Compression of Boolean Functions.
Valentine Kabanets, Antonina Kolokolova · 2013
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: • AC0, • (de Morgan) formulas, and • (read-once) branching programs, 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 super-polynomial 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/2n ϵ, for some ϵ> 0 (dependent on the size of the formula/branching program). ∗Research partially supported by an NSERC Discovery grant. †Research partially supported by an NSERC Discovery grant. 1