L.M. Pereira, J.N. Aparicio, J.J. Alferes. Hypothetical Reasoning with Well Founded Semantics. Proc. 3rd Scandinavian Conf. on AI , pp.289-300, 1991.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....called open, or abducible. A theory T in ID logic is therefore a pair (Def, Fol) where Def (the de nitional knowledge) is a set of rules as described above, and Fol (the assertional knowledge) is a set of rst order statements. The meaning of T is de ned by the extended well founded semantics [35] as follows: let M be an arbitrary two valued interpretation for the open predicates in Def. Once M is determined, Def becomes a standard logic program, with a unique well founded model [42] This model is then a model of the whole theory T if it is also a model of Fol. ID logic is a ....

L.M. Pereira, J.N. Aparicio, J.J. Alferes. Hypothetical Reasoning with Well Founded Semantics. Proc. 3rd Scandinavian Conf. on AI , pp.289-300, 1991.

....in terms of open predicates insert and retract. The key property of this theory is that its abductive solutions describe the coherent compositions. Abductive reasoning on an ID logic theory can be performed by mapping it into an abductive logic program [8] under the extended well founded semantics [24] and applying an abductive inference procedure to it. An abductive logic program (ALP) is a triple T = P ; A; IC) such that P is a logic program, the clauses of which are interpreted as de nitions for the predicates in their head, A is a set of predicates, none of which occurs in the head ....

L.M. Pereira, J.N. Aparicio, J.J. Alferes. , Hypothetical Reasoning with Well Founded Semantics , Proc. of the 3th Scandinavian Conference on AI , B. Mayoh, IOS Press, 289-300, 1991

....the logic program P [ M j= F for each F 2 T 7 Also this de nition is parametric in the choice of the logic programming semantics. All main logic programming semantics have been extended in this way (generalised completion [4] generalised stable [17] and generalised well founded semantics [33], 6] On the epistemological level, the obvious problem with these semantics is that they inherit the ambiguities and epistemological confusion of logic programming. For example, in the generalised completion semantics de ned in [4] models of an abductive program represent possible worlds; ....

L.M. Pereira, J.N. Aparicio, and J.J. Alferes. Hypothetical Reasoning with Well Founded Semantics. In B. Mayoh, editor, Proc. of the 3th Scandinavian Conference on AI. IOS Press, 1991. Well-founded semantics for abductive logic programming.

....most semantics for abductive logic programs, 3 valued completion semantics assigns a 2 valued interpretation to abducible predicates. This is also the case in the 2 valued completion semantics [8] the 2 valued generalised stable semantics [32] and the 3 valued extended well founded semantics [49]. The reason for this is explained in [11] and can be traced back to a well known argument formulated by Moore in [47] He argues that reasoning by 2 cases (i.e. something is either true or false) is crucial for reasoning 7 on uncertainty, and is one of the crucial advantages of (classical) logic ....

....completion semantics is that it is the weakest semantics known for abductive logic programming. In [14, 10] the following theorem is proven: Theorem 2. 2 If M is a Sigma model of P A wrt (2 valued completion semantics [8] generalised stable semantics [32] generalised well founded semantics [49]) justification semantics [14, 10] then M is a Sigma model of P A wrt 3 valued completion semantics. 8 As a consequence, the 3 valued completion semantics induces the weakest entailment relation: if an abductive logic program entails F according to the 3 valued completion semantics then ....

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L.M. Pereira, J.N. Aparicio, and J.J. Alferes. Hypothetical Reasoning with Well Founded Semantics. In B. Mayoh, editor, Proc. of the 3th Scandinavian Conference on AI. IOS Press, 1991.

....loops, it can be shown that the undefined predicates can have any interpretation. In other semantics such as the 3 valued completion semantics for abductive programs [4] the justification semantics for abductive programs [7] and the generalised well founded semantics for abductive logic programs [28], even for programs with loops over negation, the interpretation of the undefined predicates can be any. Despite these problems with 2 valued completion semantics, we use it here because of its declarative simplicity and its close relationship with First order Logic. 6 in [17] we use Holds=2. ....

....jsj Result 1 jNoninertial(t; a; s)j = 2 Theta jsj Result 2 One easily verifies that j:j is a level mapping. 2 Several types of semantics have been defined for open logic programs: 2 valued completion semantics [3] generalised stable semantics [20] the generalised well founded semantics [28], 3 valued completion semantics and 3 valued (direct) partial) justification semantics with FEQ [7] Due to the fact that D is acyclic and in each clause of PD , the variables of the body occur in the head, all these semantics coincide in the (weak) sense that the set of all ground atoms implied ....

L.M. Pereira, J.N. Aparicio, and J.J. Alferes. Hypothetical Reasoning with Well Founded Semantics. In B. Mayoh, editor, Proc. of the 3th Scandinavian Conference on AI. IOS Press, 1991.

....constraints. 1] proves that the resulting program is acyclic. The same holds for D 0 , and in fact for all transformed domain descriptions: Proposition 3.1 The translation D of any domain description D is acyclic. Several types of semantics have been defined for incomplete programs [3] 14] [19], 7] Due to the fact that D is acyclic and in each clause of PD , the variables of the body occur in the head, all semantics coincide in the (weak) sense that the set of all ground atoms implied by D under any of the semantics is identical. This extension of results of [1] is proven formally in ....

L.M. Pereira, J.N. Aparicio, and J.J. Alferes. Hypothetical Reasoning with Well Founded Semantics. In B. Mayoh, editor, Proc. of the 3th Scandinavian Conference on AI. IOS Press, 1991.

....where we give the definition of the model semantics of OLP FOL. For the model semantics of OLP FOL, we restrict ourselves here to Herbrand interpretations only. Under this restriction, the OLP FOL model semantics of normal open logic programs T d coincides with the extended well founded semantics [14]. Note that, because of this restriction, we do not take into account any uncertainty on the domain of discourse. In the general definition of the model semantics of OLPFOL [8] also non Herbrand interpretations are considered. For simplicity, we will only consider Herbrand interpretations here, ....

.... program T d is a Herbrand interpretation M , such that M is the well founded model [16] of the grounding of T d augmented with some ground unit clauses of open predicates representing the interpretation of the open predicates in M (i.e. M is the extended well founded model of T d as defined in [14]) A model of a theory (T d ; T c ) is a Herbrand interpretation M , such that M is a model of T d , as defined above, and M satisfies the set of FOL axioms T c (in the classical FOL sense) A formula is a consequence of a theory T iff it is true in every model of T . Note that a model is ....

L.M. Pereira, J.N. Aparicio, and J.J. Alferes. Hypothetical Reasoning with Well Founded Semantics. In B. Mayoh, editor, Proc. of the 3th Scandinavian Conference on AI. IOS Press, 1991.

.... program T d is a Herbrand interpretation M , such that M is the well founded model [14] of the grounding of T d augmented with some ground unit clauses of open predicates representing the interpretation of the open predicates in M (i.e. M is the extended wellfounded model of T d as de ned in [13]) A model of a theory (T d ; T c ) is a Herbrand interpretation M , such that M is a model of T d , as de ned above, and M satis es the set of FOL axioms T c (in the classical FOL sense) A formula is a consequence of a theory T i it is true in every model of T . Note that a model is ....

L.M. Pereira, J.N. Aparicio, and J.J. Alferes. Hypothetical Reasoning with Well Founded Semantics. In B. Mayoh, editor, Proc. of the 3th Scandinavian Conference on AI. IOS Press, 1991.

....principle of inductive de nition in the presence of positive and negative induction. In [3] it is shown how to lift the well founded semantics to ID logic 1 . In the case of the sub formalism used here, an interpretation I is a model of a de nition i it is an extended well founded model of D [9]. ID logic is a declarative logic and provides a knowledge theoretic interpretation for the formalism of Abductive logic programming [8] and Open Logic Programming First Order Logic [2] The latter can be embedded in ID logic, as theories with a set of classical logic sentences (the constraints) ....

L.M. Pereira, J.N. Aparicio, and J.J. Alferes. Hypothetical Reasoning with Well Founded Semantics. In B. Mayoh, editor, Proc. of the 3th Scandinavian Conference on AI. IOS Press, 1991. 13

.... A model of a normal logic program T d is a HI M , such that M is the wellfounded model [16] of the grounding of T d augmented with some ground unit clauses of open predicates representing the interpretation of the open predicates in M (i.e. M is the extended well founded model of T d as defined in [14]) A model of a theory (T d ; T c ) is a HI M , such that M is a model of T d , as defined above, and M satisfies the set of FOL axioms T c (in the classical FOL sense) A formula is a consequence of a theory T iff it is true in every model of T . Note that a model is necessarily 2 valued on the ....

L.M. Pereira, J.N. Aparicio, and J.J. Alferes. Hypothetical Reasoning with Well Founded Semantics. In B. Mayoh, editor, Proc. of the 3th Scandinavian Conference on AI. IOS Press, 1991.

....truth functional. The formalisms of logic programming [20] abductive logic programming (ALP) 19] open logic programming [10] when interpreted under well founded semantics, can be trivially embedded in ID logic. For example, an abductive logic program under the extended well founded semantics [25] consists of a set of abducible predicates, a set of rules de ning non abducible predicates and a set of integrity constraints. Its embedding consists of a set of FOL axioms (the integrity constraints ) and a simultaneous de nition for all non abducible predicates. The abducible predicates are ....

L.M. Pereira, J.N. Aparicio, and J.J. Alferes. Hypothetical Reasoning with Well Founded Semantics. In B. Mayoh, editor, Proc. of the 3th Scandinavian Conference on AI. IOS Press, 1991.

....open. Intuitively, they represent concepts for which no definitions are given. Partial knowledge about these predicates can be expressed in the set of FOL axioms T c . The model semantics of OLP FOL is an extension of the well founded semantics [21] and of the extended well founded semantics [19] and was defined in [11] This logic has a possible state semantics, that is, a model correspond to a state in which the problem domain might occur according to the (incomplete) expert knowledge (and not a belief set, a set of believed atoms, as in answer set semantics of Extended Logic ....

....to a Herbrand interpretation corresponds to the conventional notion of the grounding of a logic program. It trivially follows that the Herbrand model of a complete logic program is the well founded model. Also, a Herbrand model of an incomplete logic program is an extended well founded model [19]. An open logic program T d is interpreted as a definition of the defined predicates in terms of open predicates. The occurrence of an undefined fact in a model of T d reveals an ambiguity or a local inconsistency in the definition. This motivates the following definition. Definition 2.7 Given ....

L. M. Pereira, J.N. Aparicio, and J.J. Alferes. Hypothetical reasoning with well founded semantics. Technical report, AI Centre, Uninova, Portugal, 1990.

....from [McC80] and q and p are object constants corresponding to predicate constants Q and P . This coding can be viewed as a combination of the representation of normative statements in circumscription with the method used in non monotonic modal logics [McC80, MD80] It was also advocated in [PCA91]. For illustration, let us consider database T 1 from Example 10 and expand it by the following information: As a rule, professors in the computer science department have vax accounts. This rule is not applicable to Mike. He may or may not have an account. As suggested above, the first ....

Luis Pereira, Luis Caires, and Jose Alferes. Hypothetical reasoning with well founded semantics. In Proc. of the 3rd Scandinavian Conference on AI, 1991.

....two results justify the above semantics definition. Proposition 4.1 D is an acyclic program with first order constraints in the sense of [1] Corollary 4. 2 For abductive program D, COMP (D) coincides with its generalized stable model semantics [7] and generalized well founded model semantics [12]. Theorem 4.3 (Soundness) Let D be any domain description. For any value proposition Q, if COMP (D) j= Q, then D entails Q. Theorem 4.4 (Completeness) Let D be a domain description. For any value proposition Q, if D entails Q, then COMP (D) j= Q. Thus, proof of entailments in A CO is reduced ....

Pereira, L. M., Apar'icio, J. N., and Alferes, J. J., Hypothetical reasoning with well-founded semantics, In: B. Mayoh (ed.), Proc. of the 3rd Scandinavian Conference on AI, IOS Press, 1991

....and DCS, U. Nova de Lisboa 2825 Monte da Caparica, Portugal fjja lmpg fct.unl.pt Abstract Recently several authors have stressed and showed the importance of having a second kind of negation in logic programs for use in deductive databases, knowledge representation, and nonmonotonic reasoning [6, 7, 8, 9, 13, 14, 15, 24]. Different semantics for logic programs extended with : negation (extended logic programs) have appeared [1, 4, 6, 9, 11, 12, 17, 19, 24] but, contrary to what happens with semantics for normal logic programs, there is no general comparison among them, specially in what concerns the use and ....

L. M. Pereira, J. N. Apar'icio, and J. J. Alferes. Hypothetical reasoning with well founded semantics. In B. Mayoh, editor, Scandinavian Conf. on AI'91. IOS Press, 1991.

....within CRSX 36 10 Related work 37 A CRSX Review 44 1. Introduction Recently, several authors have stressed and showed the importance of having an explicit second kind of negation within logic programs, for use in deductive databases, knowledge representation, and non monotonic reasoning [3, 6, 11, 18, 16, 32, 33, 34, 43, 39]. In non monotonic reasoning with logic programming there are two main ways of giving meaning to sets of rules when a given semantics is assigned to a program defined by the set of rules. We either accept as consequences the intersection of all models identified by some semantics, which is called ....

.... as consequences the intersection of all models identified by some semantics, which is called skeptical reasoning [21, 2] or we consider one particular model identifying the consequences of a given set of assumptions this form of reasoning is called brave reasoning in [21] It has been argued [33, 34, 32, 31] that semantics with the well founded property are adequate to capture non monotonic reasoning if we interpret the least model provided by the semantics (called the Well Founded Model) as the skeptical view of the world and the other models (called Extended Stable Models) as alternative enlarged ....

L. M. Pereira, J. N. Apar'icio, and J. J. Alferes. Hypothetical reasoning with well founded semantics. In B. Mayoh, editor, Third Scandinavian Conference on Artificial Intelligence. IOS Press, 1991.

....the contradiction removal semantics coincides with WFSX. 1 Introduction Recently, several authors have stressed and showed the importance of having an explicit second kind of negation within logic programs, for use in deductive databases, knowledge representation, and nonmonotonic reasoning [2, 4, 5, 7, 6, 14, 15, 16, 23, 21]. Some proposals for extending logic programming semantics with a second kind of negation has been advanced. One such extension is the Answer Set semantics (AS) 4] which is shown to be an extension of Stable Model (SM) semantics [3] from the class of logic programs [8] to those with a second ....

L. M. Pereira, J. N. Apar'icio, and J. J. Alferes. Hypothetical reasoning with well founded semantics. In B. Mayoh, editor, Third Scandinavian Conf. on AI. IOS Press, 1991.

....the alternative ways of removing contradiction, including the MNSs. The study of the XSMs in CRS is another important subject of investigation, because the preservation of the XSM structure of the original program is useful for expressing defaults and abduction within the WFS, as shown in [8, 7]. Dealing with these models in the CRS can also be useful for dealing with belief revision and counterfactual reasoning [5] together with default reasoning and abduction. Acknowledgements We thank ESPRIT BRA COMPULOG (no. 3012) Instituto Nacional de Investiga c ao Cient ifica, Junta Nacional de ....

L. M. Pereira, J. N. Apar'icio, and J. J. Alferes. Hypothetical reasoning with well founded semantics. In SCAI'91, IOS Press, 1991.

....Well Founded Semantics [14] has been proposed as a suitable semantics for general logic programs. Its Extended Stable Models (XSM) 12, 13] version, and the inclusion of a second type of negation, have been explored as a framework 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 ....

L. M. Pereira, J. N. Apar'icio, and J. J. Alferes. Hypothetical reasoning with well founded semantics. In B. Mayoh, editor, Scandinavian Conference on AI. IOS Press, 1991.

....[ Van Gelder et al. 1980 ] has been proposed as a suitable semantics for general logic programs. Its Extended Stable Models (XSM) Przymusinska and Przymusinski, 1990, Przymusinski, 1990 ] version has been explored as a framework for formalizing a variety of forms of non monotonic reasoning [ Pereira et al. 1991d, Pereira et 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 ....

....1980 ] has been proposed as a suitable semantics for general logic programs. Its Extended Stable Models (XSM) Przymusinska and Przymusinski, 1990, Przymusinski, 1990 ] version has been explored as a framework for formalizing a variety of forms of non monotonic reasoning [ Pereira et al. 1991d, Pereira et 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 ....

[Article contains additional citation context not shown here]

L. M. Pereira, J. N. Apar'icio, and J. J. Alferes. Hypothetical reasoning with well founded semantics. In B. Mayoh, editor, Scandinavian Conference on AI'91. IOS Press, 1991.

....of a program is identified. This model is included in all the alternative contradiction removing ones. Since CRS extends WFS to deal with contradictions arising from the introduction of classical negation, and since the XSM structure of a program is useful for expressing defaults and abduction [PAA91f, PAA91e], it is important to study in what way the structure of the XSMs is affected by CRS when the WFM is contradictory and redefined by CRS as the CRWFM. Given that the CRS is useful for expressing belief revision and counterfactual reasoning [PAA91c] dealing with such models is expected to be useful ....

....contradiction. The relationship between CFXSMs in families and the (contradictory) XSMs of the original program, allows to establish results on the changes to the XSMs after contradiction removal. The CFXSMs can then be put to use for abductive, counterfactual, and default reasoning, as in [PAA91f, PAA91e, PAA91c]. We next formalize the (partial) mapping from a given contradictory XSM CX of a program P , whose contradictory WFM is M P , to a XSM FX of some family [RS] Let Sup = CX Gamma M P . If such a mapping exists the intended result is that FX results from root( RS] in the same way as CX results ....

L. M. Pereira, J. N. Apar'icio, and J. J. Alferes. Hypothetical reasoning with well founded semantics. In SCAI'91. IOS Press, 1991.

....about nixon being a pacifist 10 . The interpretation of the above results, in particular in what concerns XSMs where literals not in the WF Model become true, suggested to us a relationship between the XS Models of a program with rules like (1) and the concept of abduction within WF Semantics [9]. 3.3 Unknown possible facts Similarly to rules about which we are undecided regarding their applicability, we might be unsure about some facts. Note that this is different from not having any knowledge at all about such a fact. Consider this simple example: John and Nixon are quakers. John is a ....

L. M. Pereira, J. N. Apar'icio, and J. J. Alferes. Hypothetical reasoning with well founded semantics. Technical report, CRIA/UNINOVA, 1990.

.... program class improves the cross fertilization between logic programs and default theories, since we generalize previous results concerning their relationship [1, 2, 3, 4, 6, 13] and also because there is an increasing use of logic programming with explicit negation for nonmonotonic reasoning [6, 9, 10, 11, 17]. It also clarifies the meaning of logic programs combining both explicit negation and negation by default. In particular, it shows in what way explicit negation corresponds to classical negation in our default theory, and elucidates the use of rules in extended logic programs. Like defaults rules ....

L. M. Pereira, J. N. Apar'icio, and J. J. Alferes. Hypothetical reasoning with well founded semantics. In B. Mayoh, editor, Scandinavian Conference on AI'91. IOS Press, 1991.

....representing non monotonic problems in logic programming. 1 Introduction Recently, several authors have stressed and showed the importance of introducing an explicit second kind of negation within logic programs, for use in deductive databases, knowledge representation, and nonmonotonic reasoning [2, 4, 6, 9, 8, 17, 18, 19, 25, 22]. It has been argued [18, 19, 17, 16] that semantics with the well founded property are adequate to capture nonmonotonic reasoning if we interpret the least model provided by the semantics (called the Well Founded Model of a program) as the skeptical view of the world, and the other models (called ....

.... 1 Introduction Recently, several authors have stressed and showed the importance of introducing an explicit second kind of negation within logic programs, for use in deductive databases, knowledge representation, and nonmonotonic reasoning [2, 4, 6, 9, 8, 17, 18, 19, 25, 22] It has been argued [18, 19, 17, 16] that semantics with the well founded property are adequate to capture nonmonotonic reasoning if we interpret the least model provided by the semantics (called the Well Founded Model of a program) as the skeptical view of the world, and the other models (called Extended Stable Models) as ....

L. M. Pereira, J. N. Apar'icio, and J. J. Alferes. Hypothetical reasoning with well founded semantics. In B. Mayoh, editor, Third Scandinavian Con. on AI. IOS Press, 1991.

.... devised [11] This relationship improves the cross fertilization between logic programs and default theories, since we generalize previous results concerning their relationship [3, 4, 7, 1, 2] and there is an increasing use of logic programming with explicit negation for nonmonotonic reasoning [7, 15, 16, 13, 23]. It also clarifies the meaning of logic programs combining both explicit negation and negation by default. In particular, it shows that explicit negation corresponds exactly to classical negation in the default theory, and elucidates the use of rules in logic programs. Like defaults, rules are ....

L. M. Pereira, J. N. Apar'icio, and J. J. Alferes. Hypothetical reasoning with well founded semantics. In B. Mayoh, editor, SCAI'91. IOS Press, 1991.

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