Abstract
This paper studies some computability notions for abstract data types, and in particular compares cosemicomputable many-sorted algebras with a notion of finality to model minimal-state realizations of abstract (software) machines. Given a finite many-sorted signature Σ and a set V of visible sorts, for every Σ-algebra A with co-r.e. behavior and nontrivial, computable V-behavior, there is a finite signature extension Σ′ of Σ (without new sorts) and a finite set E of Σ′-equations such that A is isomorphic to a reduct of the final (Σ′, E)-algebra relative to V. This uses a theorem due to Bergstra and Tucker [3]. If A is computable, then A is also isomorphic to the reduct of the initial (Σ′, E)-algebra. We also prove some results on congruences of finitely generated free algebras. We show that for every finite signature Σ, there are either countably many Σ-congruences on the free Σ-algebra or else there is a continuum of such congruences. There are several necessary and sufficient conditions which separate these two cases. We introduce the notion of the Turing degree of a minimal algebra. Using the results above prove that there is a fixed one-sorted signature such that for every r.e. degree d, there is a finite set E of Σ-equations such the initial (Σ, E)-algebra has degree d. There is a two-sorted signature Σ0 and a single visible sort such that for every r.e. degree d there is a finite set E of Σ-equations such that the initial (Σ, E, V)-algebra is computable and the final (Σ, E, V)-algebra is cosemicomputable and has degree d.
The research reported here was supported in part by a gift from the System Development Foundation.
The research reported here was supported in part by Office Of Naval Research Contracts N00414-85-C-0417 and N0014-86-C-0450.
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6 References
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© 1987 Springer-Verlag Berlin Heidelberg
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Moss, L.S., Meseguer, J., Goguen, J.A. (1987). Final algebras, cosemicomputable algebras, and degrees of unsolvability. In: Pitt, D.H., Poigné, A., Rydeheard, D.E. (eds) Category Theory and Computer Science. Lecture Notes in Computer Science, vol 283. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-18508-9_25
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DOI: https://doi.org/10.1007/3-540-18508-9_25
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