Theory of linear and integer programming
Theory of linear and integer programming
Combining matching algorithms: The regular case
Journal of Symbolic Computation
Combining unification algorithms
Journal of Symbolic Computation
Efficiency of a Good But Not Linear Set Union Algorithm
Journal of the ACM (JACM)
Variations on the Common Subexpression Problem
Journal of the ACM (JACM)
Simplification by Cooperating Decision Procedures
ACM Transactions on Programming Languages and Systems (TOPLAS)
Efficient theory combination via boolean search
Information and Computation - Special issue: Combining logical systems
AI Communications - CASC
Electronic Notes in Theoretical Computer Science (ENTCS)
Automatic combinability of rewriting-based satisfiability procedures
LPAR'06 Proceedings of the 13th international conference on Logic for Programming, Artificial Intelligence, and Reasoning
Nelson-Oppen, shostak and the extended canonizer: a family picture with a newborn
ICTAC'04 Proceedings of the First international conference on Theoretical Aspects of Computing
Proof-producing congruence closure
RTA'05 Proceedings of the 16th international conference on Term Rewriting and Applications
The algebra of equality proofs
RTA'05 Proceedings of the 16th international conference on Term Rewriting and Applications
TACAS'06 Proceedings of the 12th international conference on Tools and Algorithms for the Construction and Analysis of Systems
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Constraint solvers are key modules in many systems with reasoning capabilities (e.g., automated theorem provers). To incorporate constraint solvers in such systems, the capability of producing conflict sets or explanations of their results is crucial. For expressiveness, constraints are usually built out in unions of theories and constraint solvers in such unions are obtained by modularly combining solvers for the component theories. In this paper, we consider the problem of modularly constructing conflict sets for a combined theory by re-using available proof-producing procedures for the component theories. The key idea of our solution to this problem is the concept of explanation graph, which is a labelled, acyclic and undirected graph capable of recording the entailment of some equalities. Explanation graphs allow us to record explanations computed by a proof-producing procedure and to refine the Nelson-Oppen combination method to modularly build conflict sets for disjoint unions of theories. We also study how the computed conflict sets relate to an appropriate notion of minimality.