The relatively exponential, logarithmic and circular functions in recursive function theory

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.

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From the recursive function theory newsletter

ACM SIGACT News ◽

10.1145/1008369.1008373 ◽

1978 ◽

Vol 9 (4) ◽

pp. 22-23 ◽

Cited By ~ 1

Author(s):

Howard P. Katseff

Keyword(s):

Recursive Function ◽

Function Theory

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J. R. Shoenfield. The form of the negation of a predicate. Recursive function theory, Proceedings of symposia in pure mathematics, vol. 5, American Mathematical Society, Providence1962, pp. 131–134.

Journal of Symbolic Logic ◽

10.2307/2270076 ◽

1968 ◽

Vol 33 (1) ◽

pp. 116-116

Author(s):

Ann M. Singleterry

Keyword(s):

Recursive Function ◽

Function Theory ◽

American Mathematical Society ◽

Mathematical Society ◽

Pure Mathematics

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Effective coloration

Journal of Symbolic Logic ◽

10.1017/s0022481200051549 ◽

1976 ◽

Vol 41 (2) ◽

pp. 469-480 ◽

Cited By ~ 20

Author(s):

Dwight R. Bean

Keyword(s):

Graph Theory ◽

Planar Graph ◽

Recursive Function ◽

Function Theory ◽

Yield Information ◽

Countably Infinite ◽

Degrees Of Unsolvability ◽

Infinite Planar

AbstractWe are concerned here with recursive function theory analogs of certain problems in chromatic graph theory. The motivating question for our work is: Does there exist a recursive (countably infinite) planar graph with no recursive 4-coloring? We obtain the following results: There is a 3-colorable, recursive planar graph which, for all k, has no recursive k-coloring; every decidable graph of genus p ≥ 0 has a recursive 2(x(p) − 1)-coloring, where x(p) is the least number of colors which will suffice to color any graph of genus p; for every k ≥ 3 there is a k-colorable, decidable graph with no recursive k-coloring, and if k = 3 or if k = 4 and the 4-color conjecture fails the graph is planar; there are degree preserving correspondences between k-colorings of graphs and paths through special types of trees which yield information about the degrees of unsolvability of k-colorings of graphs.

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