Efficient Interpolation for the Theory of Arrays

Hoenicke, Jochen; Schindler, Tanja

doi:10.1007/978-3-319-94205-6_36

Cited by 17 publications

(16 citation statements)

References 33 publications

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“…By itself, the theory of arrays with extensionality does not have quantifier free interpolation [28] 5 ; however, in [8] it was shown that quantifier-free interpolation is restored if one enriches the language with a binary function skolemizing the extensionality axiom (the result was confirmed -via different interpolation algorithms -in [23,37]). Such a Skolem function, applied to two array variables a, b, returns an index diff(a, b) where a, b differ (it returns an arbitrary value if a is equal to b).…”

Section: Introductionmentioning

confidence: 99%

Interpolation and Amalgamation for Arrays with MaxDiff

Ghilardi

Gianola

Kapur

2021

Lecture Notes in Computer Science

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In this paper, the theory of McCarthy’s extensional arrays enriched with a maxdiff operation (this operation returns the biggest index where two given arrays differ) is proposed. It is known from the literature that a diff operation is required for the theory of arrays in order to enjoy the Craig interpolation property at the quantifier-free level. However, the diff operation introduced in the literature is merely instrumental to this purpose and has only a purely formal meaning (it is obtained from the Skolemization of the extensionality axiom). Our maxdiff operation significantly increases the level of expressivity; however, obtaining interpolation results for the resulting theory becomes a surprisingly hard task. We obtain such results via a thorough semantic analysis of the models of the theory and of their amalgamation properties. The results are modular with respect to the index theory and it is shown how to convert them into concrete interpolation algorithms via a hierarchical approach.

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Section: Introductionmentioning

confidence: 99%

Interpolation and Amalgamation for Arrays with MaxDiff

Ghilardi

Gianola

Kapur

2021

Lecture Notes in Computer Science

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“…Craig interpolation is a commonly used technique to infer invariants or contracts in verification. Over the last 15 years, efficient interpolation techniques have been developed for a variety of logics and theories, including propositional logic [1,2], uninterpreted functions [1,3,4], first-order logic [5][6][7], algebraic data-types [8,9], linear real arithmetic [1], non-linear real arithmetic [10], Presburger arithmetic [4,11,12], and arrays [13][14][15].…”

Section: Introductionmentioning

confidence: 99%

Interpolating bit-vector formulas using uninterpreted predicates and Presburger arithmetic

Backeman

Rümmer

Zeljić

2021

Form Methods Syst Des

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The inference of program invariants over machine arithmetic, commonly called bit-vector arithmetic, is an important problem in verification. Techniques that have been successful for unbounded arithmetic, in particular Craig interpolation, have turned out to be difficult to generalise to machine arithmetic: existing bit-vector interpolation approaches are based either on eager translation from bit-vectors to unbounded arithmetic, resulting in complicated constraints that are hard to solve and interpolate, or on bit-blasting to propositional logic, in the process losing all arithmetic structure. We present a new approach to bit-vector interpolation, as well as bit-vector quantifier elimination (QE), that works by lazy translation of bit-vector constraints to unbounded arithmetic. Laziness enables us to fully utilise the information available during proof search (implied by decisions and propagation) in the encoding, and this way produce constraints that can be handled relatively easily by existing interpolation and QE procedures for Presburger arithmetic. The lazy encoding is complemented with a set of native proof rules for bit-vector equations and non-linear (polynomial) constraints, this way minimising the number of cases a solver has to consider. We also incorporate a method for handling concatenations and extractions of bit-vector efficiently.

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“…Craig interpolation is a commonly used technique to infer invariants or contracts in verification. Over the last 15 years, efficient interpolation techniques have been developed for a variety of logics and theories, including propositional logic [1], [2], uninterpreted functions [1], [3], [4], first-order logic [5], [6], [7], algebraic data-types [8], linear real arithmetic [1], non-linear real arithmetic [9], Presburger arithmetic [10], [4], [11], and arrays [12], [13], [14].…”

Section: Introductionmentioning

confidence: 99%

Bit-Vector Interpolation and Quantifier Elimination by Lazy Reduction

Backeman

Rümmer

Zeljić

2018

2018 Formal Methods in Computer Aided Design (FMCAD)

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The inference of program invariants over machine arithmetic, commonly called bit-vector arithmetic, is an important problem in verification. Techniques that have been successful for unbounded arithmetic, in particular Craig interpolation, have turned out to be difficult to generalise to machine arithmetic: existing bit-vector interpolation approaches are based either on eager translation from bit-vectors to unbounded arithmetic, resulting in complicated constraints that are hard to solve and interpolate, or on bit-blasting to propositional logic, in the process losing all arithmetic structure. We present a new approach to bitvector interpolation, as well as bit-vector quantifier elimination (QE), that works by lazy translation of bit-vector constraints to unbounded arithmetic. Laziness enables us to fully utilise the information available during proof search (implied by decisions and propagation) in the encoding, and this way produce constraints that can be handled relatively easily by existing interpolation and QE procedures for Presburger arithmetic. The lazy encoding is complemented with a set of native proof rules for bit-vector equations and non-linear (polynomial) constraints, this way minimising the number of cases a solver has to consider.

show abstract

Efficient Interpolation for the Theory of Arrays

Cited by 17 publications

References 33 publications

Interpolation and Amalgamation for Arrays with MaxDiff

Interpolation and Amalgamation for Arrays with MaxDiff

Interpolating bit-vector formulas using uninterpreted predicates and Presburger arithmetic

Bit-Vector Interpolation and Quantifier Elimination by Lazy Reduction

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