We prove a new switching lemma that works for restrictions that set only a small fraction of the variables and is applicable to 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 kDNFs instead of clauses. We also obtain an exponential separation between depth d circuits of bottom fan-in k and depth d ci...

It is generally conjectured that there is an exponential separation between Frege and extended Frege systems. This paper reviews and introduces some candidates for families of combinatorial tautologies for which Frege proofs might need to be superpolynomially longer than extended Frege proofs. Surprisingly, we conclude that no particularly good or convincing examples are known. The examples of ...

We study the Chvátal rank of polytopes as a complexity measure of unsatisfiable sets of clauses. Our first result establishes a connection between the Chvátal rank and the minimum refutation length in the cutting planes proof system. The result implies that length lower bounds for cutting planes, or even for tree-like cutting planes, imply rank lower bounds. We also show that the converse impli...

We present a simple proof of the bounded-depth Frege lower bounds of Pitassi et. al. and Krajı́ček et. al. for the pigeonhole principle. Our method uses the interpretation of proofs as two player games given by Pudlák and Buss. Our lower bound is conceptually simpler than previous ones, and relies on tools and intuition that are well-known in the context of computational complexity. This makes t...

For an arbitrary hypergraph H, let PM(H) be the propositional formula asserting that H contains a perfect matching. We show that every resolution refutation of PM(H) must have size exp ( Ω ( δ(H) λ(H)r(H)(log n(H))(r(H) + log n(H)) )) , where n(H) is the number of vertices, δ(H) is the minimal degree of a vertex, r(H) is the maximal size of an edge, and λ(H) is the maximal number of edges incid...

Ajtai [Ajt] recently proved that if for some fixed d, every formula in a Frege proof of the propositional pigeonhole principle PHP, has depth at most d, then the proof size is not less than any polynomial in n. By introducing the notion of an “approximate proof” we demonstrate how to eliminate the non-standard model theory, including the non-constructive use of the compactness theorem, from Ajt...

In 1979, Richard Stanley made the following conjecture: Every Cohen–Macaulay simplicial complex is partitionable. Motivated by questions in the theory of face numbers of simplicial complexes, the Partitionability Conjecture sought to connect a purely combinatorial condition (partitionability) with an algebraic condition (Cohen–Macaulayness). The algebraic combinatorics community widely believed...

We prove a quasi-polynomial lower bound on the size of bounded-depth Frege proofs of the pigeonhole principle PHPm n where m 1 1 polylog n n. This lower bound qualitatively matches the known quasi-polynomial-size bounded-depth Frege proofs for these principles. Our technique, which uses a switching lemma argument like other lower bounds for boundeddepth Frege proofs, is novel in that the tautol...

We present a simple proof of the bounded-depth Frege lower bounds of Pitassi et. al. and Kraj́ıček et. al. for the pigeonhole principle. Our method uses the interpretation of proofs as two player games given by Pudlák and Buss. Our lower bound is conceptually simpler than previous ones, and relies on tools and intuition that are well-known in the context of computational complexity. This makes t...

A general framework for parameterized proof complexity was introduced by Dantchev, Martin, and Szeider [2007]. There the authors show important results on tree-like Parameterized Resolution—a parameterized version of classical Resolution—and their gap complexity theorem implies lower bounds for that system. The main result of the present paper significantly improves upon this by showing optimal...