The Disconnection Tableau Calculus

Abstract

In this paper we give a comprehensive presentation of the disconnection tableau calculus, a proof method for formulas in classical first-order clause logic. The distinguishing property of this calculus is that it uses unification in such a manner that important proof-theoretic advantages of the classical (i.e., Smullyan-style) tableau calculus are preserved, specifically the termination and model generation characteristics for certain formula classes. Additionally, the calculus is well suited for fully automated proof search. The calculus is described in detail with soundness and completeness proofs, and a number of important calculus refinements developed over the past years are presented. Referring to the model-finding abilities of the disconnection calculus, we explain the extraction and representation of models. We also describe the integration of paramodulation-based equality handling. Finally, we give an overview of related methods.

Cut-Based Abduction

“…It should be possible to expand the branch-driven abduction method to the first order formulation found in Cialdea-Mayer and Pirri [6]. Another interesting result in that line, which certainly needs greater care, is to apply the branch-driven abduction ideas to modern tableau-based theorem provers, such as the disconnection calculus [17].…”

Towards automated first-order abduction: the cut-based approach

“…Resolution-based methods use in a fundamental way the easiness for dealing with quantified variables provided by the unification procedure. The combination of unification with tableau inference systems have been best realized in the literature with the disconnection tableaux [3,17]. However, such method is still based on clausal-form first-order logic and is still very much based on analytic methods.…”

Evolving model evolution

“…While we do take advantage of some of the distinguishing features of Darwin and the ME calculus, especially in the way models are constructed, our method depends neither on Darwin nor ME. Without much additional effort, we could use any other decision procedure for function-free clause logic, such as Inst-Gen [38,45] or DCTP [50].…”

“…First, because they are not decision procedures for a function-free clause set and in practice often fail to determine its satisfiability, and second, because they are usually not capable of providing a model. In contrast, recent instantiation based calculi such as ME, Inst-Gen[38], and Disconnection Calculus[50] satisfy both criteria. It re-lies on a decision procedure for the full fragment of LIA instead of a complete enumerator of LIA-unifiers.…”