Proofs as programs

Abstract

The significant intellectual cost of programming is for problem solving and explaining, not for coding. Yet programming systems offer mechanical assistance for the coding process exclusively. We illustrate the use of an implemented program development system, called PRL ("pearl"), that provides automated assistance with the difficult part. The problem and its explained solution are seen as formal objects in a constructive logic of the data domains. These formal explanations can be executed at various stages of completion. The most incomplete explanations resemble applicative programs, the most complete are formal proofs.

A Multi-level Approach to Program Synthesis

“…However, whenever a piece of code shall be extracted from the proof, it is necessary to convert the proofs generated by a search procedure back into a constructive sequent proof which, according to the proofs-as-program paradigm [2], can be turned into a program.…”

“…Thus the final program filtered body(...) can be constructed as soon as the two auxiliary algorithms O and θ are given. 2 The specification for θ as algorithm contains all conditions in which θ finally occurs, i.e. the conditions for specialization and necessity.…”

“…Since it is based on the proofs-as-programs paradigm to program synthesis [2] it allows to treat algorithm design strategies as proof procedures and to integrate a great variety of reasoning techniques on all levels of abstraction. Finally it supports interaction with a human expert (programmer) whenever the automated strategies turn out to be too weak.…”

Towards an Operational Semantics for Alloy

“…Bates and Constable [12] initiated a significant research program on the extraction of computational context, in the form of programs, from constructive proofs. This effort continues in popular proof assistants such as Coq [13].…”

Presenting intuitive deductions via symmetric simplification

“…In systems designed to support the teaching of logic a flexible facility for giving advice should know how to construct good proofs in the deductive system underlying the course. In systems where proofs are the data to be manipulated to fit other tasks such as explanation-based generalization [10] or the extraction of programs [5], the structure of proofs becomes of primary importance which, unfortunately, in practice prohibits the use of state-of-the-art theorem proving technology in such applications. Further evidence that proof presentation is a hard and important problem is that mathematicians spend a large percentage of their time analyzing and reformulating proofs.…”