Algebraically Generalized Recursive Function Theory

by H. R. Strong

The Uniformly Reflexive Structure (URS) introduced by E. G. Wagner is, for this paper, a nonassociative algebra consisting of a domain and a binary operation satisfying the following axioms:

0.

(∃ *) (∀ a) [ a • * = * • a = * ];

1.

(∃ ψ) (∀ a,b,c,d) × [ψ ≠ * & ((a ≠ * & b ≠ * & c ≠ * & d ≠ *) →

((a = d & (((ψ • a) • b) • c) • d = b) or

(a ≠ d & (((ψ • a) • b) • c) • d = c))) ] ; and

2.

(∃ α)(∀ b,c,d) [α ≠ ψ & ((b ≠ * & c ≠ * & d ≠ *) →

((α • b) • c ≠ * & ((α • b) • c) • d = (b • d) • (c • d))) ] .

Wagner showed that these structures generalize much of Recursive Function Theory (RFT).

In this paper the functions “computed” by a URS are the functions given by left multiplications by elements of the URS. A family of functions is said to form a URS if it is the family of left multiplications of some URS. Axioms for Basic Recursive Function Theory are given characterizing those families of functions which form URS’s. The Partial Metarecursive Functions and the Computable Functionals of McCarthy are shown to form URS’s.

An investigation of notions analogous to the “recursively enumerable” notion in RFT shows that if any splinter (“successor set”) of a URS is semicomputable, then all are. A partial analogue to the Rice–Myhill–Shapiro Theorem is proved for URS’s satisfying an axiom corresponding to Kleene’s “indefinite description.” Finally, a study of pairing functions leads to work analogous to Rogers’ on Gödel numberings and generalizes similar work of Wagner.