Notes on switching lemmas

TL;DR

This paper proves three variants of the switching lemma under different random restriction models, extending Razborov's approach with probabilistic methods and weighted restrictions, contributing to complexity theory.

Contribution

It introduces new proofs of switching lemmas for independent, block, and specific distribution restrictions, using weighted restrictions and probabilistic techniques.

Findings

Proved switching lemmas for three types of random restrictions.

Extended Razborov's proof method with weighted restrictions.

Used probabilistic arguments instead of counting ones.

Abstract

We prove three switching lemmas, for random restrictions for which variables are set independently; for random restrictions where variables are set in blocks (both due to Hastad [Hastad 86]); and for a distribution appropriate for the bijective pigeonhole principle [Beame et al. 94, Krajicek et al. 95]. The proofs are based on Beame's version [Beame 94] of Razborov's proof of the switching lemma in [Razborov 93], except using families of weighted restrictions rather than families of restrictions which are all the same size. This follows a suggestion of Beame in [Beame 94]. The result is something between Hastad's and Razborov's methods of proof. We use probabilistic arguments rather than counting ones, in a similar way to Hastad, but rather than doing induction on the terms in our formula with an inductive hypothesis involving conditional probability, as Hastad does, we explicitly build one function to bound the probabilities for the whole formula.