Compression of Boolean Functions

Valentine Kabanets and Antonina Kolokolova

Abstract

We consider the problem of compression for ``easy'' Boolean functions: 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 both positive and negative results. On the positive side, we show that several circuit classes for which lower bounds are proved by a method of random restrictions:

allow non-trivial compression for circuits up to the size for which lower bounds are known. On the negative side, we show that compressing functions from any class C⊆P/poly implies superpolynomial lower bounds against C for a function in NEXP; we also observe that compressing monotone functions of polynomial circuit complexity or functions computable by large-size AC0 circuits would also imply new superpolynomial circuit lower bounds.

Finally, we apply the ideas used for compression to get zero-error randomized #SAT-algorithms for de Morgan and complete-basis formulas, as well as branching programs, on n variables of about quadratic size that run in expected time 2n/2nε, for some ε>0 (dependent on the size of the formula/branching program).

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