Optimized Program Extraction for Induction and Coinduction

Abstract

We prove soundness of an optimized realizability interpretation for a logic supporting strictly positive induction and coinduction. The optimization concerns the special treatment of Harrop formulas which yields simpler extracted programs. We show that wellfounded induction is an instance of strictly positive induction and derive from this a new computationally meaningful formulation of the Archimedean property for real numbers. We give an example of program extraction in computable analysis and show that Archimedean induction can be used to eliminate countable choice.

Intuitionistic Fixed Point Logic

“…This paper introduces the formal system IFP of Intuitionistic Fixed Point Logic as a basis for program extraction from proofs that does not suffer from this limitation. Preliminary versions of the system were presented in [5,6,7,10,9].…”

Enhanced Realizability Interpretation for Program Extraction

“…The second part presented IFP, IFP' and RIFP, as well as the proof of soundness. The Soundness Theorem originally proven in [21] showed a direct link between IFP and RIFP. However, this version of the Soundness Theorem was not included here as it is a restricted version, which requires predicates to be admissible.…”

“…They also contain some additional details related to this research. [21] (by U. Berger and O. Petrovska) This paper presents the first variant of the Intuitionistic Fixed Point logic (IFP), a formal system that serves as a base for program extraction. The paper also includes the proof of the Soundness Theorem, which is subject to a special admissibility condition.…”

“…In the next section we look at RIFP, an extension of IFP, which also includes realizers. The link between IFP and RIFP was proven via the soundness theorem in [21] albeit with the admissibility condition mentioned in the introduction on page 4. To lift the restriction caused by that condition, an intermediate system of IFP' was introduced in [24].…”

“…Initially in [21] we distinguished between programs denoting elements of D and function terms denoting continuous functions on D:…”