7 - Witnessing theorems

Summary

This chapter considers various witnessing theorems, which are theorems characterizing functions definable in various systems of arithmetic in terms of their computational complexity. A prototype of such a theorem (and its proof) is the characterization of primitive recursive functions as provably total recursive functions in fragment of PA (cf. Parsons 1970, Takeuti 1975, and Mints 1976).

There are other approaches to proving witnessing theorems, for example, skolemizing the given theory by Skolem functions from a particular class and then applying Herbrand's theorem. Or there are intrigued model-theoretic constructions. I shall mention these methods too, but my opinion is that one really has to know in advance which class of functions one targets before formulating an argument while the methods based on cut-elimination (Section 7.1) and generalizing Theorem 7.2.3 help to discover the right class. This certainly was the case for all witnessing theorems discussed in this chapter.

Cut-elimination for bounded arithmetic

We first extend the sequent predicate calculus by rules allowing the introduction of bounded quantifiers and by the induction rules and then we prove the cutelimination for such a system.

The predicate calculus LK extends the propositional LK from Section 4.3 by four rules for introducing quantifiers to a sequent as in Definition 4.6.2: