A.C. Kakas, P. Mancarella, Stable theories for logic programs. Proc. ISLP'91, MIT Press 31  Home/Search   Document Not in Database   Summary   Related Articles   Check

....program is NP complete under this semantics. Thus, from the computational side, as with deduction [68, 20] the possible model semantics is more attractive than the disjunctive stable model semantics. Even the effective computation of abductive explanation has been addressed in several works [17, 15, 22, 32, 38], where suitable extensions of the SLDNF procedure have been designed for abduction (these procedures return an abductive explanation of the given query) Finally, the use of abductive logic programming for dealing with incomplete information has been discussed by Denecker and De Schreye [16] and ....

A. Kakas and P. Mancarella. Stable Theories for Logic Programs. In Proc. ILPS '91, pp. 88--100, 1991.

....whereas in P rog EK they can be counterattacked as soon as an assumption belonging to them is identi ed. 26 9 Comparisons Toni and Kakas [40] also develop abstract argumentation theoretic proof procedures, which abstract the one in [13] for computing admissibility as well as weak stability [20] and acceptability semantics [21] in the abstract case and, by instantiation, in the special cases we have considered in this paper. Their proof procedure for computing admissibility corresponds to the proof procedure in section 6 but is formulated in terms of derivations of trees (whose nodes are ....

A.C. Kakas, P. Mancarella, Stable theories for logic programs. Proc. ISLP'91, MIT Press

....abductive explanations. In [EK89] a basic query answering procedure for abductive programs based on SLDNF resolution is defined. In addition to the usual yes no answer of SLDNF , this procedure also returns an abductive explanation of the corresponding query. The idea was further developed in [KM91], Dun91a] and [DDS92] The procedure is shown to be correct w.r.t. the stable model semantics for call consistent logic programs, but (as pointed out in [EK89] not in general. This fact led to modification of the procedure to achieve correctness w.r.t. the stable model semantics [SI92] as well ....

....for call consistent logic programs, but (as pointed out in [EK89] not in general. This fact led to modification of the procedure to achieve correctness w.r.t. the stable model semantics [SI92] as well as to work on modification of the semantics to fit the inference method of the procedure [KM91], Dun91a] These methods were applied to formalizations of various benchmarks in temporal, legal and other types of reasoning [Sha89, DMB92, Esh88] There are several useful generalizations of the notion of abductive logic programs. In [Gel91] abduction is combined with reasoning with classical ....

A. Kakas and P. Mancarella. Stable theories for logic programs. In Proc. of ISLP-91, pages 88--100, 1991.

.... for formalizing a variety of forms of non monotonic reasoning [10, 11] and generalized to deal with contradiction removal and counterfactuals [7, 8, 9] The increasing role of logic programming extensions as an encompassing framework for these and other AI topics is expounded at length in [4], where they argue, and we concur, that WFS is by design overly careful in deciding about the falsity of some atoms, leaving them undefined, and that a suitable form of CWA can be used to safely and undisputably assume false some of the atoms absent from the well founded model of a program. ....

....differs from EWFM is the game example of the introduction. In this example EWFM gives the (strange) result that a is a winning position, and thus bets are raised. A similar approach based on the notion of stable negative hypotheses (built upon the notion of consistency) is introduced in [4], identifying a stable theory associated with a program P as a sceptical semantics for P , that always contains the well founded model. One example showing that their approach is still conservative is: p q q r r p s p Stable theories identify the empty set as the meaning of the ....

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A. C. Kakas and P. Mancarella. Stable theories for logic programs. In Ueda and Saraswat, editors, International Logic Programming Symposium, pages 85--100. MIT Press, 1991.

....al. 1991e ] and generalized to deal with contradiction removal and counterfactuals [ Pereira et al. 1991a, Pereira et al. 1991b, Pereira et al. 1991c ] The increasing role of logic programming extensions as an encompassing framework for these and other AI topics is expounded at length in [ Kakas and Mancarella, 1991b ] where they argue, and we concur, that WFS is by design overly careful in deciding about the falsity of some atoms, leaving them undefined, and that a suitable form of CWA can be used to safely and undisputably assume false some of the atoms absent from the well founded model of a program. ....

....s pg: Stable theories identifies the empty set as the meaning of the program; however its O Model is f sg, since it is consistent, maximal, sustainable and tenable. Kakas (personnal communication) now also obtains this model, as a result of the investigation mentioned in the conclusions of [ Kakas and Mancarella, 1991b ] 8 Conclusions We identify the meaning of a program P as a suitable partial closure of the well founded model of the program in the sense that it contains the well founded model (and thus always exists) The extension we propose reduces undefinedness (which some authors argue is a desirable ....

A. C. Kakas and P. Mancarella. Stable theories for logic programs. In Ueda and Saraswat, editors, International Logic Programming Symposium'91. MIT Press, 1991.

....Esprit BR project Compulog 2 (no. 6810) for its support. ing to every normal program, is cumulative [Dix91] and can be obtained by a bottom up fixpoint construction. Because it is a relevant semantics [Dix92] it is susceptible of top down existencial query procedures. However, as argued in [KM91, PAA92a, PAA93], the WFS is by design overly careful in deciding about the falsity of some atoms, leaving them undefined. In [PAA92a, PAA93] we introduce a suitable form of closed world assumption (CWA) and use it to safely and undisputably assume false some of the atoms absent from the well founded model of a ....

....b are false in it. However not b can be safely assumed. The EWFM is fa; c; not bg: The atom a is true in the EWFM and has no rule with a body true in it. All rules for b have a false body in the EWFM. Another semantics also extending the WFS of normal programs is the stable theories semantics [KM91]. Like O semantics, stable theories also enlarge the WFS by adding more negative assumptions. In order to avoid some unintuitive results of stable theories, Kakas and Mancarella (personal communication) modified their definition, and presented the acceptability semantics . In this recent work, ....

A. C. Kakas and P. Mancarella. Stable theories for logic programs. In Ueda and Saraswat, editors, Int. LP Symp., pages 85--100. MIT Press, 1991.

....abductive explanations. In [EK89] a basic query answering procedure for abductive programs based on SLDNF resolution is defined. In addition to the usual yes no answer of SLDNF , this procedure also returns an abductive explanation of the corresponding query. The idea was further developed in [KM91], Dun91a] and [DDS92] The procedure is shown to be correct w.r.t. the stable model semantics for call consistent logic programs, but (as pointed out in [EK89] not in general. This fact led to modification of the procedure to achieve correctness w.r.t. the stable model semantics [SI92] as well ....

....for call consistent logic programs, but (as pointed out in [EK89] not in general. This fact led to modification of the procedure to achieve correctness w.r.t. the stable model semantics [SI92] as well as to work on modification of the semantics to fit the inference method of the procedure [KM91], Dun91a] These methods were applied to formalizations of various benchmarks in temporal, legal and other types of reasoning [Sha89, DMB92, Esh88] There are several useful generalizations of the notion of abductive logic programs. In [Gel91] abduction is combined with reasoning with classical ....

A. Kakas and P. Mancarella. Stable theories for logic programs. In Proc. of ISLP-91, pages 88--100, 1991.

.... the 3 valued stable semantics (3 SS) developed in [La 92] is strictly more powerful than WFS and that of Bonatti [Bo 90] see Section 6) In spite of their many advantages, the following drawbacks have been identified for SS and WFS (e:g: see Van et al. [VGRS 88] and Kakas and Mancarella [KM 91] Although SS is elegant and quite important, i) not all logic programs possess stable models, and (ii) it assigns an unintuitive meaning to certain programs. while WFS overcomes these problems, the conclusions sanctioned by it can sometimes be quite weak (this a result of its conservative ....

.... a bilattice, he has recently extended his previous approach in a manner that extends the applicability of WFS to a rich class of logics [Fi 91] Several other researchers have considered the problem of extending SS and WFS (while avoiding their drawbacks) see Baral [Ba 91] Kakas and Mancarella [KM 91] Sacca and Zaniolo [SZ 90] and You and Li [YL 90] The relationships between our approach and these works are explored in Section 6. This paper is organized as follows. In Section 2 we define normal programs and their Kripke models. In Section 3 we develop the notion of the canonical Kripke ....

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Kakas, A.C. and Mancarella, P.: "Stable theories for logic programs, " Proc. North American Conference on Logic Programming (1991), 85-100.

....p 2HB, ffl Ab =HB not . A logic program P is a theory, P L, in such an assumption based framework. The interpretation of negative literals as abducibles was first presented in [7] and was the basis for the preferred extension semantics [4] the stable theory and acceptability semantics [11], and the argumentation theoretic interpretation for the semantics of logic programming presented in [13] The instance of the definition 2.2 of attack for the assumption based framework h(L; R) Abi for logic programming is the following: ffl Given a logic program P and sets of assumptions ....

....extension semantics is consequently more modular than stable model semantics. For example, the program fq; p not pg has no stable extension, but it has a preferred extension containing q. 3. 1 Stable theories and acceptability semantics To capture stable theory and acceptability semantics [11] we need two new notions of counterattack, different from those introduced in definition 2.3. For simplicity, we present these notions in the assumption based framework for logic programming. However, they can also be defined more generally and can be applied to any other assumption based ....

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A. C. Kakas, P. Mancarella, Stable theories for logic programs. Proc. ISLP'91, San Diego (1991)

....in [12] in two ways: it is not restricted to the logic programming case only and it is not restricted to any execution strategy. Toni and Kakas [34] also develop abstract argumentation theoretic proof procedures, which abstract the one in [12] for computing admissibility as well as weak stability [17] and acceptability semantics [18] in the abstract case and, by instantiation, in the special cases we have considered in this paper. Their proof procedure for computing admissibility corresponds to the proof procedure in section 6 but is formulated in terms of derivations of trees (whose nodes are ....

A.C. Kakas, P. Mancarella, Stable theories for logic programs. Proc. ISLP'91, MIT Press

....the number of the atoms that appear negative in the knowledge base. The algorithm follows from work on abductive extensions of logic programming in which stable models are characterized in terms of sets of hypotheses that can be drawn as additional information (Eshghi Kowalski, 1989; Dung, 1991; Kakas Mancarella, 1991). This is done by making negative atoms abductible and by imposing appropriate denials and disjunctions as integrity constraints. The work of Eshghi and Kowalski (1989) Dung (1991) and Kakas and Mancarella (1991) implies the following. Theorem 3.1 Let Pi be a knowledge base, and let H be the ....

....that can be drawn as additional information (Eshghi Kowalski, 1989; Dung, 1991; Kakas Mancarella, 1991) This is done by making negative atoms abductible and by imposing appropriate denials and disjunctions as integrity constraints. The work of Eshghi and Kowalski (1989) Dung (1991) and Kakas and Mancarella (1991) implies the following. Theorem 3.1 Let Pi be a knowledge base, and let H be the set of atoms that appear negated in Pi. M is a stable model of Pi iff there is an interpretation I over H such that 1. for every atom P 2 H, if P 2 I, then P 2 M 0 , 2. M 0 and I are consistent, and 3. M = ....

Kakas, A. C., & Mancarella, P. (1991). Stable theories for logic programs. In Saraswat, V., & Udea, K. (Eds.), ISLP-91: Proceedings of the 1991 international symposium on logic programming, pp. 85--100. MIT Press.

....A and A B, infer B as a possible explanation of A. Nonmonotonic reasoning has been explored as a form of abductive reasoning. In particular, default assumptions in logic programs have been treated as abductive hypotheses and a number of reasoning mechanisms and semantics have been proposed [7, 10, 16, 18, 19]. Chief among these is Eshghi and Kowalski s formulation of an elegant abductive proof procedure for normal programs where default assumptions are viewed as abducibles. Kakas et al. presented a comprehensive exploration of abductive logic programming [16, 17] A fundamental insight is that ....

A. Kakas and P. Mancarella. Stable theories for logic programs. In Proc. ILPS. MIT Press, 1991.

....of an answer to a query may be undefined, due to the three valued nature of the well founded semantics. They avoid negative loops in the computation by negative contexts. The inference system provides a smooth interface with P rolog. In the area of stable model semantics, Kakas and Mancarella [KM91] define a class of stable theories by treating negation in any logic program as a form of hypothesis. The definition is given in terms of a stability property on negative hypotheses that corresponds to negation as failure literals which attempt to formalize the usual understanding of a default ....

A.C. Kakas and P. Mancarella. Stable theories for logic programs. In International Symposium on Logic Programming, pages 85--100 1991.

....negation as failure (in the sense of Condition 3 in definition 5) since :U(M ) f:rg 6 M . Therefore, M is not a simple model. Superfluous extensions such as the one in the above example can be avoided by considering only maximal acceptable extensions, as suggested for stable theories in [5]; however, the resulting semantics would then not include the well founded model fs; rg of the above program, nor the model fp; qg of the following program: p :q q :p r :s s :r The model fp; qg of this program reflects the ability to choose between the alternative literals p; q and ....

A.C. Kakas and P. Mancarella. Stable theories for logic programs. In Proceedings of the International Symposium of Logic Programming. 1991.

.... these results) This paper provides a methodology for lifting these results from the definite logic program case to the normal logic program case, with respect to stable model [9] partial stable model [15] preferred extension [5] stationary expansion [14] complete scenaria [5] stable theory [10], acceptability [11] and well founded semantics [18] Most of the concrete cases obtained by applying our methodology 1 have been already shown elsewhere in the literature (see [1] for some of these results) Therefore, the main contribution of this paper lies in the general technique rather ....

.... for normal logic programming can be expressed in argumentation theoretic terms, as proved in [3, 4, 16] In particular, stable models [9] correspond to stable sets of assumptions, partial stable models [15] and preferred extensions [5] correspond to preferred sets of assumptions, stable theories [10] correspond to stable theory sets of assumptions, acceptability [11] corresponds to acceptable sets of assumptions, stationary expansions [14] and complete scenaria [5] correspond to complete sets of assumptions and wellfounded semantics [18] corresponds to the well founded set of assumptions. As ....

A.C. Kakas, P. Mancarella, Stable theories for logic programs. ILPS'91 (V. Saraswat and K. Ueda eds.) MIT Press, 85--100

....2 HB and fi 1 ; fi n 2 Lit and n 0; ffl not ff = ff, for each not ff 2 HB not . The interpretation of negative literals as assumptions in logic programming was introduced in [16, 17] and formed the basis for the admissibility semantics [10] the stable theory and acceptability semantics [28], and the argumentation theoretic interpretation for these semantics presented in [25, 11] Note that we could, equivalently, represent clauses ff fi 1 ; fi n as inference rules fi 1 ; fi n ff In this representation, the theory is empty, and a logic program is represented by ....

....1 and Delta 2 . Intuitively, however, Delta 1 and Delta 2 are both acceptable because Delta 0 attacks itself and is therefore not an acceptable attack. Two semantics, called stable theory and acceptability semantics, have been proposed for logic programming by Kakas and Mancarella [28], to deal with cases like the one in this example. These semantics can be generalised and defined more abstractly for any assumption based framework. These generalisations are straightforward, and we shall not discuss them further in this paper. A formal definition of these generalisations can be ....

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A.C. Kakas, P. Mancarella, Stable theories for logic programs. Proc. ISLP'91 MIT Press

....not attack itself and, for every extension E 0 that attacks E, E defends itself against E 0 . The notion of defence can be understood more or less liberally. In the admissibility semantics [3] E defends itself against E 0 if and only if E attacks E 0 . In the stable theory semantics [8], E defends itself against E 0 if and only if the extension consisting of all logical consequences of E [ E 0 attacks the extension consisting of all logical consequences of E 0 Gamma E. The logic program of example 4.2 has no stable models and no acceptable extensions in the sense of the ....

A. C. Kakas, P. Mancarella, Stable theories for logic programs. Proc. ISLP'91, San Diego (1991)

....by adding incrementally all acceptable hypotheses. Thus the well founded semantics is 29 minimalist and sceptical, whereas the preferred extension semantics is maximalist and credulous. A fixpoint construction of the preferred extension semantics is given in [28] Kakas and Mancarella [69, 70] propose a modification of the preferred extension semantics. Their proposal can be illustrated by the following example. Example 4.6 In the abductive framework corresponding to the program p q q q consider the set of hypotheses Delta = fp g. The only attack against Delta is E = fq ....

.... P [ Delta is consistent with I is subsumed by the new maximality condition. Like the original definition of preferred extension, the definition of stable set of hypotheses was not originally formulated in terms of attack, but is equivalent to the one presented above. Kakas and Mancarella [70] argue that the notion of defeating an attack needs to be liberalised further. They illustrate their argument with the following example. 30 Example 4.7 Consider the program P s p p q q r r p: Here the only attack against the hypothesis s is E = fp g. But although P [fs g [ ....

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Kakas, A. C., Mancarella, P., Stable theories for logic programs. Proc. ISLP '91, San Diego (1991)

....hypotheses, by adding incrementally all acceptable hypotheses. Thus the well founded semantics is minimalist and sceptical, whereas the preferred extension semantics is maximalist and credulous. A xpoint construction of the preferred extension semantics is given in [28] Kakas and Mancarella [69, 70] propose a modi cation of the preferred extension semantics. Their proposal can be illustrated by the following example. Example 4.6 In the abductive framework corresponding to the program p q q q consider the set of hypotheses = fp g. The only attack against is E = fq g, and ....

.... P [ is consistent with I is subsumed by the new maximality condition. Like the original de nition of preferred extension, the de nition of stable set of hypotheses was not originally formulated in terms of attack, but is equivalent to the one presented above. Kakas and Mancarella [70] argue that the notion of defeating an attack needs to be liberalised further. They illustrate their argument with the following example. Example 4.7 Consider the program P s p p q q r r p: Here the only attack against the hypothesis s is E = fp g. But although P [fs ....

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Kakas, A. C., Mancarella, P., Stable theories for logic programs. Proc. ISLP '91, San Diego (1991)

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A.C. Kakas, and P. Mancarella. Stable theories for Logic Programs. In Proc. Int. Symp. of Logic Programming, pages 85-100. The MIT Press, 1991.

.... In particular, many existing semantics for LP with NAF can be given an equivalent formulation in argumentation terms, e.g. stable models [17] partial stable models [40] and preferred extensions [11] stationary expansions [37] and complete scenaria [11] well founded model [48] stable theories [24] and acceptability semantics [26] and semantics equivalent to these) All of these argumentation semantics for LP carry through to any other logic for non monotonic reasoning that can also be understood in argumentation terms. For example, the stable model semantics for LP corresponds to the ....

....Section 7 compares our approach with others and section 8 concludes and discusses possible future work. 2 Argumentation semantics for logic programming In this section we review how preferred extension [11] or equivalently partial stable model [40] as proved in [25] stable theory [24], acceptability [26] and well founded [48] semantics for LP, can be formulated within an abstract argumentation framework [4, 5, 12, 20, 26, 27] In general, an argumentation framework consists of a theory P in some background monotonic logic, a set of candidate hypotheses H, and a (binary) ....

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A. C. Kakas, P. Mancarella. Stable theories for logic programs. Proceedings of the International Logic Programming Symposium, San Diego, CA (1991) MIT Press (V. Saraswat and K. Ueda, eds) 85-102

.... and problems involved in computing acceptability (and its various approximations) The proof theory and procedure for acceptability is a generalisation of proof theories and procedures for (sound) approximations of the acceptability semantics such as preferred extensions [3] and stable theories [8], which are in turn a generalisation of the Eshghi Kowalski (E K) abductive proof procedure presented in [5] In fact, for clarity of presentation, we will first give, in sections 3 and 4, the computational framework for the special cases of the preferred extension and stable theory semantics, and ....

....in the computation. As shown in [9] many semantics for LP correspond to approximations of the full acceptability relation, obtained through iterations of the acceptability specification starting from the above base cases (BC1 2) Specifically, preferred extension [3] and stable theory [8] semantics (and other semantics which are equivalent to them) are captured by the approximation at the second iteration: A ACC) Delta is acceptable to Delta 0 if for all sets of hypotheses A, if A attacks Delta Gamma Delta 0 , then there exists a set of hypotheses D such that D attacks A ....

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A. C. Kakas, P. Mancarella, Stable theories for logic programs. ISLP'91

....is maximalist and credulous. The relationship between these two semantics is further investigated in [47] where the well founded model and preferred extensions are shown to correspond to the least fixed point and greatest fixed point, respectively, of the same operator. Kakas and Mancarella [96, 97] propose an improvement of the preferred extension semantics. Their proposal can be illustrated by the following example. Example 4.6 Consider the program p q q q: Similarly to example 4.4, the last clause gives rise to a one step loop via NAF, since q depends negatively on itself ....

....against E. The empty set is the only preferred extension. However, intuitively Delta should be admissible because the only attack E against Delta attacks itself, and therefore should not be regarded as an admissible attack against Delta. To deal with this kind of example, Kakas and Mancarella [96, 97] modify Dung s semantics, increasing the number of ways in which an attack E can be defeated. Whereas Dung only allows Delta to defeat an attack E, they also allow E to defeat itself. They call a set of hypotheses Delta weakly stable if ffl for every attack E against Delta, E [ Delta attacks ....

[Article contains additional citation context not shown here]

Kakas, A. C., Mancarella, P., Stable theories for logic programs. Proc. ISLP '91, San Diego (1991)

.... In particular, many existing semantics for LP with NAF can be given an equivalent formulation in argumentation terms, e.g. stable models [17] partial stable models [40] and preferred extensions [11] stationary expansions [37] and complete scenaria [11] well founded model [48] stable theories [24] and acceptability semantics [26] and semantics equivalent to these) All of these argumentation semantics for LP carry through to any other logic for non monotonic reasoning that can also be understood in argumentation terms. For example, the stable model semantics for LP corresponds to the ....

....Section 7 compares our approach with others and section 8 concludes and discusses possible future work. 2 Argumentation semantics for logic programming In this section we review how preferred extension [11] or equivalently partial stable model [40] as proved in [25] stable theory [24], acceptability [26] and well founded [48] semantics for LP, can be formulated within an abstract argumentation framework [4, 5, 12, 20, 26, 27] In general, an argumentation framework consists of a theory P in some background monotonic logic, a set of candidate hypotheses H, and a (binary) ....

[Article contains additional citation context not shown here]

A. C. Kakas, P. Mancarella. Stable theories for logic programs. Proceedings of the International Logic Programming Symposium, San Diego, CA (1991) MIT Press (V. Saraswat and K. Ueda, eds) 85-102

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A.C. Kakas, P. Mancarella, Stable theories for logic programs. Proc. ISLP'91, MIT Press 31

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