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Undecidable problem

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


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Book ChapterDOI
07 Oct 2018
TL;DR: A new algorithm, Quic3, is presented, that extends IC3 to infer universally quantified invariants over the combined theory of LIA and Arrays, and carefully manages quantified generalization and quantifier instantiation.
Abstract: Automated program verification is a difficult problem. It is undecidable even for transition systems over Linear Integer Arithmetic (LIA). Extending the transition system with theory of Arrays, further complicates the problem by requiring inference and reasoning with universally quantified formulas. In this paper, we present a new algorithm, Quic3, that extends IC3 to infer universally quantified invariants over the combined theory of LIA and Arrays. Unlike other approaches that use either IC3 or an SMT solver as a black box, Quic3 carefully manages quantified generalization (to construct quantified invariants) and quantifier instantiation (to detect convergence in the presence of quantifiers). While Quic3 is not guaranteed to converge, it is guaranteed to make progress by exploring longer and longer executions. We have implemented Quic3 within the Constrained Horn Clause solver engine of Z3 and experimented with it by applying Quic3 to verifying a variety of public benchmarks of array manipulating C programs.

45 citations

Journal ArticleDOI
TL;DR: It is shown that unification under one-sided distributivity with (one-sided) unit element is shown to be as hard as Markov's problem (associative unification), whereas unification under two- sided distributivity, with or without unit element, is NP-hard.

45 citations

Proceedings ArticleDOI
01 May 1998
TL;DR: This paper presents two families of rule languages, the one literal languages where each update is permitted to have just one atom in its body, and the unary languages where only unary Relations may be updated, but higher arity relations may be accessed through views.
Abstract: Active database systems enhance the functionality of traditional databases through the use of active rules or `triggers'. One of the principal questions for such systems is that of termination - is it possible for the rules to recursively activate one another indefinitely, given an initial triggering event. In this paper, we study the decidability of the termination problem, our aim being to delimit the boundary between the decidable and the undecidable. We present two families of rule languages, the one literal languages where each update is permitted to have just one atom in its body, and the unary languages where only unary relations may be updated, but higher arity relations may be accessed through views. Within each of these, we identify members close to the boundary of (un)decidability. Our context is similar to the while query language and the dynamics gives an interesting contrast to Datalog with negation; our results shed insights on the power of triggers as well as comparison of the termination problem to boundedness and query containment.

45 citations

Book ChapterDOI
20 Aug 2013
TL;DR: The results establish independence atoms as an efficient subclass of embedded multivalued data dependencies which are not axiomatizable by a finite set of Horn rules, and whose implication problem is undecidable.
Abstract: We investigate the implication problem for independence atoms $X \bot Y$ of disjoint attribute sets X and Y on database schemata. A relation satisfies $X \bot Y$ if for every X-value and every Y-value that occurs in the relation there is some tuple in the relation in which the X-value occurs together with the Y-value. We establish an axiomatization by a finite set of Horn rules, and derive an algorithm for deciding the implication problem in low-degree polynomial time in the input. We show how to construct Armstrong relations which satisfy an arbitrarily given set of independence atoms and violate every independence atom not implied by the given set. Our results establish independence atoms as an efficient subclass of embedded multivalued data dependencies which are not axiomatizable by a finite set of Horn rules, and whose implication problem is undecidable.

45 citations

Proceedings ArticleDOI
29 Mar 1989
TL;DR: It is proved that while weak safety is decidable, termination is not, and it is shown that a closely related problem, the decision problem for safety with respect to functional dependencies is undecidable even for monadic programs.
Abstract: A query is safe with respect to a set of constraints if for every database that satisfies the constraints the query is guaranteed to yield a finite set of answers. We study here the safety problem for Datalog programs with respect to finiteness constraints. We show that safety can be viewed as a combination of two properties: weak safety, which guarantees the finiteness of intermediate answers, and termination, which guarantees the finiteness of the evaluation. We prove that while weak safety is decidable, termination is not. We then consider monadic programs, i.e., programs in which all intensional predicates are monadic, and show that safety is decidable in polynomial time for monadic programs. While we do not settle the safety problem, we show that a closely related problem, the decision problem for safety with respect to functional dependencies, is undecidable even for monadic programs.

45 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
2023119
2022220
2021120
2020147
2019134
2018136