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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: The undecidability of the preservation of regularity by rewrite systems is proved and fragments of the theory of ground term algebras modulo congruence generated by a set of equations which can be compiled in a terminating, confluent rewrite system which preserves regularity are studied.
Abstract: We present a collection of results on regular tree languages and rewrite systems. Moreover we prove the undecidability of the preservation of regularity by rewrite systems. More precisely we prove that it is undecidable whether or not for a set E of equations the set E(R) congruence closure of set R is a regular tree language whenever R is a regular tree language. It is equally undecidable whether or not for a confluent and terminating rewrite system S the set S(R) of ground S-normal forms of set R is a regular tree language whenever R is a regular tree language. Finally we study fragments of the theory of ground term algebras modulo congruence generated by a set of equations which can be compiled in a terminating, confluent rewrite system which preserves regularity.
90 citations
27 Feb 1997
TL;DR: It is shown that going beyond L2 by adding any one of the following leads to an undecidable logic: very weak forms of recursion, such as transitive closure or monadic fixed-point operations.
Abstract: It is a classical result of Mortimer's that L2, first-order logic with two variables, is decidable for satisfiability (whereas L3 is undecidable). We show that going beyond L2 by adding any one of the following leads to an undecidable logic:
very weak forms of recursion, such as transitive closure or monadic fixed-point operations.
cardinality comparison quantifiers.
90 citations
TL;DR: It is shown that data mining is closely related to compression and Kolmogorov complexity; and the latter is undecidable, so data mining will always be an art, where the goal will be to find better models that fit the authors' datasets as best as possible.
Abstract: Will we ever have a theory of data mining analogous to the relational algebra in databases? Why do we have so many clearly different clustering algorithms? Could data mining be automated? We show that the answer to all these questions is negative, because data mining is closely related to compression and Kolmogorov complexity; and the latter is undecidable. Therefore, data mining will always be an art, where our goal will be to find better models (patterns) that fit our datasets as best as possible.
90 citations
TL;DR: In this paper, a new method for obtaining lower bounds on the computational complexity of logical theories is presented, which extends widely used techniques for proving the undecidability of theories by interpreting models of a theory already known to be undecidable.
89 citations
Proceedings Article•
13 Dec 1999
TL;DR: In this article, the modal logic of knowledge and linear time in distributed systems with perfect recall was studied, and it was shown that this problem is undecidable for a language with operators for until and common knowledge.
Abstract: This paper studies model checking for the modal logic of knowledge and linear time in distributed systems with perfect recall. It is shown that this problem (1) is undecidable for a language with operators for until and common knowledge, (2) is PSPACE-complete for a language with common knowledge but without until, (3) has nonelementary upper and lower bounds for a language with until but without common knowledge.M odel checking bounded knowledge depth formulae of the last of these languages is considered in greater detail, and an automata-theoretic decision procedure is developed for this problem, that yields a more precise complexity characterization.
89 citations