Forcing with Random Variables and Proof Complexity

Abstract

A fundamental problem about the strength of non-deterministic computations is the problem whether the complexity class ${\cal N}{\cal P}$ is closed under complementation. The set TAUT (w.l.o.g. a subset of {0,1}^*) of propositional tautologies (in some fixed, complete language, e.g. DeMorgan language) is co${\cal N}{\cal P}$-complete. The above problem is therefore equivalent to asking if there is a non-deterministic polynomial-time algorithm accepting exactly TAUT.

Cook and Reckhow (1979) realized that there is a suitably general definition of propositional proof systems that encompasses traditional propositional calculi but links naturally with computational complexity theory. Namely, a propositional proof system is defined to be a binary relation (on {0,1}^*) P(x,y) decidable in polynomial time such that x ∈TAUT iff ∃ y, P(x,y). Any y such that P(x,y) is called a P-proof of x.

It is easy to see (viz Cook and Reckhow (1979)) that the fundamental problem becomes a lengths-of-proofs question: Is there a propositional proof system in which every tautology admits a proof whose length is bounded above by a polynomial in the length of the tautology?