Abstract
A novel representation of so-called critical-pair/completion procedures is introduced. This representation, by the means of category theory, serves to give a treatment procedures at as general a level as possible. First we give an overview of the fundamental notions in a systematic, axiomatic way, and then translate these notions into a categorical framework. The elements of the given universe (for instance terms, or polynomials) serve as objects in a category baptized CPC. Its arrows are the reductions in the universe (for instance rewrite rules, or reductions defined by polynomials from a given ideal). This category can alternatively be obtained by free generation from the underlying graph based on the reduction rules.
Substitutions (or multipliers) and embeddings into superpatterns, typical operations for all completion procedures, are viewed as functors onto the category itself.
The basic concepts involved in critical-pair/completion algorithms are modeled by well-known categorical concepts, to form a basis for the analysis of the differences and essential correspondences between the various critical-pair/completion algorithms.
Sponsored by the Austrian Ministry of Science and Research (BMWF), ESPRIT BRA 3125 ”MEDLAR”
I would like to thank Bruno Buchberger and Jochen Pfalzgraf for many helpful remarks and discussions.
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Stokkermans, K. (1993). A categorical formulation for critical-pair/completion procedures. In: Rusinowitch, M., Rémy, JL. (eds) Conditional Term Rewriting Systems. CTRS 1992. Lecture Notes in Computer Science, vol 656. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-56393-8_25
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DOI: https://doi.org/10.1007/3-540-56393-8_25
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