Topic
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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TL;DR: The aim of this work is to delimit the boundary between the decidable and the undecidable, and presents results for two broad types of variations, variations in rule syntax and variations in meta level features.
18 citations
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08 May 2019TL;DR: A semantics is introduced, based on interpreted systems, to capture the openness of the system and show how an indexed variant of temporal-epistemic logic can be used to express specifications on them.
Abstract: We study open multi-agent systems in which countably many agents may leave and join the system at run-time. We introduce a semantics, based on interpreted systems, to capture the openness of the system and show how an indexed variant of temporal-epistemic logic can be used to express specifications on them. We define the verification problem and show it is undecidable. We isolate one decidable class of open multi-agent systems and give a partial decision procedure for another one. We introduce MCMAS-OP, an open-source toolkit implementing the verification procedures. We present the results obtained using our tool on two examples.
18 citations
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11 Jan 2016TL;DR: This paper identifies the essence of this workflow to be the notion of progress measure, and formalizes it in general, possibly infinitary, lattice-theoretic terms, to a general model-checking framework, where systems are categorically presented as coalgebras.
Abstract: In the context of formal verification in general and model checking in particular, parity games serve as a mighty vehicle: many problems are encoded as parity games, which are then solved by the seminal algorithm by Jurdzinski. In this paper we identify the essence of this workflow to be the notion of progress measure, and formalize it in general, possibly infinitary, lattice-theoretic terms. Our view on progress measures is that they are to nested/alternating fixed points what invariants are to safety/greatest fixed points, and what ranking functions are to liveness/least fixed points. That is, progress measures are combination of the latter two notions (invariant and ranking function) that have been extensively studied in the context of (program) verification. We then apply our theory of progress measures to a general model-checking framework, where systems are categorically presented as coalgebras. The framework's theoretical robustness is witnessed by a smooth transfer from the branching-time setting to the linear-time one. Although the framework can be used to derive some decision procedures for finite settings, we also expect the proposed framework to form a basis for sound proof methods for some undecidable/infinitary problems.
18 citations
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TL;DR: It is shown that it is undecidable in general whether a finite, lengthreducing, and confluent string-rewriting system yields a regular set of normal forms for each regular language.
18 citations
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TL;DR: In this paper, the robust satisfiability of systems of nonlinear equations was studied and it was shown that the problem is decidable in polynomial time for every fixed n, assuming dim K ≤ 2n−3, where the threshold comes from the stable range in homotopy theory.
Abstract: We study the problem of robust satisfiability of systems of nonlinear equations, namely, whether for a given continuous function f:K → Rn on a finite simplicial complex K and α>0, it holds that each function g:K → Rn such that ║g−f║∞ ≤ α, has a root in K. Via a reduction to the extension problem of maps into a sphere, we particularly show that this problem is decidable in polynomial time for every fixed n, assuming dim K ≤ 2n−3. This is a substantial extension of previous computational applications of topological degree and related concepts in numerical and interval analysis. Via a reverse reduction, we prove that the problem is undecidable when dim K ≥ 2n−2, where the threshold comes from the stable range in homotopy theory. For the lucidity of our exposition, we focus on the setting when f is simplexwise linear. Such functions can approximate general continuous functions, and thus we get approximation schemes and undecidability of the robust satisfiability in other possible settings.
18 citations