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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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TL;DR: In this article, it was shown that the universality problem, the inclusion problem, equivalence problem, determinizability problem, complementability problem and unambiguity problem are all φ 2^1$-complete for context-free omega-languages or infinitary rational relations.
Abstract: We show that many classical decision problems about 1-counter omega-languages, context free omega-languages, or infinitary rational relations, are $\Pi_2^1$-complete, hence located at the second level of the analytical hierarchy, and ``highly undecidable". In particular, the universality problem, the inclusion problem, the equivalence problem, the determinizability problem, the complementability problem, and the unambiguity problem are all $\Pi_2^1$-complete for context-free omega-languages or for infinitary rational relations. Topological and arithmetical properties of 1-counter omega-languages, context free omega-languages, or infinitary rational relations, are also highly undecidable. These very surprising results provide the first examples of highly undecidable problems about the behaviour of very simple finite machines like 1-counter automata or 2-tape automata.
21 citations
TL;DR: In particular, Hilbert's Tenth Problem for function fields of varieties over number fields of dimension n ⩾ 1 is undecidable as mentioned in this paper, which is the first time that it has been shown.
21 citations
TL;DR: It is shown that the (infinite) tiling problem by Wang tiles is undecidable even if the given tile set is deterministic by all four corners, i.e. a tile is uniquely determined by the colors of any two adjacent edges.
21 citations
30 Aug 2021
TL;DR: This paper introduces the synchronous variant of threshold automata, and shows that many synchronous fault-tolerant distributed algorithms have a bounded diameter, although the algorithms are parameterized by the number of processes.
Abstract: Many fault-tolerant distributed algorithms are designed for synchronous or round-based semantics. In this paper, we introduce the synchronous variant of threshold automata, and study their applicability and limitations for the verification of synchronous distributed algorithms. We show that in general, the reachability problem is undecidable for synchronous threshold automata. Still, we show that many synchronous fault-tolerant distributed algorithms have a bounded diameter, although the algorithms are parameterized by the number of processes. Hence, we use bounded model checking for verifying these algorithms.
21 citations
TL;DR: This paper is a join of two separate previous, preliminary papers on NP-completeness of Bounded Second-Order Unification and Stratified Context Unification, and gives a clear account of a method of using singleton tree grammars for compression that may have potential usage for other algorithmic questions in related areas.
Abstract: Bounded Second-Order Unification is a decidable variant of undecidable Second-Order Unification. Stratified Context Unification is a decidable restriction of Context Unification, whose decidability is a long-standing open problem. This paper is a join of two separate previous, preliminary papers on NP-completeness of Bounded Second-Order Unification and Stratified Context Unification. It clarifies some omissions in these papers, joins the algorithmic parts that construct a minimal solution, and gives a clear account of a method of using singleton tree grammars for compression that may have potential usage for other algorithmic questions in related areas.
21 citations