Resolution Theorem Proving

Rewriting Conjunctive Queries over Description Logic Knowledge Bases

“…The limit N ∞ of a (possibly infinite) fair derivation from a set of Horn clauses N is defined as N ∞ = i N i . It is well known that N ∞ is saturated by R and does not depend on the derivation [4]. A set of Horn clauses N is satisfiable if and only if ∈ N ∞ .…”

“…It is well known that, if R |= F ( a), then a derivation tree T for F ( a) from R by H exists [4]. We represent such a tree T as a tuple T N , δ, λ for T N the set of nodes where we denote with t.i the i-th child of t ∈ T N and with ǫ the root node; δ is a function that maps each node t ∈ T N to a fact δ(t); and λ is a function that maps each node t ∈ T N to a clause λ(t) such that δ(t) is derived by hyperresolving each δ(t.i) with the i-th body literal of λ(t).…”

“…If P ∪ Q C ∪ A |= Q P ( a), then Q P ( a) can be derived from the set of clauses P ∪ Q C ∪ A using SLD resolution [4]. First, we nondeterministically choose a query Q i ∈ Q C and ground it by nondeterministically choosing a set of constants from A.…”

“…Resolution with free selection is a well-known calculus that can be used to decide satisfiability of a set of Horn clauses N [4]. The calculus is parameterized by a selection function S that assigns to each Horn clause C a nonempty set of atoms such that either S(C) = {H} or S(C) ⊆ {B i }.…”

Saturation Up to Redundancy for Tableau and Sequent Calculi

“…We discuss an adaptation of the technique of saturation up to redundancy, as introduced by Bachmair and Ganzinger [1], to tableau and sequent calculi for classical first-order logic. This technique can be used to easily show the completeness of optimized calculi that contain destructive rules e.g.…”