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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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Journal ArticleDOI
TL;DR: It is shown that two equal-length predicates are equivalent and are not existentially definable by equations in concatenation and by adding a further length predicate, one gets an undecidable existential theory.
Abstract: We study extensions of the decidable existential theory of concatenation over words. We show that two equal-length predicates are equivalent and are not existentially definable by equations in concatenation. By adding a further length predicate we get an undecidable existential theory. Whether the existential theory of concatenation and equal length is decidable remains unknown.

63 citations

Book ChapterDOI
01 Apr 2009
TL;DR: The reachability problem for parametric flat counter automata, in relation with the satisfiability problem of three fragments of integer arithmetic, is studied and is shown to be undecidable.
Abstract: In this paper we study the reachability problem for parametric flat counter automata, in relation with the satisfiability problem of three fragments of integer arithmetic. The equivalence between non-parametric flat counter automata and Presburger arithmetic has been established previously by Comon and Jurski. We simplify their proof by introducing finite state automata defined over alphabets of a special kind of graphs (zigzags). This framework allows one to express also the reachability problem for parametric automata with one control loop as the satisfiability of a 1-parametric linear Diophantine systems. The latter problem is shown to be decidable, using a number-theoretic argument. In general, the reachability problem for parametric flat counter automata with more than one loops is shown to be undecidable, by reduction from Hilbert's Tenth Problem. Finally, we study the relation between flat counter automata, integer arithmetic, and another important class of computational devices, namely the 2-way reversal bounded counter machines.

63 citations

01 Jan 1994
TL;DR: In this paper, it was shown that semantic query optimization can be completely done in recursive rules provided that order constraints and negated EDB subgoals appear only in the recursive rules, but not in the it's.
Abstract: Semantic query optimization refers to the process of using integrity constraints (ic ‘s) in order to optimize the evaluation of queries. The process is well understood in the case of unions of select-project-join queries (i. e., nonrecursive datalog). For arbitrary datalog programs, however, the issue has largely remained an unsolved problem. This paper studies this problem and shows when semantic query optimization can be completely done in recursive rules provided that order constraints and negated EDB subgoals appear only in the recursive rules, but not in the it’s. If either order constraints or negated EDB subgoals are introduced in it’s, then the problem of semantic query optimization becomes undecidable. Since semantic query optimization is closely related to the containment problem of a datalog program in a union of conjunctive queries, our results also imply new decidability and undecidability results for that problem when order constraints and negated EDB subgoals are used.

63 citations

Proceedings ArticleDOI
01 Jan 1986
TL;DR: Girard's techniques are applied to establish that the type-of-all-types assumption creates serious pathologies from a programming perspective: a system using this assumption is inherently not normalizing, term equality is undecidable, and the resulting theory fails to be a conservative extension of the theory of the underlying base types.
Abstract: A function has a dependent type when the type of its result depends upon the value of its argument. Dependent types originated in the type theory of intuitionistic mathematics and have reappeared independently in programming languages such as CLU, Pebble, and Russell. Some of these languages make the assumption that there exists a type-of-all-types which is its own type as well as the type of all other types. Girard proved that this approach is inconsistent from the perspective of intuitionistic logic. We apply Girard's techniques to establish that the type-of-all-types assumption creates serious pathologies from a programming perspective: a system using this assumption is inherently not normalizing, term equality is undecidable, and the resulting theory fails to be a conservative extension of the theory of the underlying base types. The failure of conservative extension means that classical reasoning about programs in such a system is not sound.

62 citations

Journal ArticleDOI
TL;DR: This paper increases the expressive power of Description Logics by allowing for more complex roles in number restrictions, and shows that concept satisfiability is decidable for a restricted logic.
Abstract: Number restrictions are concept constructors that are available in almost all implemented Description Logic systems. However, they are mostly available only in a rather weak form, which considerably restricts their expressive power. On the one hand, the roles that may occur in number restrictions are usually of a very restricted type, namely atomic roles or complex roles built using either intersection or inversion. In the present paper, we increase the expressive power of Description Logics by allowing for more complex roles in number restrictions. As role constructors, we consider composition of roles (which will be present in all our logics) and intersection, union, and inversion of roles in different combinations. We will present two decidability results (for the basic logic that extends ALC by number restrictions on roles with composition, and for one extension of this logic), and three undecidability results for three other extensions of the basic logic. On the other hand, with the rather weak form of number restrictions available in implemented systems, the number of role successors of an individual can only be restricted by a fixed non-negative integer. To overcome this lack of expressiveness, we allow for variables ranging over the non-negative integers in place of the fixed numbers in number restrictions. The expressive power of this constructor is increased even further by introducing explicit quantifiers for the numerical variables. The Description Logic obtained this way turns out to have an undecidable satisfiability problem. For a restricted logic we show that concept satisfiability is decidable.

62 citations


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