A Coinductive Approach to Proof Search through Typed Lambda-Calculi

TL;DR

This paper introduces a coinductive lambda-calculus framework for proof search that captures infinite solution spaces, providing a new semantics and finitary representations for solutions in intuitionistic logic.

Contribution

It extends the Curry-Howard paradigm to coinductive solutions, proposing a finitary system with fixed points and formal sums, and proves the existence of finite representations for infinite solution spaces.

Findings

Finitary representations exist for all coinductive solution spaces.

The approach is sound and complete with respect to coinductive semantics.

Generalizes inhabitation problems in simply-typed lambda-calculus.

Abstract

In reductive proof search, proofs are naturally generalized by solutions, comprising all possibly infinite structures generated by locally correct, bottom-up application of inference rules. We propose an extension of the Curry-Howard paradigm of representation, from proofs to solutions: to represent solutions by possibly infinite terms of a dedicated lambda-calculus. This new, comprehensive approach to proof search is exemplified with the sequent calculus LJT for intuitionistic implication logic. A finitary representation is proposed, comprising lambda-terms extended with a formal greatest fixed point, and a type system that can be seen as a logic of coinductive proofs. In the finitary system, fixed-point variables enjoy a relaxed form of binding that allows the detection of cycles through the type system. Formal sums are used to express alternatives in the search process. Moreover, formal sums are used in the coinductive syntax to define "decontraction" (contraction bottom-up) - an operation whose theory we initiate in this paper. As a semantics, we assign a coinductive lambda-term to each finitary term. The main result is the existence of an equivalent finitary representation for any full solution space expressed coinductively. This result is the main ingredient in the proof that our logic of coinductive proofs is sound and complete with respect to the coinductive semantics. These results are the foundation for an original approach to proof search, where the search builds the finitary representation of the full solution space, and the a posteriori analysis typically consisting in applying a syntax-directed procedure or function. The paper illustrates this for proof search and inhabitation problems in the simply-typed lambda-calculus, reviewing results detailed elsewhere, and including new results that obtain extensive generalizations of the so-called monatomic theorem.