Delayed Theory Combination vs. Nelson-Oppen for Satisfiability Modulo Theories: A Comparative Analysis

  • Roberto Bruttomesso
  • Alessandro Cimatti
  • Anders Franzén
  • Alberto Griggio
  • Roberto Sebastiani
Conference paper

DOI: 10.1007/11916277_36

Part of the Lecture Notes in Computer Science book series (LNCS, volume 4246)
Cite this paper as:
Bruttomesso R., Cimatti A., Franzén A., Griggio A., Sebastiani R. (2006) Delayed Theory Combination vs. Nelson-Oppen for Satisfiability Modulo Theories: A Comparative Analysis. In: Hermann M., Voronkov A. (eds) Logic for Programming, Artificial Intelligence, and Reasoning. LPAR 2006. Lecture Notes in Computer Science, vol 4246. Springer, Berlin, Heidelberg

Abstract

Many approaches for Satisfiability Modulo Theory (SMT\({\mathcal({T})})\) rely on the integration between a SAT solver and a decision procedure for sets of literals in the background theory \({\mathcal{T}} ({\mathcal{T}}-solver\)). When \({\mathcal{T}}\) is the combination \({{\mathcal{T}}_1\cup{\mathcal{T}}_2}\) of two simpler theories, the approach is typically handled by means of Nelson-Oppen’s (NO) theory combination schema in which two specific \({\mathcal{T}}\)-solver deduce and exchange (disjunctions of) interface equalities.

In recent papers we have proposed a new approach to \(({{\mathcal{T}}_1\cup{\mathcal{T}}_2})\), 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\(({{\mathcal{T}}_1\cup{\mathcal{T}}_2})\), which shows that, using state-of-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 \({\mathcal({T}-solver)}\) and for both convex and non-convex theories.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Roberto Bruttomesso
    • 1
  • Alessandro Cimatti
    • 1
  • Anders Franzén
    • 1
    • 2
  • Alberto Griggio
    • 2
  • Roberto Sebastiani
    • 2
  1. 1.ITC-IRSTPovo, TrentoItaly
  2. 2.DITUniversità di TrentoItaly

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