Proof of Recursive Unsolvability of Hilbert's Tenth Problem

Jones, James P.; Matijasevič, Yuri

doi:10.2307/2324421

Cited by 49 publications

(4 citation statements)

References 21 publications

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“…In this section, we first show that the question of emptiness is undecidable for realtime MC(const)-IAs by reduction of Hilbert's tenth problem which is known to be undecidable (see, e.g., [8,16]). The problem is to decide whether a given polynomial p(x 1 , .…”

Section: Undecidability Results For Realtime Mc(const)-iasmentioning

confidence: 99%

Iterative Arrays with Finite Inter-cell Communication

Kutrib

Malcher

2019

Cellular Automata and Discrete Complex Systems

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Iterative arrays whose internal inter-cell communication is quantitatively restricted are investigated. The quantity of communication is measured by counting the number of uses of the links between cells. In particular, iterative arrays are studied where the maximum number of communications per cell occurring in accepting computations is drastically bounded by a constant number. Additionally, the iterative arrays have to work in realtime. We study the computational capacity of such devices. One main result is that a strict and dense hierarchy with respect to the constant number of communications exists. Due to their very restricted communication, the question arises whether the usually studied decidability problems such as, for example, emptiness, finiteness, inclusion, or equivalence become decidable for such devices. However, by reduction of Hilbert's tenth problem it can be shown that all such decidability questions remain undecidable.

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Section: Undecidability Results For Realtime Mc(const)-iasmentioning

confidence: 99%

Iterative Arrays with Finite Inter-cell Communication

Kutrib

Malcher

2019

Cellular Automata and Discrete Complex Systems

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“…in the following sense: integers x > y satisfy (1) if and only if (x, y) = (a b (n), a b (n + 1)) for some integer n (see [Mat93, and [JM91]). The first step towards this result begins with the matrix identity…”

Section: Diophantine Equations From Recurrencesmentioning

confidence: 99%

Experimenting with Apéry Limits and WZ pairs

Dougherty-Bliss

Zeilberger

2021

Maple Trans.

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This article, dedicated with admiration in memory of Jon and Peter Borwein,illustrates by example, the power of experimental mathematics, so dear to them both, by experimenting with so-called Apéry limits and WZ pairs. In particular we prove a weaker form of an intriguing conjecture of Marc Chamberland and Armin Straub (in an article dedicated to Jon Borwein), and generate lots of new Apéry limits. We also rediscovered an infinite family of cubic irrationalities, that suggested very good effective irrationalitymeasures (lower than Liouville's generic 3), and that we conjectured to go down to the optimal 2. As it turned out, as pointed out by Paul Voutier (see the postscript kindly written by him), our conjectures follow from deep results in number theory. Nevertheless we believe that further experiments with our Maple programs would lead to new and interesting results.

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“…It was answered in the negative by Matiyasevich [48], based on earlier work by Davis, Putnam, and Robinson; see also [49]. A short self-contained proof, using register machines, is presented in [39]. Theorem 3.1.…”

Section: 3mentioning

confidence: 99%