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A Category-Theoretic Treatment of

Automated Theorem Proving

**Maria Paola Bonacina and Jieh Hsiang ^{#}**

University of Iowa

Iowa City, IA 52242-1419, USA

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.