Randomized feasible interpolation and monotone circuits with a local oracle

Abstract

The feasible interpolation theorem for semantic derivations from K. (1997) [16] allows to derive from some short semantic derivations (e.g. in resolution) of the disjointness of two NP sets U and V a small communication protocol (a general dag-like protocol in the sense of K. (1997) [16]) computing the Karchmer-Wigderson multi-function KW [U, V ] associated with the sets, and such a protocol further yields a small circuit separating U from V . When U is closed upwards the protocol computes the monotone Karchmer-Wigderson multi-function KW m [U, V ] and the resulting circuit is monotone. K. (1998) [18] extended the feasible interpolation theorem to a larger class of semantic derivations using the notion of a real communication complexity (e.g. to the cutting planes proof system CP).In this paper we generalize the method to a still larger class of semantic derivations by allowing randomized protocols. We also introduce an extension of the monotone circuit model, monotone circuits with a local oracle (CLOs), that does correspond to communication protocols for KW m [U, V ] making errors. The new randomized feasible interpolation thus shows that a short semantic derivation (from a certain class of derivations larger than in the original method) of the disjointness of U, V , U closed upwards, yields a small randomized protocol for KW m [U, V ] and hence a small monotone CLO separating the two sets. This research is motivated by the open problem to establish a lower bound for proof system R(LIN/F2) operating with clauses formed by linear Boolean functions over F2. The new randomized feasible interpolation applies to this proof system and also to (the semantic versions of) cutting planes CP, to small width resolution over CP of K. (1998) [17] (system R(CP)) and to random resolution RR of Buss, Kolodziejczyk and Thapen [5]. The method does not yield yet lengths-of-proofs lower bounds; for this it is necessary to establish lower bounds for randomized protocols or for monotone CLOs.Consider a propositional proof system R(LIN/F 2 ) that operates with clauses of linear equations over F 2 and combines the rules of both resolution and linear equational calculus. A line C in a proof has the form {f 1 , . . . , f k }

Some Subsystems of Constant-Depth Frege with Parity

“…Clauses (20)-(23) rule out, respectively: two x's matched to the same type-1 extra point; two type-1 extra points matched to the same x; a type-1 extra point matched both to an x and to another type-1 extra point; a type-1 extra point matched to two different type-1 extra points. Altogether, clauses (17)- (23) say that, for each pigeon i, there is a perfect matching on the set of x variables set to 1 and the type-1 extra points corresponding to the pigeon. Clauses 24…”