TY - GEN
T1 - Mining circuit lower bound proofs for meta-algorithms
AU - Chen, Ruiwen
AU - Kabanets, Valentine
AU - Kolokolova, Antonina
AU - Shaltiel, Ronen
AU - Zuckerman, David
PY - 2014
Y1 - 2014
N2 - 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 2 n/n. We get non-trivial 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 cite [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 mathcal C ⊆ P/poly would imply the circuit lower bound NEXPnotsubseteq mathcal ⊈ C. This complements Williams's result [55] that any non-trivial Circuit-SAT algorithm for a circuit class mathcal {C} would imply a super polynomial lower bound against mathcal {C} for a language in NEXP also proves such an implication in NEXP.
AB - 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 2 n/n. We get non-trivial 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 cite [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 mathcal C ⊆ P/poly would imply the circuit lower bound NEXPnotsubseteq mathcal ⊈ C. This complements Williams's result [55] that any non-trivial Circuit-SAT algorithm for a circuit class mathcal {C} would imply a super polynomial lower bound against mathcal {C} for a language in NEXP also proves such an implication in NEXP.
KW - Circuit-SAT algorithms
KW - average-case circuit lower bounds
KW - compression
KW - meta-algorithms
KW - natural property
KW - random restrictions
KW - shrinkage of de Morgan formulas
UR - http://www.scopus.com/inward/record.url?scp=84906675462&partnerID=8YFLogxK
U2 - 10.1109/CCC.2014.34
DO - 10.1109/CCC.2014.34
M3 - Conference contribution
AN - SCOPUS:84906675462
SN - 9781479936267
T3 - Proceedings of the Annual IEEE Conference on Computational Complexity
SP - 262
EP - 273
BT - Proceedings - IEEE 29th Conference on Computational Complexity, CCC 2014
PB - IEEE Computer Society
T2 - 29th Annual IEEE Conference on Computational Complexity, CCC 2014
Y2 - 11 June 2014 through 13 June 2014
ER -