n n

AifAjy y a a then thereisa z A with jz jq ja j

n

ja j such that Aj z a a

n n

Note that for an honest ptime function mapping Tal into Tal Nero de

and Remmel showed in that f is also honest ptime

For a group we will distinguish twotyp es of computability The structure

of a group G is determined by the binary op eration whichwe will denote by

G G

the addition sign since we are interested in Ab elian groups Welete

denote the additiveidentityofG However the inverse op eration denoted by

G

inv may also b e included as an inherent part of the group Thus wehavethe

following distinction

G G

e is computable Denition Agroup G is computable if G

G G G

and is fully computable if G inv e is computable

G

It is easy to see that any recursive group is also fully recursive since inv a

G G

can b e computed as the least member b of G such that a b e where the

elements of G are ordered rst by length and then lexicographically for elements

of the same length

On the other hand the fully ptime groups make up a prop er sub class of the

ptime groups as shown by Prop osition of

Denition For any complexity class and any structures

A A

g fc g g ff A A fR

iT i iU iS

i i

and

B B

B B fR g ff g fc g

iS iT i iU

i i

we say that A and B are isomorphic if there is an isomorphism f from A

onto B and computable functions F and G such that f F dA the restriction

of F to Aand f GdB

Polynomial time sets and isomorphisms

In this section we shall giveanumb er of useful lemmas ab out the relations

between the various standard universes that we will consider in our study of

feasible structures The most basic standard universe is the set where

is a nite alphab et and in particular where f g fgorfgOther

standard universes include the set Tal of tally representations of natural

numb ers the set Bin of binary representations of natural numb ers and for

any k the set B ofk ary representations of natural numb ers In recursion

k

theory all of these sets are recursively isomorphic and therefore interchangeable

For our purp oses wemust consider carefully which of these isomorphisms are

p olynomial time or even p olynomial time in one direction

First we need to explicitly dene a p olynomial time pairing function For

any nite alphab et there is a natural emb edding of into Bin given

as follows Wemay supp ose that f ng for some nLet

and for i i let

k

i i i

 k

The function is actually an isomorphism from onto Bin and has an

inverse It is also clear that for each n the set f ng is linear

time uniformly in n Thus we can normally assume that an arbitrary structure

has universe a subset of Bin

The co ding function h i for f g is now dened

 k k k

by

h i

 k k  k k

Let Q fh i f g for eachig For i k

k  k k i

k

from Q onto f g are implicitly dened bythe the pro jection functions

k

i

equation

k k k

i h

k

 k

The subscript k will normally b e omitted It is easy to see that the sets Q and

k

k

B are all linear time and that the functions andh i are all

k k

i

computable in linear time

Given two subsets A and B of f g dene

A B fha bi a A b B g

and

A B fhai a Agfhbi b B g

It is clear that if A and B are ptime then b oth A B and A B will also b e

ptime

Now for each k a natural numb er in reverse k ary form is simply a

string f k g which is either or else ends with an elementof

fk gThus the set B of kary representations of natural numbers

k

is a linear time subset of f k g Tal fgfg is of course

a linear time subset of f g but is not p olynomial time isomorphic to the

whole set This will follow immediately from Lemma b elow For a string

n

b n B with n let b e the unary representation ofn

k k k

and let Wenow state a sequence of lemmas which will b e useful for

k

our main results Most of these lemmas are proved in or

Lemma a Suppose that A is a polynomial time structure and that

is a polynomial time set isomorphism from A onto a set B Then B is

apolynomial time structure where the functions and relations on B are

denedtomake an isomorphism of the structures

b Suppose that A is an EXPTIME structure and that isapolynomial time

set isomorphism from A onto a set B Then B is an EXPTIME struc

ture where the functions and relations on B aredened to make an

isomorphism of the structures

c Suppose that A is a qtime time structure and that is a qtime time set

isomorphism from A onto a set B Then B is a qtime time structure where

the functions and relations on B aredenedtomake an isomorphism of

the structures

Pro of Wesketch the pro of for the ptime case The other cases are sim

A

ilar To simplify the pro of let us supp ose that A has one function f and

A

one relation R Observe rst that B is a p olynomial time set since b

B B

B b A The function f is p olynomial time since f b b

n

A B B

f b b The relation R is p olynomial time since R b b

n n

A

R b b

n

Next we state two lemmas which relate tally representation of a structure

to its binary representation and more generallytoitsk ary representation

Part b of the rst lemma is an improvement of Lemma of where the

computation was b ounded in p olynomial time

Lemma For each k

a Bin is linear time isomorphic to f g

b There is a linear time function p such that for al l nboth the computation

n n

of b n and the inverse computation of b n can be

k k k

k

computed in time pn

j j j j

c For each n and b n k n k

k

Pro of Wesketch the pro of of b for k The basic computation in either

direction consists of enumerating the binary numb ers from to n on one tap e

while either writing or reading n s on the other tap e The enumeration of the

binary numb ers is done by rep eatedly adding by the usual algorithm which

consists of replacing s with s while lo oking for the rst and then replacing

the rst with a If we dene hn to b e the total numberofsymb ols written

n

by this pro cedure for the numb ers from k up to k then we observe

that h and that in general hn n hn This is b ecause the

n n n

numb ers from up to are obtained by writing which takes n

n

steps and then essentially writing the numb ers from up to again while

leaving the nal on the end of each Now it is easy to see by induction that

n

Counting a slightly smaller numb er of steps for hn n for each n

returning to the b eginning of the string we see that all of the binary numbers

n n

from up to k may b e written in total time hn k

n n

For anynumber k with k the binary numb ers from to k maybe

n

written in total time k

For any subset M of lettal M ftal nn M gletbinM

fbinnn M g and for any nite k let b M fb nn M g

k k

Lemma For any nite k any M and any oracle X

X X

a tal M P b M DE X T

k

X X

b tal M NP b M NEXT

k

X X

c b M P tal M LI N

k

Pro of We give the pro of for k where b M binM The pro of of

k

b is the same as the pro of of a

X

tal M P Then there is a pro cedure with oracle a Supp ose rst that

c

X which tests whether tal n tal M in time n for some xed c and

all n To test whether binn binM we rst compute tal n

which requires time n by the pro of of Lemma Then we test whether

c

tal n tal M which requires n steps by assumption Nowbypartcof

c j j c cj j X

Lemma n so that binM DE X T

X

Next supp ose that binM DE X T Then there is a pro cedure with

cjbinnj

oracle X which tests whether binn binM in time for some

xed c and all nTo test whether tal n tal M we rst compute binn

which requires time n by the pro of of Lemma Then we test whether

cjbinnj

binn binM which requires steps by assumption Nowby

c c cjbinnj jbinnj c

minn Hence part c of Lemma n

X

tal M P

X

c Supp ose that binM P sothatwe can test binn binM in

c r r

time j j Nowlet j j r so that n Thenasinpartaabove

c

we see that we can test tal n tal M in time r Now it is clear that for

c r X

suciently large r r nThus tal M LI N

X

We note that the assumption that tal M LI N actually implies that we

j j

can test BinM intime c for some xed c and almost all

For any structure M with universe M let tal M b e the tally represen

tation of M with universe tal and relations and functions dened so that the

mapping taking n to tal n is an isomorphism from M onto tal M binM

and b M are similarly dened Lemma is easily extended to tally and

k

binary representations of relational structures M and one direction extends to

structures with functions

Lemma Let k letM be a structure with universe M and let

A tal M and B b M Then

k

a If A is ptime then B is exponential time

b If B is exponential time then A is EXPTIME

M M

c If B is exponential time and for al l functions f f n n

k

cn n 

k

for some xedconstant c and al l but nitely many k tuples

then tal M is exponential time

M M

d If B is exponential time and for al l functions f f n n n

k

c

n for some xedconstant c and al l but nitely many k tuples then

k

A is ptime

M M

e If B is polynomial time and for al l functions f f n n cn

k

n for some xedconstant c and al l but nitely many k tuples then

k

A is linear time

Pro of Wesketch the pro ofs for k

a Supp ose that A is ptime It follows from Lemma that B has a

exp onential time universe and it is easy to see that the relations of B are also

M

exp onential time For simplicity supp ose that f is a unary function The

A

general pro of can b e found on p of Supp ose tal mf tal n

c

By assumption tal mmay b e computed from tal n in time n for some

c B

xed c and all n so that m n To compute f binn werst

compute tal n from binn which takes exp onential time by Lemma Then

A c cjbinnj

wecomputetal mf tal n in Awhichtakes time n

Finallywemust compute binmfromtal m This nal computation takes

c jbinnj c

time m n Thus B is exp onential time

For parts b c and d supp ose that B is exp onential time Then we

easily see that the universe of A and the relations of A are p olynomial time For

M A

simplicity let f b e a unary function The pro cedure for computing f tal n

has three parts as ab ove First compute binn from tal n whichtakes time

B

n by Lemma Next compute binmf binn whichtakes time

cjbinnj c

n The nal and most timeconsuming part of the computation

is to compute tal m from binm whichtakes time m In general we only

c

cjbinnj c n

knowthat jbinmj n so that m and hence wecan

cn

only conclude that A is EXPTIME For part c wehave m and hence

c

A is exp onential time For part d wehave m n so that A is ptime

e It follows easily from the hyp othesis and Lemma that the universe

A

of A and the relations of A are linear time Nowlettal mf tal n and

observe that bythehyp othesis m nc The computation of binnfromtal n

takes time n by Lemma Since B is p olynomial time the computation of

B d

binmf binn takes time jbinnj for some xed d It follows as in the

pro of of Lemma c that this computation can almost always b e done in time

n Finally the computation of tal m from binm takes time m nc

Thus A is a linear time structure

Note that the hyp othesis needed for part d follows from the assumption

that for some xed constant c and all but nitely many k tuples

B

jf j cj j j j

k k

Nero de and Remmel dene in the notion of a ptime equivalencetype

and B of Tal are ptime equivalentif PET bysaying that two subsets A

there is a partial honest ptime function f with domain including A such

that f AB It is natural to extend this notion to ptime subsets of Bin

by dening two sets A and B to b e ptime isomorphic if there is a ptime

function mapping A onto B whose inverse is also ptime Wemay assume that

f and f have domain Bin since A and B assumed to b e ptime

It follows from Theorem of that for any ptime subset A of Tal

there are innitely many ptime subsets of Tal which are recursively iso

morphic to A but not ptime isomorphic to AWecannowcharacterize those

subsets of fg which are p olynomial time isomorphic to Tal and put con

ditions on those subsets of f g which are p olynomial time isomorphic to

Tal The following was proved in

Lemma a Let A be a ptime subset of Bin which is polynomial time

isomorphic to Tal and let a a list the elements of A in the stan



dardordering rst by length and then lexicographical ly Then for some

j k

jk and al l n n ja j and ja j n

n n

b Let A be a ptime subset of Tal and let a a list the elements of A



in the standardordering Then the fol lowing areequivalent

A is ptime isomorphic to Tal

k

For some k and al l n ja j n

n

n

The canonical map taking to a is ptime

n

Lemma For any innite set M of natural numbers tal M ftal n

n M g and binM fbinnn M g are not ptime isomorphic

Lemma Let B be the set of kary representations of natural numbers

k

Then

a The addition subtraction multiplication and division with remainder

functions from B B to B theorder relation on B and

k k k k

the length function from B to B are al l ptime As usual m n

k k

is set to if mn

b Bin nfg is ptime isomorphic to Bin

c For each k and for A equal to either the set B or the set f k

k

g thereisapolynomial time isomorphism from A to Bin and a

constant c such that for al l but nitely many a A jajjaj cjaj

We will frequently wanttocombine structures using disjoint unions and

direct sums Nero de and Remmel proved in that there exist ptime subsets

A B and C of Tal such that A is not ptime isomorphic to B but A C

is ptime isomorphic to B C and similarly there exist subsets X Y and Z

of Tal such that X is not ptime isomorphic to Y but X Z is ptime

isomorphic to Y Z This result of Nero de and Remmel also easily follo ws

from our next results due to Cenzer and Remmel

Lemma a Let A be a ptime subset of Tal ThenA Tal is ptime

isomorphic to Tal and A Bin is ptime isomorphic to Bin

b Let A be a nonempty ptime subset of Tal ThenA Tal is ptime

isomorphic to Tal and A Bin is ptime isomorphic to Bin

Pro of a First observethatA Tal is ptime isomorphic to the set

y the obvious isomorphism C fa a Agfn n Tal g b

Nowletc c enumerate C in increasing order Since C contains every o dd



numb er it is clear that c n It follows from Lemma that C is ptime

n

isomorphic to Tal

Next A Bin is certainly ptime isomorphic to A Tal Bin n

Tal Nowby the preceding discussion A Tal is ptime isomorphic to

Tal It follows that A Bin is ptime isomorphic to Tal Bin n

Tal which is clearly ptime isomorphic to Bin

b If A has only one element this is obvious If A has at least two elements

let a b e one of them Then A Tal is ptime isomorphic to fagTal

AnfagTal Now the rst part of this sum is obviously ptime isomorphic

to Tal and the second part is ptime isomorphic to some ptime subset of

Tal It now follows from part a that the sum is ptime isomorphic to

Tal

For the case of Bin again if A consists of single element the result is

trivial Otherwise let a b e the least elementofA and consider the following

b

mapping If b A and b a then let f hb i The idea is to dene

f so that the image of fag Bin under f is Bin n f A nfag Bin

by letting f ha binnibethenth elementofBin n f A nfag Bin

b

Observe that for any binq Bin which is not of the form for

some b A nfagwe can nd in p olynomial time in j j all strings of the form

k b

b

such that

b

b

A and

k b k  b

b b

under the usual ordering on Bin but

That is since A is a p olynomial time subset of Tal we can test eachelement

b k

of the with bj j for membership in A in b steps for some xed k

b b

and hence nd all strings of the form where q b and A in at

P

j j

k k 

j j j steps Thus in p olynomial time in j jwe can nd all most

j 

k

k b b

b

b p

p

the strings satisfying prop erties ab ove This

P

p

k

i

mean that exactly elements of Bin which are less than or equal to

i

P

p

k

i

are in the image of f A nfag Bin Thus weletf ha binq i

i

It follows that f is p olynomial time To see that f is p olynomial time

note that f ha binn binn so a similar computation starting with

binn will allowustondthe nth elementofBin n f A nfag Bin

in p olynomial time in jbinnj

It follows easily from the lemmas ab ovethatany ptime relational structure

A is recursively isomorphic to a ptime structure B suchthatA and B are not

ptime isomorphic We assume that A has universe BinAfbinaa Ag

for some A and let B haveuniverse TalAftal aa Ag Now

these structures are actually exp onentialtime isomorphic However there is a

stronger result

g thereisa set Lemma For any ptime set A fbina bina



M M A fbinm binm g such that M is ptime and the map



which takes binm to bina is ptime but there is no primitive recursive map

i i

from A into M which maps at most k elements of A to any element of M where

k is any xed nite number Furthermore M may be taken to be a subset of

Tal

Pro of Let b e the eth primitive recursive function mapping Bin into

e

Bin and for each e let t b e the total time required to test all numbers

e

up to a for mem b ership in A and to compute bina for all j i a

e i j e

t t

e e

clearly t t For each eletm sothatbinm and

e e e e

jbinm j t Itfollows that bina binm for all i esince

e e e i i

byconvention it takes at least k steps to compute an output of length k Let

A fbinm eg

e

Here is the ptime algorithm for testing whether x A First checktosee

t

that x for some n Then start to test binbin for memb ership in

AAssoonaswe nd that binniseth member of A so that binnbina

e

then compute in order a a and a a and then

e  e e  e e e

return to testing whether binn binn are in A If the total number

of steps reaches t exactly when the computation of some a has just b een

e e

completed then t t so that x binm b elongs to A Otherwise x A

e e

This argument also shows that the map whichtakes binm tobina isp

e e



time Finallylet M M Afbinm igThenM is also ptime

i

since we can clearly check in p olynomial time whether i is a square if we discover



that x binm at the end of the computation The map taking binm to

e

i



bina is ptime since we can test all numb ers a for membership in A in

i

i

 

j and therefore determine a from binm time jbinm

i

i i

A into M Supp ose nowbyway of contradiction that were a map from

e

whichwere at most k to Since each primitive recursive function has innitely

many indices wemay assume that e k Then



f bina bina g

e  e

e

must contain at least e distinct elements so that at least one of them is



binm whichcontradicts the observation ab ovethat bina binm

e i e

e

for all i e and thus establishes the result

Since Bin and Tal are primitive recursively isomorphic via the stan

dard map it follows that we could replace M in this argument with the set



M f tal m tal m g



Remark The only prop erties of the set of primitive recursive functions

needed for this result is that the primitive recursive functions is a class of total

recursive functions which can b e eectively listed Thus in the statementof

Theorem we can replace the primitive recursive functions byany class of

total recursive functions which can b e eectively listed

Remark Letting k we see that there is no primitive recursiveembed

ding of A into M A Furthermore there is no primitive recursiveemb edding

of a conite subset C of A into M A since if jA n C j k then we could

extend to a map from A into M Awhich is at most k to by mapping all

elements of A n C to some xed elementofM A

The Existence of Feasible Structures

In this section we shall survey a numb er of mo del existence results for p olyno

mial time structures In particular we will consider four existence questions for

any class C of structures

Is every recursive structure in C isomorphic to a p olynomial time struc

ture

Is every recursive structure in C recursively isomorphic to a p olynomial

time structure

Is every recursive structure in C isomorphic to a p olynomial time structure

with a sp ecied universe such as the binary or tally representation of the

natural numb ers

Is every recursive structure in C recursively isomorphic to a p olynomial

time structure with a sp ecied universe such as the binary or tally repre

sentation of the natural numb ers

The fundamental result for relational structures is due to Grigorie and

for structures with innitely many relations to CenzerRemmel Recall

that a structure with no function is said to b e relational

Theorem Every relational structureisrecursively isomorphic to a real time

structure with universe a subset of Bin andtoalinear time structure with

universe a subset of Tal

Sketch of Pro of Wemay assume without loss of generality that the struc

ture A has universe The element a is represented in the binary real time

a t

mo del by a where t is the time required to compute whether

tj 

and all tuples x x from fag R x x for all j a

j

tj  tj 

The set B f aa g is real time by the following algorithm

Given binn start to read binn from left to right If at anytime

a t

we discover that binn is not of the form for some a and

t then binn B Otherwise having read the rst in binnso

a t

that wehave found a suchthatbinn start the com

putation which tests for all j a and for all tuples x x

j  t

tj 

from fag whether R x x If the total computa

j

tj 

tion nishes in exactly t steps then binn B If the computation

either nishes in fewer steps or has not nished by t steps then

binn B

Each relation R is real time by the following algorithm

j

tj 

Given binn binn B rst compute a and t for

 k k

tj 

a  t

k k

each k j so that binn Then let t maxft t g

k 

tj 

Nowby the construction we can test whether R a a in

j 

tj 

time t

The tally representation of A has universe ftal nbinn B g and is linear

time by Lemma e

Note that Theorem allows us to conclude that if G is a recursive graph

ie G V E where V the vertex set of G is a recursive set of natural num

b ers and the edge relation E is also recursive then G is recursively isomorphic

to a p olynomial time graph G However if G has a recursive k coloring then

to conclude that G is recursively isomorphic to a p olynomial time graph with a

p olynomial time k coloring requires a stronger version of Theorem We will

present an improved version of Theorem due to Cenzer and Remmel

which is their primary to ol in their analysis of p olynomial time combinatorial



structures and classes The improved version of the theorem presented ap

plies to structures with two distinct typ es of ob jects the rst typ e b eing the

normal universe of the structure and with functions whichmapthersttyp e

into the second typ e The typ e of example that wehave in mind is a function

from the vertices of a graph into the natural numb ers which computes the degree

of a vertex or the color assigned to a vertex The universe of the graph is now

expanded by adding a ptime set which represents the natural numb ers and the

degree function or coloring now b ecomes part of the structure Naturallythe

new ob jects are not vertices and therefore are not joined to any other ob jects

by edges

Theorem Let

C C

C C A B fR g ff g

iS iT

i i

bea recursive structure such that

with C A B and B is a polynomial i A and B are disjoint subsets of C

time set

ii thereisa recursive isomorphism from Bin onto a subset of

Bin n B with a ptime inverse

iii for each i T f maps C into B

i

iv for each i S the relation R is independent of B thatisforany

i

n

x x C where n sianyj n such that x B and

n i

C

any b B R x x holds if and only if

n

i

C

x x bx x holds R

j j  n

i

v for each i T the function f is independent of B that is for any

i

n

x x C where n tianyj n such that x B and

n i

C C

any b B f x x f x x bx x

n j j  n

i i

Then thereisarecursive isomorphism of C onto a ptime structure M such

that b b for al l b B

Structures with Functions

In contrast to purely relational structures recursive structures with functions

need not b e eectively isomorphic to feasible structures The following results

are Theorems and of

Theorem Let L be the language with exactly one function symbol f which



is unary

A

a Thereisarecursive structure A A f which is not recursively isomor

phic to any primitive recursive structure

D

b There is an exponential time structure D D f which is not recursively

isomorphic to any polynomial time structure

Pro of a Let A f A f b e an eective list of all primitivere

 

cursive structures over L and let b e a list of all onetoone partial

 

recursive functions Wemust meet the following set of requirements in our

construction of A

R is not a recursive isomorphism from A to A f

ij j i i

To meet the requirements R recursively partition f g into innitely

ij

many disjoint innite recursivesetsS We then dene A so that A

ij

S

A

S f g and for all i j f maps S into S

ij ij ij

ij

A

Wenowx i j and then we dene f f on S in stages We let a a

ij 

b e some eective listing of S At stage swe shall dene f a We start by

ij s

s s

dening f a a at stage At stage s compute a If a

  

j j

s

s

or if a then we dene f a a Otherwise that is if a

 s s 

j

j

s

s

but a let x a and do the following Compute the sequence

 

j

j

s k 

x f xf f xf x where here f denotes f comp osed with itself

i i i

i i

k times Note that if were an isomorphism then it must b e the case that

j

k 

x for all k Thus if were an isomorphism then it must b e the a f

j j k

i

case that

s

f a a f xx

s 

i

s s

Thus if f xx then we dene f a a Iff x x then we

s s

i i

dene f a a Note that in either case wewillhave ensured that cannot

s  j

b e an isomorphism from A onto A f

i i

b The pro of of part a must b e mo died in several ways First let

E f b e an eective list of all ptime structures whose universe E f

 



is contained in f g over L Let b e a list of all onetoone partial

 

recursive functions whichmapf g into f g We shall build our structure

D

D f sothatD Tal andwe meet the following set of requirements

R is not a recursive isomorphism from D to E f

ij j i i

To meet the requirements R we construct D as a disjoint p olynomialtime

ij

S

union of innite ptime sets D T Dene the function Tal

ij

ij

Tal Tal Tal by tal italj tal i j and tal n

p

tal i tal j tal if tal n tal i tal j tal p

Note that tal xtal i j can b e computed from input

tal i tal j in time a x for some xed constant a and that the computation of

x

tal y tal from input tal x can b e computed in time at most b y for some

xed constant b aThus the computation of tal z tal n tal italj

from input tal ntali tal j takes at most the following number of steps

x

x  

bi j bx b b bz b z bz

For each i j we let T f n i j n gThenwe can test whether

ij

tal z tal ntalitalj p erform the computation of

tali tal j tal i tal j tal ntali tal j



for bz steps and see if the computation converges to z by that time It follows

that the sets T are uniformly ptime and that D is also ptime since

ij

tal z T nz tal z tal ntali tal j

ij

and

tal z D i j z tal z T

ij

D

Wenowx i j and dene f f on T ftal a tala g where

ij 

tal a tal n tal i tal j For each m p erform the following series of

n

a

m

steps computations for at most

a

m

steps let Start to compute tal a If this converges in less than

j 

b tal a

 j 

Checkthatb E

 i

Compute the sequence b f b b f b b f b

i   i m i m

Let s b e the least m such that the computations can b e successfully com

a

m

pleted in at most steps Assuming the existence of b a we can show

 j 

that suchans must exist That is it takes some constantamount c of time to



k

compute b Sincef is ptime f y can b e computed in time b ounded by jy j

 i i

for some xed integer kandany y f g with jy j Let c b e the time

and f if needed and let c c c Then required to compute f f

i  i i

to compute the sequence b a b f b b f b takes time

 j  i  m i m

at most

m 

k k k k k k

jb j tmc jb j jb j c jb j jb j

    

We need to show that this sequence is eventually dominated by the sequence

a a

m 

a a Wemay assume without loss of generalitythat a

 m

m

k

then jb j Nowifm is large enough so that b oth c and m are jb j

 

m m

k k

jb j tm m jb j

 

m

  m  m

Nowletm b e large enough so that jb j k and m m



 m 

m m   m m

  m m  

and tm Then k exp m Toshow

that the latter is dominated by a note rst that a and that for anym

m 

a m it follows that a exp m

m m

The denition of f now pro ceeds in stages as in part a Let s b e the least

m such that the computations describ ed ab ove can b e successfully completed

Nowfort swelet f a a To compute f a weletx a

t t s j 

s s

and compute f x Then if f xxwe dene f a a If

s s

i i

s

x x then wedene f a a Note that in either case we will have f

s 

i

ensured that cannot b e an isomorphism from D onto E f

j i i

It remains to b e seen that the computation of f can b e done in exp onential

time Given tal x D we rst compute the unique triple n i j suchthat

tal x tal ntali tal j This can b e done in p olynomial time since n i

and j are all less than x Next we p erform the computations and for

m n in turn This can b e done in exp onential time since each series

a

m

of computations is b ounded by time The remainder of the computation

of f xtakes little time We lo ok for the least nifanysuchthatthe n th

series of computations has b een successfully completed If m nthenwe

s

a

m

xx and let f tal a tal if so otherwise check to see if f

m

i

f tal a tal a

m 

We note here that the functions in the previous theorem maybetaken to

b e p ermutations of the sets A and D Then next two results showthatwecan



diagonalize over isomorphisms if the underlying language has at least two



unary function symb ol or at least one nary function symb ol with n

Theorem Let L be the language with exactly two function symbols f and



g which are unary

 A A

which is not isomorphic a Thereisa recursive structure A A f g



to any primitive recursive structure

D D 

b Thereisanexponential time structure D D f g which is not



isomorphic to any polynomial time structure

Theorem Let L be the language with exactly one function symbol h which



is binary

A 

a Thereisarecursive structure A A h which is not isomorphic to



any primitive recursive structure

D 

b Thereisanexponential time structure D D h which is not iso



morphic to any polynomial time structure

Two natural typ es of structures with functions will b e considered b elowin

A A

more detail These are p ermutation structures A f where f is a p ermuta

tion of the set A and Ab elian groups

Next we state an unpublished theorem due to H Freidman and J Remmel

whichcharacterizes when structures which are nitely generated are isomorphic

to p olynomial time structures

Denition a We say that

A A A

A A fR g ff g fc g is nitely generatedifA

ik in im

i i i

A A

equals the closureoffc g under the set of functions ff g

im in

i i

b A nitely generated structure A as above has a double exponential time

decision procedureif

A f g is double exponential time

A A

There is an algorithm which given any two terms t c c and

m

A A A A

t c c in the free term algebra generatedbyc c and



m m

dk

A A  A A

f f decides in steps for some constant k if t c c

n m

A A A A

t c c where d is equal to the maximum of the depth of t c c



m m

A A

and the depth of t c c



m

A

There is an algorithm which given a relation R and terms

i

A A A A

c t c c in the free term algebra generated t c

p

m m

dk



A A A A 

by c c and f f decides in steps for some con

m n

A A A A A

stant k if R t c c t c c holds where d is

 p

m m

i

A A

equal to the maximum of the depths of t c c for j p

j

m

A A A

g ff g fc g be a nitely Theorem Let A A fR

ik in im

i i i

generated structure Then A is isomorphic to a polynomial time model i A has

a double exponential time decision procedure

Linear Orderings

There are several results on the existence of ptime linear orderings

Theorem Grigorie Every recursive linear ordering L is isomor



phic to a real time linear ordering L Bin

L

SketchofProof Wesketch a pro of showing that L is isomorphic to a

ptime ordering There are two cases The rst case is where L has either a

recursive increasing sequence

s S s s

L  L  L

such that S is conal or S has a limit or L has a recursive decreasing sequence

D d d d

 L L  L

such that D is conal or D has a limit

In this case wehave a ptime copy of either S or D with universe Bin

and we can make a ptime copyofLn S or LnD with universe a subset of

Tal by Theorem Then we can apply Lemma to combine the two

orderings into one ptime ordering with universe Bin which is recursively

isomorphic to L

The second case is where no such sequences exist In this case L is isomorphic

to Z for some recursive ordinal HereZis the order typ e of the

integers Then there are two sub cases First if has a rst or last element

or has a pair of elements x y such y is an immediate successor of x then

L

L contains an explicit recursivecopyofD which is isomorphic to a

ptime linear ordering with universe Bin We can make a ptime copyof

Ln D with universe a subset of Tal by Theorem Then we can apply

Lemma to combine the two orderings into one ptime ordering with universe

Bin The only other sub case is that is a dense linear ordering without

endp oints so that is isomorphic to the rationals Q But in this case it is easy

to construct a ptime linear ordering with universe Bin which is isomorphic

to L

The natural question is whether the isomorphism in the previous theorem is

eective It should b e observed that the pro of is not uniform Indeed in case

we only constructed an isomorphic copy and not a recursively isomorphic copy

of L The next result due to Remmel is Theorem of whichshows that

in case the isomorphism cannot b e replaced by a recursive isomorphism

Theorem Let A f g be any innite ptime set and let L bea recursive

linear ordering which is isomorphic to Z for some linear ordering

Then thereexistsa recursive linear ordering K which is isomorphic to L but

which is not recursively isomorphic to any ptime linear ordering whose universe

is A

Sketch of Pro of

We will not give the full pro of as it requires an innite injury priority argu

ment However we will give the pro of in the case where the L is isomorphic to

since in that case a simple nite injury priority argument suces

Recall that is the ith partial recursive function and let R R be an

i 

eective list of all p olynomial time binary relations on f g For simplicity

welethA R i denote the structure with universe A and relation R whichisthe

e

restriction of R to A A

e

Let b e an eectiveenumeration of A in the usual order rst by



length and then by lexicographic order

We shall construct our desired recursive linear ordering L in stages Let

b e an eective listing of f g Atanygiven stage swe shall sp ecify



s s s s s s

two sequences a a s such that B a and b b b for some n

s s

 n  n

s s

s s s s s s

f g fa b a b a b g Moreover at stage s weshall

 n 

s

  n n

s s

dene the ordering on B B so that

L s s

s s s s s s

b b b a a a

 n n n 

s s s

s s

Our construction will ensure that for all i lim a a and lim b b

s i s i

i i

exist Moreover our construction will ensure that f g fa b a b g

 

and that

a for all i a a and b b and

i i i i

b for all i and j a b

i j

Thus f g will have order typ e To ensure that L is not

L

recursively isomorphic to A R forany ewe shall meet the following set of

e

requirements

P There exist n and m such that one of the following four conditions

ek

holds

i a or a x A

k n k n

ii b or b x A

k m k m

iii a x A and there exist n elements v v of A suchthat

k n  n

v x R for i n

i e

iv b y A and there exist m elements w w of A such

k m  m

that w y R for i m

i e

We write w y R rather than y w R in clause iv to allow for the

i e i e

p ossibilitythat R is not actually a linear ordering

e

It is easy to see that if requiremen t P is satised then is not a re

e

ek

cursive isomorphism from L f g onto A R Thus meeting all the

L e

requirements P ensures that L is not recursively isomorphic to any ptime

ek

linear ordering with universe A

Our basic strategy for meeting a requirement P where z e k is as

z

follows Let us assume that s z is a stage large enough so that requirements

P P no longer require action at any stage t s Then at stage s we

 z

t s s

for all j z and a consider a Our construction will then ensure that a

z

j j

s u

a t s unless there is a stage u s suchthat Of course if there is no

z

k

s

suchuthen a a and a will witness that requirement P is satised by

z z z

z

virtue of clause i

s u

a If x A then again we Now if there is such a stage u then let x

z

k

s

will simply ensure a a so that once again a will witness that requirement

z z

z

P is satised If x A then we will compare x to the rst n elements

z u

of A in the xed order prescrib ed ab ove with resp ect to the binary



relation R Note that since A and R are p olynomial time we can eectively

e e

make these n comparisons There are two p ossibilities

u

i There are h n of these elements v of A such that v x R

u e

denote these elements by v v

 h

ii There are h n of these elements w of A such that w x R

u e

denote these elements by w w

 h

s

In case i we will simply ensure a a But then a is preceded by exactly

z z

z

z elements in L where z n whereas x a is preceded by at least

u k z

n elements in hA R iThus is not an isomorphism from L onto

u e k

A R

e

s u

In case ii we will switch a a from the side of L to the side

z z

u

of L That is we shall let n n z and let b a

n i u u

u

n i

u

u

u

for i n n We also let b b for i n Then our

u u u

i

i

s u t

a Thus in this b construction will ensure that for all t u b

z n n

u u

case there will b e precisely n elements w namely b b suchthat

u  n

u

w b However in A R there are at least n elements x such

n L e u

u

that x b R Butn n zn so that

k n e u u u u

u

cannot b e an isomorphism from L onto A R Our construction will ensure

e

s

that a can switchfromthe side to the side of L only for the sakeof

z

s

requirements P P The usual priority argument will then showthata

 z

z

switches sides for at most nitely many s

We shall employa setofmovable markers to help us keep trackofwhich

e

requirements wehave acted on The idea is that if wehavetaken an action as

u

will witness that requirement P is satised describ ed ab ove which ensures a

z

z

u

then we will place a marker on a Thus at anygiven stage s either is

z z

z

inactiveie do es not rest on anyelement at stage sor is active ie

z z z

s s s s

gIf is active we let sx b a b rests on some element x fa

z z

n n  

s s

where x is the elementonwhich is placed

z

CONSTRUCTION

Stage

   

Let a b and declare a b Welet b e inactive for all z

 z

   

at stage

s s s s

Assume wehave dened n n a b a b so that n s Stage s

s

  n n

and

s s s s

g f g B b a b fa

 n s

n n  

Moreover assume wehave dened a linear order on B B so that

L s s

s s s s s s

a a a b b b

 n n n 

s s

Lo ok for a p s suchthat is inactive at stage s and a where p e k

p

p

k

If there is no such p then for all z let b e inactive at stage s if and only

z

if is inactive at stage sIf is active let s s In addition

z z z z

s s

s s

let a a and b b for all i n n Finallylet n n

s s

i i

i i

s s

a and b and extend our denition of to

n n L

n n

B B by declaring

s s

s s s s

b b a a

 n n 

If there is suchap let p ps es ks e k b e the least such

s

p and x xs a If xs A then pro ceed exactly as in the case

k

p

s

where ps is not dened except declare active and let s a

p p

p

If xs A then nd the rst n elements of ANow compare these

s

elements to x with resp ect to R We will then b e in one of two cases

e

Case There are h n elements v v among the rst n

s  h s

elements of A such that v x R for i h In this case we

i e

pro ceed exactly as in the case where xs A

Case Otherwise there must b e h elements w w among the rst

 h

s

s

n elements of Asuchthatw x R for all iThenweleta a for

s i e

i

i

 s s

s s

a i pand b b for all j n n Setn n pLetb

s s

ni j

ni j

s

for j n p Activate for i n p and let a

nj

pj

s s

the marker and place it on b a Removeanymarkers that were on

p z

n p

s

s s

elements among a a and make them inactive Any marker whichwas

z

p n

s s s s

active at stage s where s fa a b b g is still active at stage

z

 p  n

s and s s Markers where z p whichwere inactiveat

z z z

stage s remain inactive at stage s Finally extend to B B

L s s

by declaring

s s

s s

a a b b

n n

 

s s

This complete our construction Because A is a p olynomial time set and each

R is a p olynomial time relation it easily follows that each stage is completely

e

eective The following facts are easily proved by induction

For all s n s

s

s s s s

s fa b a b g f g For all

 n 

s

  n n

s s

Our denition of is consistent that is if i j n and stage swe

L s

declare then for all t swe declare at stage t

i L j i L j

L

Note that these facts imply that L f g is a recursive linear

ordering b ecause to decide if we simply go to stage s maxfi j g and

i j

then if and only if at stage sw e declare

i j i j

Next we provetwo lemmas which will complete the pro of that L has the

desired prop erties

s s

b exist and thereisa a and lim b Lemma For each z lim a

z z s s

z z

stage t such that either is inactive at stage s for al l s t or is active

z z z z

at stage s and s t for al l s t

z z z z

Pro of We pro ceed by induction on z e k By induction we can assume

that there is a stage uz large enough so that

s u s u

i a a and b b for all jz and s u and

j j j j

ii for each jz either is inactive at stage s for all su or for all s u

j

is active at stage s and s u

j j j

Note that our construction ensures that a marker can b e removed from

j

s

an elementatstage s only if s a for some k and we takeactionto

j

k

meet a requirement at stage s where ps k Similarly the only way

ps

s 

s

a a is if ps k and we act according to Case at stage s

k

k

t s

Moreover our construction ensures that if j n then b b for all s tIt

s

j j

s u

follows that lim b b Nowif is active at stage sthenourchoice of u

s z

z z

s u

ensures ps z for all suso that lim a a and is active for all su

s z

z z

If is not active at stage utheneither

z

s u

i a for all s uinwhich case for all s u is inactive at stage s

z

z

k

s u

ps z and a a or

z z

s u

ii There is an su suchthat a

z

k

u t

a Then our choice In case ii let t b e the least susuchthat

z

k

of u ensures that for all u s t ps z and is inactive at stage s

z

so that pt will b e dened and pt z But then at stage t

z

 t t t

b ecomes active and is placed on either a or b If is placed on a

z

z z n

t

t t t

then a a and will never b e removed from a This is b ecause can

z z

z z z

t

b e removed from a only if ps z for some st which is ruled out by

z

t

our choice of uIf is placed on b then again can never b e removed

z z

n

t

t

Thus in either case will remain active for all s t But this from b

z

n

t

s s

for all s t a means ps z for all s t sothat a

z z

Lemma Requirement P is satisedforallp

p

Pro of Let p e k and let s b e a stage such that s p and

p p

s s

p p

s s

i s s j pa a and b b and

p

j j

j j

ii s maxft t g where t is a stage such that either

p o p z

a for all s t is activeat sor

z z

b for all s t is inactiveat s

z z

e at stage s It then easily follows from our construction that if is inactiv

p p

s

p

s s s s

a and a a for all s s Thus a where a lim a then

p

p k p p s

p p p

k

Hence the requirement P is automatically satised If is activeatstages

p p p

s

then there are two p ossibilities The rst is that sa a inwhichcase

p p

p

s s

our construction guarantees that either a A or a A but there are

p p

k k

at least p elements v v A suchthatv a R for i p

 p i k p e

s

The second p ossibilityisthat sb b for some m n In this case

p m s

m p

s s

our construction ensures that b A and there are at least m elements

m

k

w w A such that w b R for i mThus in any

 m i k m e

case P is satised

p

It now follows from Lemma that a a and b b for all i and

i L i i L i

that a b for all i and j sothatL is isomorphic to By our remarks

i L j

preceding the construction of L it follows that meeting all the requirements

P ensures that L is not recursively isomorphic to any p olynomial time linear

p

ordering over A

In the general case if is a recursive linear ordering then a nite injury

priority argumentvery similar to the one given ab ovefor will prove

the theorem However the only thing that we can conclude from the fact that



Z is a recursive linear ordering is that is a linear ordering and



in that case a more complicated innite injury priority argument is required

We note that in the sp ecial case of one can make K haveuniverse

Tal when A Bin This is Theorem of

Cenzer and Remmel gave a general condition which implies that a re

lational structure is recursively isomorphic to a ptime structure with universe

Bin This condition can b e though t of as a generalization of the argument

in case of Gregorie s Theorem on linear orderings

Denition Let A bea recursive substructure with universe A of the re

cursive relational structure M with universe M ThenA is said to be a highly

recursive relatively indiscernible binary substructureofM if

Thereisarecursive map f from A to Bin which induces a ptime

f

model MLet a f bini

i

For any mary relation Rx x of Manyxedsequence

m

b b M n A with k m and any xedsequence I I mlet

k k

b b

k

R denote the m k ary relation on M A which results by substituting

I I

k

b for x for j k

I I

j j

Then for any such R I I m and b b M n A either

k k

b b b b

k k

mk

R holds for al l but nitely many elements in M A orR

I I I I

k k

mk

holds for al l but nitely many elements in M A

Furthermore there is a uniform eective algorithm which given an index for

b b

k

R and sequences b b M n A and I I wil l compute whether R

k k

I I

k

b b

k

mk

holds for al l but nitely many elements in M A or R holds for

I I

k

mk

al l but nitely many elements in M A along with with a complete list

mk

of the nitely many sequences of M A which are exceptions

Theorem Suppose that A is a highly recursive relatively indiscernible bi

nary substructure of the relational structure M Then M is recursively isomor

phic to a ptime structure with universe Bin and thereforealsorecursively

isomorphic to a ptime structure with universe Tal

Pro of There is no loss in generality in assuming the universe of M is the set

of natural numbers Let S S b e an eective list of all relations of A and



assume that there is a recursive function F such that S is an F iary relation

i

f

Let A with universe A fa a g b e isomorphic to M as describ ed ab ove



and let n A fb b gFor any x n Alettx b e the total time



needed to search all elements y x and determine if y A and to compute the

b b

k

and the nite list of exceptions common value of all relations of the form R

I I

k

S for some i x and b b are elements of M A whichare where R

i k

xtx

less than or equal to x Let x and let P fx x M n AgIt

is clear that P is a ptime subset of Tal

We dene a p olynomial time mo del D with universe Bin by dening

D

the relation S for D as follows Given an element bini search the strings

q

k

of the form for k jbinij and determine whether eachsuch string is in

b tb 

n n

P If bini P thenlet r bini binn where bini If

bini P then let r bini binmwherebiniisthemth elementof

Bin n P Note that b ecause P is a p olynomial time set there is a xed

p olynomial p such that we can compute whether x P in pjxjsteps It

P

jbinij

k

thus takes at most pj steps to search the strings of the form for

j 

b ership in P so that the function r is p olynomial time k jbinij for mem

D

Then S s s holds if either

F q 

q

f

A no s P and S r s rs holds in Mor

i q

F q 

B there is some s in P and S Z Z holds in M where Z b

i q j k

F q 

if s P and r s bink and Z a ifs P and r s bin k

j j j k j j

D

Note that in case A we can compute whether S s s holds in

F q 

q

f

time p olynomial in js j js j since M is a p olynomial time mo del In

F q 

case B let n b e the maximum value such that there is an s P with j F q

j

b tb 

n n

and r s binn If n q thens and in tb steps wecan

j j n

compute whether R s s holds by the denition of tThus in case

q

F q 

B the only cases in whichwe can not directly compute in linear time whether

D

S s s holds is if n q However it is easy to see that our assumptions

F q 

q

ensure that it takes only a nite amount of information to determine whether

D D

is a p olynomial time relation s s holds in all such cases Thus S S

F q 

q q

Finally it is easy to see that our denitions ensure that wehave dened

the map g where g bini b ifbini P and r bini binn and

n

g bini a if bini P and r bini bin n is an isomorphism from D

n

onto M

C

The mo del C with universe Tal simply has relations R dened by

C D

R tal i tali R bini bini

n n

C

The relation R is ptime since bini can b e computed from tal i in p olynomial

time and jbinij jtal ij

M

For a simple example consider a wellordering M M oftyp e

M

and let A A betherst elements of M A is a recursive set since

M

x A x w where w is the th elementofM A is certainly

recursively isomorphic to the standard ordering on Bin and is indiscernible

M

since for all a A and all x M n A a xThus by Theorem M is

recursively isomorphic to a ptime mo del with standard universe Bin This

is a sp ecial case of Theorem ab ove

Bo olean Algebras

There are some cases of structures with function symb ols where we can get some

p ositive results For example every recursive Bo olean algebra is recursively

isomorphic to a p olynomial time Bo olean algebra

Denition i The language L of Boolean algebras consists of two binary

function symbols meet and join one unary function symbol

complement and two constant symbols zero and unity A Boolean

B B B B B

for this language which algebra B is a structure B

satises the usual axioms

ii Given a linear ordering M M with a rst element the interval

algebra IntalgM is the Boolean algebra of subsets of M generatedbythe

leftclosed rightopen intervals of M ab fx a xbg

B B

The partial ordering of a Bo olean algebra B is dened by a b if and

only if b a c for some c Anelement a of a Bo olean algebra is said to b e

an atom if whenever b a either b orb a The Bo olean algebra A is said

to b e atomic if for any b there exists some atom a with a b A element

x B is atomless if x is not the zero of B and there is no atom a of B suchthat

B

a x B is said to b e nonatomic if B contains an atomless element

The following is Lemma of

Lemma For any ptime linear ordering L with a rst element the interval

algebra I ntal g L is a ptime Boolean algebra

Sketch of Pro of The nonzero elements of IntalgLaregiven the natural

L L L

representation a a a a a a where a a

 n n 

or a L and a a a and either a

n n n n n

This lemma is used to prove the following result from

Theorem Every recursive Boolean algebra B is recursively isomorphic to

a ptime Boolean algebra

SketchofProof First observe that the classical pro of that every count

able Bo olean algebra is isomorphic to the interval algebra of a countable linear

ordering is eective See Remmel Thus every recursive Bo olean algebra

is recursively isomorphic to IntalgM where M is a recursive linear ordering

However by Theorem M is recursively isomorphic to a p olynomial time

linear ordering P Theinterval algebra of P is thus recursively isomorphic to B

and is p olynomial time by Lemma

The next two theorems are unpublished results due to Cenzer and Remmel

Theorem Every innite nonatomic recursive Boolean algebraisrecur

sively isomorphic to a ptime Boolean algebra a with universe Bin b with

universe Tal

Theorem Let A f g be an innite ptime set and let B be an innite

atomic recursive Boolean algebra Then thereisa recursive Boolean algebra D

which is isomorphic to B but is not recursively isomorphic to any ptime Boolean

algebra with universe A

Graphs

Next wegivetwo applications of Theorem to recursive graphs due to Cenzer

and Remmel in

on i Agraph G V E is lo cally nite respectively lo cally Deniti

conite if for every v V the set NBv of neighbors of v is nite resp

conite

ii Agraph G V E is lo cally niteconite if for every v V either

the set NBv of neighbors of v is nite or V n NBv is nite

iii Alocal ly niteconite recursive graph G is highly recursive if thereare

algorithms for deciding whether a given v V has nite degree and for

computing NBv is the degree is nite and V n NBv if not

Theorem Every innite highly recursive local ly niteconite graph G is

recursively isomorphic to a ptime graph H with universe Bin

Sketch of Pro of If there are innitely manyvertices of G whose degree

is nite then we can construct a recursive subset of suchvertices U suchthat

G restricted to U is the empty graph Moreover we can construct U so that it

will b e the highly recursive relatively indiscernible binary substructure required

to apply Theorem If there are innitely manyvertices of nite codegree

We can construct a a complete subgraph C which will b e the highly recursive

relatively indiscernible binary substructure needed for Theorem

Theorem Every innite recursive graph G which is either local ly nite

or local ly conite is recursively isomorphic to a ptime graph H with universe

Bin

SketchofProof Supp ose that all vertices have nite codegree Then

we again pick out a complete subgraph C fv v gE now with the



additional prop ertythatv is not joined to anyvertex from f i g

i

The next result shows that the hyp otheses of the two preceding theorems

are needed

Theorem Let A be any innite polynomial time subset of f g Then

thereisarecursive graph having every vertex of either nite degreeornite

codegree which is not recursively isomorphic to any ptime graph with universe

A

Trees

A connected graph G with no cycles is said to b e a treeThevertices of T are

called nodesWe will assume that any tree T has a designated ro ot Thenany

no de v of T can b e reached from the ro ot byapath that is a sequence of edges

v v v v v Wesay that v is a successor of v It is clear

 n n

that the successor relation is recursive since the path from the ro ot to a no de

T

v may b e computed from v in a uniform fashion The partial ordering u v

whichsays that there is a path from the ro ot to v which passes through uis

also recursive On the other hand if T is a ptime tree then this computation

of the path from to v might not b e in p olynomial time in jv jsothatthe

T

successor relation and the relation need not b e ptime Thus wesaythat

T

T is ful ly ptime if b oth the successor relation and the relation are ptime

Similar notions may b e dened for any b ounded resource class

cursive if there is a recursive function which computes T is said to b e highly re

from anynode v a list of successors of v The corresp onding notion of a highly

feasible tree or more generally of a highly feasible graph is more dicult to

formulate Several inequivalent notions are studied in In particular

a graph is said to b e local ly ptime if there is a ptime function which computes

from anynodev a list of successors of v andissaidtobe highly ptime if there

is a ptime function which computes from a vertex v and a tally number tal n

a list of all vertices at distance n from v

The following results are Theorems and of

Theorem Any innite recursive tree T is recursively isomorphic to a p

time tree with universe Bin

Sketch of Pro of There are two cases First every no de of T mayhave only

nitely many successors In this case Theorem may b e applied Second

some no de v of T mayhave an innite set A of successors In this case the set A

plays the role of the highly recursive relatively indiscernible binary substructure

so that Theorem may b e applied

Theorem There is a highly recursive tree T which is not recursively iso

morphic to any ful ly primitive recursive tree P with a standard universe Bin

Theorem There is a highly recursive binary tree which is not isomorphic

as a directedgraph to any highly primitive recursive tree with universe Bin

A similar but stronger result for graphs was given in Theorem of

Theorem There is a highly recursive graph G which is not isomorphic to

any local ly primitive recursive graph

Equivalence Structures

A

Another typ e of relational structure is an equivalence structure A R where

A A

R is an equivalence relation on A A recursive equivalence structure A R

is said to b e highly recursive if the set of elements that b elong to innite equiv

alence classes is recursive and there is a recursive function f such that f ais

R R R

the cardinalityofa when a is nite so that the equivalence class a can

A

ishighly ptime if A is a ptime b e computed from a WesaythatA R

subset of Bin the set of elements that b elong to innite equivalence classes

R

is ptime and there is a ptime function f suchthatf a co des a when it is

A

nite The ful l spectrum of of A R is the set of pairs nsuch that there

A

are at least n innite equivalence classes in A R and pairs q nsuch

A

that q and there are at least n equivalence classes of size q in A R

The following results are Theorems and of

A

Theorem Any recursive equivalence structure A R is recursively iso

morphic to a ptime model with universe Bin

A

Sketch of Pro of There are two cases If all equivalence classes of A R

A

are nite then Theorem can b e applied to A R viewed as a graph If

A

A R has an innite equivalence class then this class is a highly recursiv e

relatively indiscernible substructure so that Theorem may b e applied

A

Theorem For any equivalence structure A R with ful l spectrum S

such that S fhtal q tal r i q r S g is ptime thereisahighlyre

A

cursive ptime equivalence structure Bin R isomorphic to A R

Theorem There is an innite recursive ful l spectrum S of such that no

highly primitive recursive equivalence structure with universe Bin has ful l

spectrum a subset of S

Wenow turn to the study of some structures with functions There are three

basic mo dels whichwehave b een considered rst mo dels of some fragmentof

arithmetic second Ab elian groups and third p ermutation structures

Mo dels of Arithmetic

Our rst result taken from demonstrates that the unary exp onential func

x

tion may b e adjoined to the standard mo del of arithmetic while b eing rep

resented by a ptime function

x

Theorem N S is isomorphic to a ptime structure

M

Sketch of Pro of The elements of M are terms in the language

fAIEg The natural number n is represented by the expression n dened

as follows

k

Ek

k k

mAE k mfor m

k k

mIEk mfor m

It is easy to see that the universe of M is a linear time set and that the term

n which represents n can b e computed from binn in p olynomial time It



can b e shown by induction that j nj jbinnj and that

M

j j j j j j

We note that it is an op en question whether S exp is is

n

isomorphic to a p olynomial time mo del where expm n m is the general

x x

exp onential or even whether S is isomorphic to a p oly

nomial time Bauerle proved that in the mo del of Theorem the function

x

is not p olynomial time

We also note that the mo del of Theorem can b e used to build a EXPTIME

group isomorphic to the integers under ZZ whichisnotq time iso

morphic to the standard mo del of Z where the p ositiveinteger n is co ded as

binn and a negativeinteger n is co ded as binn see section

Injection Structures

A

The simplest typ e of structure with a function is an injection structure A f

A A

where f is a onetoone mapping from A into itself If f maps A onto A then

A

wesay that A f isapermutation structureTheorbit O aofanelement

n n

otyp es of a of A is fb A n f ab f b ag There are tw

innite orbits one of typ e which is isomorphic to S and the other of typ e

Zwhich is isomorphic to ZS The order jaj of a is car dO a if O a is nite

A

and is either or Zotherwise The ful l spectrum of A f isthesetofpairs

nsuch that there are at least n orbits of typ e pairs nsuchthat

there are at least n orbits of typ e Z and pairs q nsuchthatq and

A

there are at least n orbits of size q inA f

It is easy to see that the full sp ectrum of a recursive injection structure is

always a recursively enumerable set It is shown in Theorem of that

any re sp ectrum can b e realized by a ptime injection structure Thus every

recursive injection structure is isomorphic to a ptime structure However we

knowby Theorem that this isomorphism need not b e recursive Nowwe

consider the question of when we can obtain a ptime injection structure with

a standard universe Bin orTal

The basic result here is Theorem of

A

Theorem Any recursive permutation structure A f with al l nite or

bits is recursively isomorphic to an honest ptime structure with universe a sub

set of Tal

SketchofProof The element a A is represented by tal n where

a t

binn and t is the total time required to compute the orbit of aThe

details follow as in the pro of of Theorem ab ove

Theorems and of givetwo cases in whichwe can improve this

result to obtain a standard universe

Theorem Let B Bin or Tal Any recursive injection struc

ture A f with at least one but only nitely many innite orbits is recursively

isomorphic to a ptime structure with universe B

Sketch of Pro of Let F fa A jaj is niteg and let I A n F Since

there are only nitely many orbits in I b oth F and I are recursive It is easy

to see that I f is recursively isomorphic to a ptime structure with universe

B By Theorem F f is recursively isomorphic to a ptime structure with

universe a subset of Tal The result now follows from Lemma

Theorem Let B Bin or Tal Any recursive injection structure

isrecursively A f with innitely many orbits of size q for some nite q

isomorphic to a ptime structure with universe B

Sketch of Pro of Let C fa A jaj q g and let D A n C It follows

from Theorem that D f is recursively isomorphic to a ptime structure

with universe a subset of Tal It therefore suces by Lemma to show

that C is recursively isomorphic to a ptime structure B g For B Bin

the p ermutation g is simply dened by g binnq i binnq i if

i q and g binnq q binnq The tally denition is similar

A general result on the existence of ptime injection structures is given by

Theorems and of

Theorem a For any re ful l spectrum S thereisaptimeinjection

structure A f with ful l spectrum S

b If fhtal q tal r i q r S g is ptime then A may be taken to be

either Bin or Tal

Sketch of Pro of Wesketch the pro of of part b for B Tal assuming

that all orbits will b e nite Let q q enumerate in nondecreasing order



the set of orbit sizes with rep etitions Then the p ermutation f may b e dened

by

f tal q q q r tal q q q r if rq

 k  k k

and tal q q q q if r q

 k k k

An innite orbit of typ e is given by the standard successor function on Tal

and an orbit of typ e Z is given by f tal n tal n f and

f tal n tal n Multiple innite orbits and a combination of nite

and innite orbits may then b e obtained by Lemma

Corollary Any recursive injection structure is isomorphic to a ptime

injection structure

Finallywe consider some negative results The rst is Theorem of

and deals with structures with a xed universe

Theorem Thereisarecursive set M such that no injection structure with

ful l spectrum M is isomorphic to any primitive recursive structure with universe

Bin or Tal

Sketch of Pro of Let f enumerate all primitive recursive unary functions

e

and let B Bin f Construct a set R f r r g by a diagonal

e e 

argument so that for all e either

f is not onetoone or

e

B has an innite orbit or

e

B has twodisjoint orbits of the same nite size or

e

B has an orbit of size q R

e

We establish this requirement given r r by computing enough

 e

of B to either nd two orbits of the same size or an orbit p erhaps incomplete

e

of size rr If the orbit is complete welet r r thus ensuring that

e e

B has an orbit of size r R If the orbit is incomplete wecontinue to build the

e

orbit at later stages and take a similar action when the orbit b ecomes complete

If this never happ ens then the orbit is innite so that is satised

It follows that no primitive recursivepermutation structure with all nite

orbits can have M fr r Rg as a subset of its full sp ectrum

The nal result here is Corollary of and deals with the question of

recursive isomorphism The pro of is omitted

Theorem For any recursive injection structure C f with innitely many

A

innite orbits thereisarecursive structure A f which is isomorphic to

C f but is not recursively isomorphic to any primitive recursive structure

Ab elian Groups

Wenow turn to the study of feasible versus recursive Ab elian groups The results

here are parallel to those for p ermutation structures We b egin by recalling some

basic notation For any natural number n Zn is the cyclic group of order

nFor a prime number p the group Zp istheinverse limit of the sequence

n

Zp or more concretely the set of rational numb ers with denominator equal

to a p ower of p and where the group op eration is addition mo dulo The group

Zp issaidtobe quasicyclic The additive group of rational numb ers is

denoted by Q

A A A

For any element a of an Ab elian group A A and anyinteger

A

n n a is dened recursively by a and n a a n a Then

A A

n a n aTheorder jaj of a is the least n suchthatn a

A

A is said to b e torsion if all elements have nite order and torsionfree if all

elements except the identity have innite order

We will frequently b e concerned with pro ducts of Ab elian groups

Denition For any sequence A A of Abelian groups where A

 i

A e and A f g the direct product A A is denedto

i i i i i n n

have domain

A fh i k A for i k and e g

 k i i k k

A A A

The identity of A is e and addition and subtraction are dened

A A

as fol lows for h i and h i

 m  n

h iwher e k maxfi i m i n e m

 k i i i i i

i n ni mg and for i k

for i minm n

i i i i

for n i k

i

i

for m i k

i

In particular we write G to be the direct product of a countably innite

i

number of copies of G

et B be either Bin or Tal For any complexity class Denition L

thesequence A A of groups where A A e is said to be

 n n n n n

uniformly computable over B if

i fhbnai a A g is a computable subset of B B where bnbinn

n

if B Bin and bntal nifB Tal

ii The functions F bnab a b and Gbnab a b are b oth

n n

the restrictions of computable functions from B into B where we set

F bnab Gbnab if either a or b is not in A

n

iii The function from Tal into B given by etal i e is computable

i

The following is Lemma of

Lemma LetBbe either Bin or Tal and let be one of the fol lowing

complexity classes recursive primitive recursive exponential time polynomial

time Suppose that the sequence A A e of Abelian groups is

i i i i i

computable over B Then

a The direct product A of the sequenceisrecursively isomorphic to a

computable group with universe containedinB

b If A is a subgroup of A for al l i and if thereisa computable function

i i

f such that for al l a A a A then the union A is recursively

i i i i

f a

isomorphic to a computable group with universe containedinB

c If the sequence is nite and one of the components has universe B then the

product is recursively isomorphic to a computable group with universe

B

d If the sequence is innite and if each component has universe B then the

product is recursively isomorphic to a computable group with universe

Bin

e If each component has universe Tal and thereisauniformconstant

c such that for any i and any a b A ja bj and ja bj areboth

i i i

cjaj jbj then the product is recursively isomorphic to a computable

group with universe Tal

n

i

of prime If a torsion Ab elian group G is isomorphic to a direct sum Zq

i

i

power cyclic groups then we dene the characteristic G tobe

n

m m

i

fp kq p for at least k distinct values of ig

i

Khisamiev shows in Corollary of that for any k and any torsion

Ab elian pgroup G G Zp is isomorphic to a recursive group if and

ik



only if Gisa set Wewillsay that a subset Q of is hereditary if



m k Q implies m k Q for all m k It is clear that a subset Q of

is the characteristic G for some Ab elian torsion group G if and only if

Q is hereditary and m k Q implies m is a prime p ower Therefore we will

say that any such set Q is a characteristic

The following are results and of

Theorem Each of the groups Z Zk Zp and Q are isomor

i

phic to polynomial time groups a with universe Bin and b with universe

Tal

Theorem Any nitely generatedrecursive Abelian group is recursively

isomorphic to a ptime Abelian group a with universe Bin and b with

Tal universe

The simplest torsion groups are primary groups or pgroups in whichevery

element has order a p ower of p where p is a prime In Smith characterized

the recursively categorical pgroups as follows

Theorem Smith A recursive pgroup G is recursively categorical i ei

ther

G Zp F or

in

m

G Zp Zp F where F is a nite pgroup and m and

in i

n are nonnegative integers

Corollary Any recursively categorical pgroup is recursively isomorphic

to a polynomial time group a with universe Bin and b with universe

Tal

Note that not every pro duct of cyclic groups is recursively categorical For

example consider Z Z The following fundamental result from

i i

shows that this group is recursively isomorphic to a ptime group

G G G

Theorem Any recursive Abelian torsion group G G e is

recursively isomorphic to a polynomial time group H with universe a subset of

Tal

Sketch of Pro of It suces by the remarks following Lemma to dene

H

a ptime group H with universe a subset of Bin suchthatbotha b and

H

a b have length b ounded by some constantmultiple of jaj jbj

Let A b e the subgroup generated by f k gRenumb er the elements

k

of A as a a so that the elements of A precede the elements of A nA

 k k  k

This can b e done so that the map taking i to a is a recursive isomorphism Now

i

k tk 

map a A to a where tk is the total time required to compute

k k

G

the op eration table for A Thekey to the pro of is that whenever a a a

k i j k

where i j then wehave tk tj since a will b e in the group generated

k

by fa a gFurthermore ja j k tk tj ja j jsince

 j k i

k tk

We need an eectiveversion of the Fundamental Theorem of Ab elian groups

which states that every torsion Ab elian group is a direct pro duct of primary

groups This is Lemma of

Theorem Any recursive Abelian torsion group G is recursively isomorphic

to a ptime direct product of primary groups over B where B may be taken to

be either Tal or Bin

The main result on the existence of feasible groups with standard universe

is the following theorem from

Theorem Let G be an innite recursive Abelian group with bounded or

der Then G is recursively isomorphic to a polynomial time group with universe

Tal and to a polynomial time group with universe Bin

Sketch of Pro of Let B b e either Bin orTal Wemay assume that

G is ptime by Theorem Since the orders are b ounded there is no loss

m

of generality in assuming that G is a p group for some prime pLetp be

the largest order of an elementofG Theproofisby induction on mWecan

express G as a pro duct HK where H is generated by some indep endent set

m

of elements of order p and K is maximal indep endentofH with no elements

m

of order p There are twocases

Case If H is nite then K is innite and may b e assumed to haveuniverse

B by induction The result now follows from Lemma

m

Case If H is innite then H is isomorphic to Zp and is therefore

i

recursively isomorphic to a ptime group with universe B by Theorem

Since K is recursively isomorphic to a ptime group with universe a subset of

Tal by Theorem the result again follows from Lemma

Next we state some results on characteristics The rst result here follows

from Theorem together with the theorem of Khisamiev cited ab ove



Theorem For any characteristic Q there is a ptime Abelian group



with characteristic Q

We will show in Theorem that not all recursivecharacteristics can b e

realized by ptime groups The next result shows that any ptime characteristic

can b e so realized

Theorem Let Q be a nonempty innite characteristic such that

m m

tal Q ftal p k p k Qg is a ptime set Then there exists a p

time Abelian group with characteristic Q and universe B where a B Bin

or b B Tal

Theorem Thereisarecursive characteristic M such that no recur

sive group G with characteristic Q is can be isomorphic to any primitive recursive

group with universe Bin or Tal

Pro of Let A A b e an eectiveenumeration of the primitive recursive



structures f with one binary op eration f LetQ fp p A g

e e e e

aj b e the order of a in A Construct a set R fr and for any a A let j

e e  e

r g of prime numb ers such that for any e A either

e

is not an Ab elian group with identityor

has an element of innite order or



has an element of order p for some prime por

has two subgroups of the same prime order or

has an element of prime order p Q

Now the order jaj of an element a may not b e a prime Therefore we need

e

some way to control the prime factors of jaj We will now dene a recursive

e



function such that for any q and any rq r either is divisible by p for

some prime p or r has a prime factor bigger than q Given q q is simply the

pro duct of all of the prime numbers p q

We will dene the set Q in stages Atstage swe will have s elements

s s

q q q in Q along with a certain nite subset I of of restraints

 s

which will preventnumb ers from coming into Q at stage s or at any later stage

 

Let q Q fg and I



The initial stage of the construction pro ceeds as follows Compute f



There are then two cases

Case If f then wehave Q or else A is not an Ab elian

  

group with identityThus we can ensure that Q is not the characteristic of



A by setting q and restraining from ever coming into QWeletI



Case If f thenweknow that either A is not an Ab elian group

 

with identity or jj and therefore either has a prime factor qoris



divisible byNow let q andletI f g This means that either



A will have an element of order thus satisfying part of the requirement



or wewilleventually restrain at least one of the prime factors of jj from ever



coming into Q

s

At stage s we are given q q and the set I of previous restraints

 s

s

Moreover assume by induction that for every a e I either

i jaj q and there is a prime factor qq of jaj suchthatq q for

e s s e i

any i sor

ii jaj q andjaj is divisible by the square of a prime or

e s e

iii jaj q

e s



Nowlet k q and compute i a in A for each a k and each i k

s s

This will pro duce a set of equivalence classes a where a and b are equivalent

if either b i a or a i b for some i k Note that every number a k

b elongs to some equivalence class but numb ers greater than k can also b elong

A pro duces a sequence of Nowallwe need is that the computation of i a in

s

distinct elements up until it pro duces If this is ever violated then A is not

s

an Ab elian group so that we will have satised the eth requirement In this

s s

case welet I I and wecho ose q to b e the least prime pq which

s s

s

do es not violate any of the restraints t b I That is q is the least q

s

s

such that i ii or iii ab ove are satised for all restraints in I Otherwise

there are two further cases

Case There is some equivalence class which has more than q elements

s

In this case let a b e the least suchthata has more than q elements It

s

follows that jaj q Nowputa sinto the set of restraints so that

s s

s s

I I fa sg Since we will keepthisrestraint active throughout the

construction it will b e the case that either jaj j or else it is nite and has

s



a prime factor p such that either p Q or p divides jaj

s

Then let q b e the least prime p q which do es not violate anyofthe

s s

s

restraints t b I That is q is the least q such that i ii or iii

s

s

ab ove are satised for all restraints in I

Case Each class has q or fewer elements In this case jaj q

s s s

for all a k and each equivalence class is a cyclic subgroup of A Now since

s



k q there must b e at least q dierent subgroups among the classes

s

s

Since there are no more than q p ossible orders that is numb ers b etween

s

and q for these subgroups there must b e two distinct subgroups of the

s

same order in A It follows from this that A has two distinct subgroups of

s s

some prime order and hence part of the sth requirement will b e satised

Then we again let q b e the least qq which do es not violate anyofthe

s s

s

restraints t b I

This completes the construction The set Q fq q g is recursivesince



q Q sqq q Let M Q fg

s

Now supp ose that A M for some s Then A do es not havetwo

s s

distinct subgroups of the same order so that Case do es not apply at stage

s Thus Case must apply and hence there is an element a with nite order

t

jaj qq such that a sisinI for all t sThus there must b e a

s s

stage tssuch that either condition i or ii is satised That is jaj q

s t

and there is a prime factor qq of jaj such that either

t s

i q q for any i t or

i



ii jaj is divisible by q

e



In case i wehaveq A butq Q In case ii wehaveq A

s s



but q M In either case we see that A is not a subset of M

s

Corollary Thereisarecursive torsion Abelian group A which is not

isomorphic to any primitive recursive Abelian group with universe Bin or

Tal

For groups of unb ounded order Theorem of and Theorem of

give dierent results

Theorem

a Thereisarecursive Abelian group A which is not recursively isomorphic

to any primitive recursive group

b Thereisanexponentialtime Abelian group B which is not recursively

isomorphic to any polynomialtime group

Theorem Thereisarecursive torsionfreeAbelian group which can

not beembedded into any ptime Abelian group

Uniqueness of Feasible Structures

In this section we shall survey results on feasible categoricity Again weshall

concentrate mainly on p olynomial time structures As we shall see unlikere

cursive mo del theory where there are many b eautiful classication results on

recursively categorical structures there are very few examples of p olynomial

time categorical structures even if we restrict the universe Thus most of the

results on p olynomial time categoricity are negative Recall that a structure A

with universe B is said to b e ptime categorical over B if any structure D with

universe B which is isomorphic to A is in fact ptime isomorphic to A A similar

denition can b e given for other notions of feasibilityWe note that restricting

the universe is crucial if we are to haveany p ositive results due to the following

general theorem of Cenzer and Remmel

A

Theorem For any ptime relational structur e A A fR g there

iS

i

are innitely many ptime structures B A B B fR g B

 iS 

i



B fR g which areeach recursively isomorphic to A and such that

 iS

i

for each m nthere is a ptime map taking B onetoone and onto B but

n m

there is no primitive recursive map from B into B which is at most c to

m n

for some nite number cFurthermore the universes B may be taken to be

n

for each n subsets of Tal

Sketch of Pro of Let B A and B A Given B fbinb

  n 

binb gletB M B fbinm binm g as dened

n n 

ab ove in the pro of of Lemma and dene the relations R on B to make

i n

the map taking binb tobinm an isomorphism

e e

Linear Orderings

In this subsection wesurvey results of Remmel whichwas the rst pap er

ed that there are no on p olynomial time categoricity Remmel essentially show

p olynomial categorical linear orderings over either Tal orBin

The classic backandforth metho d of Cantor whichshows that anytwo

dense linear orderings without end p oints are isomorphic is crucial to the study

of categoricity in linear orderings The key step in dening an isomorphism

between two structures requires a way to select given two elements abof one

structures an element ca an element d band an element e with aeb

Thus we are led to the following eective notion of density functions in the eort

of nding conditions which will provide some form of feasible categoricity

Denition A computable dense linear ordering L D without end

points is said to have computable density functions if thereare computable

functions f f and f such that for any x and y in D f x xf x and

a b i b a

x f x y y

i

By carefully following the backandforth argument and keeping trackofthe

numb er of steps required we obtain the following Theorems and of

Theorem Suppose L B and L B arepolynomialtime

 

dense linear orderings without endpoints with polynomialtime density functions

Then

a if B Tal L and L are doubleexponentialtime isomorphic and



b if B Bin L and L are tripleexponentialtime isomorphic



Theorem Suppose L B and L B arepolynomialtime

 

dense linear orderings without endpoints with lineartime density functions

Then

a if B Tal L and L are exponentialtime isomorphic and



b if B Bin L and L are doubleexponentialtime isomorphic



Theorem Suppose L Bin and L Bin are

 

polynomialtime dense linear orderings without endpoints with quasirealtime

density functions Then L and L are exponentialtime isomorphic



Note that the standard ordering on the dyadic rationals in the interval

is in fact a ptime linear ordering with quasireal density functions and has

universe ptime isomorphic to Bin Details are given in Theorem of

On the other hand there are ptime structures without nice density functions

as shown by Corollary of

Theorem There exist ptime dense linear ordering without end points with

universe B for B Bin and B Tal which have no primitive recursive

density functions

We note that there is a p ossible p ositive result namely one can showthat

anytwo ptime linear orderings with universe Tal whichhave quasirealtime

density functions are p olynomial time isomorphic However Ash showed that

there are no ptime linear orderings with universe Tal whichhave quasireal

time density functions see

Examination of the previous theorems shows that the complexity of the back

andforth isomorphism falls within the scop e of exp onential iteration Thus we

have the following

Theorem Suppose L B and L B arepolynomialtime

 

dense linear orderings without endpoints with qtime density functions Then

for B Bin or B Tal L and L are qtime isomorphic



The main result of improves Theorem ab oveby obtaining mo dels

with a xed universe This result shows that there really are no categorical

linear orderings

Theorem Let L be a ptime linear ordering with universe B either Tal

or Bin Then

a There exists a ptime linear ordering L with universe B which is not prim

itive recursively isomorphic to L

e exists a ptime linear ordering b If L is not recursively categorical then ther

L with universe B which is not recursively isomorphic to L

Sketch of Pro of Wejustsketch the pro of of part a If L is recursively cat

egorical then L contains a copy of a dense linear ordering without end p oints

Then by Theorems and there exist ptime orderings L and L with



universe B one having ptime density functions and one without primitivere

cursive density functions Thus L may not b e primitive recursively isomorphic

to b oth structures

Injection Structures

For injection structures Cenzer and Remmel classied in Theorem of

the recursively categorical injection structures

Theorem Arecursive injection structure A f is recursively categorical if

and only if it has only nitely many innite orbits

The feasible categoricity results for injection structures dep end on the sp ec

trum of orbits For example there is one very nice p ositive result Theorem

of

Theorem Let A A f and B B g be two nitary permutation

structur es such that al l but nitely many orbits have the same size q for some

nite q

a If A and B areboth ptime over Tal then A is ptime isomorphic to B

b If A and B areboth ptime over Bin thenA is exponential time iso

morphic to B

c If A and B areboth qtime over either Bin or Tal then A is qtime

isomorphic to B

Sketch of Pro of Wesketch the argument for Tal Wemay assume with

out loss of generality that all orbits have the same size q The desired isomor

s

phism is dened in stages inwhichweenumerate s orbits A A A

 s

and B B B of each structure by dening a sequence of elements a a

 s s

q

and b b so that A fa fa f a g and similarly for B Then

s i i i i i

n n

welet f a f b The key to measuring the complexity of this map

k i i

ping is that since each orbit has q memb ers a tal m for some m kq

k

The general negative result is analogous to Theorem ab ove for linear

orderings except that we cannot sp ecify the universe for the nonrecursively

categorical structures since as seen by Theorem there actually are some

ptime categorical structures Our next result combines Corollaries and

of

Theorem Let A be a ptime injection structure with universe B where B

is either Tal or Bin Then

a There exists an innite family A of ptime structures each recursively iso

i

morphic to A which arepairwise not primitive recursively isomorphic

b If A is not recursively categorical then there exists a ptime structure A

with universe B which is not recursively isomorphic to A

The most general result for recursively categorical structures is the following

This combines Theorems and of

Theorem Let B be either Bin or Tal and let A beaninjection

structure such that either

a A has an innite orbit or

b A has innitely many orbits of size q for some nite q and has innitely

many other orbits

Then there is an innite family A of ptime structures each with universe

i

B and isomorphic to A which arepairwise not primitive recursively isomorphic

Sketch of Pro of There are two distinct arguments We rst sketchthe

pro of in the case that A has either an innite orbit or innitely many orbits

of nite size q together with an innite set of other elements We partition

the structure into two parts The rst part B is either the innite orbit or

the innitely many orbits of size q and may b e assumed to have universe B

by Theorem and The second part C has an innite family of copies

C with universe C suchthatB cannot b e primitive recursively emb edded in

i i

h that by Theorem and for any i j C is not primitive any C and suc

i i

recursively isomorphic to C Now just let A BC

j i i

For the case of a single innite orbit we app eal directly to Lemma Here

is the construction of a copyof S with universe Bin but not primitive

recursively isomorphic to the standard structure Let m m b e the set



from Lemma where A Bin and assume m



B B B

We dene f f f in blo cks so that the k th blo ckisinthree

parts

m m m

k k k 

followed by

m m m

k  k  k

and then

m m m

k  k k

B

The unique isomorphism from S toB f mapsm to m

k k 

m and is not primitive recursiveby Lemma

k

FinallywenotethatifA has no innite orbits and the sp ectrum of A is

ptime in tally as in Theorem then the conclusion of Theorem also

applies by Theorem of

Mo dels of Arithmetic

Before fo cusing on the categoricity of torsion Ab elian groups we briey present

two results for the group Zof integers These are Theorems and

of

B B

Theorem Let B be Tal or Bin There is a ptime structure B S

isomorphic to ZS but not exponential time isomorphic

Sketch of Pro of Binary case Let binn represent n and let



n

bin represent n

Theorem There is a ful ly ptime group A isomorphic to Zbut not qtime

isomorphic

Sketch of Pro of This is a corollary of since the mo del dened there

n

is not qtime isomorphic to N To see this observe that the term E which



has length n is mapp ed to the iterated exp onential

Let BinZ b e the standard structure of Zwith universe ptime isomorphic

to Bin and similarly for TalZ

Theorem There is an EXPTIME respectively exponentialtime

group A with universe Bin Tal which is isomorphic to BinZ

TalZ but not qtime isomorphic

Torsion Ab elian Groups

The results for Ab elian groups are parallel to those given ab ove for injection

structures We b egin with the p ositive results Theorems and of

Theorem Let p be a prime and let A and B be two groups with universe

B where B Tal or B Bin both isomorphic to Zp

n

a If A and B are ptime then A and B are EXPTIME isomorphic if B

Tal and doubleexponentialtime isomorphic if B Bin

b If A and B are qtime then A and B are qtime isomorphic

SketchofProof Wesketch the pro of of a for universe Tal The

standard structure B ma y b e viewed as an innite dimensional vector space

over Zp where the general elementc c is represented by tal c c

n 

n

p c p The arbitrary structure A will have a basis dened recursively

n

by letting a b e the least element indep endentoffa a gItcanthenbe

n n

n

seen that the map taking tal c c p c p toc a c a

 n n n

is exp onential time and its inverse is EXPTIME

The next result Theorem of shows that a ptime isomorphism is

not always p ossible in Theorem

Theorem For any prime p and for B Tal or B Bin there

exist two ptime groups with universe B and which are isomorphic to Zp

i

but which are not ptime isomorphic to each other

m

The case of Zp where m requires a stronger hyp othesis The

diculty is that only elements not divisible by p can b e used for the generators

a a and these mayallbevery large This is the basis for the pro of of



Theorem ab ove What is needed is the ability to compute a divisor of an

element x which is divisible by p Let us say that the group A has recursive

divisors if there is an algorithm which for any a A determines whether a

is divisible and which computes a divisor of a if there is one if the algorithm

runs in p olynomial time then wesay that A has ptime divisors Note that the

standard mo dels of the recursiv ely categorical groups all have ptime divisors

Theorem Let p be a prime let m beniteandletA and B be

two groups with universe B B Tal or B Bin both isomorphic to

m

Zp

a If A and B are ptime and have ptime divisors then A and B are EXP

TIME isomorphic if B Tal and doubleexponentialtime isomorphic

if B Bin

b If A and B are qtime and have qtime divisors then A and B areqtime

isomorphic

The next result Theorem of shows that the hyp othesis of ptime

divisibility is needed in Theorem

Theorem Let B be either Bin or Tal letp be a prime and let

m be nite Then there exists an innite family of ptime groups G each

i

m

recursively isomorphic to Zp and having universe B such that thereis

i

no primitive recursive structurepreserving embedding from G into G for any

i j

i j

The basic noncategoricity result for torsion groups is Theorem of

Theorem For any innite recursive Abelian torsion group Athereis

an innite family A of ptime groups each recursively isomorphic to A and

i

having universe a subset of Tal which arepairwise not primitive recursively

isomorphic

It is also the case that if some pprimary comp onentofA is innite and has

b ounded order or is isomorphic to Z p then each A in Theorem may

i

b e taken to have standard universe

Next wegivetwo results for pgroups Theorems and of The

rst is the fundamental result for nonrecursively categorical pgroups and the

second is a summary of results for pro ducts of basic pgroups

Theorem Let G bearecursive pgroup which is not recursively categor

ical Then there exist ptime groups H and H both isomorphic to G but not



recursively isomorphic to each other If G has boundedorder then we may take

H and H to have universe B where either B Tal or B Bin



Theorem Let p be a prime number let B Tal or Bin and let

C be an innite recursive group which is a product of cyclic and quasicyclic

pgroups and which is not isomorphic to Zp F for any nite group F

i

and either C has a quasicyclic factor or is a product of cyclic groups such that

C is ptime in tal ly

a Then there exists an innite family A of ptime groups with universe B

i

and isomorphic to C which arepairwise not primitive recursively isomor

phic

b If C is not recursively categorical then there exist ptime groups A and

A each with universe B and isomorphic to C which arenotrecursively



isomorphic to each other

The group Q of rationals is closely related to the quasicyclic groups Zp

since Q is isomorphic to the pro duct of the quasicyclic groups Weuse

this to obtain the following result Theorem of

Theorem Let B be either Bin or Tal Then there is an innite

family of ptime groups H each with universe B and isomorphic to Q but not

i

pairwise primitive recursively isomorphic

Scott Families

Wenow consider some general syntactic conditions which lead to some feasi

ble categoricity results Nurtazin and Goncharov provided sucient

conditions to ensure that a mo del A with universe A is recursively categor

ical namely if there is a nite sequence c c of elements of A and

 k

a recursive sequence called a Scott family of recursiveexistential formulas

f x x c c ng in the extended language with names for

n m  k

c c satisfying the following two conditions

 k

Every a a A satises one of the formulas

m n

satises a a c c For each n and for anya a andd d if A

n m  k m m

and d d c c then

n m  k

A a a c c is isomorphic to A d d c c

m  k m  k

via the map which sends a to d for i to m and c to c for i k

i i i i

Several notions of feasible Scott families were develop ed in and applied

to the feasible structures wehave studied We will present one suchformulation

here

A Scott family f x x c c c ng of ptime existen

n m  k

tial formulas for a ptime mo del A with universe A satisfying and as

describ ed ab oveissaidtobe strongly ptime if there is some xed integer r

such that the following conditions are satised for each m

For any nite sequence a a of elements of Awe can compute in

m

r

time maxfmja jja jg aformula from the list suchthat

m t

a a c c c holdsinA

t m  k

For each formula x x c c andeach

t m  k

a a A if there exists a suchthatA satises

m

a a ac c c then there exists suchana with

t m  k

r

jaj m maxfja jja jg

m

For each x x c c c and each

t m  k

a a A if there exists a suchthatA satises

m

a a ac c c then we can compute an a as describ ed

t m  k

r

in in time maxfmja jja jg

m

Note that clause ab ove implies that the structure A has only nitely

manytyp es of eacharity The following theorem is proved by a careful analysis

of the backandforth metho d

Theorem If A and B possess a common strongly ptime Scott family then

A and B are ptime isomorphic if both have universe Tal and areexponential

time isomorphic if both have universe Bin

Theorem can b e proved directly from this general result Wegiveone

other corollary here which provides some additional feasible categoricity for

permutation structures

Corollary Let A Tal f and B Tal g be two isomorphic

ptime permutation structures such that for some xed integer k

i for any a and a in the same orbit

ja j jaj k

and

ii for any a a a B and any nite q if there is an orbit of size q

 m

not containing any of a a then there is such an orbit containing

 m

an element a of size

k

jaj maxfja jja jg m

 m

Then A and B are ptime isomorphic

Weaker notions of Scott families dened in include the strongly expo

nential time Scott familywhich leads to exp onential time isomorphism for uni

verse Tal and double exp onential time isomorphism for universe Bin and

the strongly EXPTIME Scott familywhich leads to EXPTIME isomorphism for

universe Tal and double exp onential time isomorphism for universe Bin

The following applications are given in

A B

Corollary Let A B and B B be two polynomial time

models of an equivalencerelation such that for some xedinteger k both

models satisfy the fol lowing

i for any a and a in the same equivalence class

ja j k jaj if B Tal

or where a binn and a binn

n k jaj n n k jaj if B Bin

and

ii for any a a B and any nite q ifthereisanequivalence class

 m

of size q not containing any of a a then thereissucha class

 m

containing an element b of size

m

jbj k maxfk ja j ja jg if B Tal

 m

or

jbjk maxfmg if B Bin

Then A and B are exponential time isomorphic if B Tal and double

exponential time isomorphic if B Bin

Corollary Let A and B be two isomorphic ptime torsion Abelian groups

with the same universe Tal such that for some xed integer k

i for any a b

ja bjk maxfjaj jbjg

and

ii for any a a in either A or B and any nite q if thereisan

 m

element of order q not in Ga a that is the subgroup generated

 m

by fa a a g then there is such an element b of size

 m

m

jbjk maxfja jja jg

 m

Then A and B are EXPTIME isomorphic if B Tal and are double expo

nential time isomorphic if B Bin

Complexity Theoretic Algebra

In this section weintro duce the second theme of our survey That is instead

of fo cusing on problems of comparing p olynomial time versus recursive mo dels

we will x a given p olynomial time mo del such as an innite dimensional vec

tor space over a p olynomial time eld or a p olynomial time atomless Bo olean

algebra and consider the internal structure of that mo del Once again weshall

use established results from recursive algebra as a guide

In recursive algebra one studies the eectivecontent of results likethefact

that every indep endent subset of a vector space V can b e extended to a basis

If the vector space V is innite dimensional then all known pro ofs of this fact

use some version of the axiom of choice eg Zorns Lemma whichisknown to

e Thus one would exp ect that it is not the case that every b e nonconstructiv

recursive indep endent set can b e extended to a recursive basis in innite dimen

sional recursivevector space Indeed Metakides and Nero de proved that

not every recursive indep endent set of a recursively presented innite dimen

sional vector space over a recursive eld could b e extended to a recursive basis

Another theme in the study of recursive algebra has b een to study the lattice

of re substructures of various recursive structures structures see the survey

article by Nero de and Remmel Nero de and Remmel b egan the study of

complexity theoretic algebra in a series of pap ers and We

survey their results as well as results byBauerle in the next two sections

The overriding paradigm of Nero de and Remmels study of complexitythe

oretic algebra was to use the admittedly awed analogy that recursiveisto

re as P is to NP toformulate natural complexity theoretic analogues of

theorems in recursive algebra For example Dekker proved that every re

subspace of a recursively presented innite dimensional vector space over a re

cursive eld with a dep endence algorithm has a recursive basis The natural

complexity theoretic analogue of Dekkers Theorem is that in a suitable p oly

nomial time innite dimensional vector space V over a p olynomial time eld

with a p olynomial time dep endence algorithm every NP subspace of V has a

basis in P It turns out that the pro of of Dekkers Theorem is not uniform in

that the pro of breaks up into two cases dep ending on whether the underlying

eld of V is nite or innite The complexity theoretic analogue of Dekkers

Theorem b ehaves very dierently in these two cases That is Nero de and Rem

mel proved that if the underlying eld is innite and has a p olynomial time

representation with certain nice prop erties then every NP subspace of V has

a basis in P However if the underlying eld is nite then Dekkers Theorem

X X

is oracle dep endent That is there is an oracle X such that P NP and

X X

every NP subspace of V has a basis in P and there is an oracle Y such

Y Y Y

and there is a subspace W of V whichisNP but has no that P NP

Y

basis in P This presents us with two general themes Sometimes the com

plexity theoretic analogue of a theorem of recursive algebra is true but must

b e proved by more delicate metho ds which takeinto account the b ounds of the

resources used in a computation Sometimes the complexity theoretic analogue

is false or oracle dep endent b ecause the pro of of the recursive algebra result

uses unb ounded resources available in a recursive construction in a crucial way

Thus complexity theoretic algebra is not just a mere translation of the results

of recursion theoretic algebra

Another problem that complicates the study of complexity theoretic algebra

is that fact that not all p olynomial time mo dels are equivalent as wehaveseen

in the previous sections That is Metakides and Nero de showed that all in

nite dimensional recursivevector spaces with an eective dep endence algorithm

are recursively isomorphic SimilarlyCantors basic back and forth argument

which shows that all countable free Bo olean algebras are isomorphic is eective

so that all recursive free Bo olean algebras are recursively isomorphic As we

have seen in the previous section it is certainly not the case that all p olynomial

time free Bo olean algebras are p olynomial time isomorphic Thus in complexity

theoretic algebra one xes a p olynomial time presented structure over a natural

universe such as the tally representation of the natural numb ers or the binary

representation of the natural numb ers and studies that particular structure In

deed Nero de and Remmel studied two basic mo dels of vector spaces the tal ly

representation of an innite dimensional vector space of a p olynomial time eld

with a p olynomial time dep endence algorithm TalV where the underlying

universe is the tally representation of the natural numb ers and the binary rep

resentation of an innite dimensional vector space of a p olynomial time eld

with a p olynomial time dep endence algorithm BinV where the underlying

universe is the binary representation of the natural numb ers Similarly they

consider a tally representation and a binary representation of the free Bo olean

algebra The results for the tally representation and standard representation of

a structure are not always the same

Another basic question in the study of complexity theoretic algebra is whether

the priority metho d whichwas so useful in the study of recursive algebra would

again playa central role In Metakides and Nero de initiated the sys

tematic study of recursion theoretic algebra and intro duced the use of the nite

injury priority metho d from recursion theory as a uniform to ol to meet algebraic

requirements Prior to that time the priority metho d has b een limited primar

ily to internal applications within recursion theory in the theory of recursively

enumerable sets and in the theory of degrees of unsolvability and their gener

alizations Recursion theoretic algebra has b een develop ed since in depth by

many authors in such sub jects as commutativeeldsvector spaces orderings

and Bo olean algebras see Crossley for references and a crosssection of re

sults b efore Recursion theoretic algebra yielded as a bypro duct a theory

of recursively enumerable substructures see the survey article Nero deRemmel

for references

Simultaneously in computer science there was a vast developmentofP and

NP problems in complexitytheory This sub ject started out as a to ol for mea

suring the relative diculties of classes of computational problems see Cobham

Co ok Hartmanis and Stearns Manypapersinthisareahave

dealt with co ding a given problem M into a calibrated problem to nd an up

p er b ound on the complexityof M and co ding a calibrated problem into a

given problem M to nd a lower b ound the complexityof M see Hop croft

and Ullman and Garey and Johnson Due to the intractabilityof

the fundamental problem P NP BakerGillSolovay b egan a line of in

quiry using diagonal arguments to pro duce sets oracles R R suchthat



R R R R

 

Typical of recentwork in this direction is the con NP P NP P

struction byYao of oracles relative to which none of the p olynomial time

hierarchy collapses and the result of Cai that this holds for oracles with

probability The BakerGillSolovayYao and Cai results are fundamental

but they do not use the priority metho d whichwas used systematically with

success in recursiontheoretic algebra

A

Priority arguments have b een used bymany authors in the study of P

A

and NP sets for recursive or recursively enumerable oracles AFor example

A

Homer and Maass used priority arguments to investigate the lattice of NP

sets Shinota and Slaman and Shore and Slaman have used priority

argument to study the structure of the p olynomial time Turing degrees relative

ts to study to a recursive oracle Downey and Fellows used priority argumen

the density of their xed parameter complexity classes Nero de and Remmel

and showed that indeed priority metho ds play a central

role in the study of complexity theoretic algebras as we will bring out in the

following sections

We will start by surveying results of Nero de Remmel and Bauerle on p oly

nomial time vector spaces

Polynomial Time Vector Spaces

In this section we shall study the structure of an innite dimensional vector

space V over a p olynomial time eld We will start by giving some basic de

nitions and dening the binary or standard representation of V and the tally

representation of V Our denitions of the standard and tally representation

of V will b e broken down into two cases dep ending on whether the underlying

eld F is nite or innite

A recursiveeldF hU AI MI i consists of a recursive subset

F F F f F

U of the natural numbers and partial recursive functions eld addition

F F

eld multiplication AI eld additiveinverse and MI eld multiplica

F F F

tiveinverse such that these op erations restricted to F turn U into a eld A

F

recursively presentedvector space V hU i consists of a recursivesub

V V V

set U of the natural numb ers and partial recursive functions vector space

V V

addition and U U U scalar multiplication which turn U into

V F V V V

avector space V is said to haveadependence algorithm if there is a uni

form eective pro cedure whichgiven any ntuple v v will determine if

 n

v v are dep endent

 n

Wesay that a recursive eld F hU AI MI i is a polynomial

F F F f F

time eld if U is a p olynomial time subset of f g and the op erations

F

AI MI are the restrictions of total p olynomial time functions We

F F F F

will always assume that U and that is the zero of F and that is the

F

multiplicative identityofF

Let V b e the innite dimensional vector space over a p olynomial time eld

F which consists of all nite sequences a a of elements of F where

n

a together with the empty sequence which is the zero of the vector space

n

The op erations on V are induced by co ordinatewise addition and scalar mul

tiplication Finally wesaythatavector v a a of V where a F

n i

for i n and a has height nWesay that the zero vector of V has

n

height

Case F is nite

Supp ose that F f k g is a nite eld where is zero of F and

isthemultiplicative identityofF The space V can b e co ded into the natu

ral numbers f g as a p olynomial time vector space in manyways

We refer to e e as the standard basis of V where e is the sequence of

 n

the length n h i with n zeros and denotes the unit of F Now

the question of whether V is p olynomial time recursive etc dep ends on how

we co de the sequences ha a iFollowing we will distinguish two

n

sp ecic p olynomial time representations of V whichwecallthetal ly and bi

nary or standard representations of V Weidentify eachvector v V with

a natural number Rv by R and

n

Rha a ia a k a k if a

n  n n

Next with a slight abuse of notation we dene maps b V B bin

k k

V Bin and tal V Tal by b v b Rv binv binRv

k k

and tal v tal Rv

Then B V consists of the set B with the op erations of vector addi

k k

tion and scalar multiplication induced by the corresp onding op erations

B B

k k

from V Similarly binV consists of the set Bin fbinv v V g

with corresp onding induced op erations and and tal V consists of the

bin bin

set Tal with the induced op erations and scalar multiplication Itis

tal tal

easy to see that B V binV andtal V are p olynomial time structures

k

V are ptime isomor and it follows from Lemma that B V andbin

k

phic We shall normally refer to either of these two structures as the standard

representation stV ofV and write the op erations as and

st st

Case F is innite

Recall the ptime co ding functions h i dened in section Now

k k

supp ose that F hU AI MI i is an innite p olynomial time eld

F F F f F

of characteristic Let and denote the zero and of F resp ectivelyFor

any p ositiveinteger nletn where there are n summands and let

n MIm Then n AI n For anyintegers n and m let nm

F F

set



Q fnm n N m N nfgg



Thus Q is a copy of the nonnegative rationals inside of F Wesay that Q is

properly embedded in F if



i Q is a p olynomial time subset of f g and



ii the map f Q f g given by f nmbinnbinm binn m

is the restriction of p olynomial time function from f g to f g

Now supp ose that F hU AI MI i is a p olynomial time eld

F F F f F



where Q is prop erly emb edded and U f g Dene bin V Bin

F

by binand

bina a ha a i for a a in F with a

n n n n n

In this case we let stV U where U fbinv v V g

b b b b

and where the op erations and are dened so that bin is an isomorphism

b b

from V onto stV It is easy to see that the op erations and are

b b

the restrictions of p olynomial time functions and that U is p olynomial time

b

isomorphic to f g We call stV the binary representation or the standard

representationofV in this case

The tally representation of V is dened by observing that that if

is any string of U other than the empty string then ends in a

 n st

Hence there is an integer n such that binn

n

Now dene a map tal V Tal by tal v tal n where n is the

v v

natural number suchthatbinv binnandlettal V U where

t t t

U ftal v v V g and the op erations and are dened so that tal is an

t t t

isomorphism from V onto tal V It is easy to see that the op erations and

t

are the restrictions of p olynomial time functions and that U is p olynomial

t t

time isomorphic to Tal Wecalltal V thetal ly representation of V in

this case

Finally we argue that b oth the standard and tally representation of V

k

have p olynomial time dep endence algorithms First the deco ding functions

i

dened in section allowustorecover the co ecients a a from any

k

vector binv ha a i stV We can then similarly recover the

k k

ws that given any co ecients from tal v by rst computing binv It follo

set v v of vectors in either of our representations of V we can recover

n

the matrix of co ecients of v v corresp onding to the expansions of those

n

vectors in terms of the standard basis e e of V in p olynomial time in



the sums of the lengths jv j jv jWe can then use Gaussian elimination

n

on the matrix of co ecients to determine whether or not fv v g is an

n

indep endent set Since Gauss elimination is p olynomial time over the co ecients

since the op erations of F are p olynomial time it follows that in eachofour

representations there is a p olynomial p such that we can decide if fv v g

n

is dep endentinpjv j jv j steps

n

We end this section with some basic denitions and notations for vector

spaces Let V b e either V stV ortal V We shall abuse notation and

let denote the zero vector for V stV and tal V even though technically

the zero vectors of the three vector spaces are distinct ob jects Given a subset

A of V weletspaceA denote the subspace of V generated by AGiven two

subspaces U and W of V weletU W denote the subspace generated by

U W We shall write W U U if WU and U are subspaces of V such

 

U fgWesay U is a complementary subspace that W U U and U

 

of W if U W V Given x V welethtx denote the heightofxWe

note that if x stV then in p olynomial time in jxjwe can pro duce the

binary representations of the integers a a suchthatx binha a i

n n

with a so that we can nd the heightofx in p olynomial time in jxj

n

Similarlyif x tal V then in p olynomial time in jxjwe can pro duce the

tally representations of the integers a a suchthatx tal ha a i

n n

with a so that we can nd the heightofx in p olynomial time in jxj

n

Subspaces and Bases over innite p olynomial time

elds

We shall see that there is a vast dierence b etween the theory of bases and

subspaces of stV ortal V when the underlying eld is innite as opp osed

to when the underlying eld is nite For example Nero de and Remmel proved

the following strengthening of Dekkers Theorem that every re subspace of

a recursively presented vector space over a recursive eld with a dep endence

algorithm has a recursive basis

Theorem

Let F beapolynomial time eld where Q is properly embedded Then

a every re subspace V of tal V has a basis in P and

b every re subspace W of stV has a basis in P

Bauerle proved the existence of simple and maximal subspaces of tal V

whichareinP To prop erly state Bauerles results we rst need some deni

tions

In the lattice E of recursively enumerable re sets of natural numbers a

re set S is simple if n S is innite and for any innite re set W W S

Are set M is maximal if n M is innite and for anyre set W M

either n W or W n M is nite The analogues of these notions in the lattice of

A A

NP sets E A for any oracle A are the following A NP set S f g is

NP

A A

NP simple if f g n S is innite and for any innite NP set W f g

A A

W S ANP set M f g is NP maximal if f g n M is innite

A

and for any NP set W M either f g n W or W n M is nite

It was shown by Homer and Maass that there exists oracles A and B

A A A B B

suchthat NP P and no NP simple sets exist and NP P and there

B

exist NP simple sets It follows from a result of Briedbart that there are

A

no NP maximal sets for any A

In the lattice LV of re subspaces of a recursively presented copyofV

a re subspace S of V is simple if the dimension of the quotientspace V S

is innite and for any innite dimensional re subspace W of V W S fg

A re subspace M is maximal if the dimension of V M is innite and for

any re subspace W M either the dimension of V W or the dimension of

WM is nite A re subspace M is supermaximal if the dimension of V M

is innite and for any re subspace W M either V W or the dimension

of WM is nite The NP analogues of these notions in stV andtal V

A

are the following Let A b e an oracle then a NP subspace S of stV

A

tal V is NP simple if the dimension of stV S tal V S is innite

A

and for any innite NP subspace W of stV tal V W S fbing

A A

W S ftal g A NP subspace M is NP maximal if the dimension

A

of stV M tal V M is innite and for any NP subspace W of stV

tal V either the dimension of stV W tal V W or the dimension

A A

of WM is nite A NP subspace M is NP supermaximal if the dimension

A

of stV M tal V M is innite and for any NP subspace W of stV

tal V either stV W tal V W or the dimension of WM is

nite

X

Nero de and Remmel intro duced a slightly weaker notion than NP

X

simple subspace which they called P simple subspace Note that in the case

of simple sets or simple subspaces we can replace the innite re set W or

the innite dimensional re subspace W by an innite recursivesetW or

an innite dimensional recursive subspace That is every innite re set W

contains an innite recursive set and every innite dimensional re subspace

V of V contains an innite dimensional recursive subspace Thus a re set S

is simple i n S is innite and for any innite recursive set W W S

Similarly an re subspace S of V is simple i the dimension of V S is innite

and for any innite dimensional recursive subspace W of V W S fgThus

A

we make the following denition Let A b e an oracle then a NP subspace S

A

of stV tal V is P simple if the dimension of stV S tal V S is

A

innite and for any innite dimensional P subspace W of stV tal V

W S fbing W S ftal g It follows from results of Nero de

and Remmel that there exists oracles A such that there exists an innite

A

dimensional NP subspace V of tal V suchthatV has no innite dimensional

A A

subspace W P Thus while a subspace W whichisNP simple is certainly

A A A

P simple it is not clear that every P simple subspace of tal V isNP

simple

Given a subspace V of stV tal V we let

D V fhv v i v v are dep endent g

n n n n

D V D V

k

n

The Turing degree of D V is called the nth dep endence degree and the Turing

n

degree of D V is called the dep endence degree of V The sets D V and D V

n

can b e dened for any subspace of a recursively presented vector space over a

recursive eld using a suitable co ding of the nite sequences of N Nero de and

Remmel proved the following

is an innite recursive Theorem Assume the underlying eld F of tal V

eld Let A A A be any eective sequence of re sets such that A

  T

A A and A is not recursive Then thereisa supermaximal

 T T  

subspace V in tal V such that D V A and D V A

T  k T k

There is a nice application of Theorem in the case where wepick A A



to b e recursiveandA to b e nonrecursive In that case the sup ermaximal space



V of Theorem is recursive so the quotientspace W tal V V is a recur

sively presented vector space suchthat

i every re indep endent set I of W is nite

ii for any xed n there is an eective pro cedure whichgiven an ntuple

w w will determine if w w are dep endent but

n n

iii W has no dep endence algorithm

Bauerle proved the following result for tal V

Theorem Let F bea polynomial time eld where Q is properly em

beddedand be any nonzeroredegree Then for any nite k thereisa

supermaximal subspace V of tal V such that

i D V D V arepolynomial time

k

ii for al l j D V PSPACE DE X T and

j

iii D V

Thus in particular there exist a p olynomial time sup ermaximal subspace

W of tal V which is of course automatically NP simple and NP maximal

Moreover if we consider the quotientspace U tal V W then it is easy to

see that U is a p olynomial time vector space That is if weidentify U with the

set of minimal elements in each equivalence class of tal V W the Q will b e a

p olynomial time set and the op erations of tal V will induce p olynomial time

op erations on U which will make it isomorphic to tal V W Thus wehave

the following

Theorem There exists a polynomial time presentedvector space U such

that the only re independent sets of U are nite

As we shall see in the next section the analogues of Theorems and

are oracle dep endent

Subspaces and Bases over nite elds

In this section we shall state several results on the relation b etween the com

plexity of a subspace V of either stV ortal V and the complexityofa

basis of that subspace when the underlying eld is nite These results turn out

to b e essential for many of the more complicated results and constructions in

p olynomial time vector spaces

Note that since the universe of stV isBin there is a natural order

on the elements of stV inherited from the standard ordering of the natural

numb ers Similarly since the universe of tal V isTal there is a natural

order on the elements of stV inherited from the standard ordering of the

natural numb ers This given we can now state some very useful denitions

for our purp oses Recall that e e is the standard basis for V Thus



n

Re k

n

We start with the denition of a height increasing basis

Denition Let V be a subspaceofstV or tal V

Cal l B a height increasing basis of V if B is a basis for V and for al l

n B has at most one element of height n

The standard height increasing basis of V B is denedbydeclaring

V

that x B i x V and thereisno y V such that yxand

V

hty htx

The standard height increasing complementary basis of V tal V

B isdenedintal V by declaring that tal e B i tal e V

n n

V V

and thereisno y V such that hty n Similarly the standard

isde height increasing complementary basis of V stV B

V

nedinstV by declaring that bine B i bine V and there

n n

V

is no y V such that hty n

We cal l the spaceB thestandard complement of V

V

There is a crucial dierence b etween stV andtal V with resp ect to

searches That is the vector of height n with the smallest R value is e and

n

n

Re k Thevector of height n with the largest R value is k e

n

k e and

n

N

X

i n

Rk e k e k k k

n

i

V given a vector v of height nwe can pro duce in p olynomial Thus in tal

time in jv j a list of all vectors of height n in tal V However in stV

given a vector v of height nittakes exp onential time in jv j to pro duce a list

of all vectors of height n in stV For this reason the relation b etween the

complexityof V B B and spaceB isvery dierentintal V thanin

V

V V

stV For this reason we shall divide this subsection into two parts one for

tal V and one for stV and discuss the relation b etween the complexityof

bases and subspaces for each case separately

Bases and Subspaces for tal V

Nero de and Remmel in studied bases of NP subspaces of tal V so we

start by listing a numb er of results from that pap er

Theorem

Let V be a subspaceoftal V

P

a If B is a height increasing basis of V then V B

T

P P

V b B V and B

V

T T

V

Pro of The key p oint here is that in our tally representation car dfx

n n

V htx ng k k k k k k Moreover

n

if hty n then jy j k Given x V such that htxnweknowthat

n

jxj k So there are at most k jxj elements of tal V with heightlessthan

or equal to htx For q xed we can run any uniform computation which take

q

at most n steps on strings of length n for all the elements of tal V of height

less than or equal to htx in p olynomial time This is b ecause

q

k jxj

q q

X X

k jxj k jxj

 q q

k jxj jy j i

i

y talV hty htx



Given these observations it is immediate from our denitions of B and B

V

V

P P

V that B V and B

V

T T

V

To prove Theorem a note that if B is a height increasing basis for V

then x V i x spacefy B hty htxg Thus to decide if x V

we simply search all the elements y in V with hty htx and pro duce all

vectors y y in fy B hty htxgWe can then use the p olynomial

k

time dep endence algorithm to determine if y spacefy B hty htxg

P

in p olynomial time in jy jThus V B

T

An immediate corollary of Theorem is the following

Corollary

i A subspace V of tal V is in P i V has a height increasing basis B in P

ii If V is a subspaceoftal V and V P thenV has a complementary

subspace W in P

P P

We note that one cannot replace by in the statement of Theorem

m

T

due to the following result of Nero de and Remmel

Theorem

m m

such that neither B V nor V B There exists a subspace V of tal V

V V

P P

Next we observe that height increasing bases in NP generate NP spaces

Theorem Suppose that A is a height increasing independent set of

tal V in NP Then spaceA NP

Pro of Note that if A is a height increasing indep endentsetthenx

spaceAix spacefy A hty htxg Thus x spaceAi

there are elements b b of height htxand F suchthat

n n

P

n

m m

b Moreover if htx m then k jxj k so that each x

i i

i

b must have length k jxjThus in nondeterministic p olynomial time wecan

i

guess b b and computations whichshowthatb A and then

n n i

P

n

verify that x b Thus spaceAisinNP if A NP

i i

i

X X

Similarly one can show that if NP coNP then wehave the following

Theorem

X X

Suppose NP coNP and V is a subspaceoftal V Then

X X

i V NP i V has a height increasing basis in NP

X X

ii V NP implies V has a complementary subspace W in NP

Our next result will allowustoshow that the prop erty of a subspace V

of tal V having a basis in P do es not necessarily tell us anything ab out the

complexityof V other than that V is recursively enumerable

Theorem

Let V bearecursively enumerable innite dimensional subspaceoftal V

Then the fol lowing areequivalent

V hasabasis C in P

V contains an innite dimensional subspace W in P

V contains an innite height increasing independent subset S in P

Another consequence of a subspace containing an innite indep endent subset

in P is the following

Theorem Let V bea recursive subspaceoftal V such that V contains

an innite height increasing independent set C in P Then if the dimension of

tal V V is innite there is an innite height increasing independent set D

in P such that V spaceD fg

Pro of Note that B is recursive Let b b b e a list of the elements



V

of B such that hb hb Letf b e a recursive function suchthat



V

n

f b Similarly let c c b e a list of the elements of C suchthat

n 

hc hc Then let d b c where

 s s tal

r s

s

X

r s hb thenumb er of steps to compute f fs

i

i

Then we claim that D fd d g is our required height increasing in



dep endentset First observethatby our denition of r s r s hb

s

so that hd hc Also it is clear that r r so that

s

r s

hd hd Thus D is a height increasing basis Moreover it is easy to



see that D is indep endentover V Thus we need only showthat D is ptime

To decide whether a given x tal V isinD we rst compute which elements

y with hy hxareinC Now C is a ptime set so that for all z wecan

m

determine whether z C in max jz j steps for some xed m Moreover if

jxj n n

hxn then x where k jxj k so that it requires at most

n

P P

k jxj k

m m  m m m m

j k jxj k jxj steps to j

j  j 

nd the elements of C of height less than or equal to hx If no elementof

height hxisinC then clearly x D If there is an element of height hx

in C then in p olynomial time in jxjwe can nd r suchthathc hx At

r

this p oint we start to compute the sequence of elements f f in order

for r steps Supp ose that end the end of r steps wehave successfully computed

f ft Note that if we are not successful in computing f by the end

of r steps then x D Otherwise see if there is some s t suchthat

s

X

r hb thenumb er of steps to compute f fs

i

i

If there is no such sthenx D and if there is suchans then x D i

x f s c Itfollows that we can decide if x D in p olynomial time in

tal r

jxj so that D is a ptime height increasing indep endent set which is indep endent

over V

Next wesho w that having a basis in P do es not restrict the degree of a

subspace other than ensuring the subspace is recursively enumerable

Theorem Let be any re degree Then therethere exists a re subspace

V in tal V such that V hasabasis in P

Pro of Let B b e an innite subset of fe n g in P and for anygiven

n

re degree let B b e an innite re subset of fe n g of degree

n

Then it is easy to see that the Turing degree of V spaceB B is



By Theorem V has a basis in P since spaceB is an innite dimensional

P subspace of V whichisin

It is also easy to construct spaces with no basis in P In fact Nero de and

Remmel gave a general construction which given any eective list of re

indep endent sets of tal V A A pro duced a subspace V of tal V such



that V A is nite for all Their construction can b e sp ecialized to provethe

i

following results

Theorem Thereisasubspace V of tal V in DE X T such that V

has no basis in P

Thereisarecursive subspace V of tal V such that V has no primitive

recursive basis

There is a subspace V of tal V which is recursive in such that for any

re independent set I I V is nite

We should also note that every re subspace has a basis which has high

complexity

Theorem

Let V be an re subspace of either tal V or stV Then V hasarecursive

basis B which is not primitive recursive

All of the results so far do not settle the question of whether every subspace

V of tal V whichisinNP has a basis in P In fact this question is oracle

dep endent Toprove the existence of an oracle B suchthatevery subspace V

B B

of tal V whichisinNP has a basis in P Nero de and Remmel proved the

following result which strengthens a similar result of Homer and Maass

B B

Theorem Thereisarecursive oracle B such that P NP and

such that every innite set X whichisptimeTuring reducible to a set Y in

B B

NP P contains an innite subset in

We note that in light of Theorem it also follows that for the oracle B

B B

of Theorem every NP subspace V of tal V has a basis in P Thus

wehave the following

B B

Theorem Thereisarecursive oracle B such that P NP and

B B

every every NP subspace V of tal V has a basis in P

Via a delayed diagonal argument Nero de and Remmel also proved the fol

lowing

Theorem Thereisarecursive oracle A such that

A

a there is an innite dimensional subspace V in NP such that V has no

A A A

and basis in P and hence NP P

A A

b NP coNP

Combining Theorems and wehave the following

Theorem Arguments valid under relativization are not sucient

to prove

P NP every subspaceoftal V in NP has a basis in P and

P NP there is a subspace V of tal V in NP which has no basis

in P

We end this section with some results of Bauerle Wesayaset A f g

X X

is P immune if there are no innite subset of A in P The next results show

that a subspace V tal V can hav e a basis in P without the standard basis

b eing in P

Theorem There exists an exponential time subspace V of tal V

which has a basis in P but for which the standard height increasing basis of V

B is P immune

V

Theorem There exists a recursive oracle A such that there exists a

A A A

subspace V of tal V which is in NP n P hasabasis in P and yet the

A

standardhieght increasing basis of V B isP immune

V

Theorem

Thereexistsarecursive oracle B such that there exists a subspace V of

B B B B

tal V in NP n P and such that for al l NP n P subspaces V of tal V

B

the standard height increasing basis B has an innite subset in P

V

be nite and V tal V IfV has an innite dimen Theorem Let F

sional subspaceinP then V has a height increasing basis D with a subset in P

P

such that B D

V

T

A A

Theorem Let A beanoracle such that NP n P subspaces of

A

tal V exist Then if V has an innite dimensional subspacein P then

A P

V has a height increasing basis D with a subset in P such that B D

V

T

Bases and Subspaces of stV

dened It will b e convenient to think of stV via the representation B V

k

ab ove The advantage is that for nonzero x B V htx jxj The

k

n n

standard basis for B V isgiven by e b k

k n k

As p ointed out in the intro duction to this section there is a signicant

dierence b etween stV andtal V with regard to searches Indeed many

of the pro ofs of the prop ositions and theorems in the previous subsection relied

on the fact that given an x tal V we could pro duce a list of all elements

tal V ofheight htx in p olynomial time in jxj This is no longer the case

n n

in stV That is if x tal V and htx nthenk jxj k

jxj

while if x stV then htx jxj so that there are k elements of height

less than or equal to htxin stV Thus in stV we can not nd all the

elements of height less than or equal to htx in a ptime height increasing set S

in p olynomial time in jxjHowever there is a sp ecial class of ptime indep endent

sets of stV whichwe call strongly ptime indep endent sets whichdohave

most of the useful prop erties p ossessed by ptime height increasing bases of

tal V

Denition Anindependent set B stV is cal led strongly ptime if

i B is a ptime set

ii B is height increasing and

iii if B fb b g where htb htb then thereisapolynomial

 

time function f such that for al l n

n

iiia f b if htb n and B has an element of height n

k k

n

iiib f if B has no element of height n

We note that condition iii allows us to nd for any x stV all elements

of b of height htx in p olynomial time in jxj That is given x stV

 htx

htx jxj and we can compute f f f in p olynomial time

n n

gAs in jxj Then fb b B htb htxg ff n jxj f

noted ab ove any ptime height increasing indep endentsetB in tal V also

has the prop erty that for any xwe can nd all elements of B of height htx

in p olynomial time in jxjThus condition iii is sp ecically designed to give

us this prop ertywhich holds for all ptime height increasing bases in tal V

automatically It is easy to see that our standard basis fe e e g of stV



is strongly ptime

Our next prop osition lists several basic prop erties of subspaces generated by

subsets of a strongly ptime basis

Theorem Let B beastrongly ptime basis of stV and suppose that

S B Then

i S P i spaceS P

ii S NP i spaceS NP

iii S coNP i spaceS coNP

P

iv S spaceS

T

P

Pro of Since S spaceS B itfollows that S spaceS and S is in

T

P NP coNP ifspaceS isinP NP coNP

n

Let f b e the ptime function suchthatf b where b is the element

n n

of height n in B Then given an x stV of height nwe can compute

n

f b f b and test b b for memb ership in S all in time

n n

p olynomial in jxjThus in p olynomial time in jxjwe can nd fs s g

k

where fb b g fy S hty htxg Moreover the fact that B is

s s

k

P

jxj

aheight increasing basis means that x b for some in F

i i

jxj

i

Now supp ose that jxj nthenwe can write x x x where all x F

n i

and and each b b b where b F Then we can solve the matrix

i i in ij

equation over F

BY X

where B b Y is a column vector of unknowns and X is the column vector

ij

x x in p olynomial time in n jnjThus in p olynomial time in jxjwe

n

P

jxj

can nd such that x b This given

k i i

i

x spaceS ifi g fs s g

i k

P

It then easily follows that spaceS S and spaceS isinP NP coNP if

T

S is in P NP coNP

Our next result is a weak analogue for stV of Theorem

Theorem Let V be a subspaceofstV with strongly ptime basis R

Then R B is a strongly ptime basis for stV and both V and spaceB

V V

arein P

Our next theorem shows that no extra condition on a height increasing basis

such as condition iii is required to generate subspaces of stV inNP

Theorem Let B be a height increasing independent set of stV which

is in NP Then spaceB is in NP

Pro of The key prop erty of a height increasing basis is that if x spaceB

then x spacefb B htb htxg That is x must b e generated

by the elements of height htxinB if x spaceB Thus to see that

spaceB NP we simply guess the elements of B of height htx say

fb b g fb B htb htxg where htb htb Now for

k k

all nonzero y stV hty jy j so jb j jxj for all i and k jxj Then we

i

p erform a nondeterministic p olynomial time computation to checkifb b

k

are all in B Finallywe use our p olynomial time dep endence algorithm to

check whether x spacefb b g Thus spaceB isinNP

k

X X

Theorem Suppose NP coNP and V is a subspaceofstV

Then

X X

i V NP i V has a height increasing basis in NP

X X

has a complementary subspace W in NP ii V NP implies V

Our next result is a weak analogue of Theorem of for stV

Theorem Let V bean re innite dimensional subspaceofstV Sup

pose that there exists an innite strongly ptime independent subset I V

Then V has a basis in P

Our next next result is the analogue of Theorem for stV

Theorem Let V bearecursive coinnite dimensional subspaceofstV

such that V contains an innite strongly ptime height increasing independent

set C Then there is an innite strongly ptime height increasing independent

set D such that V spaceD f g

Theorem Given any reTuring degree there exists an resubspace

V of stV such that V has degree and V has a basis in P

Again one can show that exists an exp onential time subspace of stV

which has no basis in P

Theorem There is a subspace V of stV such that V DE X T and V

has no basis in P

X

The semilattice of NP subspaces

X

In this section we shall study various prop erties of the lower semilattice of NP

subspaces of tal V and stV for various oracles X Our rst result shows

that in contrast to the collection of re subspaces which is closed under b oth

X

intersection and sum and hence forms a lattice the collection of NP

subspaces of either tal V andstV is only closed under intersection and

hence only forms a lower semilattice

Theorem There exist two polynomial time subspaces W and V of

tal V stV such that W V fg and W V is not recursive

Pro of The pro of that we presentbelowworks equally well for b oth tal V

and stV Thus we shall write a generic pro of where V maybeinterpreted

as either tal V orstV and the standard basis e e maybeinterpreted



as either the standard basis tal e tal e of tal V or the standard basis



ste ste of stV as appropriate



By a result of Metakides and Nero de a subspace V of V is recursive

i V is re and V has an re complementary space It is easy to see that

we can form an eective list A B A B of all pairs of re subspaces

 

W and W of V such that W W fg That is if W W is an

i j i j 

n

eective list of all re subspaces of V and W denotes the set of elements

i

enumerated into W after n steps then A B is the pair of re subspaces

i i i

given bylettingA B beW W i i k and W W fg or

i i k k

n n

letting A B bespaceW spaceW where n is the least m suchthat

i i

k

m m

W spaceW fg if W W fg space

k

k

Given the list A B A B we shall construct W and V so that

 

W V A for any i suchthatA B V Thus W V will not b e

i i i

recursive In the construction that follows we will in fact construct two ptime

height increasing disjoint indep endentsetsK and L so that W spaceK and

L will b e our desired p olynomial time subspaces Let r r be V space



a list of all prime numb ers in increasing order Our idea is to use the vectors

n n

e e e where n to help us ensure that A W V if A B

r r  r  i i i

i i i

n

V The only vectors which will b e placed into K will b e of the form e e

r r 

i i

for some i and n and the only vectors which will b e placed into L will

n

b e of the form e for some i andn In fact for any xed i either

r 

i

n

K fe e n g

r r 

i i

and

n

L fe n g

r 

i

or there will b e an m such that

n m

K fe e n g fe e g

r r  r r 

i i i i

and

n m

L fe n g fe g

r  r 

i i

Note that in the standard representation of V L will b e a p olynomial time

subset in the strongly ptime height increasing basis fste n g and K will

n

b e a p olynomial time subset of the strongly ptime height increasing indep endent

n

set fe e k is o dd and n g so that L and K themselves will b e strongly

k k 

ptime indep endent sets Thus by Theorem W and V will b e p olynomial

time subspaces of stV In the tally representation of V K and L will b e

p olynomial time height increasing indep endent sets so that by Theorem W

tal V and V will b e p olynomial time subspaces of

m m

Lweruntheenumerations K and e e Now to decide if e

 r  r r

i i i

m m

and B denote those elements enumerated of A and B for m steps Let A

i i

i i

m

into A and B resp ectively after m steps If mje j and spaceA

i i r

i

i

m m

m

m

spaceB n spaceA spaceB then we place e e into

r r 

i i i

i i

m m m

m

K and e into L i e spaceA spaceB n spaceA Otherwise

r  r

i i

i i i

m m

we place neither e e into K nor e into L Using the fact that in

r r  r 

i i i

m steps we can at most enumerate m vectors which are of length at most m

and the fact that Gaussian elimination is p olynomial time in the dimensions of

the matrix it is easy to see that b oth K and L are ptime height increasing

indep endentsets

m m

Now supp ose e e K and e LSinceA B fgweknow

r r  r  i i

i i i

that each element v spaceA spaceB has a unique expression in the form

i i

v a b with a spaceA andb spaceB By our construction it follows

i i

m m m

so that e spaceA But n spaceA spaceB that e spaceA

r i r

i i

i i i

clearly e spaceK spaceL so that A spaceK spaceL

r i

i

m m

Supp ose there is no m such that e e K and e L Then

r r r  

i i i

m m

inwhichcase spaceB either there is no m suchthate spaceA

r

i

i i

spaceA spaceB V so that we dont havetoworry ab out A and B

i i i i

m

for some m in which case e spaceK spaceLso spaceA or e

r r

i i

i

again spaceK spaceL A

i

Next wemake some observations ab out the existence of subspaces V of

tal V whichareinNP n P We note that even with the assumption P

NP the existence of NP n P subspaces requires further complexity theoretic

assumptions That is in Hartmanis proved that the existence of sparse sets

in NP n P is equivalent to the separation of deterministic and nondeterministic

exp onential time DE X T NEXT Thus if DE X T NEXT thennoNP n

P subspaces of tal V can exist even if NP P Since the existence of an

A A A A

oracle such that NP P and DE X T NEXT was proven by Wilson

in wehave the following theorem

A A A

Theorem There exists an oracle A such that NP P and no NP n

A

P subspaces of tal V exist

As a consequence of this theorem it follows that showing the existence of

NP n P subspaces is at least as hard as separating DE X T and NEXT On

the other hand it is sucient to separate DOU B DE X T and DOUBNEXT to

show the existence of NP n P subspaces

Theorem If DOU B DE X T DOU B N E X T thenNP nP subspaces

of tal V over nite elds exist

Sketch of Pro of Let A DOU B N E X T n DOU B DE X T and assume the

n

k

underlying eld F has k elements Dene A f jx An xgSince



A DOU B N E X T n DOU B DE X T it follows that A NP n P But clearly



g and and hence A is a height increasing indep endent A ftal e tale

  

subset in NP n P Itthus follows from Theorem and that spaceA is



in NP n P

A A

Corollary Thereexistrecursive oracles A such that thereare NP n P

subspaces of tal V

Furthermore Mahaney has shown that the existence of a sparse NP

P

complete set with resp ect to implies NP P Thus if P NP then

m

there cannot b e a subspace V of tal V whichisNP complete

Next we turn our attention to the question of whether NP maximal or NP

simple subspaces exist We note that Breitbart proved that if R is any innite

recursivesetinf g then there exists a set S in P such that b oth S R and

R n S are innite This results shows that there can b e no NP maximal sets

since if M NP and R f g n M is innite then certainly R is an innite

recursivesetThus there is a set S P such that b oth S R and R n S are

innite But then W S M is a set in NP such that b oth W n M and

f g n M are innite so that M is not NP maximal Nero de and Remmel

proved that the analogue of Breidbarts splitting theorem holds for recursive

subspaces of tal V and stV

Theorem Let V be an innite dimensional recursive subspaceof

tal V stV Then there exist subspaces B and B in P such that



B B fg B B tal V B B stV and both B V and

   

B V are innite dimensional

We note that unlike the set case Theorem do es not exclude the p ossi

bility of the existence of NP maximal sets That is supp ose V is an innite and

coinnite dimensional subspace of tal V Then the complementary subspace

is certainly recursive so that there exists a pair of p olyno of V spaceB

V

mial time complementary subspaces U and W so that U spaceB and

V

W spaceB are innite dimensional However in this case we can not make

V

the conclusion that V U is a NP subspace which witnesses that V is not NP

maximal for two reasons First there is no guarantee that V U is coinnite

dimensional and second in light of Theorem there is no guarantee that

U V is in NP Indeed our next results will show that there are oracles A for

A

which NP maximal sets exists Similar remarks holds for stV

X X

First weshow that the assumption that NP coNP also eliminates

X X

the p ossibility of the existence of NP simple and NP maximal subspaces of

tal V

X X X

Theorem Suppose that NP coNP and V is an NP subspaceof

X

tal V such that tal V V is innite dimensional Then V is not NP

X

simple and V is not NP maximal

X

NP so that V Pro of By Theorem it follows that spaceB

V

X X

is not NP simple To see that V is not NP maximal note that byour

X

argument in Theorem it follows that for anygiven x NP wecan

nondeterministically from an X oracle nd a list of all elements u u

s

of height htx whichareinB andalistofallelements v v of

V t

X

height htxwhicharein B Thus we can form a new NP height increasing

V

indep endentset C where x C i x u for some i s or x v for some

i k

k t It is then easy to see that b oth tal V spaceC andspaceC V are

X

innite dimensional It also follows from Theorem that spaceC NP

X

so that C witnesses that V is not NP maximal

Since Baker Gill and Solovay pro duced recursive oracles X suchthat

X X X X

NP P but NP coNP wehave the following

A A

Theorem There exists a re cursive oracle A such that NP P and

A A

thereareno NP simple or NP maximal subspaces of tal V

X

We note that the construction of Theorem do es not construct a P

subspace W such that W V fg since it is a priori p ossible that spaceB

V

X

do es not contain an innite dimensional subspace in P Thus wedonot

X

automatically rule out the p ossibility of the existence of P simple subspaces

X X

of tal V with the assumption that NP coNP We shall see a bit

A A A

later that there exist oracles A suchthatnoNP simple P simple or NP

maximal subspaces exists in tal V

It is also the case that if a subspace V of tal V has an innite height

increasing indep endent subset in P then V is not P simple or NP simple

Corollary Let V NP besubspaceoftal V such that V con

tains an innite height increasing independent set C in P Then V is not

NP simple or P simple

Pro of Wemay assume that V is coinnite dimensional since otherwise V

A A

cannot b e NP simple or P simple We can thus use the pro of of Theorem

to construct a ptime innite height increasing indep endent set D suchthat

D is indep endentover V Itfollows by Theorem that spaceD is a ptime

subspace of tal V Since D is indep endentover V spaceD V fg so

that V is not NP simple or P simple

B B

To prove that there exists a recursive oracle B suchthatNP P and

B B B

yet no NP maximal NP simple or P simple subspaces exist we can again

use the oracle from Theorem

B B

Theorem Thereisarecursive oracle B such that P NP and no

B B B

NP maximal NP simple or P simple subspaces of tal V exist

B

Pro of Let B b e the recursive oracle of Theorem Let V be a NP

subspace of tal V such that the dimension of tal V V is innite By The

orem B is ptime Turing reducible to V so that B contains an in

V V

B

nite subset E in P Thus E is an innite height increasing indep endent

B

set in P so that by Theorem spaceE is an innite dimensional sub

B

space in P Clearly spaceE V fg so that spaceE witnesses that

B B

V is not P simple or NP simple Moreover since we can test whether

tal e tale areinE in p olynomial time in jtal e jtheset E

n n 

ftal e E car dE ftal e tale giseven g is also a ptime height

n n

increasing indep endent set We claim that W spaceV E is a subspace



B

of tal V which witnesses that V is not NP maximal Note that B E

V 

Thus W V and is a height increasing basis for W and that E n E B



W

the dimensions of b oth tal V W and WV are innite Because B E is

V 

aheight increasing basis for W it follows that x W i there exists a b V

and an e spaceE such that x b e and htbhte htx Thus

 tal

given a B oracle we can nondeterministically guess b and e of length k jxj

and the computation which shows that b V and then verify in p olynomial

B

time that x b e and e spaceE Thus W NP and hence V is not

tal 

B

NP maximal

Nero de and Remmel showed that the assumption that

X X X

NP coNP also eliminates the p ossibility of the existence of NP simple

X

and NP maximal sets in stV V

X X X

Theorem Suppose that NP coNP and V is an NP subspaceof

X

stV such that stV V is innite dimensional Then V is not NP simple

X

and V is not NP maximal

As was the case for tal V we can use the BakerGillSolovay results to

prove the following

A A

Theorem There exists a r ecursive oracle A such that NP P and

A A

thereareno NP simple or NP maximal subspaces of stV

The analogue of Theorem for stV is the following

Theorem

Let V beaNP coinnite dimensional subspaceofstV such that V contains

an innite strongly ptime height increasing independent set C ThenV is not

NP simple or P simple

Pro of Use the pro of of Theorem to construct a strongly ptime innite

height increasing indep endent set D suchthatD is indep endentover V It

follows by Theorem that spaceD is a ptime subspace of V Since

D is indep endentover V spaceD V fg so that V is not NP simple or

P simple

One can again use the oracle of Theorem to prove that there is an oracle

B B B

B where no NP maximal NP simple nor P simple subspaces of stV

exist

B B

Theorem Thereisarecursive oracle B such that P NP and no

B B B

NP maximal NP simple or P simple subspaces of stV exist

X

In contrast to the set case there are oracles X for which NP maximal

subspaces of tal V andstV exists The pro of requires a priority argument

for the construction of the oracle Such arguments are easier in tal V than

in stV In tal V one can naturally follow the usual practice in oracle

X

constructions and make the desired NP maximal subspace V be given by

n

V f X j j ng

This is not p ossible in stV In stV one constructs X so that there is a

X X

NP indep endent set which generates the desired NP maximal subspace To

see the dierence b etween these twotyp e of construction we will give the full

argumentfor tal V and give just the construction for stV We note that

similar techniques are used to prove results in the standard and tally represen

tations of the free Bo olean algebra which are given in the next section

Theorem There exists an re oracle Y and a subspace V of tal V

Y Y

which is both P simple and NP maximal

Pro of We shall construct Y so that

n

M fg f n f g jj n and Y g

Y

is our desired subspace Clearly M NP

To ensure that M is coinnite dimensional wemust meet the following set

of requirements

n n

T car dfnj Y contains no strings with k jj k g j

j

Thus T says there are at least j heights n so that M contains no strings of

j

height n So meeting requirement T ensures dimV M j

j

Y

To ensure that M is P simple we shall meet the following two sets of

htx requirements Given any subset V tal V let htV fn x V

ng

Y

S IfN is an innite dimensional subspace of tal V such that

j

j

Y Y

htN n htM is innite then M N fg

j j

Y

Now supp ose that P is an innite dimensional subspace of tal V Note

i

Y

that meeting all the requirements S will ensure that either P M fg or

j

i

Y

htP htM where for anytwo sets A and B where we write A B i

i

Y

there is a nite set F suchthatA B F Now supp ose that htP

i

Y

htM and let B b e the standard height increasing basis for P By Lemma

i

i

Y

B is in P Then clearly we can mo dify B by p ossibly deleting a nite

i i

set of elements to form a new height increasing basis C such that htM fn

i

Y

x C htx ngThus C will also b e in P and by Lemma spaceC

i i i

Y Y

will also b e in P Hence if htP htM then there exists some j such

i

Y Y

that P is an innite dimensional subspace of tal V andhtP htM

j j

Y

Thus to ensure that M is P simple it will b e enough to ensure that we meet

the following set of requirements

Y

R IfP is an innite dimensional subspace of tal V then

i

i

Y

htP htM

i

Finally to ensure M is NP maximal we shall meet the following set of

requirements

Y Y

Q IfN M is innite dimensional and N M then there is an

in

i i

Y

x N suchthatx tal e M

n

i

Y Y

M is innite then meeting all the require Note that if N M and dimN

i i

Y Y

ments Q will ensure that tal e N for all n so that N tal V

n

in

i i

Y

Thus in fact M will b e NP sup ermaximal

We shall rank our requirements with those of highest priority coming rst

as T S R Q T S R Q

   

In the construction that follows we shall let Y denote the set of elements

s

enumerated into Y by the end of stage s and

Y g M fg f l f g jj

s s

We shall ensure that for each s M is a nite dimensional subspace of tal V

s

s

and that htM is contained in fsgFor any stage sweletCH fn

s s

s

n g b e set of complementary heights for M ie the set of all heights n

s



so that there are no elements of tal V of height n in M

s

Atanygiven stage swe shall pick out at most one requirement A where A

j j

will b e one of the requirements S R orQ and take an action to meet that

j j j

requirement The fact that the requirements T will b e satised follows from

j

the construction describ ed b elow For the other requirements we shall then say

that A received attention at stage s

j

The action that wetake to meet the requirement A of the form S or Q

j j j

will always b e of the same form That is we shall put some elements into Y

at stage s and p ossibly restrain some elements from entering Y for the sake

of the requirement We shall let resA s denote the set of elements that are

j

restrained from entering Y at stage s for the sake of requirement A Wesaythat

j

requirement A of the form S or Q is satised at stage s if there is a stage

j j j

s s such that A has received attention at stage s and resA s Y

j j s

The actions that wetake to meet the requirements R will b e slightly dif

j

ferent First we shall declare that all R are in a passive state at the start of

j

Y

s

our construction Wewould like to nd an element x P of height n such

j

that n htM If we can nd suchanxthenwe will restrain all y suchthat

s

n n

k jy j k plus all elements not in Y which are queried of the oracle Y

s s

Y

s

during the computation of P x from entering Y for the sake of requirement

j

R Thus if we ensure that resR s Y then M will have no elements

j j

Y Y

of height n and x P so that htP htM If wetakesuch an action for

j j

e will say that R has received attention at stage s and R at stage sthenw

j j

declare the state of R to b e active Then for all tswewillsay that an

j

active R is satised at stage tifresR s Y However if R is injured

j j t j

at some stage ts in the sense that resR s Y then R will return to

j t j

Y

a passive state If we cannot nd suchanxwe will attempt to force htP

j

to b e nite That is since we will ensure that htM fs g for

s

will have no elements of height s Recall that we are assuming that all s M

s

X

for n the run time of computations of P y for any oracle X is b ounded

j

j

maxn for any string of length n Then for n weletb b e the largest

n

n n

i such that for all k r k

n

n i k

k

Note that it is easy to see that lim b Our idea is that elements of

s s

r n n

height n in tal V are of the form where k r k Our strategy

s will b e to ensure that for all R with j b at the end of stage s for

j s

Y

s

r

forall which are in a passive state and have the prop ertythatP

j

s s

k r k we restrain all elements whicharenotinY and whichare

s

queried of the oracle Y in such computations from entering Y for the sake

s

Y

to b e nite if R is in a passive state at of R This action will force htP

j j

j

stage s for all but nitely many sFor any xed j b the maximum restraint

s

imp osed for R is if we restrained all elements not in Y which are queried of

j s

Y

s

r n

the oracle Y in some computation P with r k Since

s

j

the total numb er of steps used in all these computations is at most

s

k

X

j j s s j s j 

i k k k

i

s j 

then clearly wecouldhave restrained at most k elements from entering

Y for the sakeofR Thus at stage swewillhave restrained at most

j

b

s

X

n

n i n b  k

s

k k

i

elements for entering Y for the sake of some passive requirement R with j b

j s

n n

at stage s Hence for anygiven r with k r k wewillhave

r

h R s restrained at most elements of length r from entering Y for suc

j

CONSTRUCTION

Stages

Let Y Y so that M M fg Let resA resA for

  j j

all requirements A of the form S R or Q

j j j j

Stage s with s

Let A b e the highest priority requirement among S R Q S R Q such

j    s s s

that

Case A S and S is not satised at stage s and there exists an

j j j

with ht s such that

Y

s

a N

j

s

b ht CH and ht n and

s

j

n

f g such c for each spacef gM n M there is a string

s s n

n

that j j j j n and is not restrained from Y byany requirement

n n

of higher priority than S at stage s nor is queried of the oracle in

j n

Y

s

some xed computation of N which accepts

j

Case A R and R is not satised at stage s and there exists an

j j j

with ht s such that

Y

s

and i P

j

s

ii ht CH and ht n

s

j

Case A Q and Q is not satised at stage s and if j e n there

j j j

exists an with ht s suchthat

Y

s

I N

e

s

I I ht CH and ht maxn n and

s

j

s

m m

I I I For each spacef tal e gM n M ht n and

tal n s s

j

there is a string of length m in f g which is not restrained from

m

Y byany requirement of higher priority than Q at stage s noris

j m

Y

s

which accepts queried in some xed computation of N

e

If there is no such requirement A let Y Y Also for all requirements

j s s

A of the form S or Q and for all requirements A of the form R where

j j j j j

either R is satised at stage s orjb let resA s resA s

j s j j

Declare that a requirement R is active at stage s i R is active at stage s

j j

For any R with j b which is currently passive and has the prop ertythat

j s

Y

s r s s

P let resR s equal resR s for all k r k

j j

j

union the set of all y Y suchthaty is queried of the oracle in one of the

s

Y

r s s

s

computations P where k r k

j

If there is such a requirement A wehave three cases

j

Case A S

j j

s

Let denote the least corresp onding to S Then for each

s j

s

n

s

n gM n M pick the least string suchthatj j spacef

n s s n n

is not restrained from Y byany requirement of higher priority than S at

j

s

Y

s

stage s nor is queried of the oracle Y in the computation of N

n s

j

s

which accepts and put into Y This will ensure that if M is a nite

n s

dimensional subspace of V then M will also b e a nite dimensional sub

s

s

space of V Note that the assumption that ht CH ensures that all

s

n n

s s

That spacef gM n M have the prop ertythatht ht

s s

n n

s

is sucha must b e of the form m where m M and

tal tal s

n

s s

it must b e the case that ht ht F Then since htm ht

s s s

s

Thus htM fn n g and hence for all i j n n Let

s s

i

j i

s

resS s equal the set of all strings not in Y whic h are queried of the or

j s

s

Y

s

s

acle Y in the computation of N which accepts andsay S receives

s j

s

j

s

attention at stage s Also for all requirements A of the form S or Q and for

j j j

all requirements A of the form R where either R is satised at stage s

j j j

or jb let resA sresA s if Y resA s and let

s j j s j

Declare that a requirement R is active resA s if Y resA s

j j s j

at stage s i R is activeatstage s andY resR s For any R

j s j j

Y

r s

with j b whichiscurrently passive and has the prop ertythatP

s

j

s s

for all k r k let resR s equal resR s union the set of

j j

all y Y such that y is queried of the oracle Y in one of the computations

s s

Y

r s s

s

P where k r k

j

Case A R

j j

s

s

Let denote the least corresp onding to j and n ht We then say

s s s

s s

that R is active and receives attention at stage sWeletY Y and

j

s

n n

s s

resR s consist of all elements y with k jy jk and all elements

j

s

which are not in Y and which are queried of the oracle Y in the computa

s s

Y

s

s

resR s Y then M will have no elements tion P Note that if

j

s

j

s

Y

s s

Also for all requirements A of the form P but of height n ht

j s

j

s

S or Q and for all requirements A of the form R where j j and where

j j j j s

either R is satised at stage s orjb let resA s resA s

j s j j

For j j declare that a requirement R is active at stage s i R is active

s j j

at stage s For any R with j b whichiscurrently passive and has

j s

Y

r s s

s

forallk r k let resR s equal the prop erty that P

j

j

resR s union the set of all y Y such that y is queried of the oracle Y

j s s

Y

r s s

s

in one of the computations P where k r k

j

Case A Q

j j

s

Let j e n and denote the least corresp onding to j Then for each

s s s s s

m

s

tal e gM n M pick the least string suchthat spacef

tal n s s m

s

j j mand is not restrained from Y byany requirement of higher prior

m m

Y

s

which nor is queried in the computation of N itythanQ at stage s

e

m j

s

s

s

and put into Y Once again this will ensure that M is a nite accepts

m s

s

n httal e dimensional subspace of V Note that since ht it fol

s n

s

s s

Thus as in case the assumption that tal e ht lows that ht

tal n

s

n

s s

tal e gM n CH ensures that all spacef ht M

tal n s s s

s

n

s

Let resQ have the prop erty that ht ht s equal the set of all

j

s

strings which are not in Y which are queried of the oracle in the compu

s

Y

s

s

which accepts and say Q receives attention at stage s tation of N

e

j

s

s

Also for all requirements A of the form S or Q and for all requirements

j j j

A of the form R where either R is satised at stage s or jb let

j j j s

resA s resA s if Y res A s and let resA s

j j s j j

if Y resA s Declare that a requirement R is activeatstage s

s j j

i R is active at stage s andY resR s For any R with

j s j j

Y

r

s

j b which is currently passive and has the prop erty that P

s

j

s s

for all k r k let resR s equal resR s union the set of

j j

all y Y such that y is queried of the oracle Y in one of the computations

s s

Y

r s s

s

where k r k P

j

This completes the construction of Y

Lemma Each requirement of the form S R orQ receives attention

j j j

at most nitely often

Pro of We pro ceed by induction on j Supp ose that s is such that there is



no stage s s such that one of S R Q S R Q receives attention at

    j j j

stage s Then if there is a ts such that S receives attention at stage t then

 j 

is satised at stage t and resS t Y However by construction S

j  t j 

it is easy to see from our construction that for st resS s resS t

j  j 

and resS s Y unless some requirement of higher priority than S

j  s j 

receives attention at stage s Since this never happ ens byourchoice of s S

 j 

will b e satised for s tThus S can receive attention at most once after

j 

stage s Thus there must b e a stage s such that there is no stage s s



such that one of S R Q S R Q S receives attention at stage sA

   j j j j 

similar argumentwillshow that R can receiveattention at most once after

j 

stage s Thus there must b e a stage s such that there is no stage s s such

 

that one of S R Q S R Q S R receives attention at stage s

   j j j j  j 

Finally a similar argument will showthat Q can receiveattention at most

j 

once after stage s Thus each of the requirements S R orQ can receive

 j j j

attention only nitely often

Lemma dimtal V M is innite

Pro of Weproveby induction that dimtal V M k for all k That

is let t b e a stage such that no requirement S R Q S R Q receives

    k k k

t



is dened for attention at any stage s t SinceM is nite dimensional n

 t



i

t t

 

all i Hence M contains no strings of height n for n n n Butno

t

k

s

requirement S R orQ with jk can force elements of height n n into

j j j

k

M at any stage s Hence byourchoice of t there can b e no strings of heights



t t

 

in M Thus dimtal V M k n for n n n

k

Y

Lemma M is P simple

Y Y

Pro of First we showthatifN isasubspaceoftal V suchthathtN n

j j

Y Y

htM is innite then N M fgFor a contradiction assume N is such

j j

Y Y

htN n htM is innite and N M fg Note that since M is co that

j j

s

innite dimensional by Lemma it follows that n lim n exists for

i s

i

s

all iLets b e a stage large enough so that n n for i j andnoneofthe

 i

i

requirements S R Q S R Q receives attention after stage s

   j j j 

n

Let U denote the set of all such that there exists a requirement A among

s i



S R Q S R Q which is satised at stage s such that there

   j j j 

exists an resA s withjj nOurchoice of s ensures that if n U

i   s



then no string of length n is ever restrained from Y by a requirement of higher

priority than S which is satised at some stage t s Also our choice of s

j  

t

ensures that n n for all i j and t s Nextlett s b e such that

i   

i

s n g t maxfhty y U gf

 j  s



b jand

t



r j

r for all rt



Note that for any t t our construction ensures that the numb er of strings



t t

of length r where k r k which are restrained by some requirement

r

R with ij which is passive at stage t is less than Moreover weare

i

X

assuming that any successful computation of the oracle machine N for any

j

j

oracle X on a string of length r takes at most r steps Thus our choice

Y

x

t

of t ensures that if t t and N is string of height t then there

  

j

is at least one string f g of length x which is not restrained from

x

Y byany requirement of higher prioritythan S at stage t nor is queried

j

Y

x

t

Since of the oracle Y in some xed computation whichshows that N

t

j

Y n Y n

htN n htM is innite there must exist a N suchthatht t and



k k

Y

s

n n

ht htM Then there must b e some stage st suchthat N



j

m n

Note that at stage seach spacef gM n M has the prop erty

s s

n m

ht t and thus there is at least one string of length that ht

 m

m which is not restrained from Y byany requirement of higher prioritythan

S at stage s nor is queried of the oracle Y in some xed computation

j s

Y

s

n n

which shows that N Thus witnesses that S is a candidate to

j

j

receive attention at stage sThus either S is satised at stage s orS is

j j

highest priority requirement among S R Q S R Q which can receive

   s s s

attention at stage s In either case it follows that S will b e satised at stage

j

Y

n

s

sThus there will b e some N M nfg such that all elements which

s

j

Y

n

s

are queried of the oracle Y in some computation whichshows that N

s

j

and which are not in Y areinresS s However our choice of t ensures that

s j 

n

we can never put any elementofresS sinto Y after stage s so that will

j

Y

M fg witness that N

j

Y

n htM is innite seems Remark We note that the assumption that htN

j

Y

to b e crucial in this argument That is if we merely assume that dimN M

j

n Y

is innite then it may b e the case that whenever there exists a N such

k

Y

s

n n n

there is that ht t and M then at a stage st where N

 

k

m m n

some M such that hy ht In such a situation it is p ossible

s

n m n

that ht ismuch less than ht That is it may b e p ossible that

tal

x n

some elementin spacef gM n M has height so small that all

s s

strings of length x are queried of the oracle during any computation whichshows

Y

s

n

that N Then it will b e imp ossible to put a string of length x into Y

s

k

x

so as to ensure that M while maintaining the computation to ensure that

s

n Y

N

k

Y

is Tocontinue our pro of of the lemma wecannow assume that if P

r

Y

an innite dimensional subspace of tal V suchthatP M fg then

r

Y

htP n htM is nite By our argument preceding the construction it would

r

Y

is an innite dimensional subspace then follow that there is some j such that P

j

Y

of tal V andhtP htM We shall nowshow that there can b e no such

j

Y

j For a contradiction assume that P is an innite dimensional subspace of

j

s

Y

tal V and htP htM Let s b e a stage large enough so that n

j

i

n for i j and none of the requirements S R Q S R Q S

i    j j j j

n

receives attention after stage s Let U denote the set of all suchthat

s

there exists a requirement A among S R Q S R Q S which

i    j j j j

A s withjj nOur is satised at stage s and there exists an res

i

choice of s ensures that if n U then no string of length n is ever restrained

s

from Y by a requirement of higher prioritythanR which is satised at some

j

t

stage ts Also our choice of s ensures that n n for all i j and t s

 i 

i

Next let t b e suchthat

gfs n g t maxfhty y U

i s

jand b

t

r j

r for all rt

Nowwe claim that there can b e no stage t t at which R is satised

j

at stage t That is if R is satised at stage t there must b e some s t

j

Y

s

x

such that R receives attention at stage s and there is a P suchthat

j

j

x

q ht CH and res R sresR tcontains all strings of length

s j j

q q

r where k r k and contains all strings which are not in Y

s

Y

s

x

which are queried of the oracle Y in the computation P and

s

j

resR s Y But then our choice of tt ensures that resR s Y

j t j

x Y

But which means that M can have no strings of height q while P

j

x Y

ht then witnesses that htP M whichcontradicts our assumption that

j

Y

htP htM Thus it must b e the case that for all stages tt R

j

j

is in a passive state It follows that for all t t there can b e no r with

Y

r t t

t

since otherwise at stage t there k r k such that P

j

Y

t t t r

is some r with k r k suchthatP But then at stage

j

r

t witnesses that R is a candidate to receive attention at stage t By

j

our choice of t t itwould followthat R is the highest priority requirement

j

Q S R Q which could receive attention at stage among S R

  t t t 

t so that R would receiveattention at stage t whichwehave already

j

t t

ruled out Thus it must b e the case that for all r with k r k

Y

t r

P But then our choice of tt ensures that j b and hence all

t

j

elements which are not in Y which are queried of the oracle Y during one of

t t

Y

r t t

t

where k r k are put into resR t the computations P

j

j

Again the fact that t t ensures that resR t Y so that for all r with

j

t t Y r Y

k r k P That is P has no strings of length t

j j

Y Y

such is nite Thus there can b e no such P for any tt and hence htP

j j

Y Y

htM is an innite dimensional subspace of tal V andhtP that P

j j

Y

But this means that there can b e no r suchthatP is an innite dimensional

r

Y Y

subspace of tal V andP M fgThus M is P simple as claimed

r

Y

Lemma M is NP maximal

Pro of By our remarks preceding the construction weneedonlyshowthat

Y

we meet all the requirements Q So assume N is a subspace of tal V

en

e

Y Y

suchthatN M is innite dimensional and N M Let j e n and

e e

s



let s b e a stage such that n n for i j and none of the requirements

 i

i

S R Q S R Q S R receive attention after stage s Let

   j j j j j 

n

such that there exists a requirement A among U denote the set of all

i s



S R Q S R Q S R which is satised at stage s and there

   j j j j j 

exists an resA s withjj nOurchoice of s ensures that if n U

i   s



then no string of length n is ever restrained from Y by a requirement of higher

priority than Q which is satised at some stage t s Also our choice of s

j  

t

ensures that n n for all i j and t s Nextlett be suchthat

i  

i

t maxfhty y U gfs n g

 s  i



b jand

t



r j

r for all rt



Note that for any t t our construction ensures that the numb er of strings



t t

of length r where k r k which are restrained by some requirement

r

R with ij which is passive at stage tislessthan Moreover weare

i

X

assuming that any successful computation of the oracle machine N for any

j

j

oracle X on a string of length r takes at most r steps Thus our choice of

Y

x t

t ensures that if tt and N is string of height t then there is at

  

j

least one string f g of length x which is not restrained from Y byany

x

requirement of higher priority than Q at stage t nor is queried of the oracle Y

j t

Y

x t

in some xed computation whichshows that N

j

Y Y

Next observe that since dimN M is innite and N M itmust b e the

e e

Y

case that htN n htM is innite That is let A fa a g b e an innite



e

Y

set of elements of N which is indep endentover M Then consider some xed

e

P

q

tal e where F for i q a A and supp ose a

i j i i i q

i

i

and j j Thus hta j Now if there exists an m M suchthat

q i q

P

tal e where F for all and htmhta then m

j i

q

j

q

j

q

Y

But then a a m is an elementofN n M such hta hta

i tal i

i e i

j

q

Now if there exists an m M such that hta htm then once again

 

i

Y 

m is an elementofN n M with a there is some F such that a

tal 

e

i i



hta hta hta If wecontinue in this fashion wemust eventually

i

i i

k k

v where v M suchthathta htM That is we nd some a a

i tal k k

i i

can replace our original indep endent set A over M byaset A fa a g



where for all i a a M and hta htM But then A is an innite

i tal

i i

Y

subset of N which is indep endentover M Thus there is no nite set F

e

suchthat spaceM F A This implies that ht i g A fhta

i

must b e innite since otherwise there clearly would b e a nite set F suchthat

Y

spaceM F A But by construction htA htN n htM sothat

e

Y

htN n htM must b e innite

e

Y q Y

Since htN n htM is innite there must exist a N suchthat

e e

q q q

ht t ht nand ht ht M Then there must b e some stage



Y

s

q m q

st such that N Note that at stage seach spacef

e

 tal

m q

tal e gM n M has the prop erty that ht ht tal e

n s s tal n

q

ht t andthus there is at least one string of length m whichisnot

 m

restrained from Y byany requirement of higher priority than Q at stage s

j

nor is queried of the oracle Y in some xed computation which shows that

s

Y

s

q q

N Thus witnesses that Q is a candidate to receive attention at

j

j

stage s Hence either Q is satised at stage s or Q is highest priorityre

j j

quirement among S R Q S R Q which can receiveattention at stage

   s s s

s In either case it follows that Q will b e satised at stage sThus there will

j

Y

q q

s

b e some N such that tal e M and all elements whichare

tal n s

j

Y

q

s

queried of the oracle in some computation whichshows that N and which

j

Y are in resQ s However our choice of t ensures we can never are not

s j 

q Y

put any elementof resQ sinto Y after stage s so that N and hence

j

j

Y

requirement Q is meet Thus M will b e NP sup ermaximal and hence will b e

j

Y

NP maximal This completes the pro of of Lemma and of Theorem

We note that M constructed in Theorem has a numberofinteresting

Y Y

prop erties b esides b eing NP maximal and P simple First of all it is easy

to check that in meeting the requirements S we made no use of the fact that

j

Y Y

N was a subspace of tal V but only that N was a subset of tal V

j j

it is easy to check that in meeting the requirements R wemadeno Similarly

j

Y Y

use of the fact that P was a subspace of tal V but only that P was a

j j

subset of V Thus meeting all the requirements R ensures that there is no

j

Y

innite subset W of tal V inP suchthathtW htM Thus M do es

Y Y

not contain any innite P set and hence M do es not have a basis in P We

Y

also claim that tal V n M do es not haveany innite subsets in P That

Y Y

is supp ose that P tal V n M Now it cannot b e that ht P n htM is

j j

Y Y

and the fact that we N innite since otherwise there is an i suchthatP

i j

Y Y

met requirement S would mean that P M fgThus htP htM

i

j j

Y

Let Q htspaceA n htP Then clearly

j

Y

S fx P htx Qg

j

Y

Since meeting all is an innite set in P suchthathtS htspaceA

the requirements R rules out the existence of suchanS tal V n M do es

j

Y Y

not contain an innite set in P Recall that a set of strings S is called P

Y

Thus b oth M and tal V n M are immune if S has no innite subset in P

Y

P immune

Y

Note also that by Theorem the fact that M is NP maximal implies

Y Y Y Y

that NP coNP and hence that P NP Thus wehaveproved the

following

Corollary There exists an re oracle Y and a subspace M of tal V

such that

Y Y Y Y

P NP and NP coNP

Y

M NP

Y Y

M is P immune and hence has no basis in P

Y

tal V n M is P immune and

Y Y

M is both P simple and NP supermaximal

We next give an analogue of Theorem for stV Once again weshall

think of stV asthe k ary representation B V so that for all x stV

k

jxj htx

D

Theorem There exists a re oracle D such that there exists an NP

D

supermaximal P simple subspaceinstV

b e the set of Pro of The construction again pro ceeds in stages WeletD

s

elements enumerated into D by the end of stage s For anygiven x fk

g with jxj weletC denote the set of all strings of length jxj of

x

jxj

fk g of the form x where is any string of length jxj in

jxj

fk g Note that there are k strings in C for any x stV Let

x

C fg It is then easy to see that if x y thenC C

x y

D

We then dene A fx C D gThus A will b e in NP Our idea is

x

to dene D so that A is a height increasing indep endentsubsetofstV Then

D

by the relativized version of Theorem spaceA NP Our construction

D D

of D will ensure that spaceA is our desired P simple NP sup ermaximal

space Let A fx C D gAteach stage swe shall let B fste

s x s s n

A hasnoelement of height ng Our construction will ensure that at each stage

s

s

s A B is a height increasing basis of stV for all i and s so We dene b

s s

i

s s s s

htb g where htb b that B fb

s

 

To ensure that spaceA is coinnite dimensional wemust meet the following

set of requirements

T car dfn D contains no strings with jj n g j

j

Thus T says there are at least j heights n so that A contains no strings of

j

height n So meeting requirement T ensures dimV spaceA j

j

D

To ensure that spaceAisP simple we shall meet the following two sets

of requirements

D

S IfN is an innite dimensional subspace of stV such that

j

i

D D

htN n htspaceA is innite then spaceA N fg

i i

D

Now supp ose that P generates an innite dimensional subspace of stV

j

D

whichisinNP Note that meeting all the requirements S will ensure that ei

j

D D

htspaceA Nowsup P spaceA fg or htspace ther spaceP

i i

D D

n htspaceA htspaceA and let U htP p ose that htspaceP

i i

D

If U thenhtP htspaceA Otherwise U is a nite set so let

i

D

suchthathtx U fn n g and let x x b e elements of spaceP

i  q  q

i

n Note that any x stV is a string of the form x a a where

i

jxj

a fk gThenwe dene the full heightofx fhxfn n

j

D

jxj and a g Then it is easy to see that given any x spaceP there

n

i

P

q

exists some in F suchthatfhx x U That

q st i i

i

where jxj n and a and x a a is if x a a

q n q q n q

jxj

q q

a

q

where a then x x x b b where b so

n q st q  n

jxj

q q

a

n q

q

andx a a where that n fhx Nowifb

q q n q q n

q  q 

b

q

a then x x x c c where c b

n q st q  n n

jxj

q  q q

a

n q 

q 

so that neither n nor n is in fhx Continuing on in and c

q q n

q 

P

q

this waywe can construct our desired linear combination x suchthat

i i

i

P

q

D

U fhx x U Now let Q fx spaceP fhx

st i i

i

i

D

g It is easy to see that Q is a subspace of P and our argumentabove

i

D

shows that spaceP spacefx x g QThus Q is an innite di

q

i

mensional subspace of stV suchthathtQ htspaceA Let T be

jx j

q

the set of all y such that fhy U jy j k and there exists an

D

x P and z spacefx x gsuchthatx z y Note that

q st

i

q

spacefx x g has exactly k elements since fx x g is a heightin

q q

jx j

q

creasing basis for spacefx x g Thus given any y with jy j k in

q

p olynomial time in jy jwe can nd all y w such that w spacefx x g

st q

jx j

q

Now for any w spacefx x g htw htx jx j k so that

q q q

q j

hty w hty Thus it takeatmostk jy j steps to test all such y w

st st

D

given an oracle D But then for memb ership in P

j

D

y T i fy w w spacefx x ggP

st q

j

D

Thus it follows that T is in P and clearly T generates an innite dimen

D

generates an sional subspace of QThus there must b e some j suchthatP

j

D

spaceP htspaceA innite dimensional subspace of stV andht

j

D

Thus to ensure that spaceAisP simple it will b e enough to ensure that we

meet the following set of requirements

D

R IfP generates an innite dimensional subspace of stV then

i

i

D

htP htspaceA

i

Finally to ensure that spaceAisNP sup ermaximalwe shall meet the

following set of requirements

D D

Q IfN spaceA is an innite dimensional and N spaceA then

in

i i

D

there is an x N suchthatx ste spaceA

n

i

D D

Note that if N spaceA and dimN spaceA is innite then meeting

i i

D D

all the requirements Q will ensure that ste N for all n so that N

n

in

i i

stV

We shall rank our requirements with those of highest priority coming rst

as T S R Q T S R Q

   

As in the construction of Theorem at anygiven stage swe shall pick

out at most one requirement E where E will b e one of the requirements S R

j j j j

or Q and take an action to meet that requirement We shall then saythat E

j j

received attention at stage s The action that we take to meet the requirement

E of the form S or Q will always b e of the same form That is weshall

j j j

put some elements into D at stage s and p ossibly restrain some elements from

entering D for the sake of the requirement We shall let resE s denote the

j

set of elements that are restrained from entering D at stage s for the sakeof

requirement E Wesay that requirement E of the form S or Q is satised

j j j j

at stage s if there is a stage s s suchthatE has received attention at stage

j

s and resE s D

j s

The actions that wetake to meet the requirements R will essentially the

j

same as in the construction of Theorem First we shall declare that all R

j

are in a passive state at the start of our construction Wewould liketondan

D

s

element x P of height n suchthatn htspaceA If wecanndsuch

s

j

an x then we will restrain all y such that jy j n and y C for some

x

x stV ofheight n plus all elemen ts not in D which are queried of the oracle

s

D

s

x from entering D for the sake of requirement during the computation of P

j

R Then if we ensure that resR s D A will have no elements of height

j j

D D

n and x P so that htP htspaceA If wetakesuch an action for

j j

R at stage sthenwe will say that R has received attention at stage s and

j j

declare the state of R to b e active Then for all tswewillsay that an

j

active R is satised at stage tifresR s D However if R is injured

j j t j

at some stage ts in the sense that resR s D then R will return to

j t j

D

to a passive state If we cannot nd suchanxwe will attempt to force htP

j

b e nite That is since we will ensure that htspaceA fs g for

s

all s A will have no elements of height s Recall that we are assuming that

s

X

y for any oracle X is b ounded for n the run time of computations of P

j

j

weletd b e the largest maxn for any string of length nThenforn

n

i such that for all r

i n

n k

Note that it is easy to see that lim d Our idea is that elements

s s

of height n in stV are just the elements of length n Our strategy at the end

of stage s for s is that for all R with j d which are in a passive

j s

D

s

x for all x stV of length s state and have the prop ertythatP

j

we will restrain all elements which are not in D and which are queried in

s

such computations from entering D for the sakeofR This action will force

j

D

htP to b e nite if R is in a passive state at stage s for all but nitely many

j

j

sFor any xed j d the maximum restraint imp osed for R o ccurs if we

s j

restrained all elements not in D which are queried of the oracle D in some

s s

D

s

computation P x with jxj n and x stV Since the total

j

numb er of steps used in all these computations is at most

s

X

j i j s j s j 

k i sk s k s

i

s j 

then clearly we could have restrained at most k s elements from entering

D for the sakeofR Thus at stage swewillhave restrained at most

j

d

s

X

s i s d  s s s

s

k s k s k k k

i

elements from entering D for the sake of some passive requirement R with

j

ygiven x with jxj nwe will have j b at stage s Hence for an

s

s

restrained less than k elements of C from entering D for such R s

x j

CONSTRUCTION

Stages

Let D D so that A A LetresE resE for all

  j j

requirements E of the form S R or Q

j j j j

Stage s with s

Let E b e the highest priority requirement among S R Q S R Q such

j    s s s

that

Case E S and S is not satised at stage s and there exists an

j j j

x stV with jxj s suchthat

D

s

a x N

j

s

b jxj htspaceA and jxj jb jand

s

j

c there exists a y C suchthaty is not restrained from D byany require

x

y than S at stage s and y is not queried of the ment of higher priorit

j

D

s

oracle D in some xed computation whichshows that x N

s

j

Case E R and R is not satised at stage s and there exists an

j j j

x stV withjxj s suchthat

i jxj htspaceA and

s

D

s

ii x P

j

Case A Q and Q is not satised at stage s and if j e n there

j j j

exists an x with jxj s suchthat

D

s

I x N

e

s

I I jxj htspaceA jxj jb j and jxj n and

s

j

I I I there exists a y C suchthaty is not restrained from D byany

x ste 

st n

requirement of higher prioritythanS at stage s andy is not queried

j

D

s

of the oracle D in some xed computation whichshows that N x

e

s

If there is no such requirement E let D D Also for all requirements

j s s

or Q and for all requirements E of the form R where E of the form S

j j j j j

either R is satised at stage s orjd let resE sresE s

j s j j

Declare that a requirement R is active at stage s i R is active at stage s

j j

For any R with j d which is currently passive and has the prop ertythat

j s

D

s

P x for all x stV of length s let resR s equal resR s

j j

j

union the set of all y D suc hthaty is queried of the oracle D in one of the

s s

D

s

computations P x where x stV of length s

j

If there is such a requirement E wehave three cases

j

Case E S

j j

s

Let x denote the least x corresp onding to S Then pick the least string

s j

s

C suchthat is not restrained from D byany requirement of higher

x x x

s s s

prioritythanS at stage s nor is queried of the oracle D in the

j x s

s s

D

s

which accepts x and put into D LetresS s computation of N

s x j

s s

j

equal the set of all strings not in D which are queried of the oracle D in

s s

D

s

whichaccepts x and say S receives attention at the computation of N

s j

s

j

s

stage s Also for all requirements E of the form S or Q and for all require

j j j

ments E of the form R where either R is satised at stage s orjd

j j j s

let resE s resE s if D resE s and let resE s

j j s j j

if D resE s Declare that a requirement R is active at stage s

s j j

i R is activeatstages andD resR s For any R with

j s j j

D

s

z for j d whichiscurrently passive and has the prop erty that P

s

j

all z stV of length s let resR sequalresR s union the set

j j

of all y D suchthaty is queried of the oracle D in one of the computations

s s

D

s

P z where z stV and jz j s

j

Case E R

j j

s

Let x denote the least x corresp onding to j WethensaythatR is active

s s j

s

and receives attention at stage sWe let D D and resR s consist of

s s j

s

all elements y of length jx j which are in some C suchthatz stV and

s z

D and which are queried of the jz j jx j and all elements whicharenotin

s s

D

s

xNotethatifresR s D oracle D in the computation P

j s

s

j

s

D

then A will have no elements of height jx j but x P Also for all require

s s

j

s

ments E of the form S or Q and for all requirements E of the form R

j j j j j

where j j and where either R is satised at stage s orjd let

s j s

resE s resE s For j j declare that a requirement R is active

j j s j

at stage s i R is active at stage s For any R with j d whichis

j j s

D

s

currently passive and has the prop ertythatP xforallx stV of

j

length s let resR s equal resR s union the set of all y D such

j j s

D

s

x where that y is queried of the oracle D in one of the computations P

s

j

x stV and jxj s

Case E Q

j j

s

Let j e n and x denote the least x corresp onding to j Then pickthe

s s s s s

least string C suc hthat is not restrained from D byany require

x x x

s s s

ment of higher priority than Q at stage s nor is queried of the oracle

j x

s s

D

s

s into D LetresQ D in the computation of P x and put

e

j s s x

s

s s

consists of all strings which are not in D which are queried of the oracle

s

D

s

D in the computation of P x and say Q receives attention at stage

e

s s j

s

s

s Also for all requirements E of the form S or Q and for all requirements

j j j

E of the form R where either R is satised at stage s or jd let

j j j s

resE sresE s if D resE s and let resE s

j j s j j

if D resE s Declare that a requirement R is active at stage s

s j j

i R is activeatstages andD resR s For any R with

j s j j

D

s

x for j d which is currently passive and has the prop ertythatP

s

j

all x stV of length s let resR s equal resR s union the set

j j

of all y D suchthaty is queried of the oracle D in one of the computations

s s

D

s

x where x stV andjxj s P

j

This completes the construction of D We note that A is a height increasing

Y

indep endentsetinNP since our construction ensures that wecannever put

Y

two elements of the same heightinAThus by Theorem spaceA NP

We then havetoprove the same sequence of lemmas as in Theorem to

complete the pro of the theorem The details may b e found in

Again the spaceA constructed in Theorem has a number of interest

D D

ing prop erties b esides b eing NP sup ermaximal and P simple First of all

D

meeting all the requirements R ensures that spaceAisP immune That

j

D D

is if P is an innite subset of spaceA then certainly P generates an

i i

D

innite dimensional subspace of stV andhtP htspaceA which

i

would violate requirement R Also as in the construction of Theorem it

i

is easy to check that in meeting the requirements S wemadenouseofthe

j

D D

wasasubsetof was a subspace of stV but only that N fact that N

j j

stV We claim that stV n spaceAdoesnothaveany innite subsets in

D D

stV n spaceA Now it cannot b e that P That is supp ose that P

j

D D D

htP n htspaceA is innite since otherwise there is an i suchthatP N

j j i

D

and the fact that we met requirement S would mean that P spaceA fg

i

j

D D

Then clearly Thus htP htspaceA Let Q htspaceA n htP

j j

D

S fx P htx Qg

j

D

is an innite set in P which generates an innite dimensional subspace of

stV and htS htspaceA Since meeting all the requirements R rules

j

out the existence of suchanS stV n spaceA do es not contain an innite

D D

set in P Thus spaceAandstV n spaceAareP immune

D

Note also that by Theorem the fact that spaceAisNP maximal

D D D D

implies that NP coNP and hence that P NP Thus wehave

proved the following

Corollary There exists an re oracle D and a subspace V of stV such

that

D D D D

i P NP and NP coNP

D

ii V NP

D D

iii V is P immune and hence has no basis in P

D

iv stV V is P immune and

D D

v V is both P simple and NP supermaximal

Finally we observe that results ab out NP and P subspaces of tal V natu

rally extend to results ab out NEXT and DE X T subspaces of stV by Lemma

A

V is NEXT maximal Foratypical example say that a subspace M of st

A

if M NEXT dimstV M is innite and for any subspace W of stV

A

in NEXT containing M either dimstV W is nite or dimWM is

nite Then Theorem and Theorem show that the question of the

existence of NEXT maximal subspaces is oracle dep endent

Theorem Thereisarecursive oracles A and an re oracle B such that

the fol lowing hold

A A B B

i NEXT DE X T and NEXT DE X T

A

ii TherearenoNEXT maximal subspaces of stV

B

iii ThereisanNEXT maximal subspace W of stV

X X

In the same way all the results in this pap er ab out P and NP subspaces

X X

of tal V can b e transfered to results DE X T and NEXT subspaces of

stV

Next we consider some results on splitting theorems for tal V due to

Bauerle We note a result of Ash and Downey thatevery re subspace of

V is the direct sum of two decidable spaces In tal V the prop erty of b eing

a direct sum of two ptime subspaces is equivalenttohaving a ptime basis

Theorem A subspaceofV of tal V can be split into two polyno

mial time subspaces if and only if V has a basis in P

Note that by the results on bases and subspaces of tal V we immediately

get the following corollaries

Corollary Thereisanexponential time subspace W of tal V that can

not be split into two polynomial time subspaces

Corollary For al l re degrees there is an resubspace V of tal V

such that degV and V can be split into two polynomial time subspaces

A

A

Corollary There exists a recursive oracle A such that every NP n P

A

subspace V of tal V can be split into two P vector spaces

cursive oracle B such that Corollary Let F be nite There exists a re

B

B B

thereisa NP n P subspace V of tal V that cannot be split into two P

vector spaces

Corollary Arguments valid under relativization are not sucient to show

NP P every NP subspaceoftalV can be split into two P time sub

spaces NP P there exists an NP n P subspaceof tal V which cannot

be split into two P time subspaces

In fact Bauerle identies three typ es of splittings by p olynomial time sub

spaces of tal V

e an re vector space Denition Let V b

V al lows a P splitting if thereexistP time spaces W and W such that



W W fg and W W V WesaythatW and W P split V

  

V al lows an induced P splitting if thereexist P time spaces W and W



such that W W fg W W V dimV W dimV W

  

andW V W V V We say that W and W inducea

 

P splitting of V

V al lows an induced weak P splitting if there exist re vector spaces

W and W such that W and W have bases in P W W fg

  

W W V dimV W dimV W and W V

  

V V We say that W and W induceaweak P splitting of V W



Theorem Let V beasubspaceoftalV

i If V al lows a P splitting then V al lows an induced P splitting

ii If V al lows an induced P splitting then V al lows an inducedweak P

splitting

Theorem

i There exists an exponential time subspace V of tal V that al lows an in

duced P splitting but no P splitting

ii There exists an exponential time subspace V of tal V that does not al low

an inducedweak P splitting

This shows that three notions of Denition are increasingly weaker

We end this section we some results on Bauerle on subspaces and sup er

spaces of NP n P subspaces of tal V

Theorem Every NP n P subspace V of tal V has a nontrivial

NP n P subspace W

Theorem Let V bea subspaceoftal V such that V NP n P

If V has a nontrivial superspaceinP then V has a nontrivial superspacein

NP n P

Theorem There exists a recursive oracle C such that there exists a

vector space V tal V that satises the fol lowing properties

C

C

V NP n P

C

C

V has a nontrivial superspaceinNP n P

C

V has no nontrivial superspaces in P

Theorem Thereisarecursive oracle C such that al l nontrivial

C

C C

NP n P subspaces of tal V have nontrivial superspaces in P and in

C

C

NP n P

A A

Finally under the assumption that NP coNP

A

A

Theorem Let A beanoracle such that NP n P subspaces of

A

A

tal V exist and such that NP co NP Then the fol lowing is true

A A

For al l NP n P subspaces V of tal V their standard height increasing

A A

basis B is in NP n P

V

A A

For al l NP n P subspaces V of tal V their standard height increasing

and their standardcomplement B arein complementary basis B

V V

A A

NP n P

A A P

Every NP n P subspace V of tal V can be split into two disjoint

T

A A

incomparable NP n P subspaces

P A

The set of degrees with NP subspaces of tal V is dense

T

n

A

There exists a pair V W of NP P subspaces of tal V such that if

A P P

W thenU P V and U U

T T

P P

The set of rationals Q can beembedded in the structures of and

m

T

A

degrees of NP subspaces of tal V

Polynomial Time Bo olean Algebras

In this section we shall survey the results of Nero de and Remmel on the lower

semilattice of NP ideals of a p olynomial time presentation of the free Bo olean

algebra Again we consider two natural representations of the free Bo olean

algebra called the tally and standard representation We start by describing

these two representations

Let P denote the Bo olean algebra of all subsets of the rational left

closed rightop en interval in the rational number Q The Bo olean op era

tions of meet join and complementation on P are resp ectively intersec

tion union and relative complementin Let B b e the subalgebra

of P generated by the leftclosed rightop en intervals of the form

i j

n n

n

with n and ij For any subset S B S denotes the

subalgebra of B generated by S and I S denotes the ideal generated by

S Given a subalgebra D B welet AtD denote the set of atoms of

D

Next we dene a natural generating sequence a a for B byin



duction

a

m m

n

n

ifn and m a

n n

 m

 

Thus

a a a a

 

a a a

  

Let A fa a g Then it is not dicult to see that A A A is

n n 

a strictly increasing sequence of subalgebras such that for each n there is

a unique atom x AtA such that a splits x ie a x In

n n n n n n

n

fact one can easily showby induction that if k is of the form m with

n

m then

i i j j

n

AtA f imgf m j g

k

n n n n

m m m m

Hence a splits the atom x ofA It follows

k  n n k n n k

   

n

that A has exactly n atoms for each n so that A has exactly elements

n n

We use this generating sequence and its corresp onding sequence of subalge

bras

A A A



to dene the standard and tally representations of B

The Standard Representation of B

First we describ e a co ding of the elements of B whichwe call that stan

dardrepresentation of B Our idea is to use binary numb ers of length n

to co de the elements of A n A for n Formallywe dene a corre

n n

sp ondence s between Bin andB by induction

For the base step set

s s



For the induction step assume that the corresp ondence bink s has

k

n

and A We then extend our b een dened b etween fbink k g

n

corresp ondence to A as follows Given a binary number m of length n

n

i i

let m k where k is o dd so that binm bink and let

s a if s a

k n k n

s

m

s n a if s a

k n k n

Nowlet AtA fx x x g where x is the atom of A whichis

n n n n n

split by a Then it is easy to see that every elementof A n A is either

n n n

of the form

a x

n i

iS

or

x n a x

n n i

iS

for some set S fn gThus denes a corresp ondence b etween

n n

fk k g and A n A

n n

Indeed it is quite easy to use to recursively construct s Wewrite s

n

binn

for s in the following

n

Example Suppose binn Then s can beconstructedasfol

n

lows

s



s a since a s

   



s s a since a s

  

 

n since a s s s n a

   

 

 

s s n a n since a s

    

    

It is not dicult to showthatgiven two and in Bin with j jj j

we can nd and in Bin suchthat

s s s s s s s n s

in p olynomial time in j jFurthermore note that eachof and has length

j j since eachofs s and s b elongs to A if s A See for details

n n

It follows that if we then dene

s

s

s

Then stB Bin is a p olynomial time representation of the

s s s

countable atomless Bo olean algebra B whichwe call the standard repre

sentation of B

The Tally Representationof B

The tally representation tal B ofB can easily b e dened from the

binary representation stB to b e the isomorphic image under the map

taking binntotal n and is therefore a ptime structure by Lemma in

light of the note ab ove

Nero de and Remmel studied three basic prop erties of ideals in a recursive

Bo olean algebras Here given a Bo olean algebra B B wesay

B B B

I B is an ideal if the zero of B is in I forallx y I imply x y B

B B

and for x I and z B x z I I is a maximal ideal if for all z B either

B

z I or z I

B

Nero de and Remmel studied p olynomial time analogues of the following well

known results on re ideals in a recursive presentation of B

A In a recursive Bo olean algebra every re maximal ideal is recursive

B Every prop er recursive ideal is contained in a recursive maximal ideal

C There exists an re ideal of B which is not extendible to a recursive

ideal

alent to the prop osition that there is an re axiom WenotethatC is equiv

atizable theory whichisnotcontained in any decidable theory

First consider A The fact that every re maximal ideal of a recursive

Bo olean algebra is based on Kleenes lemma that a set which is re and co

re is automatically recursive The obvious ptime analogue of A is that every

NP maximal ideal of ptime Bo olean algebra is p olynomial time However

in this case it is a long standing op en problem whether NP coNP P

Moreover is well known results of BakerGillSolovay there exists recursive

X X X Y Y Y

oracle X and Y such that NP coNP P and NP co NP P

Y Y

but P NP Thus it should come as no surprise that the analogue of

A is oracle dep endent That is Nero de and Remmel were able of mo dify the

BakerGillSolovay constructions to prove the following

Theorem There exists a recursive oracle X such that there exist a maximal

X X

ideal I of tal B such that I NP n P

Our next corollary immediately follows from Theorem and Lemma

acle X such that thereexistsare Corollary There exists a recursive or

cursive oracle X such that there exist a maximal ideal I of stB such that

X X

I NEXT n DE X T

Theorem There exists a recursive oracle Y such that there exist a maximal

X X

ideal J of stB such that J NP n P

E E

Theorem There exists a recursive oracle E such that P NP and the

fol lowing hold

E E

i Every maximal ideal I of tal B which is in NP is in P

E E

ii Every maximal ideal J of stB which is in NEXT is in DE X T

E E

iii Every maximal ideal K of stB which is in NP is in P

Theorems then yield the following results

Theorem Arguments which remain valid under relativization to oracles do

not sucetoprove any of the fol lowing

P NP implies that every NP maximal ideal of tal B is in P

P NP implies that thereisan NP maximal ideal of tal B which is not

in P

P NP implies that every NP maximal ideal of stB is in P

P NP implies that thereisaNP maximal ideal of stB which is not in

P

Next we turn to the analogues of B In this case Nero de and Remmel proved

that the obvious analogues of B are true for b oth tal B and stB although the

argument requires a great deal more care That is Nero de and Remmel

proved the following

Theorem Every proper ideal I of stB which is in P can be extendedto

maximal ideal J of stB which is in P

Theorem Every proper ideal I of stB which is in NP coNP can be

extended to maximal ideal J of stB which is in NP coNP

Theorem Every proper ideal I of stB which is in DE X T can be extended

to maximal ideal J of stB which is in DE X T

Theorem Every proper ideal I of stB which is in NEXT coNEXT

can be extended to maximal ideal J of stB which is in NEXT coNEXT

Theorem Every proper ideal I of tal B which is in P can be extended

to maximal ideal J of tal B which is in P

Theorem Every proper I of tal B which is in NP coNP can beex

tended to maximal ideal J of tal B which is in NP coNP

Finally we turn to the analogues of C In this case the analogues are oracle

dep endent despite Theorems

A A

Theorem There exists a recursive oracle A such that P NP and

A

every proper ideal I of stB which is in NP is extendible to maximal

A

ideal J of stB which is in NP

A

every proper ideal I of tal B which is in NP is extendible to maximal



A

ideal J of tal B which is in NP



A

every proper ideal I of stB which is in NEXT is extendible to maximal

A

ideal J of stB which is in NEXT

Pro of Homer and Maass constructed a recursive oracle A suc hthat

A A A A

P NP but NP coNP Thus we can use the relativized version of

Theorem to prove part and we can use the relativized version Theorem

to prove part Finally part follows from part and Lemma

The pro of of the other direction of the oracle dep endence requires a new

construction is much more subtle than any of the previous theorems on ideals

in our ptime representation of B The actual pro ofs can b e found in

Theorem There exists a recursive oracle C and an ideal J of tal B

C

whichisinNP which is not contained in any maximal ideal of tal B

C

whichisinNP

B

Thereexistsarecursive oracle B and an ideal J of stB whichisinNP



B

which is not contained in any maximal ideal of stB which is in NP

Of course we can combine Lemma and Theorem to provethe

following

Theorem There exists a recursive oracle C and an ideal J of stB which

C

is in NEXT which is not contained in any maximal ideal of tal B which is

C

in NEXT

Conclusions and Future Directions

In this surveywehave presented the basic denitions of complexity theoretic

algebra and mo del theory and have attempted to outline the current state of

knowledge in the eld There is a great deal more which remains to b e done

We will just mention four p ossible themes for future research

First we observe that the results on complexity theoretic algebra were lim

ited to the study of ideals in the free Bo olean algebras and and subspaces of an

innite dimensional vector spaces There are many other algebraic structures

that have b een studied in recursive algebra including elds mo dules subal

gebras of Bo olean algebras subgroups of groups etc Cenzer Downey and

Remmel have recently investigated torsionfree Ab elian groups Wehave

also given complexity theoretic results in combinatorics Other related areas of

mathematics such as geometry and numb er theory should also provide fruitful

basis for investigation

Second most of our results concerned the notions of p olynomial time com

plexity with some results given on linear time and on exp onential time com

plexity There are many other interesting notions of complexity including for

example PSPACE and LOGSPACE which should provide b oth comparable

and contrasting results

Third wegave only a few results involving the imp ortant complexityhy

p otheses of theoretical computer science such as whether P NP or NP

PSPACE In the complexity theory of real functions Ko has provided

manysuch results For example he gives a condition not involving complexity

on a real function f and shows that if P NP and if f is a ptime computable

function on the unit interval which satises this condition then all ro ots of f

are ptime computable There should b e similar results in complexity theoretic

algebra

Fourth wehave only b egun the study of complexity theoretic mo del theory

with a few results on relational structures and with the general notion of Scott

sentences and categoricity If one studies the recursive mo del theory survey by

Harizanov many problems suggest themselves For example the authors

have recently investigated complexity theoretic versions of the eectivecom

pleteness theorem in Decidabilityisalsoofinterest in the study of prime

and saturated mo dels and in stability theory

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