Indexing

Author(s):  
Raymond M. Smullyan

For the remaining chapters, we will need two basic theorems in recursive function theory—the enumeration theorem of Kleene and Post and the iteration theorem of Kleene. §1. Indexing. we wish to arrange all r.e. sets in an infinite sequence ω0, ω1, . . . ,ωn , . . . (allowing repetitions) in such a way that the relation xÎ ωy is r.e. we shall take the system (Q) as our basic formalism for recursive function theory. we know that (Q) is axiomatizable and that the representable sets of (Q) are precisely the r.e. sets. we define ωi - as the set of all numbers n such that Ei[ n̅ ] is provable in (Q). Equivalently, wi- is the set of all n such that r(i,n) Î P, where r(i,n) is the Gödel number of Ei[ n̅ ] and P is the set of Gödel numbers of the provable formulas of (Q). Since r(x,y) is a recursive function and P is an r.e. set, then the relation r(x,y) Î P is r.e., and this is the relation y Î ωx. Also, every r.e. set A is represented in (Q) by some formula Ei(v1); hence A = ωi. Thus every r.e. set appears in our enumeration. we call i an index of an r.e. set A if A = ωi. we let U(x,y) be the relation x Î ωy , and we see that this relation is r.e. Indexing of r.e. Relations. For each n ³ 2, we also wish to arrange all r.e. relations of degree n an in infinite sequence . . . Ron,R1n , . . . , Rnn , . . . in such a manner that the relation Ryn(x1, . . . ,xn) is an r.e. relation among x1, . . . , xn and y. To this end, it will be convenient to use the indexing of r.e. sets that we already have and to use the recursive pairing function J(x,y) and its associated functions Jn(x1,. . . ,xn) (cf. §4, Chapter 1).

J. C. Shepherdson. Algorithmic procedures, generalized Turing algorithms, and elementary recursion theory. Harvey Friedman's research on the foundations of mathematics, edited by L. A. Harrington, M. D. Morley, A. S̆c̆edrov, and S. G. Simpson, Studies in logic and the foundations of mathematics, vol. 117, North-Holland, Amsterdam, New York, and Oxford, 1985, pp. 285–308. - J. C. Shepherdson. Computational complexity of real functions. Harvey Friedman's research on the foundations of mathematics, edited by L. A. Harrington, M. D. Morley, A. S̆c̆edrov, and S. G. Simpson, Studies in logic and the foundations of mathematics, vol. 117, North-Holland, Amsterdam, New York, and Oxford, 1985, pp. 309–315. - A. J. Kfoury. The pebble game and logics of programs. Harvey Friedman's research on the foundations of mathematics, edited by L. A. Harrington, M. D. Morley, A. S̆c̆edrov, and S. G. Simpson, Studies in logic and the foundations of mathematics, vol. 117, North-Holland, Amsterdam, New York, and Oxford, 1985, pp. 317–329. - R. Statman. Equality between functionals revisited. Harvey Friedman's research on the foundations of mathematics, edited by L. A. Harrington, M. D. Morley, A. S̆c̆edrov, and S. G. Simpson, Studies in logic and the foundations of mathematics, vol. 117, North-Holland, Amsterdam, New York, and Oxford, 1985, pp. 331–338. - Robert E. Byerly. Mathematical aspects of recursive function theory. Harvey Friedman's research on the foundations of mathematics, edited by L. A. Harrington, M. D. Morley, A. S̆c̆edrov, and S. G. Simpson, Studies in logic and the foundations of mathematics, vol. 117, North-Holland, Amsterdam, New York, and Oxford, 1985, pp. 339–352.

1990 ◽  
Vol 55 (2) ◽  
pp. 876-878
Author(s):  
J. V. Tucker

1979 ◽  
Vol 44 (2) ◽  
pp. 221-234 ◽  
Author(s):  
Luis E. Sanchis

This paper proposes a generalization of several reducibilities in the sense of recursive function theory. For this purpose two structures are introduced as models of combinatory logic and reducibilities are defined in a rather natural way by means of the application operation in each model. The first model we consider is called the graph model and was introduced by Dana Scott in [11]. Reducibilities in this model are generalizations of enumeration and Turing reducibilities. The second model is called the hypergraph model and induces reducibilities which are generalizations of hyperenumeration and hyperarithmetical reducibilities.The possibility of formulating reducibilities in this way is not new. For the graph model it is implicit in [7] and for the hypergraph model in [9]. On the other hand it is possible to look at the present method as a variant of the well-known technique of relativization in recursive function theory. We think that this does not exhaust the power of the method, which is conceptually elegant and provides a natural frame for the results of this paper.In the first part of the paper we discuss the definition and general properties of the models. Then we introduce the reducibilities in the graph model and prove several theorems which are generalizations of properties already Known for enmeration and Turing reducibilities. Next we define reducibilities in the hypergraph model and try to extend the preceding results. For this purpose we prove two theorems showing significant relations between the operators in both models. In fact we prove that each operator in the hypergraph model can be simulated on a comeager set by an operator of the graph model.


1978 ◽  
Vol 9 (4) ◽  
pp. 22-23 ◽  
Author(s):  
Howard P. Katseff

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