# Randomized feasible interpolation and monotone circuits with a local oracle

Journal of Mathematical Logic 18 (2):1850012 (2018)

# Abstract

The feasible interpolation theorem for semantic derivations from [J. Krajíček, Interpolation theorems, lower bounds for proof systems, and independence results for bounded arithmetic, J. Symbolic Logic 62 457–486] allows to derive from some short semantic derivations of the disjointness of two [Formula: see text] sets [Formula: see text] and [Formula: see text] a small communication protocol computing the Karchmer–Wigderson multi-function [Formula: see text] associated with the sets, and such a protocol further yields a small circuit separating [Formula: see text] from [Formula: see text]. When [Formula: see text] is closed upwards, the protocol computes the monotone Karchmer–Wigderson multi-function [Formula: see text] and the resulting circuit is monotone. Krajíček [Interpolation by a game, Math. Logic Quart. 44 450–458] extended the feasible interpolation theorem to a larger class of semantic derivations using the notion of a real communication complexity. 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, that does correspond to communication protocols for [Formula: see text] making errors. The new randomized feasible interpolation thus shows that a short semantic derivation of the disjointness of [Formula: see text], [Formula: see text] closed upwards, yields a small randomized protocol for [Formula: see text] 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 [Formula: see text] operating with clauses formed by linear Boolean functions over [Formula: see text]. The new randomized feasible interpolation applies to this proof system and also to cutting planes CP, to small width resolution over CP of Krajíček [Discretely ordered modules as a first-order extension of the cutting planes proof system, J. Symbolic Logic 63 1582–1596] ) and to random resolution RR of Buss, Kolodziejczyk and Thapen [Fragments of approximate counting, J. Symbolic Logic 79 496–525]. 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.

## PhilArchive

Upload a copy of this work     Papers currently archived: 94,659

Setup an account with your affiliations in order to access resources via your University's proxy server

# Similar books and articles

Interpolation by a Game.Jan Kraíček - 1998 - Mathematical Logic Quarterly 44 (4):450-458.
The mouse set theorem just past projective.Mitch Rudominer - forthcoming - Journal of Mathematical Logic.
Specializing trees and answer to a question of Williams.Mohammad Golshani & Saharon Shelah - 2020 - Journal of Mathematical Logic 21 (1):2050023.

2018-07-06

15 (#971,270)

6 months
4 (#1,053,478)