Effective coloration

1976 ◽  
Vol 41 (2) ◽  
pp. 469-480 ◽  
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.

The nonconstructive content of sentences of arithmetic

1978 ◽  
Vol 43 (3) ◽  
pp. 497-501
Author(s):  
Nicolas D. Goodman
Keyword(s):  
Natural Number ◽  
Recursive Function ◽  
Function Theory ◽  
First Order ◽  
The Standard Model ◽  
Image Position ◽  
The Relationship ◽  
Excluded Middle ◽  
Degrees Of Unsolvability ◽  
Intuitionistic Arithmetic

This note is concerned with the old topic, initiated by Kleene, of the connections between recursive function theory and provability in intuitionistic arithmetic. More specifically, we are interested in the relationship between the hierarchy of degrees of unsolvability and the interdeducibility of cases of excluded middle. The work described below was motivated by a counterexample, to be given presently, which shows that that relationship is more complicated than one might suppose.Let HA be first-order intuitionistic arithmetic. Let the symbol ⊢ mean derivability in HA. For each natural number n, let n¯ be the corresponding numeral. Let Ω be the standard model of arithmetic. Say that a sentence ϕ is true iff Ω⊨ ϕ. Now suppose ϕ(x) and Ψ(x) are formulas with only the variable x free. SupposeThen it is natural to conjecture that {n∣Ω⊨Ψ(n¯)} is recursive in {n∣Ω⊨ϕ(n¯)}.However, this conjecture is false. Consider the formula is a formalization of Kleene's T-predicate.

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
Keyword(s):  
New York ◽  
Computational Complexity ◽  
Recursive Function ◽  
Function Theory ◽  
Recursion Theory ◽  
Foundations Of Mathematics ◽  
Real Functions ◽  
Pebble Game ◽  
Logics Of Programs

How to Program an Infinite Abacus

1961 ◽  
Vol 4 (3) ◽  
pp. 295-302 ◽  
Author(s):  
Joachim Lambek
Keyword(s):  
Finite Number ◽  
Recursive Function ◽  
Computable Function ◽  
Unlimited Supply ◽  
Countably Infinite ◽  
Infinite Set

This is an expository note to show how an “infinite abacus” (to be defined presently) can be programmed to compute any computable (recursive) function. Our method is probably not new, at any rate, it was suggested by the ingenious technique of Melzak [2] and may be regarded as a modification of the latter.By an infinite abacus we shall understand a countably infinite set of locations (holes, wires etc.) together with an unlimited supply of counters (pebbles, beads etc.). The locations are distinguishable, the counters are not. The confirmed finitist need not worry about these two infinitudes: To compute any given computable function only a finite number of locations will be used, and this number does not depend on the argument (or arguments) of the function.

Reducibilities in two models for combinatory logic

1979 ◽  
Vol 44 (2) ◽  
pp. 221-234 ◽  
Author(s):  
Luis E. Sanchis
Keyword(s):  
Recursive Function ◽  
Present Method ◽  
Function Theory ◽  
Graph Model ◽  
The Other ◽  
Combinatory Logic ◽  
General Properties ◽  
Hypergraph Model ◽  
Natural Frame ◽  
Natural Way

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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