An exploration in fluid logics

Abstract

We designed and implemented a framework, programming language, and standard library for specifying and programming in categorical denotational proof languages. We encoded a number of logical systems, including several versions of the propositional calculus, and mappings between them, including a mapping based on a translation between axiomatic proofs into natural-deduction proofs, and a mapping based on the deduction theorem. We present examples illustrating how programs in this language achieve heterogeneous reasoning, and conclude with discussion of future work and applications.

Human reasoning is heterogeneous. Whether reasoning formally or informally, human reasoners frequently and effortlessly switch between many problem representations, reapplying results and techniques from one domain in another. Research in artificial intelligence, on the other hand, often produces "mechanical savants" that can solve one problem incredibly well, but are completely inapplicable to any other. In this dissertation, we set out to investigate and implement formal methods and techniques that capture the heterogeneity of formal human reasoning.

We review the ways in which combinations of logical systems have been used in mathematics and artificial intelligence, and the types of results that have been realized though mappings between logics and proof systems. We identify areas of research that are especially promising in automated reasoning and the representation of logical systems: denotational proof languages and category theory. Denotational proof languages are a family of languages that integrate deduction and computation. Category theory is an abstract branch of mathematics that provides a high level of generality and unites, among other things, many different types of logical systems.

Fusing denotational proof languages and category theory, we develop categorical denotational proof languages. These are a variant of denotational proof languages that take proofs, realized as categorical arrows, rather than propositions, as a fundamental building block. We demonstrate that category theory is a suitable formalism for representing logical systems and the mappings between them, and that categorical denotational proof languages are an effective tool for specifying the relationships between logical systems and the transformations between them, and do so in a way that promotes proof reuse.