Simplification Rules for Constrained Formula Tableaux

Abstract

Abstract. Several variants of a first-order simplification rule for nonnormal form tableaux using syntactic constraints are presented. These can be used as a framework for porting methods like unit resolution or hyper tableaux to non-normal form free variable tableaux.

Saturation Up to Redundancy for Tableau and Sequent Calculi

“…6: Theorem 3 of [6] establishes the interesting fact that under certain sensible restrictions, our calculus always permits only a finite number of inferences without intervening γ inferences. This means that we can obtain fairness simply by requiring derivations to be built in such a way that γ inferences may only be applied when there are no more possible simp inferences.…”

“…5. Completeness of such a calculus has previously been shown using proof transformation arguments [6], but the proof using saturation up to redundancy will be a lot simpler. We will start from pre-skolemized formulae that contain no existential quantifiers, to avoid discussing the soundness issues that arise in connection with free variables in δ rules.…”

“…Sometimes, proof transformation techniques can be used to cope with destructiveness, see e.g. [6], but these require plenty of creativity and are very specific to the calculus at hand.…”

Hyper Tableaux with Equality

“…In [Gie03], simplification rules and reasoning with universal variables 3 are added to the framework of [Gie02], but without equality. Putting equality aside, the most relevant contribution in [Gie03] from the viewpoint of this paper is the instantiation of the calculus there to a variant of the hyper tableau calculus [BFN96].…”

“…Putting equality aside, the most relevant contribution in [Gie03] from the viewpoint of this paper is the instantiation of the calculus there to a variant of the hyper tableau calculus [BFN96]. 4 An important difference to [BFN96] is that [Gie03] uses rigid variables for variables that are shared between positive literals in clauses. For instance, a clause like ∀x, y (p(x, y) ∨ q(x)) then is treated by β-expansion with the formulas ∀y p(X, y) and q(X), where X is a rigid variable shared between branches.…”