A First-Order Isomorphism Theorem

Home , Juris Hartmanis, Michael Sipser, Neil Immerman

Eric Allender

Department of Computer Science, Rutgers University

New Brunswick, NJ, USA 08903

[email protected]

Jos e Balc azar

U. Polit ecnica de Catalunya, Departamento L.S.I.

Pau Gargallo 5, E-08071 Barcelona, Spain

[email protected]

Neil Immerman

Computer Science Department, University of Massachusetts

Amherst, MA, USA 01003

[email protected]

Abstract

We show that for most complexity classes of interest, all sets complete under rst-

order pro jections fops are isomorphic under rst-order isomorphisms. That is, a very

restricted version of the Berman-Hartmanis Conjecture holds. Since \natural" com-

plete problems seem to stay complete via fops, this indicates that up to rst-order

isomorphism there is only one \natural" complete problem for each \nice" complexity

class.

1 Intro duction

In 1977 Berman and Hartmanis noticed that all NP complete sets that they knew of were

p olynomial-time isomorphic, [BH77]. They made their now-famous isomorphism conjecture:

namely that all NP complete sets are p olynomial-time isomorphic. This conjecture has

engendered a large amountofwork cf. [KMR90,You] for surveys.

The isomorphism conjecture was made using the notion of NP completeness via p olynomial-

time, many-one reductions b ecause that was the standard de nition at the time. In [Co o],

A preliminary version of this work app eared in Pro c. 10th Symp osium on Theoretical Asp ects of

Computer Science, 1993, Lecture Notes in Computer Science 665, pp. 163{174.

Some of this work was done while on leave at Princeton University; supp orted in part by National

Science Foundation grant CCR-9204874.

Supp orted in part by ESPRIT-I I BRA EC pro ject 3075 ALCOM and by Acci on Integrada Hispano-

Alemana 131 B

Supp orted by NSF grant CCR-9207797. 1

Co ok proved that the Bo olean satis ability problem SAT is NP complete via p olynomial-

time Turing reductions. Over the years SAT has b een shown complete via weaker and

weaker reductions, e.g. p olynomial-time many-one [Kar], logspace many-one [Jon], one-way

logspace many-one [HIM], and rst-order pro jections fops [Dah]. These last reductions,

de ned in Section 3, are provably weaker than logspace reductions. It has b een observed

that natural complete problems for various complexity classes including NC , L, NL, P,NP,

and PSPACE remain complete via fops, cf. [I87, IL , SV, Ste, MI].

On the other hand, Joseph and Young, [JY] have p ointed out that p olynomial-time, many-

one reductions maybesopowerful as to allow unnatural NP-complete sets. Most researchers

now b elieve that the isomorphism conjecture as originally stated by Berman and Hartmanis

is false.

We feel that the choice of p olynomial-time, many-one reductions in the statement of the

Isomorphism Conjecture was made in part for historical rather than purely scienti c reasons.

To elab orate on this claim, note that the class NP arises naturally in the study of logic and

can b e de ned entirely in terms of logic, without any mention of computation [Fa]. Thus

it is natural to have a notion of NP-completeness that is formulated entirely in terms of

logic. On another front, Valiant[Val] noticed that reducibility can b e formulated in algebra

using the natural notion of a pro jection, again with no mention of computation. The sets

that are complete under fops are complete in al l of these di erentways of formulating the

notion of NP-completeness.

Since natural complete problems turn out to b e complete via very low-level reductions such

as fops, it is natural to mo dify the isomorphism conjecture to consider NP-complete re-

ductions via fops. Motivating this in another way, one could prop ose as a slightly more

general form of the isomorphism conjecture the question: is completeness a sucient struc-

tural condition for isomorphism? Our work answers this question by presenting a notion of

completeness for which the answer is yes. Namely for every nice complexity class including

P,NP, etc, anytwo sets complete via fops are not only p olynomial-time isomorphic, they

are rst-order isomorphic.

There are additional reasons to b e interested in rst-order computation. It was shown

in [BIS] that rst-order computation corresp onds exactly to computation by uniform AC

circuits under a natural notion of uniformity. Although it is known that AC is prop erly

contained in NP, knowing that a set A is complete for NP under p olynomial-time or

logspace reductions do es not currently allow us to conclude that A is not in AC ;however,

knowing that A is complete for NP under rst-order reductions do es allow us to make that

conclusion.

First-order reducibility is a uniform version of the constant-depth reducibility studied in

[FSS, CSV]; sometimes this uniformity is imp ortant. For a concrete example where rst-

order reducibility is used to provide a circuit lower b ound, see [AG92].

Preliminary results and background on isomorphisms follow in Section 2. De nitions and

background on descriptive complexity are found in Section 3. The main result is stated and

proved in Section 4, and then we conclude with some related results and remarks ab out the

structure of NP under rst-order reducibilities.

One way of quantifying this observation is that since Joseph and Young pro duced their unnatural NP-

complete sets, Hartmanis has b een referring to the isomorphism conjecture as the \Berman" conjecture. 2

2 Short History of the Isomorphism Conjecture

The Isomorphism Conjecture is analogous to Myhill's Theorem that all r.e. complete sets are

recursively isomorphic, [Myh]. In this section we summarize some of the relevant background

material. In the following FP is the set of functions computable in p olynomial time.

De nition 2.1 For A; B ,wesay that A and B are p-isomorphic A B i there

exists a bijection f 2 FP with inverse f 2 FP such that A is many-one reducible to B

A B via f and therefore B A via f . 2

m m

Observation 2.2 [BH77] Al l the NP complete sets in [GJ] are p-isomorphic.

How did Berman and Hartmanis make their observation? They did it by proving a p olynomial-

time version of the Schroder-Bernstein Theorem. Recall:

Theorem 2.3 [Kel, Th. 20] Let A and B be any two sets. Suppose that thereare 1:1

maps from A to B and from B to A. Then there is a 1:1 and onto map from A to B .

Pro of Let f : A ! B and g : B ! A b e the given 1:1 maps. For simplicity assume that

A and B are disjoint. For a; c 2 A [ B ,wesay that c is an ancestor of a i we can reach

a from c by a nite non-zero numb er of applications of the functions f and/or g .Now

we can de ne a bijection h : A ! B which applies either f or g according as whether a

p oint has an o dd numb er of ancestors or not:

g a if a has an o dd numb er of ancestors

ha=

f a if a has an even or in nite numb er of ancestors

The feasible version of the Schroder-Bernstein theorem is as follows:

Theorem 2.4 [BH77] Let f : A B and g : B A, where f and g are 1:1, length-

m m

1 1 1 1

increasing functions. Assume that f; f ;g;g 2 FP where f ;g are inverses of f; g.

Then A B .

Pro of Let the ancestor chain of a string w b e the path from w to w 's parent, to w 's

grandparent, and so on. Ancestor chains are at most linear in length b ecause f and g

are length-increasing. Thus they can b e computed in p olynomial time. The theorem now

follows as in the pro of of Theorem 2.3. 2

Consider the following de nition:

De nition 2.5 [BH77] Wesay that the language A has p-time padding functions

i there exist e; d 2 FP such that

1. For all w; x 2 , w 2 A , ew; x 2 A 3

2. For all w; x 2 , dew; x = x

3. For all w; x 2 , jew; xj jw j + jxj

As a simple example, the following is a padding function for SAT,

ew; x = w ^ c ^ c ^ ^ c

1 2

jxj

where c is y _ yifthe i bit of x is 1 and y _ y otherwise, where y is a Bo olean variable

numb ered higher than all the Bo olean variables o ccurring in w .

Then the following theorem follows from Theorem 2.4:

Theorem 2.6 [BH77] If A and B areNPcomplete and have p-time padding functions,

then A B .

Finally, Observation 2.2 now follows from the following:

Observation 2.7 [BH77] Al l the NP complete problems in [GJ] have p-time padding

functions.

Hartmanis also extended the ab ovework as follows: Say that A has logspace-padding func-

tions if there are logspace computable functions as in De nition 2.5.

Theorem 2.8 [Har] If A and B areNPcomplete via logspacereductions and have

logspacepadding functions, then A and B arelogspace isomorphic.

Pro of Since A and B have logspace padding functions, we can create functions f and g

as in Theorem 2.4 that are length squaring and computable in logspace. Then, the whole

ancestor chain can b e computed in logspace b ecause each successive iteration requires half

of the previous space. 2

Here, we show that sets complete under a very restrictive notion of reducibility are isomor-

phic under a very restricted class of isomorphisms. This result is incomparable to a recent

result of [AB], showing that all sets complete under one-way logspace reductions 1-L re-

ductions are isomorphic under p olynomial-time computable isomorphisms. This work of

[AB] improves an earlier result of [A88]. Note that it is easy to prove that the class of

1-L reductions is incomparable with the class of rst-order pro jections. Other interesting

results concerning 1-L reductions may b e found in [BH90, HH]. 4

3 Descriptive Complexity

In this section we recall the notation of Descriptive Complexity whichwe will need to state

and prove our main results. See [I89] for a survey and [IL] for an extensive discussion of

the reductions we use here including rst-order pro jections.

We will co de all inputs as nite logical structures. The most basic example is a binary

string w of length n = jw j.We will represent w as a logical structure:

Aw = hf0; 1; :::;n 1g;Ri

where the unary relation Rx holds in Aw in symb ols, Aw j= Rx just if bit x of w

is a 1. As is customary, the notation jAj will b e used to denote the universe f0; 1;:::;n1g

of the structure A.We will write jjAjj to denote n, the cardinalityof jAj.

;c ;:::;c i is a tuple of input relation and constant symb ols. A vocabulary = hR :::R

1 s

We call the R 's \input relations" b ecause they corresp ond to the input bits to a b o olean

circuit. In the case of binary strings, the input relation tells us which bits are 0 and which

are 1. In the case of graphs, the input relation E tells us which edges are present.

Let STRUC[ ] denote the set of all nite structures of vo cabulary .We de ne a complexity

theoretic problem to b e any subset of STRUC[ ] for some .

For anyvo cabulary there is a corresp onding rst-order language L built up from the

symb ols of and the numeric relation symb ols and constant symb ols :=; ; BIT; 0; m,

using logical connectives: ^; _; :,variables: x; y ; z ; :::, and quanti ers: 8; 9.

First-Order Interpretations and Pro jections

In [Val], Valiant de ned the projection, an extremely low-level many-one reduction.

De nition 3.1 Let S; T f0; 1g .Ak -ary projection from S to T is a sequence of maps

fp g, n =1; 2;:::, that satisfy the following prop erties. First, for all n and for all binary

strings s of length n, p s is a binary string of length n and,

s 2 S , p s 2 T:

Second, let s = s s :::s . Then each map p is de ned by a sequence of n literals:

0 1 n1 n

hl ;l ;:::;l i where

0 1

n 1

l 2f0; 1g[ fs ; s j 0 j n 1g :

i j j

Thus as s ranges over strings of length n, each bit of p s dep ends on at most one bit of s,

p s[[i]] = l s :

n i

2 th

Here refers to the usual ordering on f0;:::;n 1g, \BITi; j " means that the i bit of the binary

representation of j is 1, and 0 and m refer to 0 and n 1, resp ectively.For simplicitywe will assume

throughout that n>1 and thus 0 6= m. These relations are called \numeric" as opp osed to the input

relations b ecause, for example, \BITi; j " and \i j " dep end only on the numeric values of i and j and

do not refer to the input. 5

Pro jections were originally de ned as a non-uniform sequence of reductions { one for each

value of n. That is, a pro jection can b e viewed as a many-one reduction pro duced bya

family fC g of circuits of depth one. The circuits consist entirely of wires connecting input

bits or negated input bits to outputs. If the circuit family fC g is suciently uniform,we

arrive at the class of rst-order projections. Recall that rst-order corresp onds to uniform

AC [BIS ]. We nd it useful to work in the framework of rst-order logic rather than in

the circuit mo del. The rest of this section presents the necessary de nitions of rst-order

reductions.

The idea of the de nition is that the choice of the literals hl ;l ;:::;l i in De nition 3.1

0 1

n 1

is given by a rst-order formula in which no input relation o ccurs. Thus the formula can

only talk ab out bit p ositions, and not bit values. The choice of literals dep ends only on n.

In order to make this de nition, wemust rst de ne rst-order interpretations. These are

a standard notion from logic for translating one theory into another, cf. [End], mo di ed

so that the transformation is also a many-one reduction, [I87]. For readers familiar with

databases, a rst-order interpretation is exactly a many-one reduction that is de nable as

a rst-order query.

De nition 3.2 First-Order Interpretations Let and be twovo cabularies, with =

hR ;:::;R ;c ;:::;c i. Let S STRUC[ ], T STRUC[ ]betwo problems. Let k be

1 s

a p ositiveinteger. Supp ose we are given an r -tuple of formulas ' 2L , i =1;:::;r,

where the free variables of ' are a subset of fx ;:::;x g. Finally, supp ose we are given

i 1 k a

an s-tuple of constant symb ols t ;:::;t from L . Let I = h' ;:::;' ;t ;:::;t i

1 s x :::x 1 r 1 s

b e a tuple of these formulas and constants. Here d = max ka .

i i

Then I induces a mapping also called I from STRUC[ ] to STRUC[ ] as follows. Let

A2STRUC[ ]beany structure of vo cabulary , and let n = jjAjj. Then the structure

I A is de ned to b e:

I A=hf0;:::;n 1g;R ;:::;R ;t ;:::;t i

1 r 1 s

where the relation R is determined by the formula ' , for i =1;:::;r as follows. Let the

i i

function h; ; i : jAj !jI Aj b e given by

k 1

hu ;u ;:::;u i = u + u n + + u n

1 2 k k k 1 1

Then,

R = fhu ;:::;u i;:::;hu ;:::;u i jAj= ' u ;:::u g

i 1 k ka i 1 ka

1+k a 1

i i

If the structure A interprets some variablesu then these may app ear freely in the the ' 's

and t 's of I , and the de nition of I A still makes sense.

Supp ose that I is a many-one reduction from S to T , i.e. for all A in STRUC[ ],

A2S , I A 2 T

Then wesay that I is a k -ary rst-order interpretation of S to T . 2

More generally,we could use closed terms which are expressions involving constants and function

symb ols. An even more general waytointerpret constants and functions is via a formula ' such that

` 8x 9!y 'x; y . However, in this pap er the simpler de nition involvin g constant symb ols suces. 6

We are now ready to de ne rst-order pro jections, a syntactic restriction of rst-order

interpretations. If each formula in the rst-order interpretation I satis es this syntactic

condition then it follows that I is also a pro jection in the sense of Valiant. In this case we

call I a rst-order pro jection.

De nition 3.3 [First-Order Pro jections] Let I = h' ;:::;' ;t ;:::;t i.beak -ary rst-

1 r 1 s

order interpretation from S to T as in De nition 3.2. Supp ose further that the ' 's all

satisfy the following projection condition:

' _ ^ _ _ ^ 3.4

i 1 2 2 e e

where the 's are mutually exclusive formulas in which no input relations o ccur, and each

or its negation. is a literal, i.e. an atomic formula P x ;:::x

j j j

In this case the predicate R hu ;:::;u i;:::;h:::;u i holds in I Aif u is true, or

i 1 k ka 1

if u is true for some 1

j j

each bit in the binary representation of I A is determined by at most one bit in the binary

representation of A.Wesay that I is a rst-order projection fop. Write S T to mean

fop

that S is reducible to T via a rst-order pro jection. 2

Example 3.5 To help the reader grasp an intuition of the way an fop reduction b ehaves,

let us describ e an example. We present here the reduction from 3-SAT, satis abilityof

CNF Bo olean expressions with exactly three literals p er clause, to 3-COL, the problem of

coloring the vertices of a graph with 3 colors under the constraint that the endp oints of

all edges get di erent colors. We use the same reduction as describ ed in section 11.4.5 of

[Man], so that the reader in need of additional help can consult it there.

The resp ectivevo cabularies for the input and output structures are as follows. To describ e

instances of 3-SAT, clauses and Bo olean variables are eachnumb ered from 0 through n

1. There are six predicates: P x; c;N x; c, i=1,2,3, indicating that variable x o ccurs

i i

p ositively or negatively in the i p osition of the clause c. The vo cabulary for the output

structures is simply a binary predicate E standing for the Bo olean adjacency matrix of the

output graph. Thus E u; v is true exactly when the edge u; v is present in the output

graph.

The output graph consists of 6 vertices p er clause and twovertices p er Bo olean variable,

plus three additional vertices usually named T , F , and R standing for true, false, and red.

Let an arbitrary 3CNF formula b e co ded by an input structure,

A = hf0; 1;:::;n 1g;P ;P ;P ;N ;N ;N i

1 2 3 1 2 3

The output structure will b e a graph with 8n + 3 relevantvertices. The easiest way for

us to co de this is to use an fop of arity2.We will assume for simplicity that n is always

greater than or equal to 9.

I A=hfha; bi :0 a; b < ng;Ei = hf0;:::n 1g;Ei

where ; E = fhx ;x i; hy ;y i jAj= 'x ;x ;y ;y g

1 2 1 2 1 2 1 2

It remains to write down the rst-order pro jection, '.To do this, we need some nitty gritty

co ding. We will let the vertices T; F , and R b e the elements h0; 0i; h1; 0i, and h2; 0i of I A

resp ectively. The formula ' will have three pieces:

'x ;x ;y ;y = x ;x ;y ;y _ x ;x ;y ;y _ y ;y ;x ;x _

1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 7

x ;x ;y ;y _ y ;y ;x ;x

1 2 1 2 1 2 1 2

Where says that there are edges b etween T; F , and R; says that vertices hx; 1i and hx; 2i

representing variable x and its negation are connected to each other and to R; and; says

that for clause C =a _ b _ d, vertices hC; 6i; hC; 7i, hC; 8i are connected to each other, and

the following edges exist: hC; 3i; hC; 6i, hC; 4i; hC; 7i,hC; 5i; hC; 8i, as well as the edges

a; hC; 3i; T; hC; 3i,

b; hC; 4i, T; hC; 4i, and d; hC; 5i; T; hC; 5i.

In case anyone really wants to see them, here are the formulas written out:

x ;x ;y ;y x = y =0 ^ x 6= y ^ x 2 ^ y 2

1 2 1 2 2 2 1 1 1 1

x ;x ;y ;y x =1 ^ y =2 ^ x = y _ x =2^ x =0^ 1 y 2

1 2 1 2 2 2 1 1 1 2 2

x ;x ;y ;y x = x =0 ^ 3 y 5 ^ 3 y 5

1 2 1 2 1 2 1 2

_ x = y ^ 3 x 5 ^ y = x + 3

1 1 2 2 2

_ [x =1 ^ y =3^ P x ;y ] _ [x =2 ^ y =3^ N x ;y ]

2 2 1 1 1 2 2 1 1 1

_ [x =1 ^ y =4^ P x ;y ] _ [x =2 ^ y =4^ N x ;y ]

2 2 2 1 1 2 2 2 1 1

_ [x =1 ^ y =5^ P x ;y ] _ [x =2 ^ y =5^ N x ;y ] _

2 2 3 1 1 2 2 3 1 1

x = y ^ x 6= y ^ 6 x 8 ^ 6 y 8

1 1 2 2 2 2

4 Main Theorem and Pro of

Theorem 4.1 Let C be a nicecomplexity class, e.g., L, NL, P, NP, etc. Let S and T be

complete for C via rst-order projections. Then S and T are isomorphic via a rst-order

isomorphism.

To prove Theorem 4.1 we b egin with the following lemma. Note the similaritybetween

Lemma 4.2 and the pro ofs of Theorems 2.4 and 2.8. For simplicity in this lemma we are

assuming that I is a single fop that maps STRUC[ ] to itself. The pro of for the case with

two fops and twovo cabularies as in Lemma 4.5 is similar.

Lemma 4.2 Let I be an fop that is 1:1 and of arity greater than or equal to two i.e.

it at least squares the size. Then the fol lowing two predicates are rst-order expressible

concerning a structure A:

a. IEA,meaning that I A exists.

b. AncestorsA;r,meaning that the length of A's maximal ancestor chain is r .

Pro of Let I = h' ;:::;' ;t ;:::;t i, where each ' is in the form of Equation

x :::x 1 r 1 s i

3.4. To prove a just observe that each bit of the relation R of A either 1 dep ends on

exactly one bit of some pre-image B sp eci ed by an o ccurrence of a literal in ' , or 2

ij i 8

it do esn't dep end on any bit of a pre-image. In case 2 a given bit of A is either \right"

or \wrong." Thus, A has an inverse i no bit of A is wrong, and no pair of bits from A

are determined by the same bit of A's preimage in con icting ways. We can check this in a

b, either they do rst-order waybychecking that for all pairs of bits from A: R a and R

not dep end on the same bit from B , or the same value of that bit gives the correct answer

b. Furthermore, the preimage B if it exists can b e describ ed uniquely by for R a and R

i i

a rst-order formula that cho oses the correct bits determined byentries of A. N.B. Since

wehave assumed that I is 1:1 every bit of I A is determined by some bit of A.

b To express AncestorsA;r, wewant to describ e the existence of an Ancestor Chain:

I I I I

A !A ! !A !A = A 4.3

r r 1 1 0

We will then assert that this is the maximal length suchchain, i.e.,

:IE A ^ 8k

r k

Equation 4.4 expresses the existence of the ancestor chain 4.3 inductively in the following

sense: Once we know that A exists, we can ascertain the value A [[p ]] of the bit at p osition

k k k

p of A ,by exhibiting a certi cate:

k k

C k; p = hA [[p ]];p ; A [[p ]];p ;:::;A [[p ]];p i

k k k k k 1 k 1 k 1 0 0 0

We can say in a rst-order sentence that C k; p isinternally consistent. That is, for all

i with k>i 0, bit p of A is determined correctly via I by bit p of A . Note

i+1 i+1 i i

that b ecause each structure A is of size at most the square ro ot of the size of A , the

i+1 i

certi cate requires only O log n bits, i.e., a constantnumber of variables, to express.

Thus, in Equation 4.4, we refer to bit p of the structure A by existentially quantifying

k k

an internally consistent certi cate C k; p . We know inductively, that since IE A , the

k k 1

bit value determined by C k; p is unique and correct. 2

Lemma 4.5 If S and T are interreducible via 1:1 fops I and J each of arity at least two,

then S and T are isomorphic via rst-order isomorphisms.

Pro of Let A b e a structure in the vo cabulary of S , and, as in the pro of of Theorem 2.3

de ne the length of the ancestor chain of A to b e the length of the longest sequence of

1 1 1 1 1 1

the form J A, I J A, J I J A;::: The argument given in Lemma 4.2

shows that there is a formula AncestorsA;r that evaluates to true i A's ancestor chain

has length r . Lemma 4.2 also shows that there is a formula computing J . The desired

isomorphism is now the function b such that the i-th bit of bA is one i the following

rst-order formula is true:

9r AncestorsA;r ^ BIT 0;r ^ I i _ :BIT0;r ^ J i

The reader who is more familiar with bit hacking on Turing machines than with rst-order formulas,

could instead convince herself that this can b e done by an alternating Turing machine running in logarithmic

time and making O 1 alternations; rst-order expressibili ty follows by [BIS]. 9

Note that this rst-order isomorphism b is not, strictly sp eaking, a rst-order interpre-

tation, since it maps some inputs to strictly shorter outputs, which is imp ossible for an

interpretation. 2

It now remains to show,

Lemma 4.6 Suppose that a problem S is complete via fops for a nicecomplexity class, C .

Then S is complete for C via fops that are 1:1 and of arity at least two.

Pro of Of course it remains to de ne \nice", but here is the pro of. Every nice complexity

class has a universal complete problem:

U = fM $w j M w using resources f r g 4.7

C C

Here f r de nes the appropriate complexity measure, e.g. r nondeterministic steps for

NP, deterministic space log r , for L, space 2 for EXPSPACE, etc.

We claim that U is complete for C via fops that are 1:1 and of arity at least two. In

order to make this claim, we need to agree on an enco ding of inputs to U that allows us to

interpret them as structures over some vo cabulary. Since all of our structures are enco ded

in binary,we will enco de $ and by 10 and 11 resp ectively, and the binary bits 0 and 1

constituting M and w will b e enco ded by 00 and 01 resp ectively.Now, as in, for example,

[I87], we consider a binary string of length n to b e a structure with a single unary predicate

over a universe of size n.Now for any given problem T 2C accepted by machine M ,we

show that T is reducible to U via a fop that is 1:1 and of arity at least two. The fop simply

maps input w to the string M $w , for an appropriate r whichwe can always taketobeat

least jw j . The fop checks that if i 2jM j then the o dd-numb ered bits are 0 and if i is even,

then the i bit is 1 i the i=2-th bit of M is 1. Similarly,if2jM j +2

then the o dd-numb ered bits are 0 and the even numb ered bits are the corresp onding bit of

w , etc.

To complete the pro of of the lemma, let T be any problem in C and let S b e as ab ove. Then

we reduce T to S via a 1:1, length squaring fop as follows. First reduce T to U as ab ove.

Next reduce U to S via the fop promised in the statement of the lemma.

It is easy to verify that, using the enco ding wehavechosen for U , it holds that for every

length n, for all i n, there are two strings x and y of length n, di ering only in p osition

i, such that x 2 U and y 62 U .

C C

Thus the fop from U cannot p ossibly ignore any of the bits in its input. But an fop cannot

pro cess several bits into one, it can only either ignore a bit or copy it, or negate it; and this

choice is made indep endently of the values of any of the bits.

It follows that the comp osition of these two fops is the 1:1 length squaring fop that we

desire. Note that an fop by de nition must have arity at least one and thus cannot b e

length decreasing on Bo olean strings. 2

From the ab ove three lemmas wehave a rst-order version of Theorem 2.3 and thus Theorem

4.1 follows. 10

We can insp ect the pro of of Lemma 4.6 to get a de nition of \nice". A complexity class

is \nice" if it has a universal complete problem via fops as in Equation 4.7. It is easy to

check that the following complexity classes, among many others, are nice and thus meet the

conditions of Theorem 4.1.

Prop osition 4.8 The fol lowing complexity classes are nice: NC , L , NL, LOGCFL,

NC ,P,NP, PSPACE, EXPTIME, EXPSPACE.

Pro of This is immediate for the Turing machine based classes: L, NL, P,NP, PSPACE,

EXPTIME, EXPSPACE. It similarly follows for the other three classes using the de nitions:

NC = ASPACE[log n] TIME[log n ], and LOGCFL = ASPACE[log n] 8TIME[log n].

5 More on the Relationship between Isomorphisms and

Pro jections

There are several questions ab out isomorphisms among complete sets that can b e answered

in the setting of rst-order computation but are op en for general p olynomial-time compu-

tation. It is not known whether one-way functions exist since their existence would imply

that P 6= NP. However, if one-way functions exist i.e., if P 6= UP then there exists a

one-way function f such that f SAT is p olynomial-time isomorphic to SAT [Ga].

jxj

Here we can b e more de nitive: the bijection f x=3x mod2 was shown in [BL ] to

b e one-way for rst-order computation, in the sense that f is rst-order expressible, but

f is not. See also [Has] for other examples. However, it is not to o hard to show that

for this choice of f , f SAT is complete for NP under rst-order pro jections, and thus it is

rst-order isomorphic to SAT.

The next result shows that the class of sets complete under rst-order pro jections is not

closed under rst-order isomorphisms. This also seems to b e the rst construction of a set

that is complete for NP under rst-order or even p oly-time many-one reductions, that is

not complete under rst-order pro jections.

Theorem 5.1 There is a set rst-order isomorphic to SAT that is not complete for NP

under rst-order projections.

Pro of Let g x b e a string of jxj bits, with bit x representing the logical AND of bits

i;j

i and j of x. Let A = fhx; g xi : x 2 SAT g. By an extension of the techniques used in

proving Theorem 4.1, it can b e shown that A is rst-order isomorphic to SAT. However a

direct argument shows that there cannot b e any pro jection even a nonuniform pro jection

from SATtoA. Sketch: For all n, one can nd bit p ositions i and j that are indep endentof

each other and are indep endentofevery other bit p osition, in the sense that for any setting

b of bit j there are twowords that di er only in bit i,having b in p osition j , such that one

of the words is in SAT and one is not. No pro jection reducing SAT to another language

can \ignore" either i or j . But since i and j are indep endent of all other bit p ositions, no

pro jection can enco de the AND of bits i and j . 2 11

A natural question that remains op en is the question of whether every set complete for NP

under rst-order many-one reductions is rst-order isomorphic to SAT. A related question

is whether one can construct a set complete for NP under p oly-time many-one reductions

that is not rst-order isomorphic to SAT. Since so many to ols are available for proving

the limitations of rst-order computation, we are optimistic that this and related questions

ab out sets complete under rst-order reductions should b e tractable. Furthermore, we hop e

that insights gleaned in answering these questions will b e useful in guiding investigations

of the p olynomial-time degrees.

Acknowledgments The authors wish to thank the organizers of the 1992 Seminar on

Structure and Complexity Theory at Schlo Dagstuhl, where this work was initiated. We

also thank Richard Beigel, Jose Antonio Medina, and two anonymous referees for comments

on an earlier draft.

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