Interpolation for data structures

Kapur, Deepak; Majumdar, Rupak; Zarba, Calogero G.

doi:10.1145/1181775.1181789

Cited by 76 publications

(93 citation statements)

References 40 publications

(69 reference statements)

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“…Kapur et al [7] prove that full QFPA is closed under interpolation (as an instance of a more general result about recursively enumerable theories), but their proof does not directly give rise to an efficient interpolation procedure. Lynch et al [8] define an interpolation procedure for linear rational arithmetic, and extend it to integer arithmetic by means of Gomory cuts.…”

Section: Related Work and Conclusionmentioning

confidence: 99%

“…In order to support expressive programming languages, much effort has been invested in the design of algorithms that compute interpolants for formulae of various theories. As a result, efficient interpolation methods are known for propositional logic, linear arithmetic over the reals with uninterpreted functions [10,1,16], datastructures like arrays and sets [7], and other theories. As for integer arithmetic, a theory particularly relevant for software, interpolating solvers have so far been reported only for restricted fragments such as difference-bound logic, and logics with linear equalities and constant-divisibility predicates.…”

Section: Introductionmentioning

confidence: 99%

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An Interpolating Sequent Calculus for Quantifier-Free Presburger Arithmetic

Brillout

Kroening

Rümmer

et al. 2010

Automated Reasoning

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Abstract. Craig interpolation has become a versatile tool in formal verification, for instance to generate intermediate assertions for safety analysis of programs. Interpolants are typically determined by annotating the steps of an unsatisfiability proof with partial interpolants. In this paper, we consider Craig interpolation for full quantifier-free Presburger arithmetic (QFPA), for which currently no efficient interpolation procedures are known. Closing this gap, we introduce an interpolating sequent calculus for QFPA and prove it to be sound and complete. We have extended the Princess theorem prover to generate interpolating proofs, and applied it to a large number of publicly available linear integer arithmetic benchmarks. The results indicate the robustness and efficiency of our proof-based interpolation procedure.

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Section: Related Work and Conclusionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

An Interpolating Sequent Calculus for Quantifier-Free Presburger Arithmetic

Brillout

Kroening

Rümmer

et al. 2010

Automated Reasoning

View full text Add to dashboard Cite

show abstract

“…Another problem we analyze is interpolation (widely used in program verification [22,23,24,16,18]). Intuitively, interpolants can be used for describing separations between the sets of "good" and "bad" states; they can help to discover relevant predicates in predicate abstraction with refinement and for over-approximation in model checking.…”

Section: Introductionmentioning

confidence: 99%

“…The first algorithms for interpolant generation in program verification required explicit constructions of proofs [19,23], which in general is a relatively difficult task. In [18] the existence of ground interpolants for arbitrary formulae w.r.t. a theory T is studied.…”

Section: Introductionmentioning

confidence: 99%

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On Interpolation and Symbol Elimination in Theory Extensions

Sofronie-Stokkermans

2016

Automated Reasoning

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Abstract. In this paper we study possibilities of interpolation and symbol elimination in extensions of a theory T0 with additional function symbols whose properties are axiomatised using a set of clauses. We analyze situations in which we can perform such tasks in a hierarchical way, relying on existing mechanisms for symbol elimination in T0. This is for instance possible if the base theory allows quantifier elimination. We analyze possibilities of extending such methods to situations in which the base theory does not allow quantifier elimination but has a model completion which does. We illustrate the method on various examples.

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Rectification of Arithmetic Circuits with Craig Interpolants in Finite Fields

Gupta

Ilioaea

Rao

et al. 2019

VLSI-SoC: Design and Engineering of Electronics Systems Based on New Computing Paradigms

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When formal verification of arithmetic circuits identifies the presence of a bug in the design, the task of rectification needs to be performed to correct the function implemented by the circuit so that it matches the given specification. In our recent work [26], we addressed the problem of rectification of buggy finite field arithmetic circuits. The problems are formulated by means of a set of polynomials (ideals) and solutions are proposed using concepts from computational algebraic geometry. Single-fix rectification is addressed -i.e. the case where any set of bugs can be rectified at a single net (gate output). We determine if single-fix rectification is possible at a particular net, formulated as the Weak Nullstellensatz test and solved using Gröbner bases. Subsequently, we introduce the concept of Craig interpolants in polynomial algebra over finite fields and show that the rectification function can be computed using algebraic interpolants. This article serves as an extension to our previous work, provides a formal definition of Craig interpolants in finite fields using algebraic geometry and proves their existence. We also describe the computation of interpolants using elimination ideals with Gröbner bases and prove that our procedure computes the smallest interpolant. As the Gröbner basis algorithm exhibits high computational complexity, we further propose an efficient approach to compute interpolants. Experiments are conducted over a variety of finite field arithmetic circuits which demonstrate the superiority of our approach against SAT-based approaches.

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Interpolation for data structures

Cited by 76 publications

References 40 publications

An Interpolating Sequent Calculus for Quantifier-Free Presburger Arithmetic

An Interpolating Sequent Calculus for Quantifier-Free Presburger Arithmetic

On Interpolation and Symbol Elimination in Theory Extensions

Rectification of Arithmetic Circuits with Craig Interpolants in Finite Fields

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