Provably total functions of Basic Arithmetic

Provably Total Functions of Arithmetic with Basic Terms

This paper presents a new characterization of provably recursive functions of first-order arithmetic. We consider functions defined by sets of equations. The equations can be arbitrary, not necessarily defining primitive recursive, or even total, functions. The main result states that a function is provably recursive iff its totality is provable (using natural deduction) from the defining set o...

Polynomially Bounded Recursive Realizability

A polynomially bounded recursive realizability, in which the recursive functions used in Kleene’s realizability are restricted to polynomially bounded functions, is introduced. It is used to show that provably total functions of Ruitenburg’s Basic Arithmetic are polynomially bounded (primitive) recursive functions. This sharpens our earlier result where those functions were proved to be primiti...

Provably Total Functions in Bounded Arithmetic Theories . . .

A Simple Proof of Parsons' Theorem

Let IΣ1 be the fragment of elementary Peano Arithmetic in which induction is restricted to Σ1-formulas. More than three decades ago, Charles Parsons showed that the provably total functions of IΣ1 are exactly the primitive recursive functions. In this paper, we observe that Parsons’ result is a consequence of Herbrand’s theorem concerning the ∃∀∃-consequences of universal theories. We give a se...

Bounded arithmetic theory for the counting functions and Toda’s theorem

In this paper we give a two sort bounded arithmetic whose provably total functions coincide with the class FP . Our first aim is to show that the theory proves Toda’s theorem in the sense that any formula in Σ∞ is provably equivalent to a Σ B 0 formula in the language of FP . We also argue about some problems concerning logical theories for counting classes.