A category-theoretic treatment of automated theorem proving

A theorem-proving strategy is given by a set of inference rules and a search plan. Search plans have been usually described either informally (e.g., a criterion to select the next inference step) or procedurally (e.g., by giving a specific algorithm). Since both the completeness and efficiency of a theorem-proving strategy depend on the search plan, a formal, abstract treatment appears desirable. We propose an approach in this direction, that allows in particular for a precise definition of how the inference rules and the search plan cooperate to generate the derivations. Theorem-proving derivations are characterized by three essential properties: soundness, relevance and proof reduction. We show that they are functoriality properties: these results clarify which parts of the structure of a theory a theorem-proving derivation is required to preserve. We close the paper with a comparison with related work and a discussion on further extensions of our approach.