A First-Order Isomorphism Theorem y Eric Allender Department of Computer Science, Rutgers University New Brunswick, NJ, USA 08903 [email protected] z Jos e Balc azar U. Polit ecnica de Catalunya, Departamento L.S.I. Pau Gargallo 5, E-08071 Barcelona, Spain [email protected] x 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. y Some of this work was done while on leave at Princeton University; supp orted in part by National Science Foundation grant CCR-9204874. z Supp orted in part by ESPRIT-I I BRA EC pro ject 3075 ALCOM and by Acci on Integrada Hispano- Alemana 131 B x 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 1 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 1 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 0 in [BIS] that rst-order computation corresp onds exactly to computation by uniform AC 0 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 0 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. 1 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. p De nition 2.1 For A; B ,wesay that A and B are p-isomorphic A B i there = 1 exists a bijection f 2 FP with inverse f 2 FP such that A is many-one reducible to B 1 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 1 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: 1 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 2 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. p 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 2 As a simple example, the following is a padding function for SAT, ew; x = w ^ c ^ c ^ ^ c 1 2 jxj th where c is y _ yifthe i bit of x is 1 and y _ y otherwise, where y is a Bo olean variable i 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, p 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.