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A Satisfiability Algorithm for AC0 961 Russel Impagliazzo, William Matthews and Ramamohan Paturi

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A Satisfiability Algorithm for AC0 961 Russel Impagliazzo, William Matthews and Ramamohan Paturi A Satisfiability Algorithm for AC0 Russell Impagliazzo ∗ William Matthews † Ramamohan Paturi † Department of Computer Science and Engineering University of California, San Diego La Jolla, CA 92093-0404, USA January 2011 Abstract ing the size of the search space. In contrast, reducing We consider the problem of efficiently enumerating the problems such as planning or model-checking to the spe- satisfying assignments to AC0 circuits. We give a zero- cial cases of cnf Satisfiability or 3-sat can increase the error randomized algorithm which takes an AC0 circuit number of variables dramatically. On the other hand, as input and constructs a set of restrictions which cnf-sat algorithms (especially for small width clauses partitions {0, 1}n so that under each restriction the such as k-cnfs) have both been analyzed theoretically value of the circuit is constant. Let d denote the depth and implemented empirically with astounding success, of the circuit and cn denote the number of gates. This even when this overhead is taken into account. For some algorithm runs in time |C|2n(1−µc,d) where |C| is the domains of structured cnf formulas that arise in many d−1 applications, state-of-the-art SAT-solvers can handle up size of the circuit for µc,d ≥ 1/O[lg c + d lg d] with probability at least 1 − 2−n. to hundreds of thousands of variables and millions of As a result, we get improved exponential time clauses [11]. Meanwhile, very few results are known ei- algorithms for AC0 circuit satisfiability and for counting ther theoretically or empirically about the difficulty of solutions. In addition, we get an improved bound on the Circuit Satisfiability. correlation of AC0 circuits with parity. This raises the question: for which classes of circuits As an important component of our analysis, we are there non-trivial Satisfiability algorithms? More extend the H˚astadSwitching Lemma to handle multiple precisely, for an algorithm that decides satisfiability for k-cnfs and k-dnfs. circuits in a class C, we express its worst-case running time as |C|2n(1−µ) where C is a circuit with n inputs. 1 Introduction We say that µ is the savings over exhaustive search and write it in terms of n and the parameters of the class C, The Circuit Satisfiability problem, deciding whether a for example, the ratio of gates or wires to the number Boolean circuit has an assignment where it evaluates to of inputs, and depth. The larger µ is, the more non- true, is in many ways the canonical NP-complete prob- trivial the algorithm can be considered. When can we lem. It is not only important from a theoretical point exploit the structural properties of C to obtain savings of view, but pragmatically, since search problems di- over exhaustive search? What are the best savings for rectly reduce to Circuit Satisfiability without increas- various circuit classes? How is the expressive power of a circuit class related to the amount of savings for its ∗Work supported by the Ellentuck Fund, Friends of the satisfiability problem? Institute for Advanced Study, and NSF grants DMS-0835373 and Williams’ recent work [20, 21] has given an addi- CCF-0832797 subcontract No. 00001583. Any opinions, findings tional motivation to look at improved algorithms for and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of Circuit Satisfiability problems. He shows that Circuit the National Science Foundation, Ellentuck Fund, or Friends of Satisfiablity algorithms for a class of circuits with even a the Institute for Advanced Study. very small improvement (µ = ω(log n/n) ) over exhaus- †This research is supported by NSF grant CCF-0947262 from tive search imply circuit lower bounds for that class. the Division of Computing and Communication Foundations. Thus, he formally proves a connection showing that the Any opinions, findings and conclusions or recommendations ex- pressed in this material are those of the authors and do not nec- same understanding of the limitations of a circuit class essarily reflect the views of the National Science Foundation. is needed both for improved algorithms and to prove 961 Copyright © SIAM. Unauthorized reproduction of this article is prohibited. lower bounds. tion with depth-3 circuits. Subsequently, [14] proposed Among the satisfiability problems for various circuit a resolution-based k-sat algorithm and obtained an im- classes, the k-sat problem has attracted the attention proved savings (by a constant factor) using a sophisti- of several researchers since Monien and Speckenmeyer cated analysis. Using the same analysis, they construct [12] first showed that it can be computed in less than 2n a fairly simple function (checking whether the input bi- n(1−µk) time in the worst-case, that is, in time |F |2 for nary string is a codeword√ of a certain code) which re- 1.282 n 0 µk > 0 where F is a k-cnf. Researchers have obtained a quires at least 2 size for any depth-3 AC cir- series of improvements for µk, with particular attention cuit. This is the best-known lower bound for depth- to the case k = 3. Currently, the best known savings for 3 circuits for any function. Although depth reduction 3-sat is about 0.614 for randomized algorithms [14, 9] techniques based on Switching Lemma [5, 16, 1] and and about 0.4157 for deterministic algorithms [18, 13]. especially the top-down technique of [7] for√ depth-3 cir- There are a suprisingly wide variety of state-of-the-art cuits prove lower bounds that are close to 2 n, it is not algorithms for k-SAT, but they all achieve savings µk of clear that these techniques√ by themselves would yield a the form Θ(1/k) [15, 18]. lower bound of 2c n for c > 1. 1 Much less is known about the nature of savings It has been a long open problem to prove 2ω(n d−1 ) for satisfiability problems for more expressive circuit size lower bounds for AC0 circuits of depth d. Since classes. Schuler [19] has shown that the satisfiability the breakthrough results of Yao and H˚astad in the of an m-clause CNF F can be determined in time mid 1980’s, there have been only modest improvements n(1− 1 ) |F |2 1+lg m . Using a bit more careful analysis, [22, 5] and only for for depth-3 circuits [7, 14]. It is Calabro et al. [3] improved this from savings 1/O (lg m) tantalizing to prove better lower bounds by exploiting m 0 to 1/O lg n , giving constant savings when m = O (n). the connection between AC satisfiability upper bounds Subsequently, Calabro et al. [4] have considered the and lower bounds. Unfortunately, we do not have any question of satisfiability of bounded-depth unbounded ideas for proving strong enough upper bounds for AC0 0 fan-in circuits over standard basis (AC circuits) and satisfiability that would imply better lower bounds. Our 0 shown that the satisfiability of AC circuits C of size cn modest goal is to obtain a satisfiability algorithm for n(1−µ) and depth d can be decided in time |C|2 where AC0 with savings sufficient enough to imply the some of d−2 d−2 µ ≥ 1/O(c2 −1 lg3·2 −2 c). Santhanam [17] has the best-known lower bounds. considered cn-size formulas F with no depth restriction and showed that the satisfiability problem for such 1.1 Our Results and Techniques Our main result n 1− 1 0 formulas can be solved in time |F |2 ( poly(c) ). More is an algorithm for AC satisfiability with the best recently, Williams [21] has shown that the satisfiability known savings. 2−Θ(d) of ACC circuits C can be solved in time |C|2n−Θ(n ). Theorem 1.1. There is a Las Vegas algorithm for 0 In this paper, we return to AC circuits and seek deciding the satisfiability of circuits C with cn further improvement in the savings for the satisfiability gates and depth d whose expected time is at most 0 n(1−µ ) algorithm for AC circuits of size cn and depth d. While poly(n)|C|2 c,d , where the savings µc,d is at least the algorithm in [4] obtains savings in terms of c and d 1 O(lg c+d lg d)d−1 . independent of n, its double exponentially small savings diminish rapidly to zero for d > 2 when c grows as a The above algorithm immediately follows from the function of n. While Williams’ algorithm [21] provides existence of an algorithm that enumerates all satisfying nontrivial savings even when c grows sufficiently large assignments by partitioning the space into sub-cubes as a function of n, its savings decreases with n even where the circuit is constant. when c and d are constants. Furthermore, it leaves open the question whether one can obtain significantly Theorem 1.2. There exists a Las Vegas algorithm better savings for AC0 circuits compared to the savings which, on input a cn gate, depth d circuit C in n vari- Williams obtained for the ACC circuits. ables, produces a set of restrictions {ρi}i which parti- n Another independent motivation is the natural con- tion {0, 1} and such that for each i, C|ρi is constant. nection to proving lower bounds. Paturi et al. [15] have The expected time and number of restrictions are both n(1−µ ) observed that the analysis that led to an improved up- poly(n)|C|2 c,d , where the savings µc,d is at least 1 per bound for k-sat can also be used to prove lower O(lg c+d lg d)d−1 .
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