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Decision procedures for algebraic data types with abstractions
 IN 37TH ACM SIGACTSIGPLAN SYMPOSIUM ON PRINCIPLES OF PROGRAMMING LANGUAGES (POPL), 2010. DECISION PROCEDURES FOR ORDERED COLLECTIONS 15 SHE75. SAHARON SHELAH. THE MONADIC THEORY OF ORDER. THA ANNALS OF MATHEMATICS OF MATHEMATICS
, 2010
"... We describe a family of decision procedures that extend the decision procedure for quantifierfree constraints on recursive algebraic data types (term algebras) to support recursive abstraction functions. Our abstraction functions are catamorphisms (term algebra homomorphisms) mapping algebraic data ..."
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We describe a family of decision procedures that extend the decision procedure for quantifierfree constraints on recursive algebraic data types (term algebras) to support recursive abstraction functions. Our abstraction functions are catamorphisms (term algebra homomorphisms) mapping algebraic data type values into values in other decidable theories (e.g. sets, multisets, lists, integers, booleans). Each instance of our decision procedure family is sound; we identify a widely applicable manytoone condition on abstraction functions that implies the completeness. Complete instances of our decision procedure include the following correctness statements: 1) a functional data structure implementation satisfies a recursively specified invariant, 2) such data structure conforms to a contract given in terms of sets, multisets, lists, sizes, or heights, 3) a transformation of a formula (or lambda term) abstract syntax tree changes the set of free variables in the specified way.
On Combining Theories with Shared Set Operations
"... Abstract. We explore the problem of automated reasoning about the nondisjoint combination of theories that share set variables and operations. We prove a combination theorem and apply it to show the decidability of the satisfiability problem for a class of formulas obtained by applying propositional ..."
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Abstract. We explore the problem of automated reasoning about the nondisjoint combination of theories that share set variables and operations. We prove a combination theorem and apply it to show the decidability of the satisfiability problem for a class of formulas obtained by applying propositional operations to quantified formulas belonging to several expressive decidable logics. 1
On Linear Arithmetic with Stars
"... Abstract. We consider an extension of integer linear arithmetic with a star operator that takes closure under vector addition of the set of solutions of linear arithmetic subformula. We show that the satisfiability problem for this language is in NP (and therefore NPcomplete). Our proof uses a gene ..."
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Abstract. We consider an extension of integer linear arithmetic with a star operator that takes closure under vector addition of the set of solutions of linear arithmetic subformula. We show that the satisfiability problem for this language is in NP (and therefore NPcomplete). Our proof uses a generalization of a recent result on sparse solutions of integer linear programming problems. We present two consequences of our result. The first one is an optimal decision procedure for a logic of sets, multisets, and cardinalities that has applications in verification, interactive theorem proving, and description logics. The second is NPcompleteness of the reachability problem for a class of “homogeneous ” transition systems whose transitions are defined using integer linear arithmetic formulas. 1
Symbolic Bounded Model Checking of Abstract State Machines
, 2009
"... Abstract State Machines (ASMs) allow modeling system behaviors at any desired level of abstraction, including a level with rich data types, such as sets, sequences, maps, and userdefined data types. The availability of highlevel data types allow state elements to be represented both abstractly an ..."
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Abstract State Machines (ASMs) allow modeling system behaviors at any desired level of abstraction, including a level with rich data types, such as sets, sequences, maps, and userdefined data types. The availability of highlevel data types allow state elements to be represented both abstractly and faithfully at the same time. In this paper we look at symbolic analysis of ASMs. We consider ASMs over a fixed state background T that includes linear arithmetic, sets, tuples, and maps. For symbolic analysis, ASMs are translated into guarded update systems called model programs. We formulate the problem of bounded path exploration of model programs, or the problem of Bounded Model Program Checking (BMPC) as a satisfiability problem modulo T. Then we investigate the boundaries of decidable and undecidable cases for BMPC. In a general setting, BMPC is shown to be highly undecidable (Σ 1 1complete); and even when restricting to finite sets the problem remains rehard (Σ 0 1hard). On the other hand, BMPC is shown to be decidable for a class of basic model programs that are common in practice. We use Satisfiability Modulo Theories (SMT) for solving BMPC; an instance of the BMPC problem is mapped to a formula, the formula is satisfiable modulo T if and only if
Ordered Sets in the Calculus of Data Structures
"... Abstract. Our goal is to identify families of relations that are useful for reasoning about software. We describe such families using decidable quantifierfree classes of logical constraints with a rich set of operations. A key challenge is to define such classes of constraints in a modular way, by ..."
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Abstract. Our goal is to identify families of relations that are useful for reasoning about software. We describe such families using decidable quantifierfree classes of logical constraints with a rich set of operations. A key challenge is to define such classes of constraints in a modular way, by combining multiple decidable classes. Working with quantifierfree combinations of constraints makes the combination agenda more realistic and the resulting logics more likely to be tractable than in the presence of quantifiers. Our approach to combination is based on reducing decidable fragments to a common class, Boolean Algebra with Presburger Arithmetic (BAPA). This logic was introduced by Feferman and Vaught in 1959 and can express properties of uninterpreted sets of elements, with set algebra operations and equicardinality relation (consequently, it can also express Presburger arithmetic constraints on cardinalities of sets). Combination by reduction to BAPA allows us to obtain decidable quantifierfree combinations
What’s Decidable About Sequences?
, 2010
"... We present a firstorder theory of sequences with integer elements, Presburger arithmetic, and regular constraints, which can model significant properties of data structures such as arrays and lists. We give a decision procedure for the quantifierfree fragment, based on an encoding into the firsto ..."
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We present a firstorder theory of sequences with integer elements, Presburger arithmetic, and regular constraints, which can model significant properties of data structures such as arrays and lists. We give a decision procedure for the quantifierfree fragment, based on an encoding into the firstorder theory of concatenation; the procedure has PSPACE complexity. The quantifierfree fragment of the theory of sequences can express properties such as sortedness and injectivity, as well as Boolean combinations of periodic and arithmetic facts relating the elements of the sequence and their positions (e.g., “for all even i’s, the element at position i has value i + 3 or 2i”). The resulting expressive power is orthogonal to that of the most expressive decidable logics for arrays. Some examples demonstrate that the fragment is also suitable to reason about sequencemanipulating programs within the standard framework of axiomatic
Symbolic Bounded Model Checking of Abstract State Machines
, 2009
"... Abstract State Machines (ASMs) allow modeling system behaviors at any desired level of abstraction, including a level with rich data types, such as sets or sequences. The availability of highlevel data types allow state elements to be represented both abstractly and faithfully at the same time. Asm ..."
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Abstract State Machines (ASMs) allow modeling system behaviors at any desired level of abstraction, including a level with rich data types, such as sets or sequences. The availability of highlevel data types allow state elements to be represented both abstractly and faithfully at the same time. AsmL is a rich ASMbased specification and programming language. In this paper we look at symbolic analysis of model programs written in AsmL with a background T of linear arithmetic, sets, tuples, and maps. We first provide a rigorous account for the update semantics of AsmL in terms of T, and formulate the problem of bounded path exploration of model programs, or the problem of Bounded Model Program Checking (BMPC) as a satisfiability modulo T problem. Then we investigate the boundaries of decidable and undecidable cases for BMPC. In a general setting, BMPC is shown to be highly undecidable, it is effectively equivalent to satisfiability in secondorder Peano arithmetic with sets (Σ1 1complete); and even when restricting to finite sets the problem is as hard as the halting problem of
Collections, Cardinalities, and Relations
"... Abstract. Logics that involve collections (sets, multisets), and cardinality constraints are useful for reasoning about unbounded data structures and concurrent processes. To make such logics more useful in verification this paper extends them with the ability to compute direct and inverse relation ..."
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Abstract. Logics that involve collections (sets, multisets), and cardinality constraints are useful for reasoning about unbounded data structures and concurrent processes. To make such logics more useful in verification this paper extends them with the ability to compute direct and inverse relation and function images. We establish decidability and complexity bounds for the extended logics. 1
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 Proceedings of TACAS’98
, 1998
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Decision Procedures for Automating Termination Proofs
"... Abstract. Automated termination provers often use the following schema to prove that a program terminates: construct a relational abstraction of the program’s transition relation and then show that the relational abstraction is wellfounded. The focus of current tools has been on developing sophistic ..."
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Abstract. Automated termination provers often use the following schema to prove that a program terminates: construct a relational abstraction of the program’s transition relation and then show that the relational abstraction is wellfounded. The focus of current tools has been on developing sophisticated techniques for constructing the abstractions while relying on known decidable logics (such as linear arithmetic) to express them. We believe we can significantly increase the class of programs that are amenable to automated termination proofs by identifying more expressive decidable logics for reasoning about wellfounded relations. We therefore present a new decision procedure for reasoning about multiset orderings, which are among the most powerful orderings used to prove termination. We show that, using our decision procedure, one can automatically prove termination of natural abstractions of programs. 1