A Generic Tool for Higher-Order Term Graph Rewriting

This thesis contains a study of Curry-Howard correspondences for Sequent Calculus formulations of Classical Logic. Starting from Gentzen’s formal definition of logical consequence (presented in the framework of his sequent calculus), we present an automated process to conservatively extend the framework with primitive logical connectives and corresponding cut-elimination rules. The key difficulty lies in deriving the main cut-elimination rules for the connective. In contrast to existing works, which employ brute-force techniques or methods using equivalences, we specify an algorithm which systematically constructs the main cut-elimination rule by operating on rows of truth-tables. (We also give a geometrical interpretation of the cut rule.) The aim is to study the computational content of the resulting formulation of classical logic. We mechanically extract from the framework, a computational term calculus inspired by the X -calculus of van Bakel, Lengrand and Lescanne. We motivate our design choices by making comparisons with existing computational calculi that hold Curry-Howard correspondences with Classical Logic. Using our process, we then build and study a number of computational calculi, focusing on their simulation properties. We find that notions of logical expressibility (the ability of a connective to logically express another) and computational expressibility (the ability of a term calculus to simulate another) do not coincide. Our (graphical and interactive) tool is a full implementation of the process we design in this thesis, but additionally serves as tool for higher-order conditional term graph rewriting in general; it also features Visser’s language of strategy combinators, allowing one to easily study complex reduction behaviour. We detail some specific implementation problems we encountered, and motivate the solutions we adopted.