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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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Book ChapterDOI
29 Aug 1988
TL;DR: In this paper, a temporal logic for reasoning about Petri nets is defined, and the fair nontermination problem is shown to be PTIME equivalent to reachability for several definitions of fairness.
Abstract: In this paper, we define a temporal logic for reasoning about Petri nets. We show the model checking problem for this logic to be PTIME equivalent to the Petri net reachability problem. Using this logic and two refinements, we show the fair nontermination problem to be PTIME equivalent to reachability for several definitions of fairness. For other versions of fairness, this problem is shown to be either PTIME equivalent to the boundedness problem or highly undecidable. In all, 24 versions of fairness are examined.

24 citations

Posted Content
TL;DR: In this paper, the authors present a theoretical program for recurison theoretic decision problems in economic analysis, which leads to answers that are ambiguous: undecidable choices, uncomputable learning processes, and algorithmically unplayable games.
Abstract: In the field of economic analysis, computability in the formation of economic hypotheses is seen as the way forward. In this book, Professor Velupillai implements a theoretical research program along these lines. Choice theory, learning rational expectations equlibria, the persistence of adaptive behaviour, arithmetical games, aspects of production theory, and economic dynamics are given recursion theoretic (i.e. computable) interpretations. These interpretations lead to new kinds of questions being posed by the economic theorist. In particular, recurison theoretic decision problems replace standard optimisation paradigms in economic analysis. Economic theoretic questions, posed recursion-theoretically, lead to answers that are ambiguous: undecidable choices, uncomputable learning processes, and algorithmically unplayable games become standard answers. Professor Velupillai argues that a recursion theoretic formalisation of economic analysisComputable Economicsmakes the subject intrinsically inductive and computational.

24 citations

Journal ArticleDOI
TL;DR: In this paper, the authors obtained similar theorems of this type but the winning conditions are extremely simple relations (polynomial equations) and the winning strategies are computable.
Abstract: Computing machines using algorithms play games and even learn to play games. However, the inherent finiteness properties of algorithms impose limitations on the game playing abilities of machines. M. Rabin illustrated this limitation in 1957 by constructing a two-person win-lose game with decidable rules but no computable winning strategies. Rabin's game was of the type where two players take turns choosing integers to satisfy some decidable but very complicated winning condition. In the present paper we obtain similar theorems of this type but the winning conditions are extremely simple relations (polynomial equations). Specific examples are given.

24 citations

Book ChapterDOI
05 Apr 2014
TL;DR: The complexity is improved and it is shown that the qualitative-analysis problems for partial-observation stochastic parity games under finite-memory strategies are EXPTIME-complete; and optimal (exponential) memory bounds for finite- memory strategies required for qualitative analysis are established.
Abstract: We consider two-player partial-observation stochastic games on finitestate graphs where player 1 has partial observation and player 2 has perfect observation. The winning condition we study are e-regular conditions specified as parity objectives. The qualitative-analysis problem given a partial-observation stochastic game and a parity objective asks whether there is a strategy to ensure that the objective is satisfied with probability 1 (resp. positive probability). These qualitative-analysis problems are known to be undecidable. However in many applications the relevant question is the existence of finite-memory strategies, and the qualitative-analysis problems under finite-memory strategies was recently shown to be decidable in 2EXPTIME.We improve the complexity and show that the qualitative-analysis problems for partial-observation stochastic parity games under finite-memory strategies are EXPTIME-complete; and also establish optimal (exponential) memory bounds for finite-memory strategies required for qualitative analysis.

24 citations

Journal ArticleDOI
TL;DR: The undecidability of the Π3-theory of the partial order of enumerable Turing degrees is shown, which shows that this theory is not Turing-reducible to Y.
Abstract: We show the undecidability of the Π3-theory of the partial order of enumerable Turing degrees. 0. Introduction. Recursively enumerable (henceforth called enumerable) sets arise naturally in many areas of mathematics, for instance in the study of elementary theories, as solution sets of polynomials or as the word problems of finitely generated subgroups of finitely presented groups. Putting the enumerable sets into context with each other in various ways yields structures whose study has for long been a mainstay of computability theory. If the sets are related in the most elementary way, namely by inclusion, one obtains a distributive lattice E with very complex algebraic properties. Another way to compare sets is to look at the information content. Turing reducibility is a very general, but the most widely accepted concept of relative computability: a setX of natural numbers is Turing-reducible to Y iff the answer to “n ∈ X?” can be determined by a Turing machine computation which can use answers to oracle questions “y ∈ Y ?” during the computation. (For more restricted notions of relative computability one would for instance place a priori 1991 Mathematics Subject Classification. 03D25,03D35.

24 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
2023119
2022220
2021120
2020147
2019134
2018136