Survey on Proof Complexity

In this survey, I will discuss two results in proof complexity. Proof complexity studies the efficiency of particular formal proof systems for proving particular formulas. The work in this area has focused on propositional logic, since results about it have a direct connection to central questions in complexity theory. In particular, we know that NP = co − NP if there is a proof system that has a small proof for every propositional tautology. If for every propositional contradiction there is a refutation of polynomial size, then we have SAT ∈ NP , using these refutations as the NP witnesses. In general, these witnesses could be arbitrary data, and they might not look anything like what we think of as “proofs” in mathematical logic. However, by studying the suitability of particular proof systems for providing these witnesses, we might learn something more fundamental about the problem. Each time we prove that some well-known proof system requires super-polynomial proofs for some contradictions, we accumulate more informal evidence that NP 6= co−NP . The first of the papers I’ll be surveying [BSW99] presents a result about the wellknown resolution proof system. The authors present a useful result that relates a property called width to the lengths of resolution proofs. They also describe a general technique for proving proof size lower bounds, based on that result, and show how to use the technique to duplicate or improve a number of past lower bound results. These include a number of exponential lower bounds for propositional tautologies, showing that resolution proofs are not acceptable NP witnesses for SAT . The second paper I will discuss [Ats04] builds up to a different kind of result. Instead of fixing a proof system for refuting propositional contradictions, this work considers the case where programs in the Datalog language are used to check a formula for unsatisfiability. They consider just how useful a fixed Datalog program can be for this task. The main result establishes that, as the size of input formulas increases, the proportion of formulas that are solutions to any fixed Datalog program that only accepts unsatisfiable formulas tends towards zero. This effectively rules out Datalog programs as well as useful tools for generating NP witnesses for SAT .