Deciding Boolean Algebra with Presburger Arithmetic

Kunčak, Viktor; Nguyen, Huu Hai; Rinard, Martin

doi:10.1007/s10817-006-9042-1

Cited by 56 publications

(74 citation statements)

References 54 publications

(63 reference statements)

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“…Cardinality constraints naturally arise in quantifier elimination for boolean algebras [21]. The quantifier-free case of Boolean Algebra with Presburger Arithmetic is described in [5,Section 11], [28] with an non-deterministic exponential time decision procedure, which is also achieved as a special case of [18,19,24], [10, Section 8, Page 90]. Recently, [16, Section 7.9] gave a nondeterministic polynomial-time algorithm for quantifier-free Boolean Algebra with Presburger Arithmetic.…”

Section: Related Workmentioning

confidence: 99%

“…We have previously considered expressive logics that can express such constraints by combining the Boolean Algebra of sets with a cardinality operator and Presburger Arithmetic [18], [16,Chapter 7]. However, the NP-hardness of these constraints potentially limits their practical use, which motivated us to find constraints that have polynomial-time algorithms.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Polynomial Constraints for Sets with Cardinality Bounds

Marnette

Kunčak²,

Rinard³

2007

Foundations of Software Science and Computational Structures

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Abstract. Logics that can reason about sets and their cardinality bounds are useful in program analysis, program verification, databases, and knowledge bases. This paper presents a class of constraints on sets and their cardinalities for which the satisfiability and the entailment problems are computable in polynomial time. Our class of constraints, based on treeshaped formulas, is unique in being simultaneously tractable and able to express 1) that a set is a union of other sets, 2) that sets are disjoint, and 3) that a set has cardinality within a given range. As the main result we present a polynomial-time algorithm for checking entailment of our constraints.

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Section: Related Workmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Polynomial Constraints for Sets with Cardinality Bounds

Marnette

Kunčak²,

Rinard³

2007

Foundations of Software Science and Computational Structures

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“…The language in Figure 1 can be seen as a generalization of quantifier-free Boolean algebra with Presburger arithmetic (QFBAPA) [8]. Given that QF-BAPA admits quantifier elimination [3,6], it is interesting to ask whether multiset quantifiers can be eliminated from constraints of the present paper. Note that a multiset structure without cardinality operator can be viewed as a product of Presburger arithmetic structures.…”

Section: Inputmentioning

confidence: 99%

“…Moreover, such constraints often contain cardinality bounds on collections. Recent work describes decision procedures for constraints on sets of objects [6,8], characterizing the complexity of both quantified and quantifier-free constraints.…”

Section: Introductionmentioning

confidence: 99%

Decision Procedures for Multisets with Cardinality Constraints

Piskač

Kunčak

Lecture Notes in Computer Science

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Abstract. Applications in software verification and interactive theorem proving often involve reasoning about sets of objects. Cardinality constraints on such collections also arise in these applications. Multisets arise in these applications for analogous reasons as sets: abstracting the content of linked data structure with duplicate elements leads to multisets. Interactive theorem provers such as Isabelle specify theories of multisets and prove a number of theorems about them to enable their use in interactive verification. However, the decidability and complexity of constraints on multisets is much less understood than for constraints on sets. The first contribution of this paper is a polynomial-space algorithm for deciding expressive quantifier-free constraints on multisets with cardinality operators. Our decision procedure reduces in polynomial time constraints on multisets to constraints in an extension of quantifier-free Presburger arithmetic with certain "unbounded sum" expressions. We prove bounds on solutions of resulting constraints and describe a polynomialspace decision procedure for these constraints. The second contribution of this paper is a proof that adding quantifiers to a constraint language containing subset and cardinality operators yields undecidable constraints. The result follows by reduction from Hilbert's 10th problem.

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“…This process does not lose completeness, yet it improves the effectiveness of the theorem proving process because the resulting formulas are smaller than the starting formula. Moreover, splitting enables Jahob to prove different conjuncts using different techniques, allowing the translation described in this paper to be combined with other translations [22,44,45]. After splitting, the resulting formulas have the form of implications A 1 ∧ .…”

Section: Translation To First-order Logicmentioning

confidence: 99%

Using First-Order Theorem Provers in the Jahob Data Structure Verification System

Bouillaguet

Kunčak²,

Wies

et al.

Lecture Notes in Computer Science

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Abstract. This paper presents our integration of efficient resolution-based theorem provers into the Jahob data structure verification system. Our experimental results show that this approach enables Jahob to automatically verify the correctness of a range of complex dynamically instantiable data structures, including data structures such as hash tables and search trees, without the need for interactive theorem proving or techniques tailored to individual data structures. Our primary technical results include: (1) a translation from higher-order logic to first-order logic that enables the application of resolution-based theorem provers and (2) a proof that eliminating type (sort) information in formulas is both sound and complete, even in the presence of a generic equality operator. Our experimental results show that the elimination of type information dramatically decreases the time required to prove the resulting formulas. These techniques enabled us to verify complex correctness properties of Java programs such as a mutable set implemented as an imperative linked list, a finite map implemented as a functional ordered tree, a hash table with a mutable array, and a simple library system example that uses these container data structures. Our system verifies (in a matter of minutes) that data structure operations correctly update the finite map, that they preserve data structure invariants (such as ordering of elements, membership in appropriate hash table buckets, or relationships between sets and relations), and that there are no run-time errors such as null dereferences or array out of bounds accesses.

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Deciding Boolean Algebra with Presburger Arithmetic

Cited by 56 publications

References 54 publications

Polynomial Constraints for Sets with Cardinality Bounds

Polynomial Constraints for Sets with Cardinality Bounds

Decision Procedures for Multisets with Cardinality Constraints

Using First-Order Theorem Provers in the Jahob Data Structure Verification System

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