Journal of Symbolic Logic

Characterization of Recursively Enumerable Sets

Jesse B. Wright

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Let $N, O$ and $S$ denote the set of nonnegative integers, the graph of the constant 0 function and the graph of the successor function respectively. For sets $P, Q, R \subseteq N^2$ operations of transposition, composition, and bracketing are defined as follows: $P^\cup = \{\langle x, y\rangle | \langle y, x\rangle \epsilon P\}, PQ = \{\langle x, z\rangle | \exists y\langle x, y\rangle \epsilon P & \langle y, z\rangle \epsilon Q\}$, and $\lbrack P, Q, R\rbrack = \cup_{n \epsilon M}(P^nQR^n)$. THEOREM. The class of recursively enumerable subsets of $N^2$ is the smallest class of sets with $O$ and $S$ as members and closed under transposition, composition, and bracketing. This result is derived from a characterization by Julia Robinson of the class of general recursive functions of one variable in terms of function composition and "definition by general recursion." A key step in the proof is to show that if a function $F$ is defined by general recursion from functions $A, M, P$ and $R$ then $F = \lbrack P^\cup, A^\cup M, R\rbrack$. The above definitions of the transposition, composition, and bracketing operations on subsets of $N^2$ can be generalized to subsets of $X^2$ for an arbitrary set $X$. In this abstract setting it is possible to show that the bracket operation can be defined in terms of $K, L$, transposition, composition, intersection, and reflexive transitive closure where $K: X \rightarrow X$ and $L: X \rightarrow X$ are functions for decoding pairs.

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J. Symbolic Logic, Volume 37, Issue 3 (1972), 507-511.

First available in Project Euclid: 6 July 2007

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Wright, Jesse B. Characterization of Recursively Enumerable Sets. J. Symbolic Logic 37 (1972), no. 3, 507--511.

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