A New Correctness Proof of the Nelson-Oppen Combination Procedure

Tinelli, Cesare; Harandi, Mehdi T.

doi:10.1007/978-94-009-0349-4_5

Cited by 100 publications

(78 citation statements)

References 20 publications

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“…satisfiable in an infinite model of T 1 ∪T 2 by the standard argument underlying the correctness of the Nelson-Oppen combination schema (see, e.g., [25,13]). …”

Section: Decidability Resultsmentioning

confidence: 99%

Decidability and Undecidability Results for Nelson-Oppen and Rewrite-Based Decision Procedures

Bonacina

Ghilardi

Nicolini

et al. 2006

Automated Reasoning

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In the context of combinations of theories with disjoint signatures, we classify the component theories according to the decidability of constraint satisfiability problems in finite and infinite models, respectively. We exhibit a theory T 1 such that satisfiability is decidable, but satisfiability in infinite models is undecidable. It follows that satisfiability in T 1 ∪ T 2 is undecidable, whenever T2 has only infinite models, even if signatures are disjoint and satisfiability in T2 is decidable.In the second part of the paper we strengthen the Nelson-Oppen decidability transfer result, by showing that it applies to theories over disjoint signatures, whose satisfiability problem, in either finite or infinite models, is decidable. We show that this result covers decision procedures based on rewriting, generalizing recent work on combination of theories in the rewrite-based approach to satisfiability.

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“…satisfiable in an infinite model of T 1 ∪T 2 by the standard argument underlying the correctness of the Nelson-Oppen combination schema (see, e.g., [25,13]). …”

Section: Decidability Resultsmentioning

confidence: 99%

Decidability and Undecidability Results for Nelson-Oppen and Rewrite-Based Decision Procedures

Bonacina

Ghilardi

Nicolini

et al. 2006

Automated Reasoning

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“…If we take as T 0 the empty theory (in the one-sorted first-order empty language with equality), then T 0 is the theory of an infinite set and condition (C3) is equivalent to stable infiniteness (by a simple argument based on compactness); thus, Theorem 5.2 reduces to the standard Nelson-Oppen result [25], [26], [31] concerning stably infinite theories over disjoint signatures. We recall from [16] that among relevant examples of theories to which Theorem 5.2 is easily seen to apply, we have Boolean algebras with operators (namely the theories axiomatizing algebraic semantics of modal logic): thus, decidability of conditional word problem transfers from two theories axiomatizing varieties of modal algebras with operators to their union (provided only Boolean operators are shared).…”

Section: Applications: Decidability Transfer Through Ultrapowersmentioning

confidence: 99%

A Comprehensive Framework for Combined Decision Procedures

Ghilardi

Nicolini

Zucchelli

2005

Frontiers of Combining Systems

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Abstract. We define a general notion of a fragment within higher order type theory; a procedure for constraint satisfiability in combined fragments is outlined, following Nelson-Oppen schema. The procedure is in general only sound, but it becomes terminating and complete when the shared fragment enjoys suitable noetherianity conditions and allows an abstract version of a 'Keisler-Shelah like' isomorphism theorem. We show that this general decidability transfer result covers as special cases, besides applications which seem to be new, the recent extension of NelsonOppen procedure to non-disjoint signatures [16] and the fusion transfer of decidability of consistency of A-Boxes with respect to T-Boxes axioms in local abstract description systems [9]; in addition, it reduces decidability of modal and temporal monodic fragments [32] to their extensional and one-variable components.

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“…Work on theorem proving has focused on decomposition for parallel implementations [8,5,15,43] and has followed decomposition methods guided by lookahead and subgoals, neglecting the types of structural properties we used here. Another related line of work focuses on combining logical systems (e.g., [32,40,3,36,44]). Contrasted with this work, we focus on interactions between theories with overlapping signatures, the efficiency of reasoning, and automatic decomposition.…”

Section: Related Workmentioning

confidence: 99%

Improving the Efficiency of Reasoning Through Structure-Based Reformulation

Amir

McIlraith

2000

Lecture Notes in Computer Science

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Abstract.We investigate the possibility of improving the efficiency of reasoning through structure-based partitioning of logical theories, combined with partitionbased logical reasoning strategies. To this end, we provide algorithms for reasoning with partitions of axioms in first-order and propositional logic. We analyze the computational benefit of our algorithms and detect those parameters of a partitioning that influence the efficiency of computation. These parameters are the number of symbols shared by a pair of partitions, the size of each partition, and the topology of the partitioning. Finally, we provide a greedy algorithm that automatically reformulates a given theory into partitions, exploiting the parameters that influence the efficiency of computation.

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A New Correctness Proof of the Nelson-Oppen Combination Procedure

Cited by 100 publications

References 20 publications

Decidability and Undecidability Results for Nelson-Oppen and Rewrite-Based Decision Procedures

Decidability and Undecidability Results for Nelson-Oppen and Rewrite-Based Decision Procedures

A Comprehensive Framework for Combined Decision Procedures

Improving the Efficiency of Reasoning Through Structure-Based Reformulation

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