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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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Cited by 24 (10 self)
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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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Cited by 11 (5 self)
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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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Cited by 8 (6 self)
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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 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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Cited by 4 (4 self)
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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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Cited by 4 (2 self)
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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
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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Cited by 4 (2 self)
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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, 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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Cited by 3 (1 self)
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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
On Bounded Reachability of Programs with Set
"... Abstract. We analyze the bounded reachability problem of programs that use abstract data types and set comprehensions. Such programs are common as highlevel executable specifications of complex protocols. We prove decidability and undecidability results of restricted cases of the problem and extend ..."
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Abstract. We analyze the bounded reachability problem of programs that use abstract data types and set comprehensions. Such programs are common as highlevel executable specifications of complex protocols. We prove decidability and undecidability results of restricted cases of the problem and extend the Satisfiability Modulo Theories approach to support analysis of set comprehensions over tuples and bag axioms. We use the Z3 solver for our implementation and experiments, and we use AsmL as the modeling language. 1
On Decision Procedures for Collections, Cardinalities, and Relations
, 2009
"... 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 ..."
Abstract
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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
MUNCH Automated Reasoner for Sets and
"... Abstract. This system description provides an overview of the MUNCH reasoner for sets and multisets. MUNCH takes as the input a formula in a logic that supports expressions about sets, multisets, and integers. Constraints over collections and integers are connected using the cardinality operator. Ou ..."
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Abstract. This system description provides an overview of the MUNCH reasoner for sets and multisets. MUNCH takes as the input a formula in a logic that supports expressions about sets, multisets, and integers. Constraints over collections and integers are connected using the cardinality operator. Our logic is a fragment of logics of popular interactive theorem provers, and MUNCH is the first fully automated reasoner for this logic. MUNCH reduces input formulas to equisatisfiable linear integer arithmetic formulas. MUNCH reasoner is publicly available. It is implemented in the Scala programming language and currently uses the SMT solver Z3 to solve the generated integer linear arithmetic constraints. 1