# An efficient computation of the extended generalized closed world assumption by support-for-negation sets

## Abstract

Closed world assumptions are one of the major approaches for non-monotonic reasoning in artificial intelligence. In [16] this formalism is applied to disjunctive logic programs, i.e. logic programs with positive disjunctive rule heads and positive atoms in the rule bodies. The disjunctive closure operator \(\mathcal{T}_P^S\) allows for the derivation of the set \(\mathcal{M}\mathcal{S}_P\) of all positive disjunctive clauses logically implied by the program *P*, the *minimal model state*. On the other hand, disjunctive clauses over negated atoms are derived in the *extended generalized closed world assumption*\(\mathcal{E}\mathcal{G}\mathcal{C}\mathcal{W}\mathcal{A}_P\). Such a clause is the negation of a conjunction of positive atoms. \(\mathcal{E}\mathcal{G}\mathcal{C}\mathcal{W}\mathcal{A}_P\) contains all conjunctions which are *false* in all minimal Herbrand models of the program.

We present efficient *δ*-iteration techniques for computing the closed world assumption \(\mathcal{E}\mathcal{G}\mathcal{C}\mathcal{W}\mathcal{A}_P\) based on an iterative computation of the set of all support-for-negation sets SN(*C*) for conjunctions *C*, i.e. certain sets of clauses which characterize \(\mathcal{E}\mathcal{G}\mathcal{C}\mathcal{W}\mathcal{A}_P :C \varepsilon \mathcal{E}\mathcal{G}\mathcal{C}\mathcal{W}\mathcal{A}_P\) iff \(\mathcal{M}\mathcal{S}_P \vDash SN(C)\). The support-for-negation sets SN(*A*) for atoms *A* are easily derived from the minimal model state \(\mathcal{M}\mathcal{S}_P\). We will propose a *bottom-up* computation deriving the support-for-negation sets of longer conjunctions from shorter ones based on an algebraic formula given by [16]: SN(C_{1} ∧ C_{2}) = SN(C_{1}) ∀ SN(C_{2}). We will present techniques for the efficient computation of these disjunctions of two clause sets and a *δ*-iteration approach for computing the support-for-negation sets of a sequence of growing minimal model states.

For disjunctive normal logic programs, i.e. logic programs with positive disjunctive rule heads and — possibly — negated atoms in the rule bodies, these operators form a basis for computing the *generalized disjunctive well-founded semantics*.

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