Satisfiability of algebraic circuits over sets of natural numbers

Glaíer, Christian; Reitwieíner, Christian; Travers, Stephen; Waldherr, Matthias

doi:10.1016/j.dam.2010.04.001

Cited by 7 publications

(27 citation statements)

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“…Meanwhile, a literature existed on satisfiability of circuit problems over sets of integers involving work of the first author [1], itself continuing a line of investigation begun in [22] and pursued in [23,24,25]. The circuits typically compute some set of integers at their unique output node and one asks for satisfiability in terms of evaluations of free set-variables at their input nodes.…”

Section: Introductionmentioning

confidence: 99%

“…The circuits typically compute some set of integers at their unique output node and one asks for satisfiability in terms of evaluations of free set-variables at their input nodes. The problems in [1] can be seen as variants of certain functional CSPs whose domain is all singleton sets of the non-negative integers and whose relations are set operations of the form: complement, intersection, union, addition and multiplication (the latter two are defined set-wise, e.g. A × B := {ab : a ∈ A ∧ b ∈ B}).…”

Section: Introductionmentioning

confidence: 99%

“…We take here Skolem Arithmetic to be the non-negative integers with multiplication (and possibly constants). In studying this problem we are able to bring to light existing results of [1] as results about their related CSPs, providing natural examples with interesting super-NP complexities (remember the constraint language here is not a reduct of Skolem Arithmetic but rather built from singleton sets of non-negative integers). In addition, we improve one of the upper bounds of [1] to a tight upper bound.…”

Section: Introductionmentioning

confidence: 99%

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Circuit satisfiability and constraint satisfaction around Skolem Arithmetic

Glaßer

Jönsson

Martin

2017

Theoretical Computer Science

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Circuit satisfiability and constraint satisfaction around Skolem Arithmetic

Glaßer

Jönsson

Martin

2017

Theoretical Computer Science

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“…(1.5) It is natural to generalize the notion of arithmetic circuits by allowing input nodes to represent variable sets of numbers [5]. Logically speaking, we enhance our language by a set V of variables which are interpreted as sets of natural numbers; arithmetic circuits correspond to the variable free terms of this language.…”

Section: Introductionmentioning

confidence: 99%

“…Furthermore, the operations f : (2 ω ) k → 2 ω definable from the given operators O can be studied [16]. In analogy to the membership problem, Glaßer et al [5] consider the complexity of…”

Section: Introductionmentioning

confidence: 99%

Complex Algebras of Arithmetic

Düntsch

Pratt-Hartmann

2009

Fundamenta Informaticae

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An arithmetic circuit is a labeled, acyclic directed graph specifying a sequence of arithmetic and logical operations to be performed on sets of natural numbers. Arithmetic circuits can also be viewed as the elements of the smallest subalgebra of the complex algebra of the semiring of natural numbers. In the present paper we investigate the algebraic structure of complex algebras of natural numbers and make some observations regarding the complexity of various theories of such algebras.

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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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Satisfiability of algebraic circuits over sets of natural numbers

Cited by 7 publications

References 2 publications

Circuit satisfiability and constraint satisfaction around Skolem Arithmetic

Circuit satisfiability and constraint satisfaction around Skolem Arithmetic

Complex Algebras of Arithmetic

Circuit Satisfiability and Constraint Satisfaction Around Skolem Arithmetic

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