Feasibly constructive proofs and the propositional calculus (Preliminary Version)

Abstract

The motivation for this work comes from two general sources. The first source is the basic open question in complexity theory of whether P equals NP (see [1] and [2]). Our approach is to try to show they are not equal, by trying to show that the set of tautologies is not in NP (of course its complement is in NP). This is equivalent to showing that no proof system (in the general sense defined in [3]) for the tautologies is “super” in the sense that there is a short proof for every tautology. Extended resolution is an example of a powerful proof system for tautologies that can simulate most standard proof systems (see [3]). The Main Theorem (5.5) in this paper describes the power of extended resolution in a way that may provide a handle for showing it is not super. The second motivation comes from constructive mathematics. A constructive proof of, say, a statement @@@@×A must provide an effective means of finding a proof of A for each value of x, but nothing is said about how long this proof is as a function of x. If the function is exponential or super exponential, then for short values of x the length of the proof of the instance of A may exceed the number of electrons in the universe. In section 2, I introduce the system PV for number theory, and it is this system which I suggest properly formalizes the notion of a feasibly constructive proof.