Compression of Boolean Functions

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(2) 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. 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: • AC, • (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 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 AC circuits would also imply new superpolynomial circuit lower bounds. Finally, we apply the ideas used for compression to get zero-error randomized #SATalgorithms for de Morgan and complete-basis formulas, as well as branching programs, on n variables of about quadratic size that run in expected time 2/2 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. ISSN 1433-8092 Electronic Colloquium on Computational Complexity, Report No. 24 (2013)