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107
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 30 (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
Proving Conditional Termination
"... Abstract. We describe a method for synthesizing reasonable underapproximations to weakest preconditions for termination—a longstanding open problem. The paper provides experimental evidence to demonstrate the usefulness of the new procedure. 1 ..."
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Cited by 24 (5 self)
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Abstract. We describe a method for synthesizing reasonable underapproximations to weakest preconditions for termination—a longstanding open problem. The paper provides experimental evidence to demonstrate the usefulness of the new procedure. 1
Deciding Presburger Arithmetic by Model Checking and Comparisons with Other Methods
 In Proceedings of FMCAD 02
, 2002
"... We present a new way of using Binary Decision Diagrams in automata based algorithms for solving the satisfiability problem of quantifierfree Presburger arithmetic. Unlike in previous approaches [5, 2, 19], we translate the satisfiability problem into a model checking problem and use the existing ..."
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Cited by 23 (1 self)
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We present a new way of using Binary Decision Diagrams in automata based algorithms for solving the satisfiability problem of quantifierfree Presburger arithmetic. Unlike in previous approaches [5, 2, 19], we translate the satisfiability problem into a model checking problem and use the existing BDDbased model checker SMV [13] as our primary engine.
Quantifier elimination by lazy model enumeration
, 2010
"... We propose a quantifier elimination scheme based on nested lazy model enumeration through SMTsolving, and projections. This scheme may be applied to any logic that fulfills certain conditions; we illustrate it for linear real arithmetic. The quantifier elimination problem for linear real arithmetic ..."
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Cited by 22 (3 self)
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We propose a quantifier elimination scheme based on nested lazy model enumeration through SMTsolving, and projections. This scheme may be applied to any logic that fulfills certain conditions; we illustrate it for linear real arithmetic. The quantifier elimination problem for linear real arithmetic is doubly exponential in the worst case, and so is our method. We have implemented it and benchmarked it against other methods from the literature.
Word Unification and Transformation of Generalized Equations
 Journal of Automated Reasoning
, 1993
"... Makanin's algorithm [Ma77] shows that it is decidable whether a word equation has a solution. The original description was hard to understand and not designed for implementation. Since words represent a fundamental data type, various authors have given improved descriptions [P'e81, Ab87, S ..."
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Cited by 21 (1 self)
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Makanin's algorithm [Ma77] shows that it is decidable whether a word equation has a solution. The original description was hard to understand and not designed for implementation. Since words represent a fundamental data type, various authors have given improved descriptions [P'e81, Ab87, Sc90, Ja90]. In this paper we present a version of the algorithm which probably cannot be further simplified without fundamentally new insights which exceed Makanin's original ideas. We give a transformation which is efficient, conceptually simple and applies to arbitrary generalized equations. No further subprocedure is needed for the generation of the search tree. Particular attention is then given to the proof that proper generalized equations are transformed into proper generalized equations. This point, which is important for the termination argument, was treated erroneously in other papers. We also show that a combination of the basic algorithm for stringunification (see [Pl72, Le72, Si75, Si78]...
Temporal Semantic Assumptions and Their Use in Databases
 IEEE TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING
, 1998
"... Temporal data explicitly stored in a temporal database are often associated with certain semantic assumptions. Each assumption can be viewed as a way of deriving implicit information from the explicitly stored data. Rather than leaving the task of deriving (possibly in nite) implicit data to applica ..."
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Cited by 20 (2 self)
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Temporal data explicitly stored in a temporal database are often associated with certain semantic assumptions. Each assumption can be viewed as a way of deriving implicit information from the explicitly stored data. Rather than leaving the task of deriving (possibly in nite) implicit data to application programs, as is the case currently, it is desirable that this be handled by the database management systems. To achieve this, this paper formalizes and studies two types of semantic assumptions: pointbased and intervalbased. The pointbased assumptions include those assumptions that use interpolation methods, while the intervalbased assumptions include those that involve different temporal types (time granularities). In order to incorporate semantic assumptions into query evaluation, this paper introduces a translation procedure that converts a user query into a system query such that the answer of this system query over the explicit data is the same as that of the user query over the explicit and the implicit data. The paper also investigates the niteness (safety) of user queries and system queries.
Decision procedures for recursive data structures with integer constraints
 In International Joint Conference on Automated Reasoning, volume 3097 of LNCS
, 2004
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Automatic modular abstractions for template numerical constraints
 Logical Methods in Computer Science
, 2010
"... We propose a method for automatically generating abstract transformers for static analysis by abstract interpretation. The method focuses on linear constraints on programs operating on rational, real or floatingpoint variables and containing linear assignments and tests. Given the specification of a ..."
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Cited by 16 (6 self)
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We propose a method for automatically generating abstract transformers for static analysis by abstract interpretation. The method focuses on linear constraints on programs operating on rational, real or floatingpoint variables and containing linear assignments and tests. Given the specification of an abstract domain, and a program block, our method transformer. It is thus a form of program transformation. In addition to loopfree code, the same method also applies for obtaining least fixed points as functions of the precondition, which permits the analysis of loops and recursive functions. The motivation of our work is dataflow synchronous programming languages, used for building controlcommand embedded systems, but it also applies to imperative and functional programming. Our algorithms are based on quantifier elimination and symbolic manipulation techniques over linear arithmetic formulas. We also give less general results for nonlinear constraints and nonlinear program constructs. 1
Cutting to the Chase  Solving Linear Integer Arithmetic
"... We describe a new algorithm for solving linear integer programming problems. The algorithm performs a DPLL style search for a feasible assignment, while using a novel cut procedure to guide the search away from the conflicting states. ..."
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Cited by 15 (5 self)
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We describe a new algorithm for solving linear integer programming problems. The algorithm performs a DPLL style search for a feasible assignment, while using a novel cut procedure to guide the search away from the conflicting states.
Generalizing DPLL to Richer Logics
"... Abstract. The DPLL approach to the Boolean satisfiability problem (SAT) is a combination of search for a satisfying assignment and logical deduction, in which each process guides the other. We show that this approach can be generalized to a richer class of theories. In particular, we present an alte ..."
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Abstract. The DPLL approach to the Boolean satisfiability problem (SAT) is a combination of search for a satisfying assignment and logical deduction, in which each process guides the other. We show that this approach can be generalized to a richer class of theories. In particular, we present an alternative to lazy SMT solvers, in which DPLL is used only to find propositionally satisfying assignments, whose feasibility is checked by a separate theory solver. Here, DPLL is applied directly to the theory. We search in the space of theory structures (for example, numerical assignments) rather than propositional assignments. This makes it possible to use conflict in model search to guide deduction in the theory, much in the way that it guides propositional resolution in DPLL. Some experiments using linear rational arithmetic demonstrate the potential advantages of the approach. 1