Gelfond, M. andPrzym usinska, H. (1986). Negation as failure: careful closure procedure. Artificial Intelligence, 30:273-- 287.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....by Minker [23] the Extended Generalized Closed World Assumption (EGCWA) by Yahya and Henschen [35] the Careful Closed World Assumption (CCWA) by Gelfond Mailing address: TU Wien, Paniglgasse 16, A 1040 Wien, Austria. Internet e mail: feiter,gottlobg vexpert.dbai.tuwien.ac. at and Przymusinska [13], and the Extended Closed World Assumption (ECWA) by Gelfond, Przymusinska, and Przymusinski [12] Circumscription was introduced by McCarthy in [22] It is known that circumscription as de ned in [36] coincides with the ECWA in the case of propositional logic [12] While much work has been ....

....T 6j= K. GCWA (Generalized CWA [23] K is a positive literal and for every positive clause B with T 6j= B, it holds that T 6j= B K. EGWCA (Extended GCWA [35] K is a conjunction of positive literals and for every positive clause B with T 6j= B, it holds that T 6j= B K. CCWA (Careful CWA [13]) K is a positive literal from P and for each clause B whose literals belong to P [ Q [ Q such that T 6j= B, it holds that T 6j= B K. ECWA (Extended CWA [12] K is an arbitrary formula not involving literals from Z and for each clause B whose literals belong to P [Q [Q such that T ....

M. Gelfond and H. Przymusinska. Negation as Failure: Careful Closure Procedure. Arti cial Intelligence, 30:273-287, 1986. 15

.... and Przymusinski [28] The Perfect Models Semantics (PERF) by Przymusinski [48] Various forms of the Closed World Assumption (CWA) including the Generalized CWA (GCWA) by Minker [43] the Extended GCWA (EGCWA) by Yahya and Henschen [67] the Careful CWA (CCWA) by Gelfond and Przymusinska [26], and the Extended CWA (ECWA) by Gelfond, Przymusinska, and Przymusinski [28] which coincides in the nite propositional case with McCarthy s circumscription (CIRC) 41, 42] and further variants such as the Disjunctive Database Rule (DDR) of Ross and Topor [55] which is equivalent to the Weak ....

....Informally, CWA adds to P each literal :x such that M(P ) 6j= x. This is not suitable for disjunctive logic programs, since the result of applying CWA may be inconsistent. 8 Several generalizations and extensions of CWA to disjunctive logic programs have been proposed in the literature, e.g. [43, 26, 67, 28], as well as improvements to some of those extensions [55, 51, 56, 13] Complexity results for these semantics in the case of nite propositional theories have been derived in [60, 11, 21, 13] In this section, we succinctly introduce various semantics and apply results of the previous section to ....

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M. Gelfond and H. Przymusinska. Negation as Failure: Careful Closure Procedure. Articial Intelligence, 30:273-287, 1986.

....# Book[doc3] # = c Book[doc3] 4. 2 Relation to other approaches Since the seminal paper by Reiter [44] many forms of closed world assumption (CWA) have been investigated (see [34, Chapter 7] for a thorough review) The proposal most similar in spirit to ours is the so called careful CWA [21], by means of which one can confine the closed world reading to a pre specified subset of predicate symbols only. Without going into the details of this and the other CWA proposals, we observe that neither careful CWA nor other forms of CWA seem suited to our program of allowing the closed world ....

Michael Gelfond and Halina Przymusinska. Negation as failure: careful closure procedure. Artificial Intelligence, 30:273--287, 1986.

....We extend the SLD resolution, the SLDNF resolution, as well as the standard model semantics to them respectively, and show that the completion of a G strati ed logic program is consistent. As a matter of fact, a lot of work has been done to achieve these goals. The careful CWA and extended CWA[13][17] generalizes the GCWA. The perfect model semantics for locally strati ed logic programs[19] generalizes the standard model semantics. After showing the completeness result of SLDNF resolution in the standard textbook [15] the author declared that the completeness of SLDNFresolution is of such ....

....It makes the GCWA more powerful. Besides, we can immediately assume all the other atoms in the same orbit of G to be negative after choosing a representative. It is easier to infer negative information in this respect. So far we don t know yet the relations between the G CWA and the careful CWA[13]. We wonder if the G CWA might be covered by the careful CWA. This problem doesn t seem theoretically important since we can also take G into account when performing the careful CWA. However, it is meaningful from the viewpoint of implementation. The G CWA has another advantage that one can modify ....

Gelfond, M., Przymusinska, H., Negation as Failure: Careful Closure of Procedure, Articial Intelligence 30, 1986, 273-287.

....introduced a consistency preserving extension of CWA to overcome this weakness called Generalized Closed World Assumption (GCWA) in [Minker, 1982] He defined negation free ground atoms where only they can be included in the final theory. The GCWA was further extended by Gelfond and Przymusinska [Gelfond and Przymusinska, 1986] and their logic was called Careful Closed World Assumption (CCWA) It allows us to restrict the effects of closing the world by specifying a set of freely chosen predicates which may be affected by the CWA rule. Gelfond, Przymusinska and Przymusinski [Gelfond et al. 1989] later proposed a ....

M. Gelfond and H. Przymusinska. Negation as failure: Careful closure procedure. Artificial Intelligence J., 30:273--287, 1986.

.... [48] several formalizations of closed world reasoning have been developed: the Generalized Closed World Assumption (GCWA) by Minker [41] the Extended Generalized Closed World Assumption (EGCWA) by Yahya and Henschen [60] the Careful Closed World Assumption (CCWA) by Gelfond and Przymusinska [18], and the Extended Closed World Assumption (ECWA) by Gelfond, Przymusinska, and Przymusinski [17] Circumscription was introduced by McCarthy in [37] It is known that circumscription coincides with the ECWA in the case of propositional logic [17] 10] deals with the complexity of closed world ....

M. Gelfond and H. Przymusinska. Negation as Failure: Careful Closure Procedure. Artificial Intelligence, 30:273--287, 1986. 37

.... the following well known approaches to closed world reasoning are considered: the Generalized Closed World Assumption (GCWA) by Minker [39] the Extended Generalized Closed World Assumption (EGCWA) by Yahya and Henschen [54] the Careful Closed World Assumption (CCWA) by Gelfond and Przymusinska [20], and the Extended Closed World Assumption (ECWA) by Gelfond, Przymusinska, and Przymusinski [19] Circumscription was introduced by McCarthy in [36] It is known that circumscription coincides with the ECWA in the case of propositional logic [19] In [12] we deal with the complexity of closed ....

M. Gelfond and H. Przymusinska. Negation as Failure: Careful Closure Procedure. Articial Intelligence, 30:273-287, 1986.

....This technique for non monotonic reasoning is closely related to circumscription [3, 10, 5] and has been defined in several ways. In its previous versions (CWA, GCWA, EGCWA [16, 14, 19] the closed world assumption was applied to all the predicates. In its more recent definitions (CCWA, ECWA [6, 5]) it is possible to vary a predicate, and the intended meaning of this variability is the same as in circumscription. Notice that the Extended Closed World Assumption (ECWA) which like circumscription minimizes some predicates P while varying other predicates Z and fixing the remaining predicates ....

....f(x) The intended meaning of b is is a bird , of ab is is abnormal , and of f is flies ) The list P of the predicates to be minimized contains only ab, while the remaining predicates b and f are the varying predicates Z. It is well known that CIRC(T ; P ; Z) j= f(Tweety) see, for example [6, 9]) The transformation we defined previously yields the following formula: Gamma(T ; f(Tweety) P ; Z) ab; b; f) j : ab 00 ; b 00 ; f 00 ) u (ab 0 ; b 0 ; f 0 j true) u Theta(ab; b; f; ab 0 ; b 0 ; f 0 ) where Theta(ab; b; f; ab 0 ; b 0 ; f 0 ) T ....

M. Gelfond and H. Przymusinska. Negation as Failure: Careful Closure Procedure. Artificial Intelligence, 30:273--287, 1986.

....inappropriate and unsafe. The generalized CWA was thus proposed by Minker[14] to overcome this problem. And later on, Gelfond et al. developed two more powerful formalizations to enhance the declarative ability of the generalized CWA, which is referred to as the careful CWA, and the extended CWA[8][9][10] In this paper we extend the CWA, the generalized CWA as well as the careful CWA to any nitely multi valued logics. We also present an example to demonstrate that the extended CWA can by no means be generalized following our treatment. 2 Basic knowledge: Multi valued logics We are ....

Gelfond, M., Przymusinska, H., Negation as Failure: Careful Closure of Procedure, Articial Intelligence 30(1986), 273-287.

....another proposal to cope with negative information based on the notion of orbits of programs, which enhances the declarative ability of GCWA. 5.1 CWA and GCWA We begin with the CWA and GCWA. For more details and the other sophisticated generalizations, the reader is referred to [22] 20] and [10]. Basically, we show that taking G(P ) into account may simplify the computation of CWA and GCWA. Deriving negative information For a ground atom A, recall that :A is said to be derivable from P under the CWA (denoted by CWA(P ) j= A) if P 6j= A, and :A is said to be derivable from P under the ....

....Besides, we can immediately assume all the other atoms in the same orbit of G(P ) to be negative after choosing a representative. it is easier to infer negative information in this respect. So far we don t know yet the relations between the OCWA and the CCWA developed by Gelfond and Przymusinska[10]. We wonder if the OCWA is covered by the careful CWA. This problem doesn t seem theoretically important since we can also take G(P ) into account when performing the careful CWA. It is meaningful from the viewpoint of practical implementation. Nevertheless, the OCWA has another advantage that one ....

Gelfond, M., Przymusinska, H., Negation as Failure: Careful Closure of Procedure, Articial Intelligence 30, 1986, 273-287.

....a comprehensive overview. We will deal with the following ones. ffl The Generalized Closed World Assumption (GCWA) by Minker [16] ffl The Extended Generalized Closed World Assumption (EGCWA) by Yahya and Henschen [30] ffl The Careful Closed World Assumption (CCWA) by Gelfond and Przymusinska [11]. ffl The Disjunctive Database Rule (DDR) by Ross and Topor [23] which is equivalent to the Weak GCWA of Rajasekar, Lobo, and Minker [21] Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, ....

....CWA (GCWA) in [16] which adds all literals :x to DB such that atom x is false in all minimal models of DB. The respective models of DB can be characterized as follows. GCWA(DB) fM 2 M (DB) 8x 2 V: MM(DB) j= x )M j= xg The Careful Closed World Assumption (CCWA) of Gelfond and Przymusinska [11] generalizes the GCWA as follows. For a partition hP ; Q; Zi of V , each literal :x, x 2 P , is added to DB such that MM(DB;P ; Z) j= x. Thus, CCWAP ;Z (DB) fM 2 M (DB) 8x 2 P: MM(DB;P ; Z) j= x )M j= xg Notice that if P = V , CCWA is identical to GCWA. It is immediate from the ....

M. Gelfond and H. Przymusinska. Negation as Failure: Careful Closure Procedure. Artificial Intelligence, 30:273--287, 1986.

....of the pure state. Typically, negative answers, i.e. negated formulae, can often not be obtained explicitly. A practically important class of database completions are closed world assumptions (CWAs) which originated with [Rei78] and were further developed by [Min82] and others (e.g. [GP86], GPP86] YH85] into different versions. The variety of completions proposed in the literature gives rise to the question which is the right one. Since Reiter s CWA lead to inconsistencies in the case of databases with indefinite information, other authors invented new CWAs which were similar ....

....1 g is a subset of Psi. Then the conditions of the previous lemma are automatically satisfied and we can conclude that these assumptions are not needed in the completed database state. They are needed in the completion process, though, to block other assumptions. This was utilized in the CWA of [GP86]. If the possible assumptions are disjunctions of atomic assumptions Psi 0 then it suffices to restrict 1 ; n in theorem 17 to such atomic assumptions (This was utilized in the CWA of [YH85] Lemma 19 Let Psi be a set of possible assumptions and Psi 0 Psi such that every 2 ....

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M. Gelfond and H. Przymusinska. Negation as failure: Careful closure procedure. Artificial Intelligence (30), 273--287, 1986.

....have a hierarchical structure that makes them suitable for efficient bottom up evaluation. Notice that transforming a query C into C implies answering the query under the assumption that the knowledge about every role is complete, like for example in [19, p. 113] On the other hand, as noted in [13], there are situations where we would like to apply the closed world assumption only to some of the concepts and the roles of the knowledge base. We argue that the use of epistemic operators as described in the previous sections is a natural way to achieve such a flexible way of interacting with ....

Gelfond, M. and Przymusinska, H. Negation as failure: Careful closure procedure. Artificial Intelligence, 30:273--287, 1986.

....reasoning. In the past decade, variations and extensions to the circumscriptive formalism has been studied by several researchers Bidoit , Bossu , Etherington , Gelfond , Hull , Imielinski , Li , Lifschitz , Minker , Moinard , Perlis , Przymusinska , Przymusinski , Reiter , Siegel and You . [40, 24, 25, 3, 5, 2, 13, 12, 23, 2, 7, 9, 11, 15, 14, 21, 32, 35, 8, 31, 10, 16, 30, 36, 33, 34, 43]. One of the major concerns in the field has been the computational aspect of circumscription. The problem of answering queries using circumscription can be stated as follows [6] Given a finite set of axioms A, a circumscriptive policy C, and a formula F, does F follow from A by the ....

M. Gelfond and H. Przymusinska. Negation as failure: Careful closure procedure. Artificial Intelligence, 30(3):273--286, 1986 December 1986.

....approximation of ordinary reasoning. Apart from AI, NMR is sometimes used to represent knowledge in a more compact fashion. Noticeably, negation through cut is commonly used among prolog programmers for writing more compact and efficient programs [41, Chap.11] Moreover, closed world reasoning [35, 15, 16, 34] allows for effective representation of implicit knowledge in relational as well as deductive databases, and has been widely used among database practitioners for many years now. Apparently, there is a mismatch between the intuition behind NMR and the theoretical results on its computational ....

.... T is defined in [35] as follows: CWA(T ) T [ f:p j T 6j= pg: 1) This rule has been refined by several authors: Minker [30] introduced the generalized CWA, Rajasekar, Lobo and Minker [34] the weak generalized CWA, Yahya and Henschen [44] the extended generalized CWA, Gelfond and Przymusinska [15] the careful CWA, Gelfond, Przymusinski and Przymusinska [16] the extended CWA. The notion of varying atoms has been used in the careful and in the extended CWA. Extended CWA, denoted as ECWA(T ; P ; Z) is the most general of all the above rules and is defined as follows: T [ f:K j 6 9B: T 6j= ....

M. Gelfond and H. Przymusinska. Negation as failure: Careful closure procedure. Artificial Intelligence Journal, 30:273--287, 1986.

....several researchers proposed more sophisticated forms of the closed world assumption. Minker [Mi82] defined the so called Generalized Closed World Assumption (GCWA) that was subsequently improved and extended by Gelfond, H.Przymusinska and T. Przymusinski ( GPP86] GPP86a] see also [YH85] and [GP86]) resulting in the so called Extended Closed World Assumption (ECWA) which was shown to be consistent and equivalent to parallel circumscription for any disjunctive deductive database. In order to obtain a syntactic equivalent of prioritized circumscription, Gelfond, H.Przymusinska and ....

Gelfond, M. and Przymusinska, H., `Negation as Failure: Careful Closure Procedure', Artificial Intelligence 30(1986), 273-287.

....assumption, or ECWA [ Gelfond et al. 1989 ] The same idea is sometimes expressed in terms of confirmation and unconfirmed formulas, or in terms of characteristic clauses. Some of the other papers belonging to this line of research are [ Bossu and Siegel, 1985 ] Lifschitz, 1985 ] Gelfond and Przymusinska, 1986 ] Baker and Ginsberg, 1989 ] Ginsberg, 1989 ] Przymusinski, 1989 ] Suchenek, 1989 ] Gelfond et al. 1990 ] and [ Inoue and Helft, 1990 ] The goal of this note is to show that the theorem from [ Gelfond et al. 1989 ] relating the ECWA to circumscription can be proved in a ....

Michael Gelfond and Halina Przymusinska. Negation as failure: Careful closure procedure. Artificial Intelligence, 30(3):273--287, 1986.

....[128] CWA(T ) T [ f:pj T 6j= pg (2) where p is a ground atom. This rule has been refined by several authors: Minker [102] introduced the generalized CWA, Rajasekar, Lobo and Minker [126] the weak generalized CWA, Yahya and Henschen [156] the extended generalized CWA, Gelfond and Przymusinska [59] the careful CWA, Gelfond, Przymusinski and Przymusinska [60] the extended CWA and the iterated CWA. The notion of fixed and varying predicates has been used in the careful, in the extended and in the iterated CWA. Extended CWA is the most general of all the above rules and is defined as ....

....the same results hold for the extended CWA as well. For what concerns other forms of closed world inference, Cadoli and Lenzerini show in the same work that many intractable cases for the extended CWA are tractable for other definitions. As an example, careful closed world reasoning (defined in [59]) is polynomial for 2CNF formulae. This can be explained by observing that the definition of careful CWA is (3) with K restricted to be a single positive literal. As a consequence the space of the potential free for negation formulae is linear in the size of the input. In view of this fact, it ....

M. Gelfond and H. Przymusinska. Negation as failure: Careful closure procedure. Artificial Intelligence Journal, 30:273--287, 1986.

....a literal is a consequence of a disjunctive program if and only if it is true in every minimal Herbrand model of the program. The syntactic counterpart for inferring negative literals, called the generalized closed world assumption, is defined as follows: We will use the terminology from [GP86]. A disjunction D of ground atoms is called essential w.r.t. theory T if T j= D and no sub disjunction of D is entailed by T . A ground atom is called free for negation in T if it does not belong to any clause essential in T . Let T be the set of negations of all ground atoms free for negation in ....

....all ground atoms free for negation in T . Then GCWA(T ) T : 40 Minker [Min82] proves that for T with finite number of constants and no function symbols, T [ GCWA(T ) classically entails a literal q iff q is true in all minimal Herbrand models of T . This result was extended to arbitrary T in [GP86, She88]. The following proposition establishes the connection between Minker s semantics and answer set semantics of disjunctive logic programs. Proposition 4.4 [Gel92b] Let Pi be a program consisting of rules of the form A 0 or : or A k A k 1 : Am (where A s are atoms) and the closed ....

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M. Gelfond and H. Przymusinska. Negation as failure: Careful closure procedure. Artificial Intelligence, 30(3):273--287, 1986.

....a literal is a consequence of a disjunctive program if and only if it is true in every minimal Herbrand model of the program. The syntactic counterpart for inferring negative literals, called the generalized closed world assumption, is defined as follows: We will use the terminology from [GP86]. A disjunction D of ground atoms is called essential w.r.t. theory T if T j= D and no sub disjunction of D is entailed by T . A ground atom is called free for negation in T if it does not belong to any clause essential in T . Let T be the set of negations of all ground atoms free for negation ....

....of all ground atoms free for negation in T . Then GCWA(T ) T : Minker [Min82] proves that for T with finite number of constants and no function symbols, T [ GCWA(T ) classically entails a literal q iff q is true in all minimal Herbrand models of T . This result was extended to arbitrary T in [GP86, She88]. The following proposition establishes the connection between Minker s semantics and answer set semantics of disjunctive logic programs. Proposition 4.4 [Gel92b] Let 5 be a program consisting of rules of the form A 0 or : or A k A k 1 : Am (where A s are atoms) and the closed world ....

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M. Gelfond and H. Przymusinska. Negation as failure: Careful closure procedure. Artificial Intelligence, 30(3):273--287, 1986.

....database as well as a first order theory obtained from it by replacing every rule A 1 or : or A n B 1 ; Bm (6) by a formula B 1 : Bm oe A 1 : A n (7) Recall that Minker s generalized closed world assumption is defined as follows. We will use the terminology from [GP86]. A disjunction D of ground atoms is called essential w.r.t. theory Pi if Pi j= D and no subdisjunction of D is entailed by Pi. A ground atom is called free for negation in Pi if it does not belong to any clause essential in Pi. Let Pi be a set of negations of all ground atoms free for ....

....a database Pi is the answer to Q by Pi . Proposition 3. For any positive disjunctive database Pi and any query Q, the GCWA answer to Q coincides with the closed world answer to Q. Proof. First let us recall that, as was proved for a finite theory Pi in [Min82] and for an arbitrary Pi in [GP86], Pi is equal to the set of negations of all ground atoms not belonging to any minimal Herbrand model of Pi. Obviously, M is a minimal model of Pi iff M is a belief set of Pi and hence, by Proposition 2 a set B = fW : W = M [ Pig is a world view of Pi . Recall, that by Corollary 1, for ....

Michael Gelfond and Halina Przymusinska. Negation as failure: Careful closure procedure. Artificial Intelligence, 30(3):273--287, 1986.

....implied by COMP(T) i.e. if COMP (T ) j= F . An important example of such a definition is Reiter s Closed World Assumption CWA(T) 21] obtained by adding to T negations of all ground atoms not provable from T. Like its more broadly applicable generalizations such as GCWA(T) and ECWA(T) [17, 26, 6, 7, 8] it is based on the idea of adding to the theory a suitably selected set of ground formulae, which are not derivable from the theory itself. Another approach, that can be called model theoretic, associates with T a set MOD(T) of one or more models of T and declares that a given sentence F is ....

....between these semantics and therefore no straightforward generalization of the results from the first part can be obtained. We prove, however, several results relating the three semantics for ground queries. The work presented in this paper can be viewed as a continuation of the work started in [6, 7, 8] (see also [11] where it was shown that, for a theory T without function symbols, with finite number of constants and the Domain Closure Assumption (DCA) 22] the Extended Closed World Assumption ECWA(T) Parallel Circumscription CIRC(T) and Domain Circumscription DCIRC(T) are all equivalent. ....

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Gelfond, M. and Przymusinska, H., "Negation as Failure: Careful Closure Procedure", Artificial Intelligence 30(1986), 273-287.

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Gelfond, M. andPrzym usinska, H. (1986). Negation as failure: careful closure procedure. Artificial Intelligence, 30:273-- 287.

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Michael Gelfond and Halina Przymusinska. Negation as failure: Careful closure procedure. Artificial Intelligence, 30(3):273--287, 1986.

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