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Roberto Sebastiani - One of the best experts on this subject based on the ideXlab platform.

  • Delayed theory combination vs. Nelson-Oppen for satisfiability modulo theories: a comparative analysis
    Annals of Mathematics and Artificial Intelligence, 2009
    Co-Authors: Roberto Bruttomesso, Alessandro Cimatti, Anders Franzén, Alberto Griggio, Roberto Sebastiani
    Abstract:

    Most state-of-the-art approaches for Satisfiability Modulo Theories $(SMT(\mathcal{T}))$ rely on the integration between a SAT solver and a decision procedure for sets of literals in the background theory $\mathcal{T} (\mathcal{T}{\text {-}}solver)$ . Often $\mathcal{T}$ is the combination $\mathcal{T}_1 \cup \mathcal{T}_2$ of two (or more) simpler theories $(SMT(\mathcal{T}_1 \cup \mathcal{T}_2))$ , s.t. the specific ${\mathcal{T}_i}{\text {-}}solvers$ must be combined. Up to a few years ago, the standard approach to $SMT(\mathcal{T}_1 \cup \mathcal{T}_2)$ was to integrate the SAT solver with one combined $\mathcal{T}_1 \cup \mathcal{T}_2{\text {-}}solver$ , obtained from two distinct ${\mathcal{T}_i}{\text {-}}solvers$ by means of evolutions of Nelson and Oppen's (NO) combination procedure, in which the ${\mathcal{T}_i}{\text {-}}solvers$ deduce and exchange interface equalities. Nowadays many state-of-the-art SMT solvers use evolutions of a more recent $SMT(\mathcal{T}_1 \cup \mathcal{T}_2)$ procedure called Delayed Theory Combination (DTC), in which each ${\mathcal{T}_i}{\text {-}}solver$ interacts directly and only with the SAT solver, in such a way that part or all of the (possibly very expensive) reasoning effort on interface equalities is delegated to the SAT solver itself. In this paper we present a comparative analysis of DTC vs. NO for $SMT(\mathcal{T}_1 \cup \mathcal{T}_2)$ . On the one hand, we explain the advantages of DTC in exploiting the power of modern SAT solvers to reduce the Search. On the other hand, we show that the extra amount of Boolean Search required to the SAT solver can be controlled. In fact, we prove two novel theoretical results, for both convex and non-convex theories and for different deduction capabilities of the ${\mathcal{T}_i}{\text {-}}solvers$ , which relate the amount of extra Boolean Search required to the SAT solver by DTC with the number of deductions and case-splits required to the ${\mathcal{T}_i}{\text {-}}solvers$ by NO in order to perform the same tasks: (i) under the same hypotheses of deduction capabilities of the ${\mathcal{T}_i}{\text {-}}solvers$ required by NO, DTC causes no extra Boolean Search; (ii) using ${\mathcal{T}_i}{\text {-}}solvers$ with limited or no deduction capabilities, the extra Boolean Search required can be reduced down to a negligible amount by controlling the quality of the $\mathcal{T}$ -conflict sets returned by the ${\mathcal{T}_i}{\text {-}}solvers$ .

  • Efficient theory combination via Boolean Search
    Information and Computation, 2006
    Co-Authors: Marco Bozzano, Roberto Bruttomesso, Alessandro Cimatti, Tommi Junttila, Silvio Ranise, Peter Van Rossum, Roberto Sebastiani
    Abstract:

    Many approaches to deciding the satisfiability of quantifier-free formulae with respect to a background theory T-also known as Satisfiability Modulo Theory, or SMT(T)-rely on the integration between an enumerator of truth assignments and a decision procedure for conjunction of literals in T. When the background theory T is the combination T1 ∪ T2 of two simpler theories, the approach is typically instantiated by means of a theory combination schema (e.g. Nelson-Oppen, Shostak). In this paper we propose a new approach to SMT(T1 ∪ T2), where the enumerator of truth assignments is integrated with two decision procedures, one for T1 and one for T2, acting independently from each other. The key idea is to Search for a truth assignment not only to the atoms occurring in the formula, but also to all the equalities between variables which are shared between the theories. This approach is simple and expressive: for instance, no modification is required to handle non-convex theories (as opposed to traditional Nelson-Oppen combinations which require a mechanism for splitting). Furthermore, it can be made practical by leveraging on state-of-the-art Boolean and SMT Search techniques, and on theory layering (i.e., cheaper reasoning first, and more often). We provide thorough experimental evidence to support our claims: we instantiate the framework with two decision procedures for the combinations of Equality and Uninterpreted Functions (EUF) and Linear Arithmetic (LA), both for (the convex case of) reals and for (the non-convex case of) integers; we analyze the impact of the different optimizations on a variety of test cases; and we compare the approach with state-of-the-art competitor tools, showing that our implemented tool compares positively with them, sometimes with dramatic gains in performance.

Abdullatif Shikfa - One of the best experts on this subject based on the ideXlab platform.

Tarik Moataz - One of the best experts on this subject based on the ideXlab platform.

Roberto Bruttomesso - One of the best experts on this subject based on the ideXlab platform.

  • Delayed theory combination vs. Nelson-Oppen for satisfiability modulo theories: a comparative analysis
    Annals of Mathematics and Artificial Intelligence, 2009
    Co-Authors: Roberto Bruttomesso, Alessandro Cimatti, Anders Franzén, Alberto Griggio, Roberto Sebastiani
    Abstract:

    Most state-of-the-art approaches for Satisfiability Modulo Theories $(SMT(\mathcal{T}))$ rely on the integration between a SAT solver and a decision procedure for sets of literals in the background theory $\mathcal{T} (\mathcal{T}{\text {-}}solver)$ . Often $\mathcal{T}$ is the combination $\mathcal{T}_1 \cup \mathcal{T}_2$ of two (or more) simpler theories $(SMT(\mathcal{T}_1 \cup \mathcal{T}_2))$ , s.t. the specific ${\mathcal{T}_i}{\text {-}}solvers$ must be combined. Up to a few years ago, the standard approach to $SMT(\mathcal{T}_1 \cup \mathcal{T}_2)$ was to integrate the SAT solver with one combined $\mathcal{T}_1 \cup \mathcal{T}_2{\text {-}}solver$ , obtained from two distinct ${\mathcal{T}_i}{\text {-}}solvers$ by means of evolutions of Nelson and Oppen's (NO) combination procedure, in which the ${\mathcal{T}_i}{\text {-}}solvers$ deduce and exchange interface equalities. Nowadays many state-of-the-art SMT solvers use evolutions of a more recent $SMT(\mathcal{T}_1 \cup \mathcal{T}_2)$ procedure called Delayed Theory Combination (DTC), in which each ${\mathcal{T}_i}{\text {-}}solver$ interacts directly and only with the SAT solver, in such a way that part or all of the (possibly very expensive) reasoning effort on interface equalities is delegated to the SAT solver itself. In this paper we present a comparative analysis of DTC vs. NO for $SMT(\mathcal{T}_1 \cup \mathcal{T}_2)$ . On the one hand, we explain the advantages of DTC in exploiting the power of modern SAT solvers to reduce the Search. On the other hand, we show that the extra amount of Boolean Search required to the SAT solver can be controlled. In fact, we prove two novel theoretical results, for both convex and non-convex theories and for different deduction capabilities of the ${\mathcal{T}_i}{\text {-}}solvers$ , which relate the amount of extra Boolean Search required to the SAT solver by DTC with the number of deductions and case-splits required to the ${\mathcal{T}_i}{\text {-}}solvers$ by NO in order to perform the same tasks: (i) under the same hypotheses of deduction capabilities of the ${\mathcal{T}_i}{\text {-}}solvers$ required by NO, DTC causes no extra Boolean Search; (ii) using ${\mathcal{T}_i}{\text {-}}solvers$ with limited or no deduction capabilities, the extra Boolean Search required can be reduced down to a negligible amount by controlling the quality of the $\mathcal{T}$ -conflict sets returned by the ${\mathcal{T}_i}{\text {-}}solvers$ .

  • Efficient theory combination via Boolean Search
    Information and Computation, 2006
    Co-Authors: Marco Bozzano, Roberto Bruttomesso, Alessandro Cimatti, Tommi Junttila, Silvio Ranise, Peter Van Rossum, Roberto Sebastiani
    Abstract:

    Many approaches to deciding the satisfiability of quantifier-free formulae with respect to a background theory T-also known as Satisfiability Modulo Theory, or SMT(T)-rely on the integration between an enumerator of truth assignments and a decision procedure for conjunction of literals in T. When the background theory T is the combination T1 ∪ T2 of two simpler theories, the approach is typically instantiated by means of a theory combination schema (e.g. Nelson-Oppen, Shostak). In this paper we propose a new approach to SMT(T1 ∪ T2), where the enumerator of truth assignments is integrated with two decision procedures, one for T1 and one for T2, acting independently from each other. The key idea is to Search for a truth assignment not only to the atoms occurring in the formula, but also to all the equalities between variables which are shared between the theories. This approach is simple and expressive: for instance, no modification is required to handle non-convex theories (as opposed to traditional Nelson-Oppen combinations which require a mechanism for splitting). Furthermore, it can be made practical by leveraging on state-of-the-art Boolean and SMT Search techniques, and on theory layering (i.e., cheaper reasoning first, and more often). We provide thorough experimental evidence to support our claims: we instantiate the framework with two decision procedures for the combinations of Equality and Uninterpreted Functions (EUF) and Linear Arithmetic (LA), both for (the convex case of) reals and for (the non-convex case of) integers; we analyze the impact of the different optimizations on a variety of test cases; and we compare the approach with state-of-the-art competitor tools, showing that our implemented tool compares positively with them, sometimes with dramatic gains in performance.

Alessandro Cimatti - One of the best experts on this subject based on the ideXlab platform.

  • Delayed theory combination vs. Nelson-Oppen for satisfiability modulo theories: a comparative analysis
    Annals of Mathematics and Artificial Intelligence, 2009
    Co-Authors: Roberto Bruttomesso, Alessandro Cimatti, Anders Franzén, Alberto Griggio, Roberto Sebastiani
    Abstract:

    Most state-of-the-art approaches for Satisfiability Modulo Theories $(SMT(\mathcal{T}))$ rely on the integration between a SAT solver and a decision procedure for sets of literals in the background theory $\mathcal{T} (\mathcal{T}{\text {-}}solver)$ . Often $\mathcal{T}$ is the combination $\mathcal{T}_1 \cup \mathcal{T}_2$ of two (or more) simpler theories $(SMT(\mathcal{T}_1 \cup \mathcal{T}_2))$ , s.t. the specific ${\mathcal{T}_i}{\text {-}}solvers$ must be combined. Up to a few years ago, the standard approach to $SMT(\mathcal{T}_1 \cup \mathcal{T}_2)$ was to integrate the SAT solver with one combined $\mathcal{T}_1 \cup \mathcal{T}_2{\text {-}}solver$ , obtained from two distinct ${\mathcal{T}_i}{\text {-}}solvers$ by means of evolutions of Nelson and Oppen's (NO) combination procedure, in which the ${\mathcal{T}_i}{\text {-}}solvers$ deduce and exchange interface equalities. Nowadays many state-of-the-art SMT solvers use evolutions of a more recent $SMT(\mathcal{T}_1 \cup \mathcal{T}_2)$ procedure called Delayed Theory Combination (DTC), in which each ${\mathcal{T}_i}{\text {-}}solver$ interacts directly and only with the SAT solver, in such a way that part or all of the (possibly very expensive) reasoning effort on interface equalities is delegated to the SAT solver itself. In this paper we present a comparative analysis of DTC vs. NO for $SMT(\mathcal{T}_1 \cup \mathcal{T}_2)$ . On the one hand, we explain the advantages of DTC in exploiting the power of modern SAT solvers to reduce the Search. On the other hand, we show that the extra amount of Boolean Search required to the SAT solver can be controlled. In fact, we prove two novel theoretical results, for both convex and non-convex theories and for different deduction capabilities of the ${\mathcal{T}_i}{\text {-}}solvers$ , which relate the amount of extra Boolean Search required to the SAT solver by DTC with the number of deductions and case-splits required to the ${\mathcal{T}_i}{\text {-}}solvers$ by NO in order to perform the same tasks: (i) under the same hypotheses of deduction capabilities of the ${\mathcal{T}_i}{\text {-}}solvers$ required by NO, DTC causes no extra Boolean Search; (ii) using ${\mathcal{T}_i}{\text {-}}solvers$ with limited or no deduction capabilities, the extra Boolean Search required can be reduced down to a negligible amount by controlling the quality of the $\mathcal{T}$ -conflict sets returned by the ${\mathcal{T}_i}{\text {-}}solvers$ .

  • Efficient theory combination via Boolean Search
    Information and Computation, 2006
    Co-Authors: Marco Bozzano, Roberto Bruttomesso, Alessandro Cimatti, Tommi Junttila, Silvio Ranise, Peter Van Rossum, Roberto Sebastiani
    Abstract:

    Many approaches to deciding the satisfiability of quantifier-free formulae with respect to a background theory T-also known as Satisfiability Modulo Theory, or SMT(T)-rely on the integration between an enumerator of truth assignments and a decision procedure for conjunction of literals in T. When the background theory T is the combination T1 ∪ T2 of two simpler theories, the approach is typically instantiated by means of a theory combination schema (e.g. Nelson-Oppen, Shostak). In this paper we propose a new approach to SMT(T1 ∪ T2), where the enumerator of truth assignments is integrated with two decision procedures, one for T1 and one for T2, acting independently from each other. The key idea is to Search for a truth assignment not only to the atoms occurring in the formula, but also to all the equalities between variables which are shared between the theories. This approach is simple and expressive: for instance, no modification is required to handle non-convex theories (as opposed to traditional Nelson-Oppen combinations which require a mechanism for splitting). Furthermore, it can be made practical by leveraging on state-of-the-art Boolean and SMT Search techniques, and on theory layering (i.e., cheaper reasoning first, and more often). We provide thorough experimental evidence to support our claims: we instantiate the framework with two decision procedures for the combinations of Equality and Uninterpreted Functions (EUF) and Linear Arithmetic (LA), both for (the convex case of) reals and for (the non-convex case of) integers; we analyze the impact of the different optimizations on a variety of test cases; and we compare the approach with state-of-the-art competitor tools, showing that our implemented tool compares positively with them, sometimes with dramatic gains in performance.