Showing papers on "Undecidable problem published in 1978"

On the decision problem and the mechanization of theorem-proving in elementary geometry

Abstract: The idea of proving theorems mechanically may be dated back to Leibniz in the 17th century and has been formulated in precise mathematical forms in this century through the school of Hilbert as well as his followers on mathematical logic. The problem consists in essence in replacing qualitative difficulties inherited in usual mathematical proofs by quantitative complexities of calculations on standardizing the proof procedures in an algorithmic manner. Such quantitative complexities of calculations, formerly far beyond the reach of human abilities, have become more and more trivial owing to the occurrence and rapid development of computers. In spite of vigorous efforts, however, researches in this direction give rise quite often to negative results in the form of undecidable mathematical theories. To cite a notable positive result, we may mention Tarski's method of proving theorems mechanically in elementary geometry and elementary algebra. The methods of Tarski as well as later ones are largely based on a generalization of Sturm theorem and are still too complicated to be feasible, even with the use of computers. The present paper, restricted to theorems with betweenness out of consideration and based on an entirely different principle, aims at giving a mechanical procedure which permits to prove quite non-trivial theorems in elementary geometry even by hands.

The Complexity of Finite Memory Programs with Recursion

Abstract: In an earlier paper (JACM, 1976) we studied the computational complexity of a number of questions of both programming and theoretical interest (eg halting, looping, equivalence) concerning the behaviour of programs written in an extremely simple programming language These finite memory programs or fmps model the behaviour of FORTRAN-like programs with a finite memory whose size can be determined by examination of the program itself The present paper is a continuation in which we extend the analysis to include ALGOL-like programs (called fmp^(rec) s) with the finite memory augmented by an implicit pushdown stack used to support recursion Our major results are the following First, we show that at least deterministic exponential time is required to determine whether a program in the basic fmpr~C model accepts a nonempty set Then we show that a model with a limited version of call-by-name requires exponential space to determine acceptance of a nonempty set, and that a more sophisticated model with rewritable conditional formal parametershas an undecidable halting problem The same lower bounds apply to the equivalence problem, which in contrast to the situation for the basic fmp model is not known to be decidable (since it is not known whether equivalence of deterministic pushdown automata is decidable)

Three universal representations of recursively enumerable sets

Abstract: In his celebrated paper of 1931 [7], Kurt Godel proved the existence of sentences undecidable in the axiomatized theory of numbers Godel's proof is constructive and such a sentence may actually be written out Of course, if we follow Godel's original procedure the formula will be of enormous lengthForty-five years have passed since the appearance of Godel's pioneering work During this time enormous progress has been made in mathematical logic and recursive function theory Many different mathematical problems have been proved recursively unsolvable Theoretically each such result is capable of producing an explicit undecidable number theoretic predicate We have only to carry out a suitable arithmetization Until recently, however, techniques were not available for carrying out these arithmetizations with sufficient efficiencyIn this article we construct an explicit undecidable arithmetical formula, F(x, n), in prenex normal form The formula is explicit in the sense that it is written out in its entirety with no abbreviations The formula is undecidable in the recursive sense that there exists no algorithm to decide, for given values of n, whether or not F(n, n) is true or false Moreover F(n, n) is undecidable in the formal (axiomatic) sense of Godel [7] Given any of the usual axiomatic theories to which Godel's Incompleteness Theorem applies, there exists a value of n such that F(n, n) is unprovable and irrefutable Thus Godel's Incompleteness Theorem can be “focused” into the formula F(n, n) Thus some substitution instance of F(n, n) is undecidable in Peano arithmetic, ZF set theory, etc

Deciding associativity for partial multiplication tables of order 3

Abstract: The associativity problem for the class of finite multiplication tables is known to be undecidable, even for quite narrow infinite subclasses of tables. We present cri- teria which can be used to decide associativity in many cases, although any effective method based on such criteria must eventually fail on a table of some size (as other- wise decidability for the general class would follow). By means of an extensive com- puter search we have been able to use the criteria successfully to solve the associativity problem for all tables of order up to 3. We find 24,733 associative tables and 237,411 nonassociative tables, and present some further statistics about how "deep" we had to search to establish the nonassociativity of a table. We also prove that there are tables of order 3 for which no "one-mountain" theorem holds (which was known previously only for order 6 examples). Our methods make use of efficient data-representations and techniques of heuristic and adaptive programming.

The undecidability of the third order dyadic unification problem

Abstract: The unification problem for expressions of third order dyadic logic is proved to be undecidable by a reduction of the problem of solving Diophantine equations. This refines an earlier result on the undecidability of the third order unification problem.

Some Results on Measure Independent Godel Speed-Ups

Abstract: We study the measure independent character of Godel speed-up theorems, in particular, we strengthen Arbib's necessary condition for the occurrence of a Godel speed-up [2, p. 13] to an equivalence result and generalize Di Paola's speed-up theorem [4]. We also characterize undecidable theories as precisely those theories which possess consistent measure independent Godel speed-ups and show that a theory τ 2 is a measure independent Godel speed-up of a theory τ 1 if and only if the set of undecidable sentences of τ 1 which are provable in τ 2 is not recursively enumerable.

Decision problems concerning parallel programming

Abstract: A notion of a correct (= deadlock free) scheduling of several recursive processes using common resources is introduced. The existence of correct schedulings is schown to be undecidable from recursive indices of the relevant processes. Further-more we isolate several cases where the more existence of a correct scheduling does not imply the existence of a computable (recursive) correct scheduling.

Decision problems concerning parallel programming

Abstract: A notion of a correct (= deadlock free) scheduling of several recursive processes using common resources is introduced. The existence of correct schedulings is schown to be undecidable from recursive indices of the relevant processes. Further-more we isolate several cases where the more existence of a correct scheduling does not imply the existence of a computable (recursive) correct scheduling.