Undecidable problem

About: Undecidable problem is a research topic. Over the lifetime, 3135 publications have been published within this topic receiving 71238 citations.

CWA-solutions for data exchange settings with target dependencies

Abstract: Data exchange deals with the following problem: given an instance over a source schema, a specification of the relationship between the source and the target,and dependencies on the target, construct an instance over a target schema that satisfies the given relationships and dependencies. Recently - for data exchange settings without target dependencies - Libkin (PODS'06) introduced a new concept of solutions based on the closed world assumption (so calledCWA-solutions), and showed that, in some respects, this new notion behaves better than the standard notion of solutions considered in previous papers on data exchange. The present paper extends Libkin's notion of CWA-solutions to data exchange settings with target dependencies. We show that, when restricting attention to data exchange settings with weakly acyclic target dependencies, this new notion behaves similarly as before: the core is the unique "minimal" CWA-solution, and computing CWA-solutions as well as certain answers to positive queries is possible in polynomial time and can be PTIME-hard. However, there may be more than one "maximal" CWA-solution. And going beyond the class of positive queries, we obtain that there are conjunctive queries with (just) one inequality, for which evaluating the certain answers is coNP-hard. Finally, we consider the EXISTENCE-OF-CWA-SOLUTIONS problem: while the problem is tractable for data exchange settings with weakly acyclic target dependencies, it turns out to be undecidable for general data exchange settings. As a consequence, we obtain that also the EXISTENCE-OF-UNIVERSAL-SOLUTIONS problem is undecidable in genera.

On correct procedure parameter transmission in higher programming languages

Abstract: The paper starts with the observation that in ALGOL 60 no specifications for formal procedure parameters are prescribed, whereas ALGOL 68 demands complete specifications. As a consequence, no ALGOL 68 program accepted by the compiler can have wrong parameter transmissions at run time whereas ALGOL 60 programs may have them. The property of ALGOL 60 programs to have only correct parameter transmissions obviously is undecidable if all data, conditional statements, etc. have to be taken into consideration (Theorem 1) and it is unfair to demand that the compiler should decide that property by a finite process. Therefore, we investigate this question of decidability under a much fairer condition, namely without taking into consideration any data or conditions and by giving all procedure calls occurring in the same block "equal rights" (Section IV, p. 123). Even this fairer problem turns out to be algorithmically unsolvable, in general (Theorem 5), but it is solvable as soon as the programs do not have global formal procedure parameters (Theorem 3). Analogous answers can be given to the problems of formal equivalence of programs and of formal reachability, formal recursivity, and strong formal recursivity of procedures (Theorems 8---11). Procedures which are not strongly formally recursive have great importance in compilation techniques as is shown in Section X.

On the extension problem for partial permutations

Abstract: A family of pseudovarieties of solvable groups is constructed, each of which has decidable membership and undecidable extension problem for partial permutations. Included are a pseudovariety U satisfying no non-trivial group identity and a metabelian pseudovariety Q. For each of these pseudovarieties V, the inverse monoid pseudovariety Sl*V has undecidable membership problem. As a consequence, it is proved that the pseudovariety operators *, **, ?, =, = n , and P do not preserve decidability. In addition, several joins, including A V U, are shown to be undecidable.

Symbolic Context-Bounded Analysis of Multithreaded Java Programs

Abstract: The reachability problem is undecidable for programs with both recursive procedures and multiple threads with shared memory. Approaches to this problem have been the focus of much recent research. One of these is to use context-bounded reachability, i.e. to consider only those runs in which the active thread changes at most ktimes, where kis fixed. However, to the best of our knowledge, context-bounded reachability has not been implemented in any tool so far, primarily because its worst-case runtime is prohibitively high, i.e. O(nk), where nis the size of the shared memory. Moreover, existing algorithms for context-bounded reachability do not admit a meaningful symbolic implementation (e.g., using BDDs) to reduce the run-time in practice. In this paper, we propose an improvement that overcomes this problem. We have implemented our approach in the tool jMoped and report on experiments.

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.