Mining Circuit Lower Bound Proofs for Meta-algorithms

Ruiwen Chen, Valentine Kabanets, Antonina Kolokolova, Ronen Shaltiel, David Zuckerman

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract / Description of output

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(2n) 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 2n/n. We get nontrivial compression for functions computable by AC0 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 n2. 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 clas- C would imply a superpolynomial lower bound against C for a language in NEXP1.
Original languageEnglish
Title of host publicationIEEE 29th Conference on Computational Complexity, CCC 2014, Vancouver, BC, Canada, June 11-13, 2014
PublisherInstitute of Electrical and Electronics Engineers (IEEE)
Pages262-273
Number of pages12
DOIs
Publication statusPublished - 2014

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