Nathan Segerlind

The Complexity of Propositional Proofs

Propositional proof complexity is the study of the sizes of propositional proofs, and more generally, the resources necessary to certify propositional tautologies. Questions about proof sizes have connections with computational complexity, theories of arithmetic, and satisfiability algorithms. This is article includes a broad survey of the field, and a technical exposition of some recently developed techniques for proving lower bounds on proof sizes.

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

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: The 2n to n weak pigeonhole principle requires exponential size to refute in Res(k) for

Lower Bounds for Lovász-Schrijver Systems and Beyond Follow from Multiparty Communication Complexity

k \leq \sqrt{\log n / \log \log n }

A Strong Direct Product Theorem for Corruption and the Multiparty Communication Complexity of Disjointness

. For each constant k, there exists a constant w > k so that random w-CNFs require exponential size to refute in Res(k). 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.

Memoization and DPLL: formula caching proof systems

We prove that an

Lower bounds for lovász-schrijver systems and beyond follow from multiparty communication complexity

\omega(\log^4 n)

Exponential Lower Bounds and Integrality Gaps for Tree-Like Lovász-Schrijver Procedures

lower bound for the three-party number-on-the-forehead (NOF) communication complexity of the set-disjointness function implies an

Switching lemma for small restrictions and lower bounds for k-DNF resolution

n^{\omega(1)}

Formula Caching in DPLL

size lower bound for treelike Lovasz-Schrijver systems that refute unsatisfiable formulas in conjunctive normal form (CNFs). More generally, we prove that an

Counting axioms do not polynomially simulate counting gates

n^{\Omega(1)}