Enhanced Realizability Interpretation for Program Extraction

This thesis presents Intuitionistic Fixed Point Logic (IFP), a schema for formal systems aimed to work with program extraction from proofs. IFP in its basic form allows proof construction based on natural deduction inference rules, extended by induction and coinduction. The corresponding system RIFP (IFP with realiz-ers) enables transforming logical proofs into programs utilizing the enhanced re-alizability interpretation. The theoretical research is put into practice in PRAWF1, a Haskell-based proof assistant for program extraction.

Program extraction applied to monadic parsing

This article outlines a proof-theoretic approach to developing correct and terminating monadic parsers. Using modified realizability, we extract formally verified and terminating programs from formal proofs. By extracting both primitive parsers and parser combinators, it is ensured that all complex parsers built from these are also correct, complete and terminating for any input. We demonstrate the viability of our approach by means of two case studies: we extract (i) a small arithmetic calculator and (ii) a non-deterministic natural language parser. The work is being carried out in the interactive proof system Minlog.

Reduction Strategies for Program Extraction

We introduce Pure Type Systems with Pairs generalising earlier work on program extraction in Typed Lambda Calculus. We model the process of program extraction in these systems by means of a reduction relation called o-reduction, and give strategies for Bo-reduction which will be useful for an implementation of a proof assistant. More precisely, we give an algorithm to compute theo-normal form of a term in Pure Type System with Pairs, and show that this defines a prejection from Pure Type Systems with Pairs to standart Pure Type Systems. This result shows that o-reduction is an operational description of aprgram extraction that is independent of the particular Typed Lambda Calculus specified as a Pure Typoe System. For B-reduction, we define weak and strong reduction strategies using Interaction Nets, generalising well-know efficient strategies for the l-calculus to the general setting of Pure Type Systems.

Program extraction in exact real arithmetic

The importance of an abstract approach to a computation theory over general data types has been stressed by Tucker in many of his papers. Berger and Seisenberger recently elaborated the idea for extraction out of proofs involving (only) abstract reals. They considered a proof involving coinduction of the proposition that any two reals in [−1, 1] have their average in the same interval, and informally extract a Haskell program from this proof, which works with stream representations of reals. Here we formalize the proof, and machine extract its computational content using the Minlog proof assistant. This required an extension of this system to also take coinduction into account.