S. Kraus, D. Lehmann and M. Magidor (1990) `Nonmonotonic reasoning, preferential models and cumulative logics', Artificial Intelligence 44:167--207.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....default reasoning from conditional knowledge bases in System . The literature contains several different proposals for default reasoning and extensive work on its desired properties. The core of these properties are the rationality postulates of System proposed by Kraus, Lehmann, and Magidor [KRA 90] It turned out that these rationality postulates constitute a sound and complete axiom system for several classical model theoretic entailment relations under uncertainty measures on worlds. More precisely, they characterize classical model theoretic entailment under preferential structures [SHO ....

....turned out that these rationality postulates constitute a sound and complete axiom system for several classical model theoretic entailment relations under uncertainty measures on worlds. More precisely, they characterize classical model theoretic entailment under preferential structures [SHO 87, KRA 90] infinitesimal probabilities [ADA 75, PEA 89] possibility measures [DUB 91] and world rankings [SPO 88, GOL 92] They also characterize an entailment relation based on conditional objects [DUB 94] A survey of all these relationships is given in [BEN 97, GAB 98] In this paper, we show that ....

[Article contains additional citation context not shown here]

KRAUS S., LEHMANN D., MAGIDOR M., "Nonmonotonic reasoning, preferential models and cumulative logics", Artif. Intell., vol. 14, num. 1, 1990, p. 167--207.

....calculus characterisation of a nonmonotonic logic provide it research partially sponsored by PICD CAPES and LOGIA ProTeM CC CNPq 1 L. J. of the IGPL, Vol. 0 No. 0, pp. 1 10 0000 c fl Oxford University Press with a much more tractable definition to be used in the proof of meta properties [8], comparing with fixed point characterisations or semantic ones usually available. A sequent calculus for a paraconsistent default logic will be presented in this paper. The strategy adopted to apply sequent calculus for this logic without deforming it too much has been to confine the disturbing ....

Kraus,S., Lehmann,D. & Magidor,M. `Nonmonotonic reasoning, preferential models and cumulative logics', Artificial Intelligence 44, 1990. 167-207.

.... known at a point in time spontaneously, that is without any inference rule prescribing their truth We should have a way to make sure that this does not happen: we want the models to have minimal information in the sense that nothing becomes known if there are no rules saying so (see also [Sh88] [KLM90]) This leads to the following notion of minimal models: Definition 3.5 (Minimal Temporal Models) A temporal epistemic model is called a minimal model of a theory Th if it is a model of Th and for any model of Th, if N M M. A minimal model of a theory is a model for which there are no ....

Kraus, S., D. Lehmann, M. Magidor: "Nonmonotonic Reasoning, Preferential models and cumulative logics"; A.I. 44 (1990), 167 - 207

.... committing to, may be sufficient for the agent to draw the required (defeasible) conclusions (within the context of that view) One may focus on the intersection of the different possible sets of beliefs for the agent; this could be described by a nonmonotonic inference operator, e.g. as in [KLM 90] A disadvantage of this (sceptical) approach may be that hypothetical conclusions that are possible within one of the belief sets may be lost due to the restriction to the common beliefs. So, the agent may not be able to draw the required conclusions, only taking into account the beliefs common ....

....be defined in the following manner: for each X L (X) m Mod m is minimal in Mod(X) Preferential semantics essentially provides a level 1 description, abstracting from lower levels. Also approaches with non singleton belief states specified by preference relations exist (cf. KLM 90] VOO 93] Belief state operator formalizing default logic Default logic (cf. REI 80] can also be formalized using a belief state operator. Let D be a set of defaults. For X L let D) denote the set of (Reiter) extensions of the default theory X, D. The following belief state operator ....

[Article contains additional citation context not shown here]

Kraus, S., Lehmann, D., Magidor, M.: "Nonmonotonic Reasoning, Preferential Models and Cumulative Logics", Artificial Intelligence 44 , 1990, pp. 167 - 207.

....adding some machinery for inheritance with overriding from default reasoning The literature contains several different approaches to default reasoning and extensive work on the desired properties. The core of these properties are the rationality postulates proposed by Kraus, Lehmann, and Magidor [47]. These rationality postulates constitute a sound and complete axiom system for several classical model theoretic entailment relations under uncertainty measures on worlds. In detail, they characterize classical model theoretic entailment under preferential structures [74, 47] infinitesimal ....

....Lehmann, and Magidor [47] These rationality postulates constitute a sound and complete axiom system for several classical model theoretic entailment relations under uncertainty measures on worlds. In detail, they characterize classical model theoretic entailment under preferential structures [74, 47], infinitesimal probabilities [1, 69] possibility measures [21] and world rankings [75, 39] They also characterize an entailment relation based on conditional objects [22] A survey of all these relationships is given in [9] Recently, Friedman and Halpern [28] showed that many approaches ....

[Article contains additional citation context not shown here]

Kraus, S., D. Lehmann, and M. Magidor: 1990, `Nonmonotonic reasoning, preferential models and cumulative logics'. Artif. Intell. 14(1), 167--207.

....above would still hold when we restrict to such a class. Whether such classes exist, and what these classes are, of course depends on the particular nonmonotonic logic considered, and we will focus here on an important class of nonmonotonic logics: the class of preferential logics (see [Sh87] [KLM90]) Such logics are based on a monotonic logic (such as propositional logic, predicate logic or modal logic) augmented with a preference order on its models. The nonmonotonic consequences of a formula are those formulae which are true in all models of which are minimal in the preference order (a ....

S. Kraus, D. Lehmann, M. Magidor, "Nonmonotonic Reasoning, Preferential Models and Cumulative Logics", Artificial Intelligence 44, 1990, pp. 167-207

....both to define a nonmonotonic consequence relation and (at least implicitly) to represent generic statements as well, have often opted for more constraints on the modal frames and the selection function. Were we to follow them, would reflect properties attributed by Gabbay (1985) and Kraus, Lehmann and Magidor (1990) to a nonmonotonic consequence relation. We would thus strengthen the basic logic of . For instance, we can force to obey all the constraints of a rational conditional (Nute 1980) by adding certain constraints on . This rational conditional obeys the following principles: CUT: A B) ....

Kraus, S., D. Lehmann, M. Magidor (1990): "Nonmonotonic Reasoning, Preferential Models and Cumulative Logics", Artificial Intelligence, no.44, pp.167-207.

....styles of explanatory reasoning that can be produced with the different parameters described so far, may be studied by logical means. That is, by analyzing their logical structural properties. This line of analysis is proposed and developed by Flach[11] building on the approach proposed in [12] for non monotonic consequence relations. Ours is built as proposed for dynamic styles of inference[13] 2 . However, these two approaches have the same focus. Indeed, it would be interesting to compare the 2 Though in this paper we have not presented such results. different notions of ....

S. Kraus, D. Lehmann, M. Magidor. Nonmonotonic Reasoning Preferential Models and Cumulative Logics. Artificial Intelligence, 44 (1990) 167--207.

....Fagin, Halpern and Megiddo in [13] to provide a logic for reasoning about probabilities. As for the new rules, introduced to capture common sense principles, these turn out to have direct analogues in non monotonic logic as presented by Gabbay [14] and Makinson [15] and Kraus Lehmann Magidor [16], 17] The plan of this paper is quite simple. We shall consider a sequence of roughly increasing uncertain proof systems and for each we shall prove a completeness theorem. 2 A Basic Proof System for Uncertain Reasoning Let L = fp 1 ; p n g be a nite language for the sentential calculus ....

....A similar case can be made against transitivity. In what follows we shall use the monotonicity and transitivity of I without explicit mention. 2. The rules AND and RWE are the exact analogues of the rules by the same names in the non monotonic logics studied by Kraus, Lehmann and Magidor in [16]. Their rule of left logical equivalence corresponds to K 1 j K 2 ; VK 1 = VK 3 K 3 j K 2 which by Theorem 2.1 (ii) holds for any proof system extending I. 3. Since all the proof systems we shall consider include I we may, in view of Theorem 2.1(ii) henceforth tacitly assume, whenever ....

[Article contains additional citation context not shown here]

S.Kraus, D.Lehmann & M.Magidor, "Non-monotonic Reasoning, Preferential Models and Cummulative Logics", Articial Intelligence, Vol.44,1990, pp167-202

....2. 2 Conditional logic The original purpose of conditional logic is to provide a formal tool for the analysis of subjunctive conditional in natural language[39] Recently, there have been a number of applications of conditional logic to nonmonotonic reasoning and belief revision in AI research[29, 22]. The syntax of conditional logic is an extension of the propositional language with a binary connective and the following formation rule, ffl if f and g are wffs, then f g is also a wff. As for the semantics, there are some competitive paradigms which lead to different systems of ....

....5.1 Formulation of nonmonotonic inference relations Since the pioneering works of Gabbay[21] was published, there have been vast amounts of literatures on the topic about the properties of general nonmonotonic inference relations. One of the most important among them is the work by Kraus et al.[29, 31]. They introduce the following properties for a nonmonotonic inference relation j . Let f; g; h be wffs of the propositional language L throughout this and the next subsections. R) f j f (LLE) j= f j g; f j h g j h (RW) j= f oe g; h j f h j g (CM) f j g; f j h f g j h (AND) f j ....

S. Kraus, D. Lehmann, and M. Magidor. "Nonmonotonic reasoning, preferential models and cumulative logics". Artificial Intelligence, 44:167--207, 1990.

....not in its preferential closure, it is always possible to repair the set of defaults so as to produce the desired conclusion. 1 INTRODUCTION It seems that there is now a general agreement about the foundations of exception tolerant reasoning which have been proposed by Lehmann and his colleagues (Kraus et al. 1990; Lehmann and Magidor, 1992) and by G rdenfors and Makinson (1994) who provide basic systems of postulates for nonmonotonic consequence relations. From these proposals two systems are particularly emerging: on the one hand the System P (P for preferential) which offers a basic core for commonsense ....

....entailment la Shoham (1988) since it can be checked that (a b) a b) w.a p(w) a) 0 w w.b , i.e. the preferred models of a (which maximize p) are models of b. It has been recently established (Dubois and Prade, 1995) that the inferential power of . is exactly the one of Kraus, Lehmann and Magidor (1990) system P. 2.3. Characterizing (D,W) In this section, some of noticeable features of the structure of the set (D,W) are pointed out. For this aim, we associate to each possibility distribution p its qualitative counterpart, denoted by p and called qualitative possibility distribution, defined ....

[Article contains additional citation context not shown here]

Kraus S., Lehmann D., Magidor M. (1990) Nonmonotonic reasoning preferential models and cumulative logics.

....5. 2 GENERAL PROPERTIES OF NONMONOTONIC OPERATIONS One way of characterizing the nonmonotonic inferences generated by a neural network is to study them in terms of the general postulates for nonmonotonic logics that have recently been introduced in the literature (Gabbay 1985, Makinson 1989, 1991, Kraus, Lehmann, and Magidor 1990, Makinson and G rdenfors 1990, G rdenfors 1991) We shall present some of these postulates and determine whether they are satisfied for a function [ a determined by a transition function in a neural network. It follows immediately from the definition of [ a that # satisfies the property ....

....13: 1.00 0.09 1.00 1.00 14: 1.00 1.00 1.00 1.00 15: 1.00 1.00 1.00 1.00 For the disjunction operation it does not seem possible to show that any genuinely new postulates for nonmontonic inferences are fulfilled. The following special form of transitivity is a consequence of Cumulativity (cf. Kraus, Lehmann, and Magidor (1990), p. 179) If a b # a and a # g, then a b # g This principle is thus satisfied whenever Cumulativity is. The general form of Transitivity, i.e. if a # b and b # g, then a # g, is not valid for all a, b, and g, as can be shown by the first example above. Nor is the following principle generally ....

Kraus, S., Lehmann, D., and Magidor, M. (1990): "Nonmonotonic reasoning, preferential models and cumula-tive logics," Artificial Intelligence 44, 167-207.

.... by the partial order between models of a representing preferential structure, second, by an order on the set of formulas of the language defined by the j relation: ff :fi ff fi iff ff j fi: This or similar relations are basic to many abstract approaches to nonmonotonic reasoning (see [KLM90], LM92] BB94] FH95] Sch97 4] We recall this now. 2.3.1 PREFERENTIAL LOGICS AND PARTIAL ORDERS The system P correponds to logics defined by preferential models (see [KLM90] LM92] we recall it first. Definition 2.7 (Axioms of P and R; see [Gab85] KLM90] LM92] Strictly speaking ....

....fi iff ff j fi: This or similar relations are basic to many abstract approaches to nonmonotonic reasoning (see [KLM90] LM92] BB94] FH95] Sch97 4] We recall this now. 2.3. 1 PREFERENTIAL LOGICS AND PARTIAL ORDERS The system P correponds to logics defined by preferential models (see [KLM90], LM92] we recall it first. Definition 2.7 (Axioms of P and R; see [Gab85] KLM90] LM92] Strictly speaking i.e. in object language , the axioms are rules, and a system X of ff j fi0s is said to satisfy P (or R) iff it is closed under the corresponding rules. 1. Right Weakening ....

[Article contains additional citation context not shown here]

S.Kraus, D.Lehmann, M.Magidor, "Nonmonotonic Reasoning, Preferential Models and Cumulative Logics", Artificial Intelligence 44 (1990), p.167-207

....employment as a tool to clarify and localise some essential features of practical reasoning, this calculus will be used to prove some meta properties of IDL in a very elegant and simple way. In fact, some rules are an almost straightforward expression of some properties such as OR (L) and S (R ) [9] and Cumulativity (Cut and LW 2 ) the last one proved in appendix A. In a subsequent paper [12] we prove, by using IDL sequent rules, that IDL consequence relation satisfies the minimal properties required to a cumulative nonmonotonic relation, namely: reflexivity, cut, cautious monotonicity, ....

....used in the task of designing a uniform semantics that encompasses both aspects. A Cumulativity Cut and Cautious Monotonicity has been considered as desirable properties any nonmonotonic consequence relation should satisfy. They may be expressed together by the Cumulativity property [10] [9]: Gamma C( Gamma;D ) C( Gamma;D ) C( D ) We will now prove that IDL really satisfies cumulativity and this prove may be used to give the correctness proof for Cut and LW 2 rules [11] Before presenting the proof of Cut and Cautious Monotonicity, some definitions and lemmas must be ....

Kraus,S., Lehmann,D. & Magidor,M. `Nonmonotonic reasoning, preferential models and cumulative logics', Artificial Intelligence 44, 1990. 167-207.

.... but allows for an attractive formalisation of the inference patterns underlying hypothetical reasoning in terms of conjectural consequence relations (section 4) This formalisation is a first step towards a logic of discovery , strongly related to recent theories about nonmonotonic reasoning [4]. Finally, I will discuss the results of the foregoing analysis in section 5. 2 PEIRCE S THEORIES OF ABDUCTION Peirce was a very prolific thinker and writer, but virtually nothing was published during his life. His collected works [5] therefore reflect, first and foremost, the evolution of his ....

....of hypothesis formation and the extra logical process of hypothesis evaluation. In this section I will present an axiomatic account of the logic of hypothesis formation. The approach has been inspired by the seminal work of Kraus, Lehmann Magidor on the formalisation of nonmonotonic reasoning [4]. The main concept in their framework is the notion of a consequence relation , which they treat as a binary metalevel predicate. Analogously, I will employ the notion of a conjectural consequence relation, denoted by : a metalevel statement a b means that b is a possible conjecture (or ....

[Article contains additional citation context not shown here]

S. Kraus, D. Lehmann & M. Magidor, `Nonmonotonic reasoning, preferential models and cumulative logics', Art. Int. 44: 167--207 (1990).

....defaults, is in its rational closure, but not in its preferential closure (in the sense of System P) it is always possible to repair the set of defaults so as to produce the desired conclusions. 1 INTRODUCTION Foundations of commonsense reasoning have been proposed by Lehmann and his colleagues [7], 8] and by G rdenfors and Makinson [5] who provide basic systems of postulates for nonmonotonic consequence relations. From these proposals two systems are particularly emerging: on the one hand the System P (P for preferential) which offers a basic core for commonsense reasoning but provides a ....

....p) are models of b, which will be Reasoning with Uncertainty 653 S. Benferhat, D. Dubois and H. Prade denoted by a . p b, i.e. D,W) a b iff p (D,W) a. p b. It has been recently established [4] that the inferential power of . is exactly the one of Kraus, Lehmann and Magidor [7] system P (interpreting a b as a b where is a nonmonotonic consequence relation) Example 1 Let us consider the following triangle example: generally, birds fly , generally, penguins do not fly , all penguins are birds , symbolically written as D= b f, p f , and W= p b . We are ....

[Article contains additional citation context not shown here]

S. Kraus, D. Lehmann and M. Magidor, 'Nonmonotonic reasoning preferential models and cumulative logics', Artificial Intelligence, 44, 167-207 (1990).

....was also used in the author s work on a semantics for defaults in a first order framework. As our system there was held deliberately weak, however, we did not postulate any coherence properties between individual weak filters they were just mentioned as an extension to cover the KLM axioms (see [12]) For motivation, consider e.g. the Friedman Halpern approach. The soft or defeasible rule # # # can be read as: # # # is more plausible or more probable than ### and formalized by ## # ### (in some order ) This order can be seen (and was created as such) as an abstraction of ....

....models. Some connections will be hinted at briefly below, for more details the reader is referred to the quoted articles. The axioms play also a central role in the articles of Ben David Ben Eliahu and Friedman Halpern, see Section 1.5 below. Definition 1. 3 (Axioms of P and R, see [8] [12], 13] Strictly speaking i.e. in object language , the axioms are rules, and a system X of # # ##s is said to satisfy P (or R) i# it is closed under the corresponding rules. 1. Right Weakening (RW) # # #, # # # # # # # 2. Reflexivity: # # # 3. AND: # # #, # ....

[Article contains additional citation context not shown here]

S.Kraus, D.Lehmann, M.Magidor, "Nonmonotonic Reasoning, Preferential Models and Cumulative Logics", Artificial Intelligence 44:167--207, 1990. 772 Filters and Partial Orders

....for every subalgebra V of V including the range of R, A, h, V, R) S iff (A, h, V , R) S. The main question now is whether can be axiomatized in a natural way, reminiscent of the results for propositional default conditionals, which are grasped by Lehmann s rationality postulates [KLM 90] 4. DEFAULT QUANTIFIER DEDUCTION For practical purposes, we need a proof theory for our ranking measure semantics, i.e. we are looking for a recursive axiomatization of . If possible, it should reflect our intuitions about the default quantifier as a first order counterpart to the rational ....

S. Kraus, D. Lehmann, M. Magidor. Nonmonotonic reasoning preferential models and cumulative logics. Artificial Intelligence, 44: 167-207,1990.

.... perspective it is natural to extend the scope of the enquiry: a probabilistic inference rule, like entropy maximization, formally defines a nonmonotonic inference relation j that can be studied with respect to the same formal properties as have been investigated for nonmonotonic logics (Kraus, Lehmann Magidor 1990), Gabbay 1985) Conversely, a concept of representation independence, framed entirely in terms of formulas and entailment relations, may be applied to a large class of nonmonotonic logics, not only probabilistic ones. In this paper we will develop the necessary tools for the investigation of ....

....develop the necessary tools for the investigation of representation independence of nonmonotonic inference relations, and take some steps towards clarifying the degree of representation (in )dependence of existing nonmonotonic logics. 2 THE LOGICAL BACKBONE In a similar spirit as Gabbay (1985) Kraus et al. 1990) and Makinson (1994) we will take a very general and abstract view of nonmonotonic logics. The definition we here give of what a nonmonotonic logic is puts into focus two elements that usually are either assumed only implicitly for a nonmonotonic logic, or not deemed necessary at all: the ....

Kraus, S., Lehmann, D. & Magidor, M. (1990), `Nonmonotonic reasoning, preferential models and cumulative logics', Artificial Intelligence 44, 167-- 207.

....holds, then one should conclude ffi after the sequence of revisions that differ from the two revisions only in that step i is a revision by the disjunction ff fi; since knowing which of ff or fi is true cannot be crucial. This property is an analogue for the left argument of the Or property of [KLM90]. Similarly, if one concludes ffi from a revision by a disjunction, one should conclude it from at least one of the disjuncts. This property is an analogue for the left argument of the Disjunctive Rationality property of [KLM90] studied in [Fre93] It is easy to see that the property (C1) of ....

....property is an analogue for the left argument of the Or property of [KLM90] Similarly, if one concludes ffi from a revision by a disjunction, one should conclude it from at least one of the disjuncts. This property is an analogue for the left argument of the Disjunctive Rationality property of [KLM90], studied in [Fre93] It is easy to see that the property (C1) of Darwiche and Pearl [DP94] i.e. K ff) ff fi) K (ff fi) is not satisfied by all revisions defined by pseudo distances. Section 2 will precisely characterize those revisions that are defined by pseudo distances. 11 Notice ....

[Article contains additional citation context not shown here]

S. Kraus, D. Lehmann, M. Magidor, "Nonmonotonic Reasoning, Preferential Models and Cumulative Logics", Artificial Intelligence, Vol. 44, No. 1-2, pp. 167-207, 1990

No context found.

S. Kraus, D. Lehmann and M. Magidor (1990) `Nonmonotonic reasoning, preferential models and cumulative logics', Artificial Intelligence 44:167--207.

No context found.

S. Kraus, D. Lehmann, M. Magidor. `Nonmonotonic Reasoning, Preferential Models and Cumulative Logics'. Artificial Intelligence, 44 (1990) 167--207.

No context found.

KRAUS S., LEHMANN D., MAGIDOR M., "Nonmonotonic Reasoning, Preferential Models and Cumulative Logics", Artificial Intelligence, vol. 44, 1990, p. 167--207.

No context found.

Kraus, S., D. Lehmann, and M. Magidor, (1990): "Nonmonotonic reasoning, preferential models and cumulative logics", Artificial Intelligence 44, 167-- 207.

No context found.

S. Kraus, D. Lehmann, M. Magidor, "Nonmonotonic Reasoning, Preferential Models and Cumulative Logics", in: Artificial Intelligence 44, 1990, pp. 167-207

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC