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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 has been shown that the set of Sierpinski-Zygmund functions is 2(c)-strongly algebrable (and thus, thus, 2c)-lineable).
38 citations
TL;DR: In this article, the authors studied the decidability of almost sure disclosure in Markov decision processes with partial or complete observation hypotheses for the schedulers and showed that all questions are decidable with complete observation and ω-regular secrets.
38 citations
01 Jan 2017
TL;DR: It follows that subtype checking in Java is undecidable, which answers a question posed by Kennedy and Pierce in 2007 and also follows that Java's type checker can recognize any recursive language, which improves a result of Gill and Levy from 2016.
Abstract: This paper describes a reduction from the halting problem of Turing machines to subtype checking in Java. It follows that subtype checking in Java is undecidable, which answers a question posed by Kennedy and Pierce in 2007. It also follows that Java's type checker can recognize any recursive language, which improves a result of Gill and Levy from 2016. The latter point is illustrated by a parser generator for fluent interfaces.
38 citations
01 Dec 1985
TL;DR: It is shown that unification in the equational theory defined by the one-sided distributivity law x × (y+z)=x×y+x×z is decidable and that unification is undecidable if the laws of associativity x+y)+z and unit element 1×x=x× 1=x are added.
Abstract: We show that unification in the equational theory defined by the one-sided distributivity law x × (y+z)=x×y+x×z is decidable and that unification is undecidable if the laws of associativity x+(y+z)=(x+y)+z and unit element 1×x=x× 1=x are added. Unification under one-sided distributivity with unit element is shown to be as hard as Markov's problem, whereas unification under two-sided distributivity, with or without unit element, is NP-hard. A quadratic time unification algorithm for one-sided distributivity, which may prove interesting since available universal unification procedures fail to provide a decision procedure for this theory, is outlined. The study of these problems is motivated by possible applications in circuit synthesis and by the need for gaining insight in the problem of combining theories with overlapping sets of operator symbols.
38 citations
TL;DR: The Identity Correspondence Problem (ICP) is introduced: whether a finite set of pairs of words can generate an identity pair by a sequence of concatenations and it is proved that ICP is undecidable by a reduction of Post's Correspondence problem via several new encoding techniques.
Abstract: In this paper we study several closely related fundamental problems for words and matrices. First, we introduce the Identity Correspondence Problem (ICP): whether a finite set of pairs of words (over a group alphabet) can generate an identity pair by a sequence of concatenations. We prove that ICP is undecidable by a reduction of Post's Correspondence Problem via several new encoding techniques. In the second part of the paper we use ICP to answer a long standing open problem concerning matrix semigroups: "Is it decidable for a finitely generated semigroup S of integral square matrices whether or not the identity matrix belongs to S?". We show that the problem is undecidable starting from dimension four even when the number of matrices in the generator is 48. From this fact, we can immediately derive that the fundamental problem of whether a finite set of matrices generates a group is also undecidable. We also answer several questions for matrices over different number fields. Apart from the application to matrix problems, we believe that the Identity Correspondence Problem will also be useful in identifying new areas of undecidable problems in abstract algebra, computational questions in logic and combinatorics on words.
38 citations