Maria Paola Bonacina and Jieh Hsiang#
Department of Computer Science
University of Iowa
Iowa City, IA 52242-1419, USA
#Department of Computer Science
National Taiwan University
Taipei, Taiwan 107
In this paper we apply category theory to investigate the mathematical structure of theorem proving derivations.
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.
Keywords: theorem proving, search, category theory, simplification, contraction
Received September 29, 1994; revised July 27, 1995.
Communicated by Wei-Pang Yang.
*Supported in part by the National Science Founndation with grant CCR-94-08667 and by the GE Foundation Faculty Fellowship to the University of Iowa.
Supported in part by grant NSC 84-2213-E-002-011 of the National Science Council of the Republic of China.