Tableaux and Related Methods

A Model Generation Style Completeness Proof for Constraint Tableaux with Superposition

“…[12]): It is clear from our completeness proof that an arbitrary number of ext-literals with the same instantiation, though with different histories, might be needed.…”

“…A number of publications would follow, providing variations on the theme, like what is known as 'basic ordered paramodulation' in the resolution community, or a version with universal variables (see e.g. [12]), or hyper tableaux [4] with equality. Curiously enough, this has not happened!…”

“…where it is sound to instantiate different occurrences of a free variable differently. If that is the case, one calls the free variable 'universal' for the formula in which it occurs, see [12]. As has been recognized in [5], using universal variables is crucial for efficient equality handling in tableaux.…”

Unified Classical Logic Completeness

“…The price is usually a lack of rigor, in particular a gap between the definition of derivation trees and its use in the completeness argument. This lack of rigor should not be taken lightly, as it may lead to serious ambiguities or errors: In the context of a tableau completeness proof development, Hähnle [14] first performs an implicit switch from finite to possibly infinite tableaux, and then claims that tableau chain suprema exist by wrongly invoking Zorn's lemma [14,Definition 3.16]. 6 The completeness theorem has been mechanized before in proof assistants.…”

“…Nondestructive tableaux [14]-including those presented in Bell and Machover [1] and in Fitting [8]-are usually persistent when regarded as derivation systems. After an application of Theorem 6, the argument for interpreting the abstract model is similar to that for Gentzen systems (Lemma 7).…”