The Structure of Logarithmic Advice



Complexity Classes

Jose L Balcazar

Department of Software LSI

Universitat Politecnicade Catalunya

Pau Gargallo E Barcelona Spain

balquilsiupces

Montserrat Hermo

Department of Software LSI

Universidad del Pas Vasco PO Box

E San Sebastian Spain

jiphehumsiehues

May

Abstract

A nonuniform class called here FullPlog due to Ko is studied It corresp onds

to p olynomial time with logarithmically long advice Its imp ortance lies in the struc

tural prop erties it enjoys more interesting than those of the alternative class Plog

sp ecically its introduction was motivated by the need of a logarithmic advice class

closed under p olynomialtime deterministic reductions Several characterizations of

FullPlog are shown formulated in terms of various sorts of tally sets with very small

information content A study of its inner structure is presented by considering the

most usual reducibilitie s and lo oking for the relationships among the corresp onding

reduction and equivalence classes dened from these sp ecial tally sets



Partially supp orted by the EU through the ESPRIT Long Term Research Pro ject ALCOM

IT and through the HCM Network CHRXCT COLORET by the Spanish DGICYT through

pro ject PB KOALA and by Acciones Integradas HispanoAlemanas HAB and ALB

Introduction

Nonuniform complexity classes were essentially introduced in where the main rela

tionships to uniform classes were already shown In order to capture characteristics of

nonuniform mo dels of computation in which xed input lengths are compulsory in the

denition of nonuniform classes a b ounded amount of extra information the advice

dep endent of the length of the input is provided The two most natural families of b ound

functions for the advice information are p olynomials as in the class Pp oly and loga

rithms as in the class Plog In structural terms the class Pp oly has a high interest

due to its many characterizations and has b een studied in considerable depth

However Plog not b eing closed under the most usual reducibilities its structural study

has less interest We consider in this pap er a variant prop osed by KerI Ko in and

also treated marginally in and of the complexity class Plog in which closure under

p olynomial time reductions is obtained via a more restrictive condition in the denition

Our aim is to argue here that this relative of Plog that we call FullPlog not only is

closed under reducibilities but also has an internal structure worth study comparable to

that of Pp oly with many similarities and a few interesting dierences

This internal structure has also consequences for the learnability of certain circuit ex

pression classes of logarithmic Kolmogorov complexity The reason is that FullPlog char

acterizes the concepts that can b e describ ed by these representation classes similarly to

the characterization of Pp oly by p olynomial size circuits This asp ect of FullPlog is

treated in depth elsewhere

We must mention in passing another motivation to study this class In a mo del

of neural networks is describ ed and the language recognition p ower of these networks

is characterized in terms of the types of numbers employed as weights Also a precise

corresp ondence is establish b etween the choice of integer rational or real weights and the

resp ective classes of languages When the computation time of the networks is constrained

to b e p olynomial in the input size the classes recognized by the resp ective nets are regular

P and Pp oly It may b e argued that any net with real weights that is computationally

feasible to implement must admit a short description of its realvalued weights Thus

setting a logarithmic b ound on the resourceb ounded Kolmogorov complexity of the real

weights it is proved in that the languages recognized corresp ond to the class FullPlog

This pap er is structured in two dierent parts The rst one shows several characteriza

tions of FullPlog as a reduction class The pro ofs are quite interesting in that they require

to dene another technical variant of the class based on prexclosed advice words and to

introduce a new pro of technique by which information is selected at a doubly exp onential

rate skipping all information corresp onding to intermediate advice words This kind of

argument has b een heavily used in As applications of this technique we obtain sev

eral characterizations of FullPlog as the reduction class of sp ecial sets tally sets whose

words follow a given regular pattern and tally sets that are regular in a resourceb ounded

Kolmogorov complexity sense

The second part fo cuses on the inner structure of FullPlog trying to understand the

p ower of this sort of sp ecial tally sets when they are used as oracles To do that we

explain some of the relationships which exist among reductions classes dened from them

In addition we describ e how equivalence classes dened in terms of the same tally sets are

related

The results of sections to app eared in preliminary form in The results of

section app eared in preliminary form in Almost all the results describ ed here are

from

Preliminaries

An alphab et is any nonempty nite set We use here the alphab ets f g and fg

Given any alphab et a nite string or word over is a nite sequence of symbols from

We denote words by lower case Latin letters such as x y Any set nite of innite

of these strings is called a language The language of all p ossible nite strings over is

denoted by There is a sp ecial string in that is the unique word consisting of zero

symbols We call it

Given a set A we indicate the cardinality of A with the expression jAj In the same

way when we refer to the number of symbols in some string x we use jxj

Regarding words x means the substring formed by x from the ith symbol on up

ij

to and including the j th symbol As our strings are usually sequences over f g

sometimes we sp eak ab out the j th bit of x instead of the j th symbol

A natural op erator the concatenation is dened among the strings of Given two

words x y the concatenation of x with y pro duces the new word xy A prex of a word

y is any word z such that for some word w y z w The notation z v y denotes the fact

that z is a prex of y

Let A b e a language over some alphab et The set of all strings x in A whose length is

n

less than or equal to n jxj n is denoted by A When we refer to the strings of A that

n

have exactly length n we use the notation A We express that the set A is included in

the set B by writing A B but when the inclusion is strict we say that A B The

complement of a language A is denoted by A when is known we omit it leaving

just A Given any with at least two symbols say and the join or marked union of

two languages A and B over is

A B fx j x Ag fx j x B g

Denition A language is tal ly if and only if it is included in fg

The class of all tally sets is denoted by Tally The upp er case initial is used to express

the complete class while the lower case initial means the prop erty of b elonging to Tally

Next let us dene the concept of characteristic sequence of a particular set

Denition Given a language A

A

Its characteristic sequence is an innite string of f g such that for all n

A

the nth bit of is if the nth nite string of ordered by lengths and in

lexicographical order within each length b elongs to A Otherwise this bit is

n

A

Its characteristic sequence up to a particular length n is a nite string fullling

n n

A n A A

that j j and v

A

When the set A is a tally set is the characteristic sequence relative to fg that

n

means we only take into account the words for any n In the same way slightly abusing

n

the notation when we use sets L such that L f j n INg the characteristic sequence

L

will b e relative to the language formed by a p ower of many s More generally

arbitrary innite strings are denoted by Greek letters as

Throughout the pap er all logarithms are to base

Dierent Types of Reductions

Our uniform computational mo del is the multitape Turing machine with a readonly input

tap e An oracle Turing machine has an additional writeonly oracle tap e The machines

can b e deterministic DTM or nondeterministic NDTM

We work with classes of languages recognized by oracle Turing machines that work in

p olynomial time Since the access to the oracle can b e restricted in dierent ways the

following reducibilities can b e obtained

Denition A language A is r reducible to B in p olynomial time A P B if and

r

only if there exists an r restricted deterministic p olynomialtime oracle Turing machine M

such that A is the language recognized by M querying oracle B

Now we explain the meaning of r for a xed reduction

When r is the Turing reduction r T there are no restrictions on the oracle

machine M

When r is the truthtable reduction r tt the machine M is able to write down

on a separate tap e all the queries to b e made during the computation b efore the rst

word is queried

A particular case of the truthtable reduction is the k truthtable reduction written

r k tt when k Now there exist p olynomialtime computable functions f and

h such that for all x f x is a list of k strings f x hx x x i hx is a truth

k

table with k variables and x A if and only if the truthtable hx evaluates to true

on the b o olean k tuple hx B x B x B i

k

Set A is b ounded truthtable reducible r btt to set B if there is a p ositive k such

that A is k tt reducible to B

When r is the manyone reduction r m it is only allowed to make a unique query

to the oracle in the last step of the computation in such a way that M accepts if

and only if the answer is YES An equivalent denition of the manyone reduction

is as follows A B if and only if there exists a p olynomialtime function f such

m

that for all x

x A f x B

We say that the Turing reduction is an adaptive pro cess b ecause the queries dep end on

the answers given by the oracle on the previous steps of the computation Conversely the

truthtable approach is nonadaptive since the queries are completely indep endent of the

answers Actually the idea of the truthtable reduction is sometimes describ ed in terms of

making all the queries to the oracle at the same time in parallel

Given a class of languages G the class of sets that are r reducible to some set in G is

expressed by writing P G Since we only work with NDTMs that access to the oracle

r

without any restriction NPA identies the class of languages that are Turing reducible

to A via a NDTM

We also dene the classes of sets that are not only reducible to but also inter

reducible with

Denition Given a family of languages G a set A is in the class E G when there

r

exists B G such that A P B and B P A

r r

Ab out Functions

There exist some functions with a sp ecial prop erty known as honesty

Denition A function f is honest if and only if for every value y in the range of f there

k

is an x in the domain of f such that f x y and jxj jy j for some xed constant k

It is not known whether there are p olynomial time computable honest functions whose

inverses are not p olynomial time computable In fact this op en problem is equivalent to

determining whether P NP

Theorem see P NP if and only if every honest partial function computable in

p olynomial time has an inverse computable in p olynomial time

We need to enco de several words into one in such a way that b oth computing the

enco ding and recovering the co ded words can b e easily done We have chosen a pairing

function h i Given x and y the word hx y i is obtained by duplicating

each bit of x app ending to this the word y inserting a in b etween Assuming that the

lengths of x and y are n and m resp ectively the length of the pairing function applied to

x y is jhx y ij n m The computations of h i and its inverses needs very little

resources

The pairing function can b e applied to tuples as follows hx y z i hhx y i z i and so

on However sometimes we use a dierent function in order to enco de a nite or innite

sequence of strings Namely the notation

x x x x

n

means that we app end together all words in the sequence fx g duplicating each bit of

i in

each x separating each x from the next by the mark

i i

Kolmogorov Complexity

Fix a Universal Turing machine U The Kolmogorov complexity of a string w resp the

Kolmogorov complexity relative to y is the length of the shortest program resp pair

hprogram y i which when given as input to U will lead U to write down w as output

Hartmanis and Sipser mo died the original idea of Kolmogorov complexity

to include the running time or space used by the Universal Turing machine in order to

pro duce an output Ko in followed the same approach but applying it to the notion

of innite sequences with resp ect to p olynomialtime and space complexity The sets of

b ounded Kolmogorov complexity strings Kf n g n is dened as follows

Denition

Kf n g n fx j y jy j f jxj U y x in at most g jxj stepsg

Nonuniform classes

The basic nonuniform complexity classes are Pp oly and Plog The conditions for a

language L to b e in some of these classes are

Denition

L Pp oly i there exist B P and a p olynomial p such that

n w jw j pn such that x jxj n x L hx w i B

n n n

L P log i there exists B P and a constant c such that

n w jw j c log n such that x jxj n x L hx w i B

n n n

It is easy to see that P Plog Pp oly since tally sets are in Plog see for

T

more details Since Plog Pp oly see for instance we have

Theorem P Plog is not included in Plog

T

Moreover one can see that even P Plog is not included in Plog see again

m

1

Preliminary versions of circulated simultaneously to and

The classes FullP=log and PrefP=log

Since the logarithmic analog to Pp oly is not closed under most usual reducibilities an

alternative approach was introduced by Ko Kos class although with a dierent name

is introduced in the following denition

Denition A set A is in FullPlog if

n w jw j c log n x jxj n x A hx w i B

n n n

where B P and c is a constant

Note that the dierence resp ect to Plog lies in the range of the x quantier

The idea is quite natural For instance if the denition of Pp oly is changed according

to this we obtain the same class Pp oly That is to say

Denition A set A is in FullPpoly if

c

n w jw j n x jxj n x A hx w i B

n n n

where B P and c is a constant

Prop osition FullPpoly Pp oly

Pro of By denition FullPpoly Pp oly Now supp ose that A Pp oly That means

c

n w jw j n x jxj n x A hx w i B

n n n

where B P and c is a constant Denote by v the concatenation of all the advice words

n

w w w

n

v w w w

n n

o

For all n the length of v is in n and v can b e used as advice word by all the lengths

n n

up to n Therefore A FullPpoly ut

It is easy to see that FullPlog is closed under p olynomialtime Turing reducibility

Prop osition P FullPlog FullPlog

T

Pro of The nontrivial inclusion is P FullPlog FullPlog Supp ose that A P C

T T

with C FullPlog and that the Turing reduction is done via a DTM M which works

q

in time n In order to decide whether a xed word x is in A simulate the computation

of the machine M on input x with the extra information of the advice word for the set C

q

and the length jxj Each time a query is made to C the answer is given using the advice

q

word That means for all input x the advice word for C and the length jxj can b e used

as advice word for A and the length jxj The size of these advice words is a function in

q

Olog n Olog n therefore A FullPlog ut

As a result of the closure of FullPlog under Turing reducibility the closure of the

class under the other common p olynomialtime reducibilities is also obtained and since by

Theorem Plog is not closed under p olynomial reducibilities we get

Corollary Plog FullPlog

Actually also their restrictions to tally sets are dierent in Theorem we precisely

characterize the tally sets in FullPlog as those of low Kolmogorov complexity

In order to characterize the class FullPlog we present a technical variant of full log

arithmic advice in which the advice words corresp onding to various lengths are not inde

p endent but highly correlated

Denition A set A has prex logarithmic full advice briey A PrefPlog if A is

in FullPlog via an innite sequence of advice words w having the additional prop erty

n

that for all n m w is a prex of w

n m

Thus each advice is simply an extension with some extra bits of the previous advice

In the limit therefore the sequence of advice words converges towards a unique innite

word such that for all n w the rst c log n bits of Observe also that here

n c log n

the advice length is so tightly b ounded that for most values of n the corresp onding advice

w do es not have ro om to include one more bit than its predecessor Indeed c log n only

n

c

increases by one when n increases by a multiplicative factor of Thus very frequently

w w and only exp onentially often can w b e a prop er prex of w

n n n n

Of course the denition can b e straightforwardly rephrased to apply to other b ounds on

the advice length or to other uniform complexity classes For instance a similar denition

for p olynomial advice gives exactly Pp oly

Denition A set A is in PrefPp oly when A is in Pp oly via an innite sequence of

p olynomial advice words w such that for all n m w is a prex of w

n n m

Prop osition PrefPp oly Pp oly

Pro of The construction of Prop osition yields a sequence of advice words having the

requested prex prop erty ut

Characterization of PrefP=log

This section shows that PrefPlog can b e characterized by p olynomialtime Turing reduc

tion classes of regularly structured tally sets as well as using b ounded query machines

Theorem The following classes of languages are the same

S

k

i P L where L f j k INg

L

T

S

ii P L via p olynomialtime machines whose queries have lengths at most Olog n

L

T

where L fg

S

P B via p olynomialtime machines whose queries have lengths at most iii

B

T

log log n O

iv PrefPlog

Observe that no constant factors are allowed on the term log log n in part iii only

additive constants can b e accommo dated in the b ound Part i is quite interesting in

that it shows that PrefPlog is the reduction class of tally sets exhibiting a high degree of

regularity All query b ounds mentioned assume as usual that n is the length of the input

Pro of The pro ofs that i implies ii and that ii implies iii are simple and similar b oth

are tantamount to a change of scale in the oracle set Let A b e a set in P L with

T

k k

k

L f j k INg Dene L f j Lg It is easy to see that A P L querying

T

k

k

when required Observe that the length of the queries is now logarithmic instead of

k o

since k Olog n whenever n Now we rep eat the argument if A P L with

T

k

Olog n length queries dene B fk j L g Again A P B and the maximum

T

length of the oracle queries is log c log n O log log n

To prove that iii implies iv we will employ the characteristic function of B as an

innite word limiting the sequence of advice words Let A b e a set with A P B

T

via a DTM M By hypothesis the lengths of the queries are smaller than or equal to

d log log n Therefore the number of dierent queries that M can make is b ounded

dlog log n

by c log n for an appropriate constant c Moreover these are the rst c log n

words So we dene the advice w as the characteristic sequence of B up to the element

n

in place c log n With this information each query to oracle B can b e answered Thus

A PrefPlog

Finally the pro of of iv implies i is essentially a converse of the comp osition of the

three arguments Supp ose that A PrefPlog where the innite word is the limit of

the sequence of advice words Let L b e the tally set

k

L f j the k th bit of is g

i

Now A P L by simply querying the words for i to i c log jxj to extract the

T

necessary advice of length c log jxj from the tally oracle and then using it ut

From now on we denote the following classes of tally sets as Tally

k

j k INgg Tally fL j L f

Hence parts i and iv in the theorem characterize PrefPlog as P Tally

T

Characterization of FullP=log

In this section one of the main contributions of this pap er is presented b oth in results by

relating FullPlog to the classes already describ ed and in technical contents by explaining

a technique which consists of selecting information separated by a doubly exp onential gap

Several examples of the application of this technique are presented in Now we will

give two of them b oth giving somewhat surprising characterizations of FullPlog The

rst one shows that FullPlog equals the seemingly more restrictive class of sets with

logarithmic prexclosed advice and the second one will show that FullPlog equals a less

restrictive class dened in terms of Kolmogorovregular tally sets

Theorem FullPlog PrefPlog P Tally

T

Equivalently whenever a set is decidable in p olynomial time with full logarithmic ad

vice then it is p ossible to construct equivalent advice words for the set within of the same

logarithmic length b ounds and ob eying the restriction that each advice word is a prex of

all the following ones

Pro of By denition PrefPlog is a sub class of FullPlog The relevant part of the

theorem is of course the converse inclusion Supp ose that A FullPlog This means that

there is a set B P and a sequence of advice words fw j n INg with jw j Olog n so

n n

that x m jxj x A hx w i B We will use the result in the previous section

m

characterizing PrefPlog as P Tally Thus we will dene a tally set L containing only

T

words of length a p ower of and will prove that A P L

T

Since FullPlog is closed under p olynomialtime Turing reducibility it is strictly smaller

than Plog and therefore our pro of now must unavoidably exploit the prop erty that one

advice can help all the smaller lengths The main idea is to keep only the information

corresp onding to some selected advice strings instead of storing all of them in the oracle

set Of course we have to select for the oracle innitely many advice words but the fact

that each of them is go o d for all the words smaller than the length it is designed for allows

us to select them arbitrarily far apart

We must nd a balance b etween two contradictory restrictions If we select advice

strings for the oracle to o frequently then they will need to o many bits and some of them

will b e enco ded to o far away in the oracle but if we select them to o separated then for

some words the nearest valid advice would b e to o long to b e extracted from the oracle by

a p olynomialtime machine

It turns out that there is a way of skipping advice words for which the balance is

satisfactory We will enco de in the oracle all the advice words corresp onding to lengths

m

for all m IN and skip all the intermediate ones Each bit of each of the selected

m

advice strings will b e enco ded by the presence or absence of a word of the form in

the set L

Let k b e a constant such that jw j k log n for all n and without loss of generality

n

assume that jw j k log n by padding out each jw j with a sux word from up to the

n n

desired length

m

The advice words corresp onding to the length resp ectively have size k

k

m

resp ectively k k We use the rst k p owers of two from until to enco de the

advice for length The empty string is not used here The second advice string to

store has length k so this information needs k p owers of two use the next ones from

k k k

until In general the information of the advice corresp onding to the length

m

is enco ded in the tally set L from the element

m

k k k k

up to

m

k k k k

So let L b e

P

i

k p

im

m

m

L f is g j p k such that the pth bit of w

m m

jxj We prove rst that A P L On input x nd an integer m such that

T

m m

jxj this selection ensures This can b e done in p olynomial time Since

that log log jxj m log log jxj

P

i

k p

im

m

Now for each value of p from to k ask whether L and in this

m

which now can b e used to decide whether x A way obtain all the bits of the advice w

in p olynomial time It remains to b e seen whether the queries can b e asked in p olynomial

time it suces to prove that they are p olynomially long

The number of queries is b ounded by k log jxj A b ound on the length of the oracle

queries is

m m m

k k k k k k

As m log log jxj the queries have length at most

d d

log log jxj log log jxj

k d k

jxj

for appropriate constants d and d So A P L ut

T

m

Let us We have chosen to keep those advice strings corresp onding to the length

briey describ e how crucial the arithmetic prop erties of the double exp onential are for

this pro of Naively it may seem that a single exp onential separation should suce but

this fails b ecause for each advice there are logarithmically many smaller advice words of

logarithmic length to b e enco ded ie a total of log n bits when distributed over the tally

log n

set they cover a broad region up to length n which cannot b e scanned in p olynomial

time Surprisingly as describ ed ab ove a double exp onential works However if we would

try to select advice strings with triply exp onential gaps skipping all advice words except

m

those corresp onding to lengths then these advice strings are to o large although there

m m

jxj and are fewer of them the rst appropriate m would b e such that

log n

straightforward computation shows that the corresp onding advice might b e n long

We give now a second application of the doubly exp onential skip technique Considering

the previous results it is clear in what sense the tally sets used exhibit a regularity their

words can app ear at only selected sp ecic places such as p owers of Many other similar

notions of tally sets with regularities can b e prop osed but among them there is one that is

particularly natural regularity could b e dened in terms of resourceb ounded Kolmogorov

complexity We could consider tally sets that are regular in the sense that there is a short

say logarithmic way of describing their characteristic function and a resourceb ounded

say p olynomial time algorithm to recover it Observe that the tally sets used in the pro of

of Theorem fulll this regularity prop erty

In principle the class obtained would b e larger since it is conceivable that some tally sets

are Kolmogorov regular but enco de more information than a set having such an extreme

k

regularity as implied by the sup erset f j k INg We will show that mo dulo p olynomial

time Turing reducibility this is not the case the reduction class of Kolmogorovregular

sets is again FullPlog As b efore L denotes a tally set

Theorem The following two classes coincide

i FullPlog

S

P L where there exists a p ositive constant c dep ending only on L such that ii

L

T

n

L c

for all n Kc log n n

Pro of

Again we use the characterization of the class FullPlog as P Tally which follows

T

from the previous result Then it is easy to see that i implies ii to construct the

characteristic sequence of a tally set L up to a xed length we only need to know which

k

ones among the logarithmically many words of the form are in L these are the only

L

p otential nonzeros in Thus given as a logarithmically long seed the characteristic

k

L

function of L relative to f j k INg we can easily print out an initial segment of in

time p olynomial in the length of the output

To see that ii implies i again we apply the doubly exp onential skip technique Ob

serve rst that an easier pro of seems p ossible Consider A P L where L is Kolmogorov

T

regular we can show that A can b e accepted with the help of a short advice On input

q

jxj

x the maximum oracle query is for some q To decide whether x A it suces to

L q

know an initial segment of the sequence up to jxj bits recall that the characteristic

sequence of a tally set is taken with resp ect to fg We can obtain this sequence in

q

p olynomial time from a seed of size log jxj Olog jxj which we take as an advice word

It follows that A Plog However this do es not prove that A FullPlog It may b e

the case that together with x we get a seed for an advice creating an exp onential part of

the characteristic function which is much more than we need but the relevant part of it

may take to o long to b e constructed and then there is no way to decide x in p olynomial

time

We resort again to a doubly exp onential skip for a given length n select as advice not

m

a single seed but a sequence of them corresp onding to lengths of the form up to the

q L

smallest one allowing us to construct n bits of This one corresp onds to

m m

q

n

i

q

so that m log log n log log n O For length the length of the seed is

i

i

c log c and thus as b efore the total length of the sequence of seeds selected for the

P

i m

advice is c c Olog n Now the diculty explained ab ove can b e avoided

im

If together with x we get the advice for a much longer length we can scan it and select

m

L q

a seed large enough to create up to jxj but not much more there is one there for

m m m

q q

with jxj which implies jxj only quadratically longer Therefore

A FullPlog ut

From now on we denote the class of Kolmogorovregular sets as Lowtally on the basis

of their low Kolmogorov complexity That is to say L Lowtally if and only if L Tally

n

L c

and there exist a p ositive constant c such that for all n Kc log n n

Thus for p olynomialtime machines tally oracle sets have exactly the same p ower as

lowtally oracle sets Again we have a phenomenon like the one discussed previously longer

and longer prexes of the characteristic function of the tally oracle which require log n

new bits linearly often to b e describ ed can b e replaced by a much simpler oracle which

exp onentially often adds a constant number of bits

The inner structure of FullPlog

FullPlog has b een characterized in previous sections in terms of the reduction class to tally

sets with very small information content We fo cus here on the most usual reducibilities

and investigate the corresp onding reduction classes to these sp ecial tally sets In addition

to other results we show that using tally languages as oracles there are more sets that can

b e recognized under Turing or equivalently truthtable reducibility than under b ounded

truthtable reductions The analogous problem for lowtally sets remains op en

To provide some context let us mention the pap er by Bo ok and Ko There

the classes of sets that can b e reduced to sparse and tally sets under dierent notions of

reducibilities are studied On the other hand Tang and Bo ok and Allender and Watan

ab e studied sets that are not only reducible to arbitrary tally and sparse languages but

also interreducible with them With the same approach we consider here reduction and

equivalence classes to tally and lowtally sets and study the relationships b etween them

although the truthtable and the Turing equivalence classes ie degrees of arbitrary tally

sets do not coincide in the case of using our restricted tally sets the problem of deciding

whether they are dierent is at least as dicult as solving P NP In the same way it is

argued that separating the class of languages that are b ounded truthtable equivalent to

lowtally sets from the class of languages that are mequivalent to the same class of sets is

also a dicult task However the relationship b etween these two degrees when tally sets

are used as oracles is clear b oth degrees dier

As a consequence of our results we can present a rather complete read not to o

incomplete map of the inner structure of FullPlog

Relationships among Reduction Classes

In this subsection we fo cus on the reduction classes dened from tally and lowtally sets

and explain some of the relationships that exist among them

The Turing and truthtable reducibilities in p olynomial time are equivalent when tally

sets are used as oracle Using the same arguments it is easy to see that the following

holds

Prop osition

P Lowtally P Tally P Lowtally P Tally

tt tt

T T

Pro of Supp ose that T is a tally set and A LM T where M runs in time b ounded by

q m m

n Let m b e such that n The number of p ossible nontrivial ie p otentially

answered yes queries to T made by M on input of size n is b ounded by mq Therefore

there is a logarithmic quantity of p otential queries to the oracle which we know in advance

We can make all these queries at the b eginning of the computation This fact implies that

tally sets pro duce the same information under truthtable than under Turing reductions

By the characterization of FullPlog the reduction classes P Lowtally and P Tally

T T

coincide From the ab ove argument P Tally P Tally therefore P Lowtally

tt

T T

P Tally But since the class Tally is included in the class Lowtally P Tally

tt tt

P Lowtally P Tally and all four coincide ut

tt

T

We now study the relationship b etween manyone and b ounded truthtable reductions

over tally and lowtally sets As we shall see next b oth reducibilities applied to lowtally

languages have the same p ower however the b ehaviour of them over tally sets is dierent

Prop osition P Lowtally P Lowtally

m

btt

Pro of The pro of is based on the fact that the b o olean closure of any class P A is

m

precisely the class P A and follows Bo ok and Kos steps in Applying this

btt

fact if we show that P Lowtally is closed under b o olean op erations then we obtain that

m

P Lowtally P Lowtally

m

btt

n

T d

Let T b e any lowtally set therefore for all length n Kd log n n for some

n

n

T T

constant d The same seed for can b e used to pro duce that is to say

n

d T

Kd log n n

Then P Lowtally is closed under complementation

m

Closure under intersection follows from the following fact Let T and T b e lowtally

A B

sets Supp ose that f reduces A T and g reduces B T Then we can dene the set

m A m B

hnmi n m

T f j T and T g whose characteristic sequence can b e also constructed

A B

hnmi

in p olynomial time from logarithmic seeds The function h such that hx when

n m

f x T and g x T or hx otherwise testies that A B T ut

A B m

The b ehaviour of tally sets is completely dierent as we p oint out in next results

Although is equivalent to for all k holds that and do not

m

tt k tt k tt

hnmi n

coincide Essentially the analogous argument using hx for f x and

m hnmi

hnmi nm

g x breaks down b ecause j j is ab out which is not p olynomial

n m

in

Prop osition P Tally P Tally

m

tt

Pro of Let T b e a tally set in Tally and assume that A P T via a Turing machine

tt

M We dene another tally set L containing information ab out T and its complement

m m m m

L f j T g f j T g

Using as oracle the set L and a new machine N we can see that A P L N works as

m

m

wlog we assume that all queries M but whenever M makes a query of the form

made by M are of this form it has to change as follows

Supp ose that the computation of M is indep endent from the answer given by T In

this case N do es not ask and follows the same steps as M

Supp ose that M rejects when the answer is NO and accepts when the answer is YES

m

Now N makes the query to L and accepts if and only if the answer is YES

If M rejects when the answer is YES otherwise accepts then N makes the query

m

to L and accepts if and only if the answer is YES

N testies that A P L and therefore A P Tally ut

m m

The following theorem states that there exists an strict hierarchy among the reductions

classes P Tally when k

k tt

Theorem k P Tally P Tally

k tt k tt

We construct a language in P Tally P Tally by diagonalization Let

k tt k tt

ff g b e an enumeration of the p olynomialtime computable functions that for any string

i iIN

x yields a list of k strings Let fh g b e an enumeration of all p olynomialtime

j j IN

k

computable functions that for any single string x yields one of the k tt conditions

We can enumerate all of the k tt reduction machines as fM g where on input

ij ij IN

x M computes the list f x hx x x i and the k tt condition h x

ij i k j

We assume for now that k is an o dd number The even case needs just slight syntactic

changes in the pro of The language we want to construct is dened in terms of a particular

tally set T and therefore it is denoted by LT

r

r

r r r r

k s

k

LT f j s s k T and r r r g

k

At stage m hq i j i we expand the set LT constructed so far adding a word in

m

such a way that LT cannot b e k reducible via f whose running time is b ounded

m i

q

by the function n and h to any tally set At the b eginning T is the empty set In

j

order to do this we study the function f b ecause we use dierent strategies dep ending

i

on whether it is injective

Similarly to f x hx x x i denote f y hy y y i to dene that x

i k i k

and y are alike for T under f if the two b o olean vectors

i

T T

f x hx T x T x T i and f y hy T y T y T i

i k i k

coincide

Stage m hq i j i

q

Consider M with running time b ounded by the function n

ij

r

r

r r r

k

k

Let w or ds fw f g j w

m

m m

r r r g and

k

k k

k

k tt conditions t For each of the

let G fx w or ds j h x tg

t m j

Cho ose any t such that jG j is maximum

t

if f is not injective on G

i t

r t

r t

k k

then lo ok for two dierent words G

t

t t r r

k k

f such that f

i i

r r

k

g T T f

m m

r r t t

k k

else lo ok for two dierent words G

t

that are alike under f for all T Tally

i

r

r

k

T T f g

m m

end if

q

Given the function f working in time n at stage m hq i j i either there exists two

i

dierent words w v G such that f w f v In this case the words added to T

t i i m

witnesses that the set LT is not k reducible to any set via f and h or for all

m i j

words w v G f w f v The key p oint now is the cardinality of G

t t

m

k

On the one hand the number of dierent words in the set w or ds is exactly

m

k

k m m

k

which is

k

k k

k

On the other hand there are k tt conditions t there must exist some t

k m

k m

whose G has at least words Therefore the G chosen in the algorithm

t t

k

k

k

k m

has at least dierent words From now on we work with this G

t

In the case that the function is injective on G we need the following lemma recall

t

T

is dened as f from ab ove that for each tally set T the function

i

T

f x hx T x T x T i

i k

Lemma If f is injective on G then there exists at least two words x y G that are

i t t

T T

alike under f for all tally sets T f f x y

i i i

Applying this lemma as we add to T the corresp onding strings in order that x

m

LT without adding the corresp onding ones for y we conclude that the set LT can

m m

not b e k tt reducible to any tally set via M

ij

Now we prove the lemma

m

Pro of The length of the largest string in the set w or ds is b ounded by Since M

m ij

q

works in time b ounded by the function n the lengths of the queries made by M are

ij

m m j

q q

b ounded by The strings of the tally sets are of the form with j IN

j

m m

but M only can query words with j q Therefore there are q

ij

many dierent words in the tally sets that could b e queried by M

ij

Now we show that there must exist two dierent words x and y of G fullling that

t

f x f y b ecause the function f is injective on G but for each l such that l

i i i t

j

m

with j q That is either k if x y then x and y are not of the form

l l l l

x and y are dierent and then b oth strings are outside of any tally set or x y This

l l l l

T T

ensures that for any tally set T x y f f

i i

Fix a string outside of any tally set for example the string Supp ose that each

j

with time that M on any input in G queries a string that is not of the form

ij t

m

j q this string is always Then we can show that the amount of dierent

j

words x in G with the prop erty that M x makes at least one query of the form

t ij

m

with j q is fewer than the cardinality of G

t

On the one hand the quantity of strings x in G such that M x makes a unique

t ij

j

k

m m

with j q is at most q In the same way query of the form

the number of strings x in G such that M x makes exactly two queries that could b e

t ij

k

m

in some tally set is q In general the number of strings x in G such

t

that M x makes exactly i queries i k that could b e in some tally set is

ij

k

m i k m

q Using that q m so that m for large enough m the total

i

amount of strings with the ab ove prop erty is

k

m i m k k m k

q q o

i

i

k m

On the other hand jG j Therefore there must exist at least two words x

t

j

m

with j q and y in G such that when M x makes a query of the form

t ij

then M y makes the same query otherwise b oth query outside every tally set This

ij

T T

implies that f x f y ut

i i

We show now how the lowtally sets provide more information than the tally sets used

as oracles under bttreduction

Theorem P Tally P Lowtally

btt btt

Pro of We prove the stronger fact that the class Lowtally is not included in P Tally

k tt

for any constant k ie we nd a set L Lowtally such that given a p olynomialtime

computable function f that for any word x yields a list of k strings it is not the case that

L P Tally

k tt

In order to dene such a lowtally set L for each i we denote by I the following set

i

i i

n

I f j n j with j g

i

and we dene the set L in such a way that for all i there is exactly one word in I

i

that b elongs to set L Actually this unique word is called x and fullls that

i

i

i

j

i

with j x

i i

i

Hence in each I there are p ossibilities of choosing this string

i

If we get L of this form then we ensure that L Lowtally To know the characteristic

m m

Let g n sequence of L up to length n we lo ok for the integer m fullling

b e a function that on each i g i expresses what x is chosen given the information j

i i

i

jj j

i

g i j

i

n

i L

we can use as seed the following word s so that jg ij In order to obtain

s g g g m g m

that is formed by concatenating each g i for i m The length of s is exactly jsj

n

d m i m L

Kd log n n d is a constant that is b ounded by log n Thus

i

and L Lowtally

We construct the set L by diagonalization Let ff g b e an enumeration of the

i iIN

p olynomialtime computable functions that for any string x yields a list of k strings Let

fh g b e an enumeration of all p olynomialtime computable functions that for any single

j j IN

k

k tt conditions We can enumerate all of the k tt reduction string x yields one of the

machines as fM g where on input x M computes the list f x hx x x i

ij ij IN ij i k

and the k tt condition h x

j

m

j

In each stage m hq i j i we are adding to the set L one word of the form

m

j in such a way that L cannot b e k reducible via f and h to any tally

i j

set As in the diagonalization of Theorem in the case of having an injective function

f we can nd two words alike under f for all tally sets so if only one of these words is

i i

included in the set L then we diagonalize over Tally

Stage m hq i j i

q

Consider M with running time b ounded by the function n

ij

m m

p

Let I f j p j with j g

m

k

For each of the k tt conditions t let G fx I j h x tg

t m j

m

Cho ose any t such that jG j

t

k

if f is not injective on G

i t

p p p p

then lo ok for I p p such that f f

m i i

p

L L f g

m m

p p

else lo ok for G p p alike under f for all T Tally

t i

p

L L f g

m m

end if

p

When the function is not injective on G the selection of ensures that L is not k tt

t

T

f b e the reducible to any set via M In the other case for each tally set T again let

i ij

function such that

T

f x hx T x T x T i

i k

A similar result to Lemma holds

Lemma If f is injective on G then there exist at least two words x y in G such

i t t

T T

that for any tally set T x y f f

i i

Applying this lemma if we add to L the string x without adding y then we

m

conclude that the set L cannot b e k tt reducible to any tally set via M

ij

The pro of of the lemma follows the same steps of Lemma The largest string in

m m

m

m

the set I is whose length is Therefore there are q many dierent

m

words in the tally sets that could b e queried by M

ij

j

Fixing a word outside the tally for all queries made by M that are not of the form

ij

m

with j q the amount of dierent words x in G with the prop erty that M x

t ij

makes at least one query that could b e in some tally set is fewer than the cardinality of

G Namely this amount is

t

k

m i m k k

q q

i

i

m m

While the cardinality of G is greater than or equal to Therefore since

t k k

m k

q for large enough m there must exist at least two words x and y in G such

t

T T

f x f y ut that

i i

This result together with the fact that in P Tally there exist languages that are

btt

not lowtally sets ensure that b oth classes are incomparable This is not the case when

Turing or equivalently truthtable reduction is used b ecause the class Lowtally is strictly

included in P Tally The relationship b etween Lowtally and P Tally is explained

T T

in more detail in the following theorem but b efore let us introduce a lemma which will

b e helpful later on

Lemma For each lowtally set L there is a tally set T such that L P T and

tt

T P L

tt

Pro of Let L b e any lowtally set By denition of the class Lowtally there exists a constant

c such that for every length n there exists a seed s with js j c log n that pro duces

n n

n

L

in p olynomial time Without loss of generality we consider that every seed s has

n

exactly c log n number of bits Moreover if there would b e many dierent seeds for the

same prex of the characteristic sequence we choose only the rst seed in lexicographical

n

L

order The idea is to enco de the seeds s into a tally set T in order to pro duce

n

The way of enco ding these seeds is again based on the doubly exp onential skip tech

m

nique and consists on keeping only the information s corresp onding to n with

n

m IN

m

m

For all m the seed corresp onding to length has size c So as in Theorem

let T b e

i

c p

im

m

m

j p c such that the pth bit of s is g T f

n

On the one hand L P T to decide whether a word of the form is in L we

tt

n

L

can generate easily querying T The steps to follow are exactly those given in the

m m

pro of of Theorem rst nd an integer m such that n and then obtain

m

querying the tally set T It is easy to see that the sizes of the queries the seed s

are p olynomially long and there are p olynomially many queries Furthermore they are

nonadaptive as required

n

the following steps suce to decide On the other hand T P L On input

tt

n

T whether

Lo ok for the number m fullling

m m

c n c

m

n c p m

Thus with p c

m

querying in a nonadaptive Find the characteristic sequence of L up to length

m

n

is p olynomial in b ecause way L Note that

n

n c

m

m m

c

c n

c

m

L

is known check which is the rst seed in lexicographical order that When

pro duces in time b ounded by an appropriate p olynomial this characteristic sequence

m

among all the p ossible seeds of length less than or equal to c As the number of

m

c n

seeds is b ounded by this pro cess can b e done in p olynomial time in

n

In the ab ove seed if the pth bit is then the word is in the set T otherwise

n

is not in T

This shows that T P L ut

tt

Now we present the relationship b etween lowtally sets and the class FullPlog Full-P/log

Pbtt (Lowtally) = Pm (Lowtally)

Pbtt (Tally2)

Lowtally

Pm (Tally2)

Tally2

Figure Reduction classes to sp ecial tally sets

Theorem Lowtally Tally FullPlog

Pro of The class Lowtally is included in Tally and by the previous lemma it is also

included in P Tally FullPlog Conversely we see that any tally set in P Tally

T T

is in particular a lowtally language Supp ose that L P T via a DTM M that works

T

c

in time n Here c is a constant and T is a tally set The characteristic sequence of L

c

up to n can b e generated from M and the characteristic sequence of T up to length n

c

m n

T

j m INg and M is a xed DTM the information is relative to the set f Since

needed is logarithmic in n ut

Up to now we have not b een able to show a precise relationship b etween the class

FullPlog and P Lowtally P Lowtally

m

btt

Figure describ es the relationships among reduction classes The arrows mean inclu

sions the b oldface arrow relates classes whose exact relationship remains op en while the

others mean that the inclusions are strict The prop er innite hierarchy of k tt reduction

classes b etweem mreduction and bttreduction is not shown

Relationships among Equivalence Classes

We move now to the study of the relationships b etween the classes of languages that are

equivalent to tally sets and lowtally sets under various notions of reducibility

The rst problem is to determine whether the mequivalence classes to tally and

lowtally sets are dierent The answer is provided by following lemma from Tang and

Bo ok relating reducibility and interreducibility

Lemma Let C and C two classes of sets and let and two reducibilities

r s

with r s fm btt tt Tg If P C P C then E C E C

r s r s

Pro of The pro of is based on the op erator Supp ose that there exists a set A

P C P C Since A P C there exists a set C C such that A C Thus

r s r r

A C C and C A C so A C E C If E C E C then there exists a

r r r r s

set D C such that A C D but this implies that A D and this is imp ossible ut

s s

Combining the ab ove lemma with the results obtained in the previous section we get

the following consequences

Corollary

E Tally E Lowtally

m m

E Tally E Lowtally

btt btt

E Tally E Tally

m

btt

Corollary For all reducibilities with r fm bttg and with s fT ttg we

r s

have E Tally E Tally

r s

We have also the corresp onding extension to all k ttreductions to Tally

As P Tally P Tally P Lowtally and P Lowtally coincide the ab ove argu

tt tt

T T

ment do es not work neither in the case of Turing nor in the case of truthtable equivalence

Indeed the equalities hold as well

Theorem

E Lowtally E Tally and E Lowtally E Tally

tt tt

T T

Pro of E Tally E Lowtally b ecause Tally Lowtally The other inclusion is

T T

not trivial Supp ose that A is Turing equivalent to a lowtally set L That means that A

P L and L P A By Lemma there exists a tally set T such that L P T

tt

T T

and T P L Since A P L A P P T thus A P T Conversely

tt tt

T T T

since T P L and again by transitivity T P A

tt

T

As Lemma is in terms of truthtable reduction using the same argument as b efore

we can prove that E Lowtally E Tally ut

tt tt

The problem of whether E Lowtally is equal to E Lowtally is now studied

m

btt

Actually we show rst that distinguishing equivalence and reducibility to lowtally sets

under the manyone reduction would imply that P NP in the same way that it is not

p ossible to separate equivalence and reducibility to sparse sets for manyone reductions

if P NP for more information see As a consequence of this result we also obtain

that separating E Lowtally and E Lowtally b ecomes a dicult task to o

m

btt

Theorem P NP P Lowtally E Lowtally

m m

Pro of The inclusion E Lowtally P Lowtally is obvious So it is only neces

m m

sary to see the converse let L and LT b e sets such that L LT via g where LT is a

m

lowtally set and L We dene the set LT using the metho d from and following

the steps of as follows

hlmi m

LT f j y jy j l g y LT g

On the one hand L LT via a function h dened in this way

m

hjy jjg y ji

hy

Indeed h can b e calculated in p olynomial time and for all y

y L hy LT

hjy jjg y ji

When y L the word is in LT by the denition of LT itself Moreover if

hjy jjg y ji

hy LT then g y LT so that y L

On the other hand since h is honest and using the hypothesis that P NP by

Theorem it is p ossible to compute an inverse function of h denote it by f in p olynomial

time Thus f fullls that

hlmi hlmi

LT f L

Therefore LT L

m

To nish this pro of we show that in our context LT is itself a lowtally set proving that

n d

there exists a constant d such that for all n LT Kd log n n From the hypothesis

that P NP we get the seeds for LT that are also seeds for LT so that when we take a

n n

seed for LT then we can pro duce LT The algorithm is as follows

n

LT

input a seed s pro ducing

n

LT

pro duce from s

for i to n do

let i hl mi

m

guess y such that jy j l g y

m

if LT

hlmi

LT then

hlmi

else LT

end if

end for

n

output LT

If P NP then this algorithm is in P ut

This result can b e lifted to bttreductions

Theorem P NP P Lowtally E Lowtally

btt btt

Pro of By Prop osition

P Lowtally P Lowtally

m

btt

By the hypothesis that PNP and Theorem

P Lowtally E Lowtally

m

btt

but it is clear that

E Lowtally E Lowtally

m

btt

Therefore P Lowtally E Lowtally but the other inclusion also holds thus b oth

btt btt

classes coincide ut

As a whole under the assumption that P NP all the classes E Lowtally

m

E Lowtally P Lowtally and P Lowtally coincide

m

btt btt

In order to present the relationship b etween the truthtable and the Turing equivalence

classes we fo cus on a new approach that studies the complexity of pro ducing advice words

for sets A in FullPlog relative to A itself For instance there exists a characterization of

E Tally according to this

T

Theorem The following facts are equivalent

i A E Tally

T

ii A has a family of logarithmic advice words that can b e obtained in p olynomial time

making queries to A

Pro of First we prove i ii

Let A b e a set in E Tally That is there exists a tally set L fullling A P L

T T

j k

and L P A Let n and n b e resp ectively the p olynomials that b ound the running

T

j

n

suces to decide time of the machines that query oracle L and oracle A resp ectively L

j

n

L

which words of length n are in A Therefore is a logarithmic advice word for length

n and b ecause L P A there exists a p olynomialtime algorithm that with input n in

T

j

n

L

unary constructs the advice word querying A

Second we show ii i Let A b e a set recognized by a family of logarithmic advice

words fw j n INg which can b e obtained in p olynomial time querying A

n

Using again the doubly exp onential skip technique we can keep only some selected

w s in order to enco de them into a tally set T in such a way that A E T The

n

T

denition of T is as in Lemma the following

i

c p

im

m

m

is g j p c and pth bit of w T f

Note that in Lemma we deal with logarithmic seeds instead of logarithmic advice words

but the denition of T is the same Following similar steps as in that lemma it is easy

to see that

A P T since given x as input the advice word w which can b e pro duced

jxj

T

querying T suces to decide whether x A

n

T P A using the hypothesis to decide whether is in T suces to lo ok for

T

m

n c p m

the value m such that with p c and then to check the pth

m

which is pro duced querying A bit of w

Therefore A E T and this implies that A E Tally ut

T T

A similar statement can b e obtained if instead of using adaptiveness and Turing reduc

tions nonadaptive queries and truthtable reductions are considered

Theorem The following facts are equivalent

i A E Tally

tt

ii A has a family of logarithmic advice words that can b e obtained in p olynomial time

making queries to A in a nonadaptive way

The next theorem provides an easy upp er b ound on the complexity of pro ducing loga

rithmic advice words for sets in FullPlog

Theorem For every set A FullPlog there exists a family of advice words for A

that can b e obtained in p olynomial time making logarithmically many queries to NPA

Pro of Supp ose A FullPlog Then

n w jw j c log n x jxj n x A hx w i B

n n n

where B P and c is a constant

For each n we can construct w by a prexsearch algorithm querying an oracle in

n

NPA which is identied in more detail next

Denition Let y b e a word such that jy j c log n We say that y is go o d for n in

the sense of b eing a correct advice if and only if

u juj n hu y i B u A

Let GA b e the following oracle set

n

GA fhz i j jz j c log n and y z v y jy j c log n and y is go o d for n g

GA and GA are resp ectively in coNPA and NPA Note that they dep end b oth on

A and B Given the length n in unary a go o d advice word corresp onding to n can b e

GA logarithmically many times ut pro duced in p olynomial time querying

Logarithmically many queries to NPA implies at most a p olynomial number of

dierent queries which moreover can b e computed in p olynomial time Therefore all of

them can b e asked at the b eginning of the computation and the following holds Full-P/log

E TT (Lowtally) = E (Tally2)

E tt (Lowtally) = E tt (Tally2)

Ebtt (Tally2) Ebtt (Lowtally)

E m (Tally2) Em (Lowtally)

Tally2 Lowtally

Figure Equivalence classes to sp ecial tally sets

Corollary For every set A FullPlog there exists a family of advice words for A

that can b e obtained in p olynomial time making queries to NPA in a nonadaptive way

Wagner used a similar argument in where the p ower of p olynomialtime machines

with restricted access to an NP oracle was studied

The notions of instance complexity and the class IClog p oly of sets of strings with

low instance complexity were introduced in we do not need the precise denition here

only two known prop erties Sp ecically the fact that FullPlog is included in IClog p oly

together with the fact that IClog p oly is in the rst level of the extended low hierarchy

EL allow us to show that

Theorem The following statements are equivalent

i A FullPlog

ii A has a family of logarithmic advice words that can b e obtained in p olynomial time

making queries to A SAT in a nonadaptive way to A

Pro of The nontrivial direction is from i to ii and it is a consequence of previous results

from Actually in it was proven that IClog p oly is in the rst level of the

extended hierarchy That is NPA PA SAT for all sets A in IClog p oly Moreover

the pro of of this shows that for each language L in NPA there exists a deterministic

algorithm that decides L in p olynomial time querying A SAT and although it has an

adaptive access to SAT the queries to A are made in a nonadaptive way

Therefore ii holds since FullPlog is included in IClog p oly ut

Now we apply the hypothesis that P NP all the queries made by the algorithm to

oracle SAT NP can b e replaced by a p olynomialtime computation This together with

previous results suce to see the following

Theorem FullPlog E Tally E Tally if P NP

tt

T

Figure presents the results regarding equivalence classes Now the discontinuous ar

rows app ear when under assumption that P NP the inclusions turn out to b e equalities

There is again an op en question indicated by the b oldface arrow note that it matches and

actually renes its corresp onding op en question arrow in gure We b elieve that a

dierent to olkit is necessary to close our two remaining fully op en questions On the other

hand forthcoming work by the authors jointly with H Buhrman and constituting the

archival version of reduces optimally the strength of the necessary complexitytheoretic

condition P NP used in several of our theorems

Acknowledgment

We would like to thank Ricard Gavalda for helpful discussions including alternative

pro ofs of theorem and for p ointing out that a pro of in of the fact that P Lowtally

m

is dierent from P Lowtally was erroneous

T

References

E Allender and L Hemachandra Lower Bounds for the Low Hierarchy Journal of

the Association for Computing Machinery

E Allender L Hemachandra M Ogiwara and O Watanabe Relating Equivalence

and Reducibility to Sparse Sets SIAM Journal on Computing

E Allender and O Watanabe Kolmogorov Complexity and Degrees of Tally Sets

Information and Computation

2

A direct pro of exists although we prefer to mention previous results

V Arvind Y Han L Hemachandra J Ko ebler A Lozano M Mundhenk M Ogi

wara U SchoningR Silvestri and T Thierauf Reductions to Sets of Low Informa

tion Content In Complexity Theory pages Cambridge University Press eds K

AmbosSpies S Homer U Schoning

V Arvind J Koblerand M Mundhenk Upp er Bounds on the Complexity of Sparse

and Tally Descriptions Mathematical Systems Theory

J Balcazar R Bo ok and U Schoning Sparse Sets Lowness and Highness SIAM

Journal on Computing

J Balcazar H Buhrman and M Hermo Learnability of KolmogorovEasy Circuit

Expression Via Queries In Computational Learning Theory Proc Eurocolt volume

pages Lecture Notes in Articial Intelligence SpringerVerlag

J BalcazarJ Dazand J Gabarro Structural Complexity I Second Edition Texts

in Theoretical Computer Science SpringerVerlag

J BalcazarR GavaldaH Siegelmann and E Sontag Some Structural Complexity

Asp ects of Neural Computation In Proc Structure in Complexity Theory th annual

conference pages IEEE Computer So ciety Press

J BalcazarM Hermo and E Mayordomo Characterizations of Logarithmic Advice

Complexity Classes In Proc of the IFIP th World Computer Congress volume

pages NorthHolland

J Balcazar and U Schoning Logarithmic Advice Classes Theoretical Computer

Science

R Bo ok and K Ko On Sets Truthtable Reducible to Sparse Sets SIAM Journal

on Computing

P P

J Castro and C Seara Complexity Classes Between and Informatique

k k

Theorique et Applications

J Hartmanis Generalized Kolmogorov Complexity and the Structure of Feasible

Computations In Proc th IEEE Symposium on Foundations of Computer Science

pages

M Hermo Degrees and Reducibilities of Easy Tally Sets In Proc th Symposium on

Mathematical Foundations of Computer Science volume pages Lecture

Notes in Computer Science SpringerVerlag

M Hermo Nonuniform Complexity Classes with SubLinear Advice Functions Do c

toral Thesis

R Karp and R Lipton Some Connections b etween Nonuniform and Uniform Com

plexity Classes In Proc th ACM Symposium on Theory of Computing pages

K Ko On the Notion of Innite Pseudorandom Sequences Theoretical Computer

Science

K Ko On Helping by Robust Oracle Machines Theoretical Computer Science

J Kobler U Schoning and K Wagner The Dierence and TruthTable Hierarchy

for NP Informatique Theorique et Applications

T Long and M Sheu A Renement of the Low and High Hierarchies Mathematical

Systems Theory

S Mahaney Sparse Complete Sets for NP Solution of a Conjecture of Berman and

Hartmanis Journal of Computer and System Sciences

P Orp onen K Ko U Schoningand O Watanabe Instance Complexity Journal of

the Association for Computing Machinery

H Siegelmann and E Sontag Analog Computation via Neural Networks Theoretical

Computer Science

M Sipser A ComplexityTheoretic Approach to Randomness In Proc th ACM

Symposium on Theory of Computing pages

S Tang and R Bo ok Reducibilities on Tally and Sparse Sets Informatique Theorique

et Applications

K Wagner Bounded Query Classes SIAM Journal on Computing