NP Satisfiability for Arrays as Powers

Raya, Rodrigo; Kunčak, Viktor

doi:10.1007/978-3-030-94583-1_15

Cited by 7 publications

(7 citation statements)

References 41 publications

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“…Finally, although quite challenging, it would be interesting to extend our interpolation results also to array theories combined with cardinality constraints, similar to those introduced, e.g., in [2], [27].…”

Section: Further Related Work and Conclusionmentioning

confidence: 95%

General Interpolation and Strong Amalgamation for Contiguous Arrays

Ghilardi¹,

Gianola²,

Kapur³

et al. 2022

Preprint

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Interpolation is an essential tool in software verification, where first-order theories are used to constrain datatypes manipulated by programs. In this paper, we introduce the datatype theory of contiguous arrays with maxdiff, where arrays are completely defined in their allocation memory and for which maxdiff returns the max index where they differ. This theory is strictly more expressive than the array theories previously studied. By showing via an algebraic analysis that its models strongly amalgamate, we prove that this theory admits quantifier-free interpolants and, notably, that interpolation transfers to theory combinations. Finally, we provide an algorithm that significantly improves the ones for related array theories: it relies on a polysize reduction to general interpolation in linear arithmetics, thus avoiding impractical full terms instantiations and unbounded loops.

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Section: Further Related Work and Conclusionmentioning

confidence: 95%

General Interpolation and Strong Amalgamation for Contiguous Arrays

Ghilardi¹,

Gianola²,

Kapur³

et al. 2022

Preprint

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“…Our work is related to a long tradition of research in decision procedures for set constraints. These have found applications in the verification and analysis of programs [1,2,27,34]. In [18], the authors studied the complexity of deciding set constrains in the style of Jònnson and Tarski's framework of Boolean algebra with operations.…”

Section: Related Workmentioning

confidence: 99%

“…We believe that such results are of interest because they compose with other constructions that preserve NP membership. In particular, in a recent analysis of array theories [34] we observed that the fragment of combinatory array logic [12] corresponds to the theory generated by a power structure with an arbitrary index set and with QFBAPA constraints on the index set. Given that [34] shows NP complexity for such product, it is natural to ask how far we can extend NP satisfiability results.…”

Section: Introductionmentioning

confidence: 99%

“…In particular, in a recent analysis of array theories [34] we observed that the fragment of combinatory array logic [12] corresponds to the theory generated by a power structure with an arbitrary index set and with QFBAPA constraints on the index set. Given that [34] shows NP complexity for such product, it is natural to ask how far we can extend NP satisfiability results. The non-linear constraints we present in this paper can be applied to the case when the index set I is a power J n , because image constraints with functions on subsets of J n reduces to non-linear constraints whose complexity we consider.…”

Section: Introductionmentioning

confidence: 99%

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NP Decision Procedure for Monomial and Linear Integer Constraints

Raya¹,

Hamza²,

Kunčak³

2022

Preprint

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Motivated by satisfiability of constraints with function symbols, we consider numerical inequalities on non-negative integers. The constraints we consider are a conjunction of a linear system Ax = b and a conjunction of non-convex constraints of the form xi ≤ x n j . We show that the satisfiability of these constraints is in NP. As a consequence, we obtain NP completeness for an extension of quantifier-free constraints on sets with cardinalities (QFBAPA) with function images S = f [P n ]. We also present related hardness results and consequences of complexity for dual, convex, constraints.

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“…We believe that such results are of interest because they compose with other constructions that preserve NP membership. In particular, in a recent analysis of array theories [45] we observed that the fragment of combinatory array logic [13] corresponds to the theory generated by a power structure with an arbitrary index set subject to QFBAPA constraints. Given that [45] shows a NP complexity bound for such product, it is natural to ask how far we can extend NP satisfiability results.…”

Section: Introductionmentioning

confidence: 99%

On the Complexity of Convex and Reverse Convex Prequadratic Constraints

Raya¹,

Hamza²,

Kunčak³

EPiC Series in Computing

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Motivated by satisfiability of constraints with function symbols, we consider numerical inequalities on non-negative integers. The constraints we address are a conjunction of a linear system Ax = b and an arbitrary number of (reverse) convex constraints of the form xi ≥ xdj (xi ≤ xdj ). We show that the satisfiability of these constraints is NP-complete even if the solution to the linear part is given explicitly. As a consequence, we obtain NP- completeness for an extension of certain quantifier-free constraints on sets with cardinalities and function images.

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NP Satisfiability for Arrays as Powers

Cited by 7 publications

References 41 publications

General Interpolation and Strong Amalgamation for Contiguous Arrays

General Interpolation and Strong Amalgamation for Contiguous Arrays

NP Decision Procedure for Monomial and Linear Integer Constraints

On the Complexity of Convex and Reverse Convex Prequadratic Constraints

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