On Decidability Within the Arithmetic of Addition and Divisibility

Bozga, Marius; Iosif, Radu

doi:10.1007/978-3-540-31982-5_27

Cited by 15 publications

(16 citation statements)

References 10 publications

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“…the modal µ-calculus, such a problem is undecidable by Theorem 6. However for LTL it seems conceivable that this problem can be translated into a sentence of a decidable fragment of Presburger arithmetic with divisibility, similar to those studied in [4].…”

Section: Resultsmentioning

confidence: 99%

Model Checking Succinct and Parametric One-Counter Automata

Göller

Haase

Ouaknine

et al. 2010

Lecture Notes in Computer Science

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Abstract. We investigate the decidability and complexity of various model checking problems over one-counter automata. More specifically, we consider succinct one-counter automata, in which additive updates are encoded in binary, as well as parametric one-counter automata, in which additive updates may be given as unspecified parameters. We fully determine the complexity of model checking these automata against CTL, LTL, and modal µ-calculus specifications.

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Section: Resultsmentioning

confidence: 99%

Model Checking Succinct and Parametric One-Counter Automata

Göller

Haase

Ouaknine

et al. 2010

Lecture Notes in Computer Science

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“…Among the decision procedures for full PA, (Chaieb and Nipkow, 2003) is the only proof-generating version, and is based on (Cooper, 1972). Decidable fragments of arithmetic that go beyond PA are described in (Bozga and Iosif, 2005;Bruyére et al, 1994). Reasoning about Sets.…”

Section: Related Workmentioning

confidence: 99%

Deciding Boolean Algebra with Presburger Arithmetic

Kunčak¹,

Nguyen

Rinard

2006

J Autom Reasoning

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Abstract. We describe an algorithm for deciding the first-order multisorted theory BAPA, which combines 1) Boolean algebras of sets of uninterpreted elements (BA) and 2) Presburger arithmetic operations (PA). BAPA can express the relationship between integer variables and cardinalities of unbounded finite sets, and supports arbitrary quantification over sets and integers.Our original motivation for BAPA is deciding verification conditions that arise in the static analysis of data structure consistency properties. Data structures often use an integer variable to keep track of the number of elements they store; an invariant of such a data structure is that the value of the integer variable is equal to the number of elements stored in the data structure. When the data structure content is represented by a set, the resulting constraints can be captured in BAPA. BAPA formulas with quantifier alternations arise when verifying programs with annotations containing quantifiers, or when proving simulation relation conditions for refinement and equivalence of program fragments. Furthermore, BAPA constraints can be used for proving the termination of programs that manipulate data structures, as well as in constraint database query evaluation and loop invariant inference.We give a formal description of an algorithm for deciding BAPA. We analyze our algorithm and show that it has optimal alternating time complexity, and that the complexity of BAPA matches the complexity of PA. Because it works by a reduction to PA, our algorithm yields the decidability of a combination of sets of uninterpreted elements with any decidable extension of PA. When restricted to BA formulas, the algorithm can be used to decide BA in optimal alternating time. Furthermore, the algorithm can eliminate individual quantifiers from a formula with free variables and therefore perform projection onto a desirable set of variables.We have implemented our algorithm and used it to discharge verification conditions in the Jahob system for data structure consistency checking of Java programs; our experience suggest that a straightforward implementation of the algorithm is effective on non-trivial formulas as long as the number of set variables is small. We also report on a new algorithm for solving the quantifier-free fragment of BAPA.

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“…Among the decision procedures for full PA, [9] is the only proofgenerating version, and is based on [11]. Decidable fragments of arithmetic that go beyond PA include [6,21]. Boolean Algebras.…”

Section: Related Workmentioning

confidence: 99%

An Algorithm for Deciding BAPA: Boolean Algebra with Presburger Arithmetic

Kunčak

Nguyen

Rinard

2005

Lecture Notes in Computer Science

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Abstract. We describe an algorithm for deciding the first-order multisorted theory BAPA, which combines 1) Boolean algebras of sets of uninterpreted elements (BA) and 2) Presburger arithmetic operations (PA). BAPA can express the relationship between integer variables and cardinalities of a priory unbounded finite sets, and supports arbitrary quantification over sets and integers. Our motivation for BAPA is deciding verification conditions that arise in the static analysis of data structure consistency properties. Data structures often use an integer variable to keep track of the number of elements they store; an invariant of such a data structure is that the value of the integer variable is equal to the number of elements stored in the data structure. When the data structure content is represented by a set, the resulting constraints can be captured in BAPA. BAPA formulas with quantifier alternations arise when verifying programs with annotations containing quantifiers, or when proving simulation relation conditions for refinement and equivalence of program fragments. Furthermore, BAPA constraints can be used for proving the termination of programs that manipulate data structures, and have applications in constraint databases. We give a formal description of a decision procedure for BAPA, which implies the decidability of BAPA. We analyze our algorithm and obtain an elementary upper bound on the running time, thereby giving the first complexity bound for BAPA. Because it works by a reduction to PA, our algorithm yields the decidability of a combination of sets of uninterpreted elements with any decidable extension of PA. Our algorithm can also be used to yield an optimal decision procedure for BA through a reduction to PA with bounded quantifiers. We have implemented our algorithm and used it to discharge verification conditions in the Jahob system for data structure consistency checking of Java programs; our experience with the algorithm is promising.

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On Decidability Within the Arithmetic of Addition and Divisibility

Cited by 15 publications

References 10 publications

Model Checking Succinct and Parametric One-Counter Automata

Model Checking Succinct and Parametric One-Counter Automata

Deciding Boolean Algebra with Presburger Arithmetic

An Algorithm for Deciding BAPA: Boolean Algebra with Presburger Arithmetic

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