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: A complexity analysis of planning under uncertainty is presented, showing the "probabilistic classical planning" problem to be formally undecidable and any problem statement where the agent operates over an infinite or indefinite time horizon, and has available only probabilistic information about the system's state.
273 citations
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01 Jan 1992TL;DR: In this paper, the problem of integrating Reiter's default logic into terminological representation systems is considered, and it turns out that such an integration is less straightforward than we expected, considering the fact that the terminological language is a decidable sublanguage of first-order logic.
Abstract: We consider the problem of integrating Reiter's default logic into terminological representation systems. It turns out that such an integration is less straightforward than we expected, considering the fact that the terminological language is a decidable sublanguage of first-order logic. Semantically, one has the unpleasant effect that the consequences of a terminological default theory may be rather unintuitive, and may even vary with the syntactic structure of equivalent concept expressions. This is due to the unsatisfactory treatment of open defaults via Skolemization in Reiter's semantics. On the algorithmic side, we show that this treatment may lead to an undecidable default consequence relation, even though our base language is decidable, and we have only finitely many (open) defaults. Because of these problems, we then consider a restricted semantics for open defaults in our terminological default theories: default rules are applied only to individuals that are explicitly present in the knowledge base. In this semantics it is possible to compute all extensions of a finite terminological default theory, which means that this type of default reasoning is decidable. We describe an algorithm for computing extensions and show how the inference procedures of terminological systems can be modified to give optimal support to this algorithm.
259 citations
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01 Jan 1979
TL;DR: A number of Petri Net problems are shown to be recursively equivalent to the Reachability Problem for Vector Addition Systems, and the equality of Reachability Sets and the equivalence of two Petri Nets in terms of their language-generating capability are recursive undecidable.
Abstract: An understanding of the mathematical properties of Petri Nets is essential when one wishes to use Petri Nets as an abstract model for concurrent systems The decidability of various problems which arise in this context is an important aspect of this question The fact that these problems also arise in the context of other mathematical theories, such as commutative semigroups, closure under linear relations, Matrix Context-Free grammars, or Weak Counter Automata, provides further motivation The Reachability Problem for Vector Addition Systems - whose decidability is still an open question - is of central importance We show that a number of Petri Net problems are recursively equivalent to this problem These include the Liveness Problem (eg can a system reach a deadlocked state?), the persistence problem (can a given transition ever be disabled by the firing of another transition?), and the membership and emptiness problems for certain classes of languages generated by Petri Nets The power of the unrestricted Petri Net model is illustrated by various undecidable equivalence results In particular, we show that the equality of Reachability Sets and the equivalence of two Petri Nets in terms of their language-generating capability are recursive undecidable It is hoped that the constructions used to prove our results will shed some light on the source of the complexities of the unrestricted Petri Net model, and may eventually permit us to achieve an optimal balance between representational transparency and analytical power of the Petri Net model
258 citations
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TL;DR: In this article, the problem of integrating Reiter's default logic into terminological representation systems is considered, and it turns out that such an integration is less straightforward than we expected, considering the fact that the terminological language is a decidable sublanguage of first-order logic.
Abstract: We consider the problem of integrating Reiter's default logic into terminological representation systems. It turns out that such an integration is less straightforward than we expected, considering the fact that the terminological language is a decidable sublanguage of first-order logic. Semantically, one has the unpleasant effect that the consequences of a terminological default theory may be rather unintuitive, and may even vary with the syntactic structure of equivalent concept expressions. This is due to the unsatisfactory treatment of open defaults via Skolemization in Reiter's semantics. On the algorithmic side, we show that this treatment may lead to an undecidable default consequence relation, even though our base language is decidable, and we have only finitely many (open) defaults. Because of these problems, we then consider a restricted semantics for open defaults in our terminological default theories: default rules are applied only to individuals that are explicitly present in the knowledge base. In this semantics it is possible to compute all extensions of a finite terminological default theory, which means that this type of default reasoning is decidable. We describe an algorithm for computing extensions and show how the inference procedures of terminological systems can be modified to give optimal support to this algorithm.
258 citations
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TL;DR: The problem of deciding whether a Petri net is persistent is reducible to reachability, partially answering a question of Keller, and it is shown that the controllability problem requires exponential space, even for 1-bounded nets.
256 citations