Category Theory and Computer Science pp 158-181 | Cite as

# Final algebras, cosemicomputable algebras, and degrees of unsolvability

## 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 Σ_{0} 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.

## Keywords

Function Symbol Recursive Function Critical Pair Constant Symbol Abstract Data Type## Preview

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## 6 References

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