Normalisation techniques in proof theory and category theory

Abstract

The word problem for the free categories with some structure generated by a category X can be solved using proof-theoretical means. These free categories give a semantics in which derivations of GENTZEN's propositional sequent calculus can be interpreted by means of arrows of those categories. In this thesis we describe, implement and document the cut-elimination and the normalization techniques in proof theory as outlined in SZABO [1978]: we show how these are used in order to solve, mechanically, the word problem for the free categories with structure of : cartesian, bicartesian, distributive bicartesian, cartesian closed, and bicartesian closed. This implementation is extended by a procedure to interpret intuitionistic propositional sequent derivations as arrows of the above categories. Implementation of those techniques has forced us to modify the techniques in various inessential ways. The description and the representation in the syntax of our implementation of the above categories is contained in chapters 1 - 5, where each chapter describes one theory and concludes with examples of the system In use to represent concepts and solve simple word problems from category theory ( of various typos ). Appendix 1 contains some apparent printing errors we have observed in the work done by SZABO. The algorithms used in the proof of the cut-elimination theorems and normalization through chapters 1 - 5 are collected in appendices 2 - 4. Appendices 5 - 8 concern the implementation and its user manual.