A Switching Lemma for Small Restrictions and Lower Bounds for k-DNF Resolution

Abstract

We prove a new switching lemma that works for restrictions that set only a small fraction of the variables and is applicable to formulas in disjunctive normal form (DNFs) with small terms. We use this to prove lower bounds for the Res(k) propositional proof system, an extension of resolution which works with k-DNFs instead of clauses. We also obtain an exponential separation between depth d circuits of bottom fan-in k and depth d circuits of bottom fan-in k + 1.Our results for Res(k) are as follows:1. The 2n to n weak pigeonhole principle requires exponential size to refute in Res(k) for k ≤ log n/ log log n. 2. For each constant k, there exists a constant w > k so that random w-CNFs require exponential size to refute in Res(k). 3. For each constant k, there are sets of clauses which have polynomial size Res(k + 1) refutations but which require exponential size Res(k) refutations.

On Approximate Majority and Probabilistic Time

“…We have recently found out that (Razborov 2002-2003 improved the parameters of the above switching lemma by Segerlind et al (2004). Using Razborov's switching lemma and making a few observations, one can prove our lower bound (Theorem 1.1) (up to different constants).…”

“…One can apply this switching lemma of Segerlind et al (2004) to prove that small depth-3 circuits for approximate majority on n bits require bottom fan-in at least Ω( √ log n); however, we were unable to apply their results to circuits with bigger bottom fan-in, such as (log n)/2.…”

“…The proof of Lemma 2.3 is an inductive argument inspired by a recent switching lemma by Segerlind et al (2004), which we discuss later in this section. A similar argument is also a component of a technically intricate lower bound on the round complexity of protocols for two-party random selection by Sanghvi & Vadhan (2008).…”

“…Recently, Segerlind et al (2004) proved a new switching lemma that assigns values to much fewer variables than does Håstad's switching lemma. One can apply this switching lemma of Segerlind et al (2004) to prove that small depth-3 circuits for approximate majority on n bits require bottom fan-in at least Ω( √ log n); however, we were unable to apply their results to circuits with bigger bottom fan-in, such as (log n)/2.…”

Lower Bounds for k-DNF Resolution on Random 3-CNFs

“…For Res(k) systems random CNFs were considered by Atserias et al (2002), who proved exponential lower bounds for Res(2) on random 3-CNFs. It was shown by Segerlind et al (2004) that random O(k 2 )-CNFs are hard for Res(k) as long as k ≤ log n/ log log n. As one of the open questions they ask whether this bound can be improved for random 3-CNFs. In the present work we give a positive answer to this question and prove that random 3-CNFs are indeed hard for Res(k) for the same range of k. Our improvement comes from carefully constructed random cc 20 (2012) k-DNF resolution on random 3-CNFs 599 restrictions.…”