Results 11  20
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98
Complete Functional Synthesis
"... Synthesis of program fragments from specifications can make programs easier to write and easier to reason about. To integrate synthesis into programming languages, synthesis algorithms should behave in a predictable way—they should succeed for a welldefined class of specifications. They should also ..."
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Cited by 29 (12 self)
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Synthesis of program fragments from specifications can make programs easier to write and easier to reason about. To integrate synthesis into programming languages, synthesis algorithms should behave in a predictable way—they should succeed for a welldefined class of specifications. They should also support unbounded data types such as numbers and data structures. We propose to generalize decision procedures into predictable and complete synthesis procedures. Such procedures are guaranteed to find code that satisfies the specification if such code exists. Moreover, we identify conditions under which synthesis will statically decide whether the solution is guaranteed to exist, and whether it is unique. We demonstrate our approach by starting from decision procedures for linear arithmetic and data structures and transforming them into synthesis procedures. We establish results on the size and the efficiency of the synthesized code. We show that such procedures are useful as a language extension with implicit value definitions, and we show how to extend a compiler to support such definitions. Our constructs provide the benefits of synthesis to programmers, without requiring them to learn new concepts or give up a deterministic execution model.
Solving quantified verification conditions using satisfiability modulo theories
 In CADE
, 2007
"... Abstract. First order logic provides a convenient formalism for describing a wide variety of verification conditions. Two main approaches to checking such conditions are pure first order automated theorem proving (ATP) and automated theorem proving based on satisfiability modulo theories (SMT). Trad ..."
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Cited by 29 (1 self)
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Abstract. First order logic provides a convenient formalism for describing a wide variety of verification conditions. Two main approaches to checking such conditions are pure first order automated theorem proving (ATP) and automated theorem proving based on satisfiability modulo theories (SMT). Traditional ATP systems are designed to handle quantifiers easily, but often have difficulty reasoning with respect to theories. SMT systems, on the other hand, have builtin support for many useful theories, but have a much more difficult time with quantifiers. One clue on how to get the best of both worlds can be found in the legacy system Simplify which combines builtin theory reasoning with quantifier instantiation heuristics. Inspired by Simplify and motivated by a desire to provide a competitive alternative to ATP systems, this paper describes a methodology for reasoning about quantifiers in SMT systems. We present the methodology in the context of the Abstract DPLL Modulo Theories framework. Besides adapting many of Simplify’s techniques, we also introduce a number of new heuristics. Most important is the notion of instantiation level which provides an effective mechanism for prioritizing and managing the large search space inherent in quantifier instantiation techniques. These techniques have been implemented in the SMT system CVC3. Experimental results show that our methodology enables CVC3 to solve a significant number of benchmarks that were not solvable with any previous approach. 1
Splitting on Demand in SAT Modulo Theories
 In Proc. LPAR’06, volume 4246 of LNAI
, 2006
"... Abstract. Lazy algorithms for Satisfiability Modulo Theories (SMT) combine a generic DPLLbased SAT engine with a theory solver for the given theory T that can decide the Tconsistency of conjunctions of ground literals. For many theories of interest, theory solvers need to reason by performing inte ..."
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Cited by 27 (6 self)
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Abstract. Lazy algorithms for Satisfiability Modulo Theories (SMT) combine a generic DPLLbased SAT engine with a theory solver for the given theory T that can decide the Tconsistency of conjunctions of ground literals. For many theories of interest, theory solvers need to reason by performing internal case splits. Here we argue that it is more convenient to delegate these case splits to the DPLL engine instead. This can be done on demand for solvers that can encode their internal case splits into one or more clauses, possibly including new constants and literals. It results in drastically simpler theory solvers, and can lead, we believe, to more efficient overall systems. We present this in an improved version of DPLL(T), a general SMT architecture for the lazy approach, and formalize and prove it correct in an extension of Abstract DPLL Modulo Theories, a framework for modeling and reasoning about lazy algorithms for SMT. A remarkable additional feature of the architecture, also discussed in the paper, is that it naturally includes an efficient NelsonOppenlike combination of multiple theories and their solvers. 1
An algorithm for deciding BAPA: Boolean Algebra with Presburger Arithmetic
 In 20th International Conference on Automated Deduction, CADE20
, 2005
"... Abstract. We describe an algorithm for deciding the firstorder 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 u ..."
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Cited by 26 (13 self)
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Abstract. We describe an algorithm for deciding the firstorder 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. 1
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 23 (11 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.
MathSAT: Tight integration of SAT and mathematical decision procedures
 Journal of Automated Reasoning
, 2005
"... Abstract. Recent improvements in propositional satisfiability techniques (SAT) made it possible to tackle successfully some hard realworld problems (e.g. modelchecking, circuit testing, propositional planning) by encoding into SAT. However, a purely boolean representation is not expressive enough ..."
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Cited by 22 (2 self)
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Abstract. Recent improvements in propositional satisfiability techniques (SAT) made it possible to tackle successfully some hard realworld problems (e.g. modelchecking, circuit testing, propositional planning) by encoding into SAT. However, a purely boolean representation is not expressive enough for many other realworld applications, including the verification of timed and hybrid systems, of proof obligations in software, and of circuit design at RTL level. These problems can be naturally modeled as satisfiability in Linear Arithmetic Logic (LAL), i.e., the boolean combination of propositional variables and linear constraints over numerical variables. In this paper we present MATHSAT, a new, SATbased decision procedure for LAL, based on the (known approach) of integrating a stateoftheart SAT solver with a dedicated mathematical solver for LAL. We improve MATHSAT in two different directions. First, the top level procedure is enhanced, and now features a tighter integration between the boolean search and the mathematical solver. In particular, we allow for theorydriven backjumping and learning, and theorydriven deduction; we use static learning in order to reduce the number of boolean models that are mathematically inconsistent; we exploit problem clustering in order to partition
Decidability and undecidability results for NelsonOppen and rewritebased decision procedures
 In Proc. IJCAR3, U. Furbach and
, 2006
"... Abstract. In the context of combinations of theories with disjoint signatures, we classify the component theories according to the decidability of constraint satisfiability problems in arbitrary and in infinite models, respectively. We exhibit a theory T1 such that satisfiability is decidable, but s ..."
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Cited by 22 (16 self)
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Abstract. In the context of combinations of theories with disjoint signatures, we classify the component theories according to the decidability of constraint satisfiability problems in arbitrary and in infinite models, respectively. We exhibit a theory T1 such that satisfiability is decidable, but satisfiability in infinite models is undecidable. It follows that satisfiability in T1 ∪ T2 is undecidable, whenever T2 has only infinite models, even if signatures are disjoint and satisfiability in T2 is decidable. In the second part of the paper we strengthen the NelsonOppen decidability transfer result, by showing that it applies to theories over disjoint signatures, whose satisfiability problem, in either arbitrary or infinite models, is decidable. We show that this result covers decision procedures based on rewriting, complementing recent work on combination of theories in the rewritebased approach to satisfiability. 1
Fast and Flexible Difference Constraint Propagation for DPLL(T)
 IN PROC. SAT, VOLUME 4121 OF LNCS
, 2006
"... In the context of DPLL(T), theory propagation is the process of dynamically selecting consequences of a conjunction of constraints from a given set of candidate constraints. We present improvements to a fast theory propagation procedure for difference constraints of the form x − y ≤ c. These improve ..."
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Cited by 22 (1 self)
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In the context of DPLL(T), theory propagation is the process of dynamically selecting consequences of a conjunction of constraints from a given set of candidate constraints. We present improvements to a fast theory propagation procedure for difference constraints of the form x − y ≤ c. These improvements are demonstrated experimentally.
Delayed theory combination vs. NelsonOppen for satisfiability modulo theories: A comparative analysis
 IN PROC. LPAR’06, VOLUME 4246 OF LNAI
, 2006
"... Many approaches for Satisfiability Modulo Theory (SMT(T)) rely on the integration between a SAT solver and a decision procedure for sets of literals in the background theory T (Tsolver). When T is the combination T1 ∪ T2 of two simpler theories, the approach is typically handled by means of Nelson ..."
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Cited by 21 (7 self)
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Many approaches for Satisfiability Modulo Theory (SMT(T)) rely on the integration between a SAT solver and a decision procedure for sets of literals in the background theory T (Tsolver). When T is the combination T1 ∪ T2 of two simpler theories, the approach is typically handled by means of NelsonOppen’s (NO) theory combination schema in which two specific Tsolvers deduce and exchange (disjunctions of) interface equalities. In recent papers we have proposed a new approach to SMT(T1 ∪ T2), called Delayed Theory Combination (DTC). Here part or all the (possibly very expensive) task of deducing interface equalities is played by the SAT solver itself, at the potential cost of an enlargement of the boolean search space. In principle this enlargement could be up to exponential in the number of interface equalities generated. In this paper we show that this estimate was too pessimistic. We present a comparative analysis of DTC vs. NO for SMT(T1 ∪T2), which shows that, using stateoftheart SATsolving techniques, the amount of boolean branches performed by DTC can be upper bounded by the number of deductions and boolean branches performed by NO on the same problem. We prove the result for different deduction capabilities of the Tsolvers and for both convex and nonconvex theories.