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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.


Papers
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Journal ArticleDOI
TL;DR: A symbolic reduction system that can handle hashing functions, symmetric keys, and public keys is introduced that can be regarded as a variant of syntactic unification which is compatible with certain set-membership constraints.

47 citations

01 Jan 2014
TL;DR: In this paper, the computational complexity of deciding entailment in separation logic with general inductive predicates was studied and it was shown that entailment is in general undecidable and ExpTime-hard.
Abstract: We establish foundational results on the computational complexity of deciding entailment in Separation Logic with general inductive predicates whose underlying base language allows for pure formulas, pointers and existentially quantified variables. We show that entailment is in general undecidable, and ExpTime-hard in a fragment recently shown to be decidable by Iosif et al. Moreover, entailment in the base language is Π2P-complete, the upper bound even holds in the presence of list predicates. We additionally show that entailment in essentially any fragment of Separation Logic allowing for general inductive predicates is intractable even when strong syntactic restrictions are imposed. © 2014 Springer-Verlag.

47 citations

Book ChapterDOI
TL;DR: In this paper, the authors give sketches of classical undecidability results in number theory, like Godel's first Incompleteness Theorem (that the first order theory of the integers in the language of rings is undecidable), Julia Robinson's extensions of this result to arbitrary number fields and rings of integers in them, as well as to the ring of totally real integers.
Abstract: In these lecture notes we give sketches of classical undecidability results in number theory, like Godel’s first Incompleteness Theorem (that the first order theory of the integers in the language of rings is undecidable), Julia Robinson’s extensions of this result to arbitrary number fields and rings of integers in them, as well as to the ring of totally real integers, and Matiyasevich’s negative solution of Hilbert’s 10th problem, i.e., the undecidability of the existential first-order theory of the integers. As Hilbert’s 10th problem is still open for the rationals (i.e., the question whether the existential theory of the field of rational numbers is decidable) we also present a sketch of the fact that there is a universal definition of the ring of integers inside the field of rationals. In terms of complexity this is the simplest definition known so far. If one had an existential definition instead then Hilbert’s 10th problem over the rationals would reduce to that over the integers (and hence be, as expected, unsolvable), but, modulo a widely believed in conjecture in number theory, we also indicate why there should be no such existential definition. We conclude with a list of nice open questions in the area.

47 citations

Journal ArticleDOI
TL;DR: An approximation method based on interval arithmetic that uses a generalization of the notion of cylindrical decomposition—as introduced by G. Collins to efficiently give approximate information on problems that are too hard for current exact methods.
Abstract: This paper applies interval methods to a classical problem in computer algebra. Let a quantified constraint be a first-order formula over the real numbers. As shown by A. Tarski in the 1930's, such constraints, when restricted to the predicate symbols <, = and function symbols +, ×, are in general solvable. However, the problem becomes undecidable, when we add function symbols like sin. Furthermore, all exact algorithms known up to now are too slow for big examples, do not provide partial information before computing the total result, cannot satisfactorily deal with interval constants in the input, and often generate huge output. As a remedy we propose an approximation method based on interval arithmetic. It uses a generalization of the notion of cylindrical decomposition—as introduced by G. Collins. We describe an implementation of the method and demonstrate that, for quantified constraints without equalities, it can efficiently give approximate information on problems that are too hard for current exact methods.

47 citations

Journal ArticleDOI
TL;DR: A semantics is given that captures parameterised, generic multi-agent systems and three notable classes that represent different ways in which the agents may interact among themselves and with the environment are identified.

47 citations


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