We 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(χ(p) - 1)-coloring, where χ(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. (shrink)

We consider equational theories for functions defined via recursion involving equations between closed terms with natural rules based on recursive definitions of the function symbols. We show that consistency of such equational theories can be proved in the weak fragment of arithmetic S 1 2 . In particular this solves an open problem formulated by TAKEUTI (c.f. [5, p.5 problem 9.]).

We study the fragment of Peano arithmetic formalizing the induction principle for the class of decidable predicates, $I\Delta_1$ . We show that $I\Delta_1$ is independent from the set of all true arithmetical $\Pi_2-sentences$ . Moreover, we establish the connections between this theory and some classes of oracle computable functions with restrictions on the allowed number of queries. We also obtain some conservation and independence results for parameter free and inference rule forms of $\Delta_1-induction$ . An open problem formulated by J. (...) Paris is whether $I\Delta_1$ proves the corresponding least element principle for decidable predicates, $L\Delta_1$ (or, equivalently. the $\Sigma_1-collection$ principle $B\Sigma_1$ ). We reduce this question to a purely computation-theoretic one. (shrink)

Abstract State Machines (ASMs) provide a formal method for transparent design and specification of complex dynamic systems. They combine advantages of informal and formal methods. Applications of this method motivate a number of computability and decidability problems connected to ASMs. Such problems result for example from the area of verifying properties of ASMs. Their high expressive power leads rather directly to undecidability respectively uncomputability results for most interesting problems in the case of unrestricted ASMs. Consequently, it is rather natural to (...) ask whether there exist expressive classes of ASMs for which we can prove positive decidability and computability results. In this work, we introduce such a class of ASMs. The concept is similar to the one of the guarded fragment of first-order logic. We analyze the expressive power of this class and prove that it is stronger than Datalog LITE and the guarded fragment of first-order fixed point logic. Some decidability and computability results have been proven in earlier works. (shrink)