Provably total functions of Basic Arithmetic

Abstract

It is shown that all the provably total functions of Basic Arithmetic BA, a theory introduced by Ruitenburg based on Predicate Basic Calculus, are primitive recursive. Along the proof a new kind of primitive recursive realizability to which BA is sound, is introduced. This realizability is similar to Kleene's recursive realizability, except that recursive functions are restricted to primitive recursives.

Polynomially Bounded Recursive Realizability

“…And finally our main theorem is a characterization of provably total functions of BA (cf. Corollary 4.5 of [13]).…”

“…Let ϕ x be the (unique) unary recursive function whose program has the (Gödel) code x (cf. Soare [14] and [13]). 1 Take , to be a fixed pairing function (such as x, y = 1 2 (x + y)(x + y + 1) + y) with the projections π 1 and π 2 , that is, π 1 ( x, y ) = x and π 2 ( x, y ) = y.…”

“…It can be shown that if n, k (and m, k ) P-realize(s) the hypothesis (hypotheses) of the above rules, then the function f (and g) P-realize(s) the conclusion of the rule (cf. [13]).…”

“…A new realizability by primitive recursive functions was introduced by Salehi [13]. It was applied to Basic Arithmetic BA, a theory built on Basic Logic, a subintuitionistic logic in which the modes ponens rule is not valid in its general form (see Ruitenburg [12] where BA was introduced).…”

“…In this paper, a result of [13], that provably total functions of BA are primitive recursive, is sharpened by applying a realizability by polynomially bounded recursive functions, abbreviated as P-realizability. Informally speaking, a formula is Prealizable if and only if it is efficiently verifiable, that is to say, its truth can be justified by polynomially computable functions.…”