Provably Total Functions of Arithmetic with Basic Terms

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...

Provably total functions of Basic Arithmetic

It is shown that all the provably total functions of Basic Arithmetic , 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 is sound, is introduced. This realizability is similar to Kleene’s recursive realizability, except that recursive functions are restricted to primitive recurs...

Provably Total Functions in Bounded Arithmetic Theories . . .

Classifying the Provably Total Functions of PA

We give a self-contained and streamlined version of the classification of the provably computable functions of PA. The emphasis is put on illuminating as good as seems possible the intrinsic computational character of the standard cut elimination process. The article is intended to be suitable for teaching purposes and just requires basic familiarity with PA and the ordinals below ε0. (Familiar...

The Provably Total Search Problems of Bounded Arithmetic

We give combinatorial principles GIk, based on k-turn games, which are complete for the class of NP search problems provably total at the kth level T k 2 of the bounded arithmetic hierarchy and hence characterize the ∀Σ̂1 consequences of T k 2 , generalizing the results of [22]. Our argument uses a translation of first order proofs into large, uniform propositional proofs in a system in which th...