Reductions in Circuit Complexity: An Isomorphism Theorem and a Gap Theorem. y Manindra Agrawal Eric Allender Department of Computer Science Department of Computer Science Indian Institute of Technology Rutgers University Kanpur 208016 P.O. Box 1179 India Piscataway, NJ 08855-117 9 USA [email protected] [email protected] Steven Rudich Computer Science Department Carnegie Mellon University Pittsburgh, PA 15213 USA [email protected] Abstract 0 We show that al l sets that arecomplete for NP under non-uniform AC reductions 0 are isomorphic under non-uniform AC -computable isomorphisms. Furthermore, these 0 sets remain NP-complete even under non-uniform NC reductions. More general ly, we show two theorems that hold for any complexity class C closed un- 1 der uniform NC -computable many-one reductions. Gap: The sets that arecomplete 0 0 for C under AC and NC reducibility coincide. Isomorphism: The sets complete for 0 C under AC reductions are al l isomorphic under isomorphisms computable and invert- 0 ible by AC circuits of depth three. Our Gap Theorem does not hold for strongly uniform reductions: we show that there 0 1 0 are Dlogtime-uniform AC -complete sets for NC that are not Dlogtime-uniform NC - complete. 1 Intro duction The notion of complete sets in complexity classes provides one of the most useful to ols currently available for classifying the complexity of computational problems. Since the mid-1970's, one of the most durable conjectures ab out the nature of complete sets is the Berman-Hartmanis conjecture [BH77], which states that all sets complete for NP under This researchwas done while visiting the University of Ulm under an Alexander von Humb oldt Fellowship. y Supp orted in part by NSF grant CCR-9509603. p olynomial-time many-one reductions are p-isomorphic; essentially this conjecture states that the complete sets are all merely di erent enco dings of the same set. Two sets A and B are considered p-isomorphic if there is a 1-1 function from A onto B that is p olynomial- time computable and p olynomial-timeinvertible. Although the isomorphism conjecture was originally stated for the NP-complete sets, subsequentwork has considered whether the complete sets for other complexity classes C and under other notions of reducibility collapse to an isomorphism typ e. In this pap er, we prove such an analogue of the Berman-Hartmanis 0 conjecture in a very natural setting: all sets complete for NP under AC reductions are 0 isomorphic under AC -computable isomorphisms. Most of the results in this pap er concern 0 0 0 0 non-uniform AC and NC reductions. In all instances, AC and NC refer to non-uniform circuits unless otherwise indicated. One ma jor ingredient of our pro of of this result is 0 a `gap' theorem: we show that all sets complete for NP under AC reductions are also 0 complete under NC reductions. In full generality, the two main theorems of this pap er can b e stated as follows. 1 For any class C closed under uniform NC -computable many-one reductions: 0 Isomorphism Theorem: The sets complete for C under AC reductions are all isomorphic 0 under AC -computable isomorphisms. 0 0 Gap Theorem: The sets that are complete for C under AC and NC reducibility coincide. The remainder of the intro duction will present a more detailed description of these theorems and previous related work. 1.1 The Isomorphism Theorem The Berman-Hartmanis conjecture has inspired a great deal of work in complexity theory, and we cannot review all of the previous work here. For an excellent survey, see [KMR90]. Wedowant to call attention to two general trends this work has taken, regarding 1 one-way functions, and 2 more restrictive reducibilities. One-way functions in the worst case sense are functions that can b e computed in p olynomial-time, but whose inverse functions are not p olynomial-time computable. Be- ginning with [JY85] see also [KMR95, Se92, KLD86], among others many authors have noticed that if worst case one-way functions exist, then the Berman-Hartmanis conjecture might not b e true. In particular, if f is one-way, nob o dy has presented a general technique for constructing a p-isomorphism b etween SAT and f SAT, even though f SAT is clearly NP-complete. Rogers [Ro95] do es showhow to construct such isomorphisms relativetoan oracle. However, the fo cus of our work is on non-relativized classes. Note also that it has b een shown in [KMR88, JPY94] that there are non-complete degrees where isomorphisms can b e constructed; however the fo cus of our work is on complete degrees. An even stronger notion than one-way functions is that of average case one-way func- tions: these are p olynomial-time computable functions whose inverse can b e eciently computed only on a negligible fraction of the range. Advances in the theoretical founda- tions of cryptographyhave shown that average case one-way functions can b e used to con- struct \pseudo-random" functions that are computable in p olynomial-time [HILL90 ]. For 2 an excellent distillation of the most imp ortant results in this area we recommend [Lu96]. Intuitively,iff is a p olynomial-time, random-like function, f SAT is NP-complete, but has no apparent structure to facilitate the construction of an isomorphism to SAT. Kurtz, Mahaney, and Royer [KMR95] are able to make this intuition technically precise in the random oracle setting. They show that when f is a truly random function, f SAT is not isomorphic to SATeven when f is given as an oracle. This suggests that a pseudo-random f might similarly guarantee that no isomorphism to f SAT is p ossible. As we argue b elow, the results in this pap er greatly undermine our con dence in this approach to resolving the Berman-Hartmanis conjecture. Our Isomorphism Theorem negates the ab oveintuition in two imp ortant sp ecial cases. 0 Firstly, it is easy to observe that thereare worst-case one-way functions in uniform NC if there are any one-way functions at all. Thus, if the worst-case one-way functions are suciently easy to compute, the intuition that worst-case one-way functions cause the isomorphism conjecture to fail is incorrect. 0 Secondly,we prove that all sets complete for NP under AC reductions are complete 0 under reductions that are computable via depth two AC circuits, and these sets are all 0 isomorphic to SAT under isomorphisms computable and invertible byAC circuits of depth 0 three. But it is known that there are functions computable in AC that are average-case 0 hard to compute for AC circuits of depth three, and that this allows one to pro duce output 0 that app ears pseudorandom to AC circuits of depth three [Ni92]. Using these to ols, one can construct one-one functions f , many of whose output bits lo ok random to depth three circuits. Although one might b elieve that for such a function f , f SAT should app ear 0 random to AC circuits of depth three, there is nonetheless a reduction from SATtof SAT computable in depth two, and an isomorphism computable and invertible in depth three. Nevertheless, we refrain from concluding that our results indicate that the isomorphism 0 conjecture is true. What is true in the AC settings need not hold for the much more general p olynomial-time settings. There has b een previous work on versions of the isomorphism conjecture for more restric- tive reducibilities see e.g., [Ag94, ABI93]. In fact, in [Ag94] a class of reductions, 1-NL, was presented such that the 1-NL-complete sets in natural unrelativized complexity classes are all 1-NL-isomorphic. However, all such reductions can b e inverted in p olynomial time. For instance, the 1-L and 1-NL reductions in [Ag94, Ag95] and earlier work, and the rst-order pro jections considered in [ABI93] have this prop erty. A p ossible exception is the so-called \1-omL reducibility" considered in [Ag94], which shares the non-invertibility prop ertyof 0 0 NC and AC reductions considered here. However 1-omL reducibility is a rather contrived reducibilityinvented solely for the purp ose of proving the \collapse" result in [Ag94], and the pro of of that result relies heavily on the invertibility of the related 1-L and 1-NL reduc- tions. That is not the case with the results presented in this pap er. Additionally,we show that the sets that are complete under the reducibilities considered in [Ag94] are in fact com- 0 plete under NLOG-uniform pro jections, and hence are also complete under NC reductions. Thus the complete sets considered in [Ag94] are a sub class of the sets for whichwe present isomorphisms. One of the ma jor goals of the work presented here is to correct a shortcoming of the results presented in [ABI93]. In [ABI93], it is shown that, for essentially any natural complexity class C , the sets complete for C under rst-order pro jections are all isomorphic 3 under rst-order isomorphisms. The shortcoming of [ABI93] to whichwe refer is this: the complete degree under rst-order pro jections is properly containedinthe isomorphism typ e of the complete sets. In order to improve the result in [ABI93] to obtain a true analog of the Berman-Hartmanis conjecture [BH77] it would b e necessary to show that the complete degree under rst-order reductions coincides exactly with the rst-order isomorphism typ e of the complete sets.