Mining Circuit Lower Bound Proofs for Meta-algorithms

Abstract

Abstract-We show that circuit lower bound proofs based on the method of random restrictions yield non-trivial compression algorithms for "easy" Boolean functions from the corresponding circuit classes. The compression problem is defined as follows: given the truth table of an n-variate Boolean function f computable by some unknown small circuit from a known class of circuits, find in deterministic time poly(2 n ) a circuit C (no restriction on the type of C) computing f so that the size of C is less than the trivial circuit size 2 n /n. We get nontrivial compression for functions computable by AC 0 circuits, (de Morgan) formulas, and (read-once) branching programs of the size for which the lower bounds for the corresponding circuit class are known.These compression algorithms rely on the structural characterizations of "easy" functions, which are useful both for proving circuit lower bounds and for designing "meta-algorithms" (such as Circuit-SAT). For (de Morgan) formulas, such structural characterization is provided by the "shrinkage under random restrictions" results [52], [21], strengthened to the "high-probability" version by [48], [26], [33]. We give a new, simple proof of the "high-probability" version of the shrinkage result for (de Morgan) formulas, with improved parameters. We use this shrinkage result to get both compression and #SAT algorithms for (de Morgan) formulas of size about n 2 . We also use this shrinkage result to get an alternative proof of the recent result by Komargodski and Raz [33] of the average-case lower bound against small (de Morgan) formulas.Finally, we show that the existence of any non-trivial compression algorithm for a circuit class C ⊆ P/poly would imply the circuit lower bound NEXP ⊆ C. This complements Williams's result [55] that any non-trivial Circuit-SAT algorithm for a circuit class C would imply a superpolynomial lower bound against C for a language in NEXP 1 .

Fourier Concentration from Shrinkage

“…This class of formulas is also lossily compressible in the sense of [9]. That is, given the truth table of a Boolean function f : {0, 1} n → {1, −1} which is promised to be computable by an unknown de Morgan (read-once) formula of size s, we can compute in deterministic time 2 O(n) , a Boolean circuit C of size about n s 1/Γ+ , where Γ is the corresponding shrinkage exponent, such that C agrees with f on all but exp(−n Ω(1) ) fraction of n-bit inputs.…”

“…In the language of [27,9], this means that we have a deterministic lossy-compression algorithm for the class of de Morgan formulas of sub-quadratic size. 1 Similarly, we get, for read-once de Morgan formulas on n inputs, a lossy-compression algorithm producing a circuit of size at most exp(n 1/3+ ) which agrees with the formula on all but at most exp(−n Ω( ) ) fraction of inputs.…”

“…On the other hand, circuit lower bounds are implied by nontrivial algorithms for SAT [30,29,52,53] or Polynomial Identity Testing [26]. It has also been observed that techniques used to prove existing circuit lower bounds are often useful for designing learning algorithms [36], SAT algorithms [57,43,44,23,5,9,10], and pseudorandom generators (PRGs) [8,24,16,48] for the same class of circuits. In particular, the method of random restrictions, useful for proving lower bounds against AC 0 circuits [14,56,17] and de Morgan formulas [46,2,18,43,33,34], turns out to be also useful for designing such algorithms for the same circuit class.…”

An Improved Deterministic #SAT Algorithm for Small De Morgan Formulas

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We give a deterministic #SAT algorithm for de Morgan formulas of size up to n 2.63 , which runs in time 2 n−n Ω(1) . This improves upon the deterministic #SAT algorithm of [3], which has similar running time but works only for formulas of size less than n 2.5 . Our new algorithm is based on the shrinkage of de Morgan formulas under random restrictions, shown by Paterson and Zwick [12].

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Beating Brute Force for (Quantified) Satisfiability of Circuits of Bounded Treewidth

“…Fomin and Kratsch [15], Robson [31] or Williams [38]). Indeed, the algorithm of Santhanam [33] for QBF satisfiability and of Chen et al [8] for counting satisfying assignments of a formula are both based on memoization. Therefore our main contribution is to show that, when based on the lower bounds of Neciporuk and de Oliveira Oliveira, memoization can be made to work surprisingly well for (quantified) satisfiability of bounded treewidth circuits.…”