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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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Journal ArticleDOI
TL;DR: Kim and Roush as mentioned in this paper showed that Hilbert's tenth problem for the purely transcendental function field ℂ(t 1,t 2 ) is undecidable for any dimension greater than or equal to 2 over an algebraically closed field of characteristic zero.
Abstract: Let K be the function field of a variety of dimension greater than or equal to 2 over an algebraically closed field of characteristic zero. Then Hilbert's tenth problem for K is undecidable. This generalizes the result by Kim and Roush, 1992, that Hilbert's tenth problem for the purely transcendental function field ℂ(t 1 ,t 2 ) is undecidable.

16 citations

Journal ArticleDOI
TL;DR: A general theorem for avoiding undecidable problems in computability theory is proposed by introducing a new class of recursive functions on different axiomatizations of numbers by way of a well‐formed formula of a first‐order predicate calculus.
Abstract: In this article we intend to analyze a chaotic system from the standpoint of its computation capability. to pursue this aim, we refer to a complex chaotic dynamics that we characterize via its symbolic dynamics. We show that these dynamic systems are subjected to some typical undecidable problems. Particularly, we stress the impossibility of deciding on a unique invariant measure. This depends essentially on the supposition of the existence of a fixed universal grammar. the suggestion is thus of justifying a contextual redefinition of the grammar as a function of the same evolution of the system. We propose on this basis a general theorem for avoiding undecidable problems in computability theory by introducing a new class of recursive functions on different axiomatizations of numbers. From it a series expansion on n algebraic fields can be defined. In such a way, we are able to obtain a very fast extraction procedure of unstable periodic orbits from a generic chaotic dynamics. the computational efficiency of this algorithm allows us to characterize a chaotic system by the complete statistics of its unstable cycles. Some examples of these two techniques are discussed. Finally, we introduce the possibility of an application of this same class of recursive functions to the calculus of the absolute minimum of energy in neural nets, as far as it is equivalent to a well-formed formula of a first-order predicate calculus. © 1995 John Wiley & Sons, Inc.

16 citations

Journal ArticleDOI
TL;DR: This paper provides here the first full proof, and shows that the general snake problem is actually PSPACE-complete, and establishes a resemblance between snake problems and classical tiling problems, considering the corresponding bounded, unbounded and recurring cases.

16 citations

Book ChapterDOI
25 Aug 2003
TL;DR: Engeler’s Lemma for Σ-definability over the reals without the equality test is proved, which relatesΣ- definability with definability in the constructive infinitary language \(L_{\omega_1 \omega}\), and a relation over the real numbers is ηdefinable if and only if it is defined by a disjunction of a recursively enumerable set of quantifier free formulas.
Abstract: In this paper we study the expressive power and algorithmic properties of the language of Σ-formulas intended to represent computability over the real numbers In order to adequately represent computability, we extend the reals by the structure of hereditarily finite sets In this setting it is crucial to consider the real numbers without equality since the equality test is undecidable over the reals We prove Engeler’s Lemma for Σ-definability over the reals without the equality test which relates Σ-definability with definability in the constructive infinitary language \(L_{\omega_1\omega}\) Thus, a relation over the real numbers is Σ-definable if and only if it is definable by a disjunction of a recursively enumerable set of quantifier free formulas This result reveals computational aspects of Σ-definability and also gives topological characterisation of Σ-definable relations over the reals without the equality test

16 citations

Book ChapterDOI
13 Dec 2004
TL;DR: It follows that the vector reachability problem is undecidable for a semigroup generated by 7 integral matrices of dimension 4, and new classes of counter automata are defined that lie on the border between decidability and undecidability.
Abstract: The main result of this paper is the reduction of PCP(n) to the vector reachability problem for a matrix semigroup generated by n 4 × 4 integral matrices. It follows that the vector reachability problem is undecidable for a semigroup generated by 7 integral matrices of dimension 4. The question whether the vector reachability problem is decidable for n = 2 and n = 3 remains open. Also we show that proposed technique can be applied to Post’s tag-systems. As a result we define new classes of counter automata that lie on the border between decidability and undecidability.

16 citations


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