The Decision Problem for Exponential Diophantine Equations

Davis, Martin; Putnam, Hilary; Robinson, Julia

doi:10.2307/1970289

Cited by 274 publications

(135 citation statements)

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“…By the Matiyasevich-Robinson-Davis-Putnam theorem [41,42], there exists an n ∈ N and a multivariate polynomial p with integer coefficients such that for every A ∈ Σ 1 there exists an a ∈ N such that x ∈ A ⇐⇒ ∃y ∈ N n , p(a, x, y) = 0.…”

Section: Recasting Circuit Results For Cspsmentioning

confidence: 99%

Circuit Satisfiability and Constraint Satisfaction Around Skolem Arithmetic

Glaßer

Jönsson

Martin³

2016

Pursuit of the Universal

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The full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. AbstractWe study interactions between Skolem Arithmetic and certain classes of Circuit Satisfiability and Constraint Satisfaction Problems (CSPs). We revisit results of Glaßer et al. [1] in the context of CSPs and settle the major open question from that paper, finding a certain satisfiability problem on circuits-involving complement, intersection, union and multiplication-to be decidable. This we prove using the decidability of Skolem Arithmetic. Then we solve a second question left open in [1] by proving a tight upper bound for the similar circuit satisfiability problem involving just intersection, union and multiplication. We continue by studying first-order expansions of Skolem Arithmetic without constants, (N; ×), as CSPs. We find already here a rich landscape of problems with non-trivial instances that are in P as well as those that are NP-complete.

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Section: Recasting Circuit Results For Cspsmentioning

confidence: 99%

Circuit Satisfiability and Constraint Satisfaction Around Skolem Arithmetic

Glaßer

Jönsson

Martin³

2016

Pursuit of the Universal

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“…He was incorrect. Two papers together, one by Davis, Putnam, and Robinson [18] and one by Matijasevic [45] showed that H10 / ∈ DEC. They essentially showed that if this could be solved then the Halting problem could be solved.…”

Section: If X /mentioning

confidence: 99%

Classifying Problems into Complexity Classes

Gasarch

2014

Advances in Computers

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A fundamental problem in computer science is, stated informally: Given a problem, how hard is it?. We measure hardness by looking at the following question: Given a set A whats is the fastest algorithm to determine if "x ∈ A?" We measure the speed of an algorithm by how long it takes to run on inputs of length n, as a function of n. For example, sorting a list of length n can be done in roughly n log n steps.Obtaining a fast algorithm is only half of the problem. Can you prove that there is no better algorithm? This is notoriously difficult; however, we can classify problems into complexity classes where those in the same class are of roughly the same complexity.In this chapter we define many complexity classes and describe natural problems that are in them. Our classes go all the way from regular languages to various shades of undecidable. We then summarize all that is known about these classes.

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“…The first steps toward the eventual negative solution of the entire (unrestricted) form of Hilbert's tenth problem, were taken in 1961 by Julia Robinson, Martin Davis and Hilary Putnam [2]. They proved that every recursively enumerable set, W can be represented in exponential diophantine form 9 x v ) = 0.…”

Section: Undecidable Diophantine Equations By James P Jonesmentioning

confidence: 99%

Undecidable diophantine equations

Jones¹

1980

Bull. Amer. Math. Soc.

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The Decision Problem for Exponential Diophantine Equations

Cited by 274 publications

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Circuit Satisfiability and Constraint Satisfaction Around Skolem Arithmetic

Circuit Satisfiability and Constraint Satisfaction Around Skolem Arithmetic

Classifying Problems into Complexity Classes

Undecidable diophantine equations

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