H. Przymusinska and T. Przymusinski. Stationary default extensions. Fundamenta Informaticae, 21(1-2):67-87, 1994.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....xpoints of the operator E as extensions of . Further, in view of Proposition 5.5, we call xpoints of the operator E , partial extensions of . One can show that consistent partial extensions of a default theory are in one to one correspondence with stationary extensions of de ned in [PP94]. We also note that the operator E coincides with the operator de ned on sets of interpretations and proposed by Guerreiro and Casanova [GC90, Lif90, MT93] Guerreiro and Casanova simply rephrased the original de nition of the operator (which works on theories and can be restricted, ....

....to as weak extensions) Kripke Kleene semantics provides an approximation for the skeptical reasoning under the semantics of expansions. The operator E also leads to the operator E . Consistent xpoints of this operator yield partial extensions (stationary extensions in the terminology of [PP94]) Fixpoints of the related operator E correspond to extensions by Reiter. The least xpoint of the operator E results in the well founded semantics for default logic and approximates the skeptical reasoning under the semantics of extensions. The relationships between the operators of ....

H. Przymusinska and T. Przymusinski. Stationary default extensions. Fundamenta Informaticae, 21(1-2):67-87, 1994.

....E Delta defined on the set of belief pairs. We show that the operator E Delta gives rise to three other operators and that the fixpoints of these operators yield several semantics for default theories. Among these semantics are the semantics of extensions by Reiter, the stationary semantics [PP94], the well founded semantics for default logic [BS91] it approximates skeptical reasoning under extensions) the semantics of weak extensions [MT89a] and the Kripke Kleene semantics (it approximates skeptical reasoning under weak extensions) Our results settle the issue of the relationship ....

....that the fixpoints of E st Delta can be regarded as 4 valued (3 valued, in the case of consistent fixpoints) generalizations of an extension of a default theory. We will therefore call them partial extensions. It turns out that partial extensions coincide with stationary extensions defined in [PP94]. Our next result describes monotonicity properties of the operators E st Delta and E st Delta . Theorem 4.6 Let Delta be a default theory. Then, the operator E st Delta is v antimonotone and the operator E st Delta is kn monotone. Theorem 4.6 implies that the operator E st ....

H. Przymusi'nska and T. Przymusi'nski. Stationary default extensions. Fundamenta Informaticae, 21(1-2):67--87, 1994.

....E Delta defined on the set of belief pairs. We show that the operator E Delta gives rise to three other operators and that fixpoints of these operators yield several semantics for default theories. Among these semantics are the semantics of extensions by Reiter [13] the stationary semantics [12], the wellfounded semantics for default logic [1] the semantics of weak extensions [9] and the Kripke Kleene semantics (which approximates skeptical reasoning under weak extensions) Our results settle the issue of the relationship between autoepistemic and default logics. We show that the ....

.... they correspond to stationary 11 weak extensions partial extensions Kripke Kleene fixpoint E Delta E Delta partial extensions well founded semantics oe Reiter s extensions oe E st Delta E st Delta Gamma Gamma Psi Gamma Gamma Gamma Gamma Psi R extensions [12]. The operator E st Delta is kn monotone. Consequently, it has a least fixpoint. This fixpoint coincides with the well founded semantics for default logic described in [1] We will refer to it as the well founded fixpoint of E st Delta . A complete discussion of relations between fixpoints ....

H. Przymusi'nska and T. Przymusi'nski. Stationary default extensions. Fundamenta Informaticae, 21(1-2):67--87, 1994.

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