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: This paper uses a novel encoding of regular role inclusions using bounded-stack pushdown automata--that is, FAs extended with a stack of bounded size to solve the open problem of answering conjunctive queries over OWL 2 EL knowledge bases.
Abstract: OWL 2 EL is a popular ontology language that supports role inclusions--axioms of the form S1...Sn ⊆ S that capture compositional properties of roles. Role inclusions closely correspond to context-free grammars, which was used to show that answering conjunctive queries (CQs) over OWL 2 EL knowledge bases with unrestricted role inclusions is undecidable. However, OWL 2 EL inherits from OWL 2 DL the syntactic regularity restriction on role inclusions, which ensures that role chains implying a particular role can be described using a finite automaton (FA). This is sufficient to ensure decidability of CQ answering; however, the FAs can be worst-case exponential in size so the known approaches do not provide a tight upper complexity bound.
In this paper, we solve this open problem and show that answering CQs over OWL 2 EL knowledge bases is PSpace-complete in combined complexity (i.e., the complexity measured in the total size of the input). To this end, we use a novel encoding of regular role inclusions using bounded-stack pushdown automata--that is, FAs extended with a stack of bounded size. Apart from theoretical interest, our encoding can be used in practical tableau algorithms to avoid the exponential blowup due to role inclusions. In addition, we sharpen the lower complexity bound and show that the problem is PSPACE-hard even if we consider only role inclusions as part of the input (i.e., the query and all other parts of the knowledge base are fxed). Finally, we turn our attention to navigational queries over OWL 2 EL knowledge bases, and we show that answering positive, converse-free conjunctive graph XPath queries is PSPACE-complete as well; this is interesting since allowing the converse operator in queries is known to make the problem ExpTime-hard. Thus, in this paper we present several important contributions to the landscape of the complexity of answering expressive queries over description logic knowledge bases.
38 citations
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15 Dec 2003TL;DR: It is shown that the model-checking problem of TCTL extended with parameters is undecidable over discrete-timed automata with only one parametric clock.
Abstract: In this paper, we study the model-checking and parameter synthesis problems of the logic TCTL over discrete-timed automata where parameters are allowed both in the model and in the property. We show that the model-checking problem of TCTL extended with parameters is undecidable over discrete-timed automata with only one parametric clock. The undecidability result needs equality in the logic. When equality is not allowed, we show that the model-checking and the parameter synthesis problems become decidable.
38 citations
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13 Apr 2010
TL;DR: In this paper, the Incomputable, the Non-constructive and the Undecidable in Mathematical Economics have been discussed from an Algorithmic Complexity Theory from an Elementary Standpoint.
Abstract: Part I: Foundations 1. The Incomputable, the Non-constructive and the Undecidable in Mathematical Economics 2. Advanced Computational Complexity Theory from an Elementary Standpoint 3. Economic Dynamics and Computation - Recursion Theoretic Foundations for the Icarus Tradition 4: Let's Take the Con out of Mathematical Economics Part II: General Equilibrium Theory 5. Effectivity and Constructivity in Economic Theory 6. Algorithmic Foundations of Computable General Equilibrium Theory 7. Uncomputability and Undecidability in Economic Theory Part III: Methodology 8. The Unreasonable Ineffectivity of Mathematics in Economics 9. A Constructive Interpretation of Sraffa's Mathematical Economics 10. The Computable Alternative in the Mathematization of Economics Part IV: Simon's Behavioural Economics - A Computable Vision 11. Computable Rationality 12. Boundedly Rational Choice and Satisficing Decisions 13. Simon's Epicurean Adventures - A Prolegomena. Appendix 1 to Part IV: Artificing a Rationally Unbounded Life. Appendix 2 to Part IV: The Logic of Discovery, Problem Solving and Retroduction. Appendix 3 to Part IV: Herbert Simon's Letter on Computable Economics Part V: Inductive Reflections 14. De-Mystifying Induction, Falsification and other Popperian Extravaganzas 15. Re-reading Jevons's Principles of Science: Induction Redux 16. Impossibility of Effectively Computable Inductive Policies in a Complex Dynamic Economy. Part VI: Concluding Notes 17. Epilogue - A Research Program for the Algorithmic Social Sciences
37 citations
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20 Jan 2007TL;DR: In this article, the authors give constructions of small probabilistic and MO-type quantum automata that have undecidable emptiness problem for the cut-point languages, which is a special case of the problem we consider in this paper.
Abstract: We give constructions of small probabilistic and MO-type quantum automata that have undecidable emptiness problem for the cut-point languages.
37 citations
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13 Dec 2006TL;DR: This work considers timed games extended with cost information, and proves computability of the optimal cost and of e-optimal memoryless strategies in timed games with one clock.
Abstract: We consider timed games extended with cost information, and prove computability of the optimal cost and of e-optimal memoryless strategies in timed games with one clock. In contrast, this problem has recently been proved undecidable for timed games with three clocks.
37 citations