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Axiomatic enumeration and parametrization: A category-theoretic approach
Abstract
The enumeration and parameterization (s-m-n) theorems are central results in classical recursion theory, and were used by Wagner and Strong to provide an axiomatic presentation of the theory which emphasizes the algebraic links with combinatory logic. On the other hand, several category-theoretic models have been constructed to reflect diverse aspects of computability. We review that material in an introductory chapter. We then use partial cartesian categories (pCC) as a natural framework to carry on an axiomatization derived from Wagner and Strong's. The category-theoretic context yields an algebraic calculus which we apply to obtain Kleene's recursion theorems. The dichotomy between partial and total is emphasized by the pCC framework and shown to be at the heart of the combinatorial representation. Finally, generalizing a construction of Freyd and Scedrov, we give a characterization of enumeration and parameterization in a pCC with liftings and (total) equalizers; here our algebraic methodology profits from the concept of residual in a preordered groupoid.
Subject Area
Mathematics
Recommended Citation
Zalamea, Fernando, "Axiomatic enumeration and parametrization: A category-theoretic approach" (1991). Doctoral Dissertations Available from Proquest. AAI9120962.
https://scholarworks.umass.edu/dissertations/AAI9120962