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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01 Jan 1997
TL;DR: In this article, the complexity of the validity problem of arrow logic and cylindersignric modal logic has been investigated, and several natural expansions of these, interpreted on a range of (relativised) model classes.
Abstract: We consider two families of modal logics of relations: arrow logic and
cylindric modal logic and several natural expansions of these, interpreted
on a range of (relativised) modelclasses. We give a systematic study
of the complexity of the validity problem of these logics, obtaining
price tags for various features as assumptions on the universe of the
models, similarity types, and number of variables involved. The general
picture is that the process of relativisation turns an undecidable logic
into one whose validity problem is exptimecomplete. There
are interesting deviations to this though, which we also discuss. The
numerous results in this paper are all directed to obtain a better
understanding why relativisation can turn an undecidable modal logic
of relations into a decidable one. We connect the semantic way of
``taming logic'' by relativisation with the syntactic approach of
isolating decidable socalled guarded fragments by showing that validity
of loosely guarded formulas is preserved under relativisation.
19 citations
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TL;DR: A recasting of Jevon's sceptical vision of induction in the light of MDL; and a complexity interpretation of an undecidable question in dynamics are suggested.
Abstract: Rissanen's fertile and pioneering minimum description length principle (MDL) has been viewed from the point of view of statistical estimation theory, information theory, as stochastic complexity theory - i.e., a computable approximation of Kolomogorov Complexity - or Solomonoff's recursion theoretic induction principle or as analogous to Kolmogorov's sufficient statistics. All these - and many more - interpretations are valid, interesting and fertile. In this paper I view it from two points of view: those of an algorithmic economist and a dynamical system theorist. From these points of view I suggest, first, a recasting of Jevon's sceptical vision of induction in the light of MDL; and a complexity interpretation of an undecidable question in dynamics
19 citations
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TL;DR: A variant of Post's Correspondence Problem is considered where two different index words are allowed provided that one of them can be obtained from the other by permuting a fixed number of subwords, and it is shown that this variant is undecidable.
Abstract: A variant of Post's Correspondence Problem is considered where two different index words are allowed provided that one of them can be obtained from the other by permuting a fixed number of subwords. It is shown that this variant is undecidable. Post's Correspondence Problem is also extended to circular words, doubly infinite words and doubly infinite powers of words, and shown to be undecidable in all these extensions.
19 citations
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01 Jun 2017TL;DR: In this article, the authors characterize the complexity of liveness verification for parameterized systems consisting of a leader process and arbitrarily many anonymous and identical contributor processes, and show that the problem is decidable and has a surprisingly low complexity: it is NP-complete when all processes are finite-state machines, and is in NEXPTIME (and PSPACE-hard) when they are pushdown machines.
Abstract: We characterize the complexity of liveness verification for parameterized systems consisting of a leader process and arbitrarily many anonymous and identical contributor processes. Processes communicate through a shared, bounded-value register. While each operation on the register is atomic, there is no synchronization primitive to execute a sequence of operations atomically. We analyze the case in which processes are modeled by finite-state machines or pushdown machines and the property is given by a Buchi automaton over the alphabet of read and write actions of the leader. We show that the problem is decidable, and has a surprisingly low complexity: it is NP-complete when all processes are finite-state machines, and is in NEXPTIME (and PSPACE-hard) when they are pushdown machines. This complexity is lower than for the non-parameterized case: liveness verification of finitely many finite-state machines is PSPACE-complete, and undecidable for two pushdown machines. For finite-state machines, our proofs characterize infinite behaviors using existential abstraction and semilinear constraints. For pushdown machines, we show how contributor computations of high stack height can be simulated by computations of many contributors, each with low stack height. Together, our results characterize the complexity of verification for parameterized systems under the assumptions of anonymity and asynchrony.
19 citations
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19 citations