## Delayed theory combination vs. Nelson-Oppen for satisfiability modulo theories: A comparative analysis (2006)

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Venue: | IN PROC. LPAR’06, VOLUME 4246 OF LNAI |

Citations: | 20 - 7 self |

### BibTeX

@INPROCEEDINGS{Bruttomesso06delayedtheory,

author = {Roberto Bruttomesso and Alessandro Cimatti and Anders Franzen and Alberto Griggio and Roberto Sebastiani},

title = {Delayed theory combination vs. Nelson-Oppen for satisfiability modulo theories: A comparative analysis},

booktitle = {IN PROC. LPAR’06, VOLUME 4246 OF LNAI},

year = {2006},

pages = {527--541},

publisher = {Springer}

}

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### Abstract

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 (T-solver). When T is the combination T1 ∪ T2 of two simpler theories, the approach is typically handled by means of Nelson-Oppen’s (NO) theory combination schema in which two specific T-solvers 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 stateof-the-art SAT-solving 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 T-solvers and for both convex and non-convex theories.

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Citation Context ...(T1 ∪ T2) problem. The approach relies on a decision procedure able to decide the satisfiability of sets of literals in T1 ∪ T2, that is typically based on an integration schema like NelsonOppen (NO) =-=[11]-=- (or its variant due to Shostak [13]): the Ti-solvers are combined by means of a structured exchange of (disjunctions of) interface equalities (ei j’s). Unfortunately from a practical point of view th... |

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Citation Context ...face equalities, which have to be managed within the decision procedure by means of case 2 These are standard techniques implemented in most SAT solvers in order to build the boolean conflict clauses =-=[14]-=-. 5sv3 = h(v0) v4 = h(v1) v6 = f (v2) v7 = f (v5) ¬(v6 = v7) RESET5 Branch 1 Branch 2 v0 = v1 v0 = v1 v0 = v1 〈ei j-deduction〉 v3 = v4 v2 = v5 v3 = v4 〈ei j-deduction〉 v2 = v5 〈ei j-deduction〉 v3 = v4... |

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Citation Context ...ies, in this paper we always deal with only two theories T1 and T2. The discourse generalizes to more than two theories.sA prominent approach to SMT(T ) which underlies several systems (e.g., CVCLITE =-=[2]-=-, DLSAT [8], DPLL(T)/BarceLogic [10], MATHSAT [4], TSAT++ [1], ICS/YICES [9]), is based on extensions of SAT technology: a SAT engine is modified to enumerate boolean assignments, and integrated with ... |

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Citation Context ...es on a decision procedure able to decide the satisfiability of sets of literals in T1 ∪ T2, that is typically based on an integration schema like NelsonOppen (NO) [11] (or its variant due to Shostak =-=[13]-=-): the Ti-solvers are combined by means of a structured exchange of (disjunctions of) interface equalities (ei j’s). Unfortunately from a practical point of view this schema poses some challenges. Fir... |

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Citation Context ...ent are kept separate in this description only for the sake of simplicity. In practice, the enumeration of truth assignments is carried out by means of efficient implementations of the DPLL algorithm =-=[15]-=-, where a partial assignment µ p is built incrementally, each time selecting an unassigned literal l (literal selection), called decision literal, according to some heuristic criterion, adding it to µ... |

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Citation Context ...ith only two theories T1 and T2. The discourse generalizes to more than two theories.sA prominent approach to SMT(T ) which underlies several systems (e.g., CVCLITE [2], DLSAT [8], DPLL(T)/BarceLogic =-=[10]-=-, MATHSAT [4], TSAT++ [1], ICS/YICES [9]), is based on extensions of SAT technology: a SAT engine is modified to enumerate boolean assignments, and integrated with a decision procedure for sets of lit... |

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Citation Context ...ourse generalizes to more than two theories.sA prominent approach to SMT(T ) which underlies several systems (e.g., CVCLITE [2], DLSAT [8], DPLL(T)/BarceLogic [10], MATHSAT [4], TSAT++ [1], ICS/YICES =-=[9]-=-), is based on extensions of SAT technology: a SAT engine is modified to enumerate boolean assignments, and integrated with a decision procedure for sets of literals in the theory T (T -solver). The a... |

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Citation Context ...nd T2. The discourse generalizes to more than two theories.sA prominent approach to SMT(T ) which underlies several systems (e.g., CVCLITE [2], DLSAT [8], DPLL(T)/BarceLogic [10], MATHSAT [4], TSAT++ =-=[1]-=-, ICS/YICES [9]), is based on extensions of SAT technology: a SAT engine is modified to enumerate boolean assignments, and integrated with a decision procedure for sets of literals in the theory T (T ... |

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Citation Context ...F but can be very expensive for LA(Z).) Third, in case of non-convex theories (e.g., LA(Z)), a backtrack search must be used to take care of the disjunctions that need to be managed. In recent papers =-=[3, 6]-=- we have proposed a novel approach to SMT(T1 ∪ T2), called Delayed Theory Combination (DTC). The main idea is to avoid the integration schema between T1 and T2, and tighten the connection between each... |

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Citation Context ... that Strategy 1 has been conceived only for mimicking NO, and by no means it is assumed to be the most efficient strategy for DTC. (E.g., Step 3.(ii) can be substituted with a weakened version of EP =-=[4]-=-, and more efficient literal selection strategies might be preferable to Step 3.(i) and (iii).) Some alternatives are currently under investigation, and their theoretical properties and practical perf... |

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(Show Context)
Citation Context ...F but can be very expensive for LA(Z).) Third, in case of non-convex theories (e.g., LA(Z)), a backtrack search must be used to take care of the disjunctions that need to be managed. In recent papers =-=[3, 6]-=- we have proposed a novel approach to SMT(T1 ∪ T2), called Delayed Theory Combination (DTC). The main idea is to avoid the integration schema between T1 and T2, and tighten the connection between each... |

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Citation Context ...r future work. As far as the ¬ei j-minimality hypothesis is concerned, we notice that, at least for theories like EUF and LA(Q), there are known decision procedures that fulfill this requirement (see =-=[12]-=- and [4] respectively.) For other theories, the problem of ¬ei jminimization opens a novel research branch. 10 However, we remark that DTC works also when the Ti-solvers are not ¬ei j-minimal, at the ... |

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Citation Context ...s paper we always deal with only two theories T1 and T2. The discourse generalizes to more than two theories.sA prominent approach to SMT(T ) which underlies several systems (e.g., CVCLITE [2], DLSAT =-=[8]-=-, DPLL(T)/BarceLogic [10], MATHSAT [4], TSAT++ [1], ICS/YICES [9]), is based on extensions of SAT technology: a SAT engine is modified to enumerate boolean assignments, and integrated with a decision ... |

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Citation Context ...educe the search space by allowing the T -solvers to explicitly return truth values for unassigned literals, which can be unit-propagated by the SAT solver. The interested reader is pointed to, e.g., =-=[1, 4, 10, 5]-=- for details and further references. 2.3 Nelson-Oppen’s schema Given two signature-disjoint stably infinite theories T1 and T2, the Nelson-Oppen combination schema [11], in the following referred to a... |

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