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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.
Automating Separation Logic Using SMT
"... Abstract. Separation logic (SL) has gained widespread popularity because of its ability to succinctly express complex invariants of a program’s heap configurations. Several specialized provers have been developed for decidable SL fragments. However, these provers cannot be easily extended or combine ..."
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Abstract. Separation logic (SL) has gained widespread popularity because of its ability to succinctly express complex invariants of a program’s heap configurations. Several specialized provers have been developed for decidable SL fragments. However, these provers cannot be easily extended or combined with solvers for other theories that are important in program verification, e.g., linear arithmetic. In this paper, we present a reduction of decidable SL fragments to a decidable firstorder theory that fits well into the satisfiability modulo theories (SMT) framework. We show how to use this reduction to automate satisfiability, entailment, frame inference, and abduction problems for separation logic using SMT solvers. Our approach provides a simple method of integrating separation logic into existing verification tools that provide SMT backends, and an elegant way of combining SL fragments with other decidable firstorder theories. We implemented this approach in a verification tool and applied it to heapmanipulating programs whose verification involves reasoning in theory combinations.
Sets with Cardinality Constraints in Satisfiability Modulo Theories
"... Abstract. Boolean Algebra with Presburger Arithmetic (BAPA) is a decidable logic that can express constraints on sets of elements and their cardinalities. Problems from verification of complex properties of software often contain fragments that belong to quantifierfree BAPA (QFBAPA). In contrast to ..."
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Abstract. Boolean Algebra with Presburger Arithmetic (BAPA) is a decidable logic that can express constraints on sets of elements and their cardinalities. Problems from verification of complex properties of software often contain fragments that belong to quantifierfree BAPA (QFBAPA). In contrast to many other NPcomplete problems (such as quantifierfree firstorder logic or linear arithmetic), the applications of QFBAPA to a broader set of problems has so far been hindered by the lack of an efficient implementation that can be used alongside other efficient decision procedures. We overcome these limitations by extending the efficient SMT solver Z3 with the ability to reason about cardinality (QFBAPA) constraints. Our implementation uses the DPLL(T) mechanism of Z3 to reason about the toplevel propositional structure of a QFBAPA formula, improving the efficiency compared to previous implementations. Moreover, we present a new algorithm for automatically decomposing QFBAPA formulas. Our algorithm alleviates the exponential explosion of considering all Venn regions, significantly improving the tractability of formulas with many set variables. Because it is implemented as a theory plugin, our implementation enables Z3 to prove formulas that use QFBAPA constructs with constructs from other theories that Z3 supports, as well as with quantifiers. We have applied our implementation to the verification of functional programs; we show it can automatically prove formulas that no automated approach was reported to be able to prove before. 1
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
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
Combinations of theories for decidable fragments of firstorder logic
"... Abstract. The design of decision procedures for firstorder theories and their combinations has been a very active research subject for thirty years; it has gained practical importance through the development of SMT (satisfiability modulo theories) solvers. Most results concentrate on combining deci ..."
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Abstract. The design of decision procedures for firstorder theories and their combinations has been a very active research subject for thirty years; it has gained practical importance through the development of SMT (satisfiability modulo theories) solvers. Most results concentrate on combining decision procedures for data structures such as theories for arrays, bitvectors, fragments of arithmetic, and uninterpreted functions. In particular, the wellknown NelsonOppen scheme for the combination of decision procedures requires the signatures to be disjoint and each theory to be stably infinite; every satisfiable set of literals in a stably infinite theory has an infinite model. In this paper we consider some of the bestknown decidable fragments of firstorder logic with equality, including the Löwenheim class (monadic FOL with equality, but without functions), BernaysSchönfinkelRamsey theories (finite sets of formulas of the form ∃ ∗ ∀ ∗ ϕ, where ϕ is a functionfree and quantifierfree FOL formula), and the twovariable fragment of FOL. In general, these are not stably infinite, and the NelsonOppen scheme cannot be used to integrate them into SMT solvers. Noticing some elementary results about the cardinalities of the models of these theories, we show that they can nevertheless be combined with almost any other decidable theory. 1
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 Proceedings of TACAS’98
, 1998
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Decision Procedures for the Temporal Verification of Concurrent Lists
"... Abstract. This paper studies the problem of formally verifying temporal properties of concurrent datatypes. Concurrent datatypes are implementations of classical data abstractions, specially designed to exploit the parallelism available in multiprocessor architectures. The correctness of concurrent ..."
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Abstract. This paper studies the problem of formally verifying temporal properties of concurrent datatypes. Concurrent datatypes are implementations of classical data abstractions, specially designed to exploit the parallelism available in multiprocessor architectures. The correctness of concurrent datatypes is essential for the overall correctness of the client software. The main difficulty to reason about concurrent datatypes is due to the simultaneous use of unstructured concurrency and dynamic memory. The first contribution of this paper is the use of deductive temporal verification methods, in particular verification diagrams, enriched with reasoning about dynamic memory. Proofs using verification diagrams are decomposed into a finite collection of verification conditions. Our second contribution is a decision procedure mixing memory regions, pointers and lisplike lists with locks, that allows the automatic verification of the generated verification conditions. We illustrate our techniques proving safety and liveness properties of lockcoupling concurrent lists. 1
On Deciding Functional Lists with Sublist Sets
"... Abstract. Motivated by the problem of deciding verification conditions for the verification of functional programs, we present new decision procedures for automated reasoning about functional lists. We first show how to decide in NP the satisfiability problem for logical constraints containing equal ..."
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Abstract. Motivated by the problem of deciding verification conditions for the verification of functional programs, we present new decision procedures for automated reasoning about functional lists. We first show how to decide in NP the satisfiability problem for logical constraints containing equality, constructor, selectors, as well as the transitive sublist relation. We then extend this class of constraints with operators to compute the set of all sublists, and the set of objects stored in a list. Finally, we support constraints on sizes of sets, which gives us the ability to compute list length as well as the number of distinct list elements. We show that the extended theory is reducible to the theory of sets with linear cardinality constraints, and therefore still in NP. This reduction enables us to combine our theory with other decidable theories that impose constraints on sets of objects, which further increases the potential of our decidability result in verification of functional and imperative software. 1
Combination of disjoint theories: beyond decidability
"... Abstract. Combination of theories underlies the design of satisfiability modulo theories (SMT) solvers. The NelsonOppen framework can be used to build a decision procedure for the combination of two disjoint decidable stably infinite theories. We here study combinations involving an arbitrary first ..."
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Abstract. Combination of theories underlies the design of satisfiability modulo theories (SMT) solvers. The NelsonOppen framework can be used to build a decision procedure for the combination of two disjoint decidable stably infinite theories. We here study combinations involving an arbitrary firstorder theory. Decidability is lost, but refutational completeness is preserved. We consider two cases and provide complete (semi)algorithms for them. First, we show that it is possible under minor technical conditions to combine a decidable (not necessarily stably infinite) theory and a disjoint finitely axiomatized theory, obtaining a refutationally complete procedure. Second, we provide a refutationally complete procedure for the union of two disjoint finitely axiomatized theories, that uses the assumed procedures for the underlying theories without modifying them. 1