Moore, R. C.: Semantic Considerations on Non-Monotonic Logic. Arti cial Intelligence, Vol. 25, 1985, pp. 75-94.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....model of civil pleading , where claims have to be proven by two players, the plainti and the defendant. One criterion current in logic approaches to AI distinguishes between monotonic and nonmonotonic logics. The latter class includes Reiter s default logic [91] and Moore s autoepistemic logic [75, 76, 58], which di er in whether and how they allow given assumptions to be defeated by evidence. Especially the latter distinction is nothing new to AI Law researchers. Yuan [112] as already Marek and Truszczinski [64] reformulates default and autoepistemic logic in terms of each other: he shows that ....

Moore, R. C.: Semantic Considerations on Non-Monotonic Logic. Arti cial Intelligence, Vol. 25, 1985, pp. 75-94.

.... For example, the static semantics, the well founded circumscriptive semantics, and the disjunctive wellfounded semantics [2] and the skeptical well founded semantics [21] are representatives of the skeptical semantical category; and the answer set semantics and the stable extension semantics [12] (based on the autoepistemic translation of logic programs) are representatives of the stable semantical category. For the partial stable semantical category, there are the partial stable model semantics [13] the regular model semantics [18] and the maximal stable model semantics [6] These ....

.... seat :Bdriving The answer set semantics, which adopts the minimal model based default negation, of 1 has two answer sets, one conclude fdriving; fixing car; notf lying; notresering seatg and the other ff lying; reserving seat; notdriving; notfixing carg: The stable extension semantics [12], which is consistency based, of A 1 , on the other hand, contains a unique stable extension which concludes fdriving f lying; fixing car; reserving seat; Bdriving; Bf lyingg Let A 1m be obtained from A 1 by adding instantiated formulas of axiom : B , i.e. A 1m = A 1 [ f:driving ....

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R. C. Moore. Semantic considerations on non-monotonic logic. AI, 25:75-94, 1985.

.... of the stationary completion of a normal program, while defined using a slightly different language, is equivalent to the notion of a well founded completion introduced in [Prz91c] Stationary expansions of a normal program P also include as a subset the class ST B of all stable expansions of P [Moo85], which are well known to be in one to one correspondence with stable models [Gel87] Theorem 3.3 Let P be an arbitrary normal program. A stationary expansion P of P is stable if and only if it satisfies the condition: P j= A or P j= not A; for any proposition A ; i.e. if it completely ....

.... instead of using the predicate minimization operator j= min (based on circumscription or on GCWA) we use the following inference operator j= cwa (based on CWA [Rei78] T j= cwa A j T j= A T j= cwa :A j T 6j= A; for any proposition A, we will obtain stable expansions of logic programs [Moo85], which are well known to be in one to one correspondence with stable models [Gel87] Thus, the only difference between stable expansions (or stable models) and stationary expansions (or partial stable models) is the fact that the former are obtained by using CWA as the default formalism for ....

R.C. Moore. Semantic considerations on non-monotonic logic. Journal of Artificial Intelligence, 25:75--94, 1985.

.... is on the table) gives B is on the table (normally) These conclusions are exactly the same as the proposed answer in [Lifschitz, 1989] which also shows that it can be derived using default logic [Reiter, 1980] default logic with abnormality predicates [McCarthy, 1986] autoepistemic logic [Moore, 1985], circumscriptive theory [Lifschitz, 1988] closed world assumption [Reiter, 1978a] ATMS [de Kleer, 1986a] predicate completion [Clark, 1978] THEORIST [Poole, 1988] etc. 2 Problems A2 to A5 in [Lifschitz, 1989] can be represented and reasoned in a similar way as A1 by NLBN3.1 with ....

....our goal in making NLBN closely resemble human symbolic belief representation and reasoning. 155 All these show that, given this set of benchmark problems, a NLBN compares well with many formal nonmonotonic belief representation systems such as default logic [Reiter, 1980] autoepistemic logic [Moore, 1985], circumscriptive theory [Lifschitz, 1988] and TMS [Doyle, 1979] Unlike some of the formalisms such as circumscriptive theory [McCarthy, 1986] no additional abnormal relation is necessary to aid the reasoning process although NLBN3.1 is capable of handling these problems with the additional ....

Robert C. Moore. Semantic considerations on non-monotonic logic. Artificial Intelligence J., 25:75--94, 1985.

....AX: model fa; not :a; not b; bg which is not a stationary model of P: Note however that a is in the model because it is a logical consequence of AX: irrespective of the program. 2. 2 The parametrizeable schema Stable Models Semantics [5] has a one to one correspondence with stable expansions [10], and the latter can be obtained simply by replacing j= CIRC by j= CWA in the definition of stationary expansion of normal programs. As with the stationary semantics of extended programs, a generic definition of stable semantics for extended programs can also be obtained, with P j= CWA :L ....

R. C. Moore. Semantics considerations on nonmonotonic logic. Artificial Intelligence, 25:75--94, 1985.

....categories. For example, the static semantics, the well founded circumscriptive semantics, the disjunctive well founded semantics [2] and the skeptical well founded semantics [22] are representatives of the skeptical semantical category; the answer set semantics and the stable extension semantics [13] (based on the autoepistemic translation of logic programs) are representatives of the stable semantical category. For the partial stable semantical category, there are the partial stable model semantics [14] the regular model semantics [18] and the maximal stable model semantics [5] These ....

.... Stable Partial Stable Well founded Circums Stable Circums Partial stable criptive Semantics [21] criptive Semantics [21] Model Semantics [14] Static Semantics Answer Set Regular Semantics [15] Semantics [9] 18] Disjunctive Well Stable Extension Maximal Stable founded Semantics [2] Semantics [13] Model Semantics [5] Skeptical Well Partial stable founded Semantics [22] Assumption Semantics [17] Regularly justified Set Semantics [22] Both the static and the well founded circumscriptive semantics were defined based on the same idea of minimal model based negative introspection. The specific ....

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R. C. Moore. Semantic considerations on non-monotonic logic. AI, 25:75--94, 1985.

....(Krishnaprasad, 1989) For instance, many logical formalisations which are modified forms of first order Chapter 2. Inheritance Hierarchies 35 predicate calculus (FOPC) have been presented, such as default logic (Reiter, 1980) circumscription (McCarthy, 1980) autoepistemic logic (Moore, 1984; Moore, 1985) and other formalisms. Unfortunately, inheritance problems involving pre emption and contradiction do not get much attention. For instance, the conclusion of the statement, Clyde elephant grey:thing, is considered, logically, to be true, namely Clyde is a grey.thing by the logical rule of ....

....of classical logic and throwing in as many formulae of the form M p as possible. Chapter 3. Current approaches to nonmonotonic reasoning 60 is false. i.e. M(sun shining) is not inconsistent with : sun shining) i.e. NML has been shown to be inconsistent. Autoepistemic Logic Moore (Moore, 1984; Moore, 1985) adopts autoepistemic logic (AEL) AEL is a more extended and more successful proposal than McDermott s NML. AEL can be defined as a logic of an agent s own belief (rather than consistency) such as if Losy is a cat and if you don t believe that Losy is not a mammal, then Losy is a mammal ....

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Moore, R. C. (1985). Semantic considerations on nonmonotonic logic. Artificial Intelligence, 25:75--94.

.... and deductive databases, such as the stable semantics, well founded semantics, and stationary semantics, propose even more sophisticated meanings for default negation, closely related to more general nonmonotonic formalisms such as Default Logic, DL (Reiter 1980) AutoEpistemic Logic, AEL (Moore 1985), and AutoEpistemic logic of Beliefs, AEB (Przymusinski 1994a; 1995) 1.1 Default negation does not suffice Although default negation proved to be quite useful in various domains and application frameworks, there are at least two important reasons why it is not the Partially supported by ....

Moore, R. 1985. Semantic considerations on nonmonotonic logic. Journal of Artificial Intelligence 25:75--94.

.... was first made in [BF87] In [EK89] negation as failure of normal programs was first formalized as abduction, and in [Dun91] was extended to capture both stable models [GL88] and the well founded semantics (WFS) of normal logic programs [GRS91] The view of logic programs as autoepistemic theories [Moo85] first appeared in [Gel87] which envisages every literal not L of logic programs as LL, i.e. not L has the epistemic reading: there is no reason to believe in L 1 . In [Bon92] a variety of translations of negation as failure by belief literals are studied, in order to show how different ....

.... and answer sets) is f:ag: A suitable translation between extended programs with answer sets semantics and reflexive AEL theories was proposedindependently in [LS93b] and [MT93] Reflexive AEL, introduced in [Sch91] views the operator L as is known instead of the is believed of Moore s AEL [Moo85] 2 . The translation renders an objective literal A (resp. A) as LA (resp. L A; where denotes classical negation) i.e. A is known to be true (resp. A is known to be false ) and renders not L as L LL, i.e. it is known that L is not known . In [LS93b, MT93] the authors prove that the ....

R. Moore. Semantics considerations on nonmonotonic logic. Artificial Intelligence, 25:75--94, 1985.

....In this paper we assume a first order belief language L, which is a usual first order language that contains a set of objective predicate symbols and a set of belief predicate symbols. A usual first order formula is referred to as an objective formula. In the language of autoepistemic logic [11], belief formulas are of the form LOE where OE is a formula and L is a special symbol in the alphabet of the language. For the purpose of this paper, we only need belief predicates. The name of an n nary belief predicate consists of two parts: a normal predicate symbol and a prefix L, such as Lp(x ....

R.C. Moore. Semantic considerations on non-monotonic logic. AI, 25:75--94, 1985.

.... programs and deductive databases, such as the stable semantics, well founded semantics, and stationary semantics, propose even more sophisticated meanings for default negation, closely related to more general non monotonic formalisms such as Default Logic, DL [Rei80] AutoEpistemic Logic, AEL [Moo85], and AutoEpistemic logic of Beliefs, AEB [Prz94a, Prz95] 1.1 Default negation does not suffice Although default negation proved to be quite useful in various domains and application frameworks, there are at least two important reasons why it is not the only form of negation that is needed in ....

R.C. Moore. Semantic considerations on non-monotonic logic. Journal of Artificial Intelligence, 25:75--94, 1985.

.... the possible world semantics provides an excellent formal tool for modeling knowledge and belief and that nonmonotonic reasoning is based on reasoning about belief, none of those prominent nonmonotonic reasoning semantics, such as Reiter s default theory [12] Moore s autoepistemic stable sets [9], and the well founded and stable semantics for logic programming [2, 3] can be characterized by the standard possible model semantics. Especially, we have yet to find out whether these prominent nonmonotonic reasoning semantics can be characterized in the possible world semantics by just ....

R. C. Moore. Semantic considerations on non-monotonic logic. AI, 25:75--94, 1985.

....many researchers as an important step towards the ultimate solution of the problem of semantics. Unfortunately, while providing an elegant formalism, Gelfond s approach has some important drawbacks which are direct consequences of the choice of the specific autoepistemic logic, namely Moore s AEL [Moo85]. Stable models of logic programs often do not exist and even when they do exist the resulting semantics does not always coincide with the expected (intended) meaning of the program (see [VGRS90, PP90, Prz94] for a more detailed discussion) Moreover, the semantics obtained in this way is ....

R.C. Moore. Semantic considerations on non-monotonic logic. Journal of Artificial Intelligence, 25:75--94, 1985.

....P = P [ f D F : b P j= min F g; 8) where F is an arbitrary objective formula. We recall that all theories are assumed to be closed under the usual propositional consequence. Although the definition of static expansions is similar to the definition of Moore s stable autoepistemic expansions [Moo85], it uses satisfaction by default D F instead of logical satisfaction L F and it does not specify when a default proposition D F is false in the expansion. As a result, properties of static expansions are very different from the properties of stable autoepistemic expansions. We also note that, in ....

....Definition 7. 1 A theory E is called an extended static expansion of a program P if it satisfies the following fixed point equation: E = P [ fD F : E j= min Fg [ fL F : E j= Fg [ f:L F : E 6j= Fg: Here by L F we denote the epistemic belief operator as defined in Moore s autoepistemic logic AEL [Moo85]. Observe that the second part of the definition is identical to the definition of stable expansions while the first part coincides with the definition of static expansions. Naturally, we assume now that the extended language L is now closed under the nested use of both operators D F and L F . ....

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R.C. Moore. Semantic considerations on non-monotonic logic. Journal of Artificial Intelligence, 25:75--94, 1985.

.... semantics [14] well founded semantics [34] partial stable or stationary semantics [27] and static semantics [31] propose even more sophisticated meanings for default negation, closely related to more general nonmonotonic formalisms such as Default Logic, DL [33] AutoEpistemic Logic, AEL [19], and AutoEpistemic logic of Beliefs , AEB [29] Under all of these semantics, however, default negation not significantly differs from classical negation : For example, the formula charged(x) guilty(x) oe acquitted(x) says that a person charged with a crime should be acquitted if he or ....

R.C. Moore. Semantic considerations on non-monotonic logic. Journal of Artificial Intelligence, 25:75--94, 1985.

....developed for general non monotonic theories can be fruitfully employed, resulting in powerful and natural semantical paradigms for logic programming. The well known stable semantics (SS) 1 Gelfond and Lifschitz [GL 88] can be viewed as an application of Moore s autoepistemic (AE) reasoning [Mo 85] developed for general non monotonic theories, to logic programs (also see [Ge 87] The success of this technology transfer is evident from the fact that the stable semantics is one of the most popular semantical paradigms for logic programming. Interestingly, there is a significant potential ....

Moore, R.C.: "Semantic consideration on non-monotonic logic," Artificial Intelligence 25 (1985), 75-94.

....definitions of some of them. It also naturally leads to new, even more expressive and flexible formalizations and semantics. Keywords: Non Monotonic Reasoning, Logics of Knowledge and Belief, Semantics of Logic Programs and Deductive Databases. 1 Introduction Moore s autoepistemic logic AEL (Moore 1985) is obtained by augmenting classical propositional logic with a modal operator L. The intended meaning of the modal atom LF is F is provable or F is logically derivable (in the stable autoepistemic expansion) Thus Moore s modal operator L can be viewed 3 To appear in the Proceedings of ....

....representation framework turns out to be rather simple and yet quite powerful. We prove that many of the recently introduced non monotonic formalisms and semantics for normal and disjunctive logic programs are isomorphically embeddable into AELB. In particular this applies to autoepistemic logic (Moore 1985); circumscription (McCarthy 1980; Lifschitz 1985) CWA (Reiter 1978) GCWA (Minker 1982) ECWA (Gelfond, Przymusinska, Przymusinski 1989) epistemic specifications (Gelfond 1992) stable, well founded, stationary and static semantics of normal and disjunctive logic programs (Gelfond Lifschitz ....

Moore, R. 1985. Semantic considerations on nonmonotonic logic. Journal of Artificial Intelligence 25:75--94.

.... programs, the perfect model semantics, has been shown (see [Prz88b] to be equivalent to suitable forms of all four major formalizations of non monotonic reasoning in AI McCarthy s circumscription [McC80, McC86] Reiter s closed world assumption CWA [Rei78] Moore s autoepistemic logic [Moo85] and Reiter s default theory [Rei80] thus establishing a close link between the areas of logic programming and non monotonic reasoning and describing a relatively large class of theories for which natural forms of different non monotonic formalisms coincide. Originally, it seemed that no ....

R.C. Moore. Semantic considerations on non-monotonic logic. Journal of Artificial Intelligence, 25:75--94, 1985.

....program rule: H B 1 ; Bn ; not C 1 ; not Cm with the default rule: B 1 ; Bn : C 1 ; Cm H where denotes classical negation. Moreover, from the results of [Gel87] it follows directly that stable models are equivalent to Moore s autoepistemic expansions [Moo85] of the theory obtained by replacing in P every default literal not A by BA and then reinterpreting the rule connective as material implication. In spite of the strong relationship between logic programming and non monotonic reasoning, in the past these research areas were developing largely ....

R. Moore. Semantics considerations on nonmonotonic logic. Artificial Intelligence, 25:75--94, 1985.

....Lu is Moniz Pereira DCS, U.Nova de Lisboa and CRIA Uninova 2825 Monte da Caparica, Portugal (lmp fct.unl.pt) Teodor C. Przymusinski Department of Computer Science University of California Riverside, CA 92521, USA (teodor cs.ucr.edu) 1 Introduction Moore s autoepistemic logic, AEL [Moo85], was obtained by augmenting classical propositional logic with a modal operator L. The intended meaning of the modal atom LF in the stable autoepistemic expansion T is F is known in T , or, more precisely, F is logically derivable from T . Indeed, a formula LF belongs to the expansion T if ....

R.C. Moore. Semantic considerations on non-monotonic logic. Journal of Artificial Intelligence, 25:75--94, 1985.

....T is any belief theory b T which satisfies the following fixed point condition: b T = Cn ( T [ f BF : b T j= min F g ) 13) where F ranges over all formulae of L B . 2 Although the definition of static expansions is similar to the definition of Moore s stable autoepistemic expansions [Moo85], it uses minimal model entailment b T j= min F instead of the logical implication b T j= F and it does not specify when a belief atom BF is false in the expansion. As a result, properties of static expansions are significantly different from the properties of stable autoepistemic expansions. ....

.... T is called a static autoepistemic expansion of a theory T if it satisfies the following fixed point equation: b T = Cn (T [ fBF : b T j= min Fg [ fLF : b T j= Fg [ f:LF : b T 6j= Fg) 2 Here by LF we denote the autoepistemic belief operator as defined in Moore s autoepistemic logic AEL [Moo85]. Observe that the second part of the definition is identical to the definition of stable autoepistemic expansions while the first part coincides with the definition of static expansions. Naturally, we assume now that the extended language L B;L allows both operators BF and LF . The resulting ....

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R.C. Moore. Semantic considerations on non-monotonic logic. Journal of Artificial Intelligence, 25:75--94, 1985.

....reasoning, semantics of logic programs, disjunctive logic programs and deductive databases. 1 Introduction In recent years, various approaches to non monotonic reasoning and different semantics for normal and disjunctive logic programs have been proposed, including: ffl Autoepistemic Logic [Moo85]; ffl Circumscription, CWA, GCWA, ECWA, etc. McC80, Rei78, Min82, GPP89] ffl Circumscriptive Epistemic Logic [Prz91a] ffl Epistemic Specifications [Gel92] ffl Stable Semantics (with classical negation) GL88, GL90] ffl Well Founded and Stationary Semantics [VGRS90, Prz90] ffl ....

....to use prioritized circumscription in the definition of circumscriptive epistemic logic [YYar] 3 Static Expansions and Static Epistemic Logic In this section we introduce static expansions E of a propositional theory T in the language K. They will combine stable autoepistemic expansions of Moore [Moo85] and stable circumscriptive expansions of Przymusinski [Prz91a] For simplicity, any propositional theory T expressed in the language K will be called an epistemic theory. We will also assume that all theories are implicitly closed under logical consequence. Definition 3.1 A theory E is called a ....

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R.C. Moore. Semantic considerations on non-monotonic logic. Journal of Artificial Intelligence, 25:75--94, 1985.

....which set of propositions to believe according to its knowledge. Moore s AE logic is a powerful framework for this kind of introspective reasoning which means that an agent is capable not only of reasoning about the world, but also of reasoning about its own knowledge and beliefs about the world [16]. Introspective reasoning by rational agents is characterized by the concept of a stable expansion. By a stable expansion of an AE theory E we mean a set T of formulas such that T = fOEjE [ fK j 2 Tg [ f:K j 62 Tg j= S OEg; 1) where S is the propositional calculus in the modal language [16] ....

....[16] Introspective reasoning by rational agents is characterized by the concept of a stable expansion. By a stable expansion of an AE theory E we mean a set T of formulas such that T = fOEjE [ fK j 2 Tg [ f:K j 62 Tg j= S OEg; 1) where S is the propositional calculus in the modal language [16]. The terms fKOEjOE 2 Tg and f:KOEjOE 62 Tg express the positive and negative introspection of an agent 1 This research is partially supported by NSERC grand OGP42193. The paper appears in Journal of Automated Reasoning, 13(1) 69 82, 1994. Modified on December 1994. 2 Li Yan Yuan ....

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R. C. Moore. Semantic considerations on non-monotonic logic. AI, 25:75--94, 1985.

....had been recognized, work has begun on finding formal foundations of non monotonic reasoning. Several formalizations of non monotonic reasoning have been proposed, among which the best known are McCarthy s circumscription [McC80, McC86] Reiter s default theory [Rei80] Moore s autoepistemic logic [Moo85] (reconstruction of non monotonic logics of McDermott and Doyle [MD80, McD82] and Reiter s closed world assumption [Rei78] and its extensions, e.g. Minker s GCWA [Min82] All of these formalizations are obtained by augmenting a classical first order logic with some mechanism, which by ....

....In order to show the equivalence between the well founded semantics of logic programs and the 3 valued autoepistemic semantics, we need to translate logic programs into autoepistemic theories, which we will call autoepistemic logic programs. We will denote by L the autoepistemic belief symbol [Moo85], also called the belief operator. Definition 7.10 [Gel87] Let P be a logic program. The autoepistemic translation P of P consists of all clauses of the form A B 1 ; Bm ; LC 1 ; LC n ; for all possible ground instances A B 1 ; Bm ; C 1 ; C n of ....

R.C. Moore. Semantic considerations on non-monotonic logic. Journal of Artificial Intelligence, 25:75--94, 1985.

....theories of action. They are Reiter s default logic, McCarthy s circumscription and Shoham s preferential logic. There are some other elegant non monotonic logics such as McDermott and Doyle s modal non monotonic logic [McDermott and Doyle, 1980] Moore s autoepis8 temic logic [Moore, 1984, Moore, 1985] and Poole s default logic [Poole, 1988] We will not discuss these logics here because their applications in modeling state change are not as broad as the first three logics. We will also discuss several action theories which are constructed based on the these logics. 1.3.1 Default Logic Reiter ....

R. Moore. Semantic considerations on non-monotonic logic. Artificial Intelligence J., 25:75--94, 1985.

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