"Probabilistic Default Logic Based on Relevance- and Irrelevance Assumptions", in: D. Gabbay et al. (eds.), Qualitative and Quantitative Practical Reasoning (LNAI 1244), Springer, Berlin 1997, S. 536 - 553.
Dov Mo Gabbay Rudolf Kruse Andreas Nonnengart Hans Jiirgen Ohlbach (Edso)
Qualitative and Quantitative Practical Reasoning First International Joint Conference on Qualitative and Quantitative Practical Reasoning, ECSQARU-FAPR'97 Bad Honnef, Germany, June 9-12, 1997 Proceedings
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Volume· Editors Dav M. Gabbay Hans Jilrgen Ohlbach Imperial College of Science. Technology and Medicine, Dept. of Computing 180 Queen's Gate, London SW7 2AZ, U.K. E-mail: (dg/h.ohlbach)@doc.ic.ac.uk Rudolf Kruse Otto-von-Guericke-UniversiUit, FakulUit fUr Informatik UniversiUitsplatz 2, D-39106Magdeburg, Germany E-mail:
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Die Deutsche Bibliothek - CIP-Einheitsaufnahme Qualitative and quantitn.tive practical reasoning: proceedings / Firstlnternational Joint Conference on Qualitative and Quantitative Practical· Reasoning, ECSQARU FAPR '97, Bad Honner, Germany, June 9 - 12, 1997. Dov Gabbay ... (ed.). - Berlin; Heidelberg; New York; Barcelona; Budapest; Hong Kong; London; Milan; Paris; Santa Clara; Singapore; Tokyo: Springer, 1997 (Lecture notes in computer science i Vol. 1244 : Lecture notcs in
artificial intelligence) ISBN 3-540-63095-3 CR Subject Classification (1991): 1.2. E4.1 ISBN 3-540-63095-3 Springer-Verlag Berlin Heidelberg New York This work Is subject to copyright. AU rights are reserved, whether the whole Of part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or In any other way, and storage in data banks. Duplication of this publlcation or paris thereof Is permitted only under the provisions oflhe German Copyright Law of September 9. 1965, in Its current velslon, and permission for use must always be obtained from Springer. Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer.Verlag Bertin Heidelberg 1997
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Preface It has become apparent in the last decades that human practical reasoning demands more than traditional deductive logic can offer. From both a philosophical and an engineering perspective the analysis and mechanisation of human practical reasoning requires a subtle understanding of pragmatics, dialectics, and linguistics, even psychology. Philosophers, software engineers, and AI researchers have similar ambitions in this respect; they all try to deepen our understanding of human reasoning and argumentation. Various aspects of human practical reasoning have resulted in the devel()pment of non-monotonic logics, default reasoning, modal logics, belief function theory, Bayesian networks, fuzzy logic, possibility theory, and user modelling approaches, to name a few. These are new and active areas of research with many practical applications and many interesting, as yet unsolved, theoretical problems.
This volume contains the accepted and the invited papers for the ECSQARU jFAPR '97, the first international joint conference on quantitative and qualitative practical reasoning. The three predecessors of ECSQARU '97 were sponsored and organized by the consortium of DRUMS (Defeasible Reasoning and Uncertainty rVlanagement Systems, ESPIDT III BRA 6156). The goal of this project, which involved 21 European universities and research organizations, was to develop techniques in the fields of belief change, non-monotonic deduction, inconsistency in reasoning, abduction, efficient inference algorithms, and dynamic reasoning with partial models. FAPR 196 was sponsored by MEDLAR, the European project on practical reasoning, which involved 15 major European groups in mechanised deduction in qualitative practical reasoning. The purpose of the ECSQARU jFAPR is to introduce these communities to each other, compare the current state of research, and make research available to all researchers involved. This year particular attention was directed to special tutorials and invited sessions which were organised by leading researchers from the "quantitative" and the "qualitative communities" respectively. We are indebted to the program committee for their effort and thought in organizing the program, to the invited speakers, and to the presenters of the tutorials. Moreover, we gratefully acknowledge the contribution of the many referees who were involved in the reviewing process. Special thanks go to Christine Harms who ensured that the event ran smoothly.
March 1997
Dov M. Gabbay Rudolf Kruse Andreas Nonnengart Hans Jiirgen Ohlbach
537
Probabilistic Default Logic Based on Irrelevance and Relevance Assumptions Gerhard Schurz lnst. f. Philosophie, Abtlg. Logik und Wissenschaft-stheorie. UniversiUit Salzburg""
Abstract. This paper embeds default logic into an extension of Adams' probability logic, called the system ptDP. Default reasoning is fur· nished with two mechanisms: one generates (ir)relevance assumptions, and the other propagates lower probability bounds. Together both mech· anisms make default reasoning probabilistically reliable. There is an exact correspondence between Poole-extensions and ptDP·extensions. The procedure for ptDP·entailment is comparable in complexity with Poole's procedure.
1
- A mechanism of generating the minimal probabilistic default assumptions which must be made in order to derive the conclusion safely - this is called the assumption generation mechanism. A mechanism of propagating approximately tight lower probability bounds from the premises to the conclusion - this is called the lo-wer bound propa· gation mechanism.
Introduction and Motivation
Default reasoning is reasonirtg from uncertain laws, like "Birds normally can fiy" . We formalize these laws as 'B ::::} F, where::::} stands for uncertain implication and B,F are associated with an invisible variable x (Bx,Fx). Uncertain laws have two characteristics. First, they have exceptions (e.g., penguins don't fly), and second, we do not know the conditional probability of flying animals among birds. "Ve just assume that this probability is high - otherwise we would not be justified in calling this the normal case. vVe call p(B/A) the conditional probability associated with A ::::} B. Default reasoning starts from the intuitive principle that one may detach Ga from F ::::} G and Fa as long as ,Ga is not 'implied' by one's knowledge base. Hence the nonmonotonic character of default reasoning. But this principle is ambiguous. As a result, various different systems have been developed (cf. [4]). How are these systems justifiecl? In .this respect, clear criteria are often missing. The major criterion suggested here is reliability. For, in contrast to deductive inferences, default inferences are uncertain: they. do not preserve truth (from premises to conclusion) for .all of their individual instances. Still, they should at least preserve truth for most of their individual instances. Since uncertain laws are true iff their associated conditional probabilities are high, we can explicate the requirement of reliability as follows:
(Reliability:) If the conditional probabilities associated with the uncertain laws used as premises are high, then the conditional probability of the conclusion given all factual knowledge is high, too. *
The probability concept 'p(_)' which underlies this approach is statistical (as opposed to subjective) probability: this is essential for the tie between reliability and a high predictive success rate. It is important that the premises need not be identical with the uncertain laws contained in the knowledge base, but may be derived from them with help of probabilistic default assumptions (irrelevanceand relevance assumptions). Standard default logics do not satisfy the requirement of reliability. E.g.} this is demonstrated by the ConJunction Pmblem: given a set of uncertain laws with the same antecedent A =} BIt ... I A =} Bn (and without conflicting laws), all standard default logics wlll reason from A to the conjunction /\l 0 such that for all p E JI{C, V, L), if (a') holds and (b') VD E V[P{D) 21 -
(aJ C is €-inconsistent or (b) some s:; C yields A=? B, which means that the truth valuation confirming C* verifies A => B and (ii)
€-entails A
{A,
~
B iff either
i ~ n}
J
each truth valuation falsifying A =? B falsifies C*. Lemma 2 C. is €-inconsistent iff some nontrivial subset of.c is not confirmable iff for some consistent A E BLang, .c f-" A=> 1-. Lemma 3 If C is f.-consistent, then
c. f-E L implies C-" f- L-.
Proof of lemma 3: If C is €-consistent, then every L derivable from C by the rules ofP" is derivable without use of the rule (€EFQ), because this rule applies only to laws A => -L with consistent antecedent, and whenever .c implies such a law it is €-inconsistent by lemma 2. All rules of P" distinct from (€EFQ) are propositionally valid for the material counterparts of the uncertain laws. Hence L- is derivable from £- -by rules of propositional logic. Q.E.D.
Proof of theorem 3: For th.3.1: Let C = {A, =? B, 1 1 ~ i ~ n}. Given C~ U:F I- Aa, then (a): :Fx A A' .1 by tho 3.1. Since:F is consistent J lemma 2 implies that 1JF (C:c) is €-inconsistent. For 2{i): V:F(C':,:) is co-consistent by tho 4."1 and assumption and thus satisfies PO. Next we prove that 1J~{C!r) satisfies P2 (and thus, Pl). For reductio, assume (A => B) E C~, and for some literal LT E LT{B), M U F I- ~LTa holds (Le. P!! is violated). Now (A => B) E St{L') for some tranformed law L'. Thus LtlB consequent either contains LT as an elementary disjunct, or it contains an elementary disjunct of the form -.( -LT /\ G), where -F := o f and --.F := F. In both cases M U F I- ...,DS must hold for some elementary disjunct DB of Lt'g consequent. But this is excluded by def. 3.1; contradiction. Finally, we prove that every .r-consistent M-extension M' satisfies Pl. Again, assume for reduction that (a) M'UF I- ~Ba for some (A => B) E £~. Since MuF I- Aa (by def. 3.1), also (b) M' U F I- Aa must hold. But (a)+{b) imply that M' is F-inconsistent; a contradiction. For 3:-By applications of inference axiom (Irr) and transposition step (3). For 4: M -il-.{C~)~ by prop. logic, and 1J~{C~) is ,-consistent by tho 4.1; so th.4.3 follows by th.3. For 5: t: E POOLE{{C,F)) iff t: = Cn{C- UF) for some inaximal materialization £~ of (C"F) iff (i): t: = t:[1J~{£~)[ by tho 3, since (C~)~ + C~ and VF(£~) is t;-consistent (Le. is a Poolean update) by tho 4.1. It remains to show that VJ:{C~) is maximal. Assume for reductio that for some L E (£* \ £), 1JF{{CU {L})~) would be ,-consistent. Then {£U {L})~ UF must be consistent by tho 4.1. Because (£ U {L})~ properly extends M, M would then not be a maximal materialization; a contradiction. - So V:F(£~) is a maximal Poolean update, whence (i) holds iff t: E PtDP{{C',F)) (def. 3.3-4). For 6(a}: By tho 2, C U £~ U :r~{C~) ,+ -implies the same laws as C U C~ U V:F(Cj-). To prove 6(a) we first have to prove: (A): C U £~ U 1J~{£~) is ,-consistent. For reductio, assume that the negation of (A) is· true. Then by lemma 2, there is a nonempty subset X c £ U £~ U 1J~{C~) such that (B): X~ I- I\Y, where Y = {~P I (P => Q) E X}. Now note that (C): C~ I- [CU£~U1JF{C~)I~ must hold, because (£~)~ I- (1JF{£~))~ and (£~)~ + £~ (by prop. logic). (C) and (B) imply (D): C~ I- I\Y. For each element of Y E y, either (i): Y = ~A for (A =? B) E C, or (Ii): Y = ~A or Y = ~(Fx 1\ A) for (A. => B) E £~. Case (Ii) implies that M U.r I-- -,Aa, but this is impossible because also M U:F I-- Aa holds by def. 3.1 and M U:F is consistent by assumption. So only case (i) can be true. This implies that . 1' . ~ .c. and hence that C would be t;-inconsistent (by lemma 2). But this contradicts the assumption that M U :F is consistent and th.4.1. Right-to-Ieft of th.4.6{a) is eMY, since £ U N F {£) u:r~{C~) 1-, 1JF{£~) by applications of (In) and transposition step (3). Concerning left-ta-right: Assume
(I): C U £~ U 1JF{C~) I-. Fx => A The premises of (1) are ,-consistent by (A) above. Hence by lemma 3, (2): £~ U (C~)~ U (1JF{£~»)- I- Fx --> A. (2) and (C) above imply that £~ I- Fx -> A and hence (3): M U F I- Aa. (3) implies by th.3.1 that 1JF{C~) 1-, F"x => A . For 6(b}: The antecedent of 6{b) implies 1JF{£~) 1-, Fx =? A by th.4.6{a). Every D E V:F(.c.~) results from a unique L E £ (by transposing and updating) such that for all p E II(C,N,I) satisfying all irrelevance assumptions in I:F('c) , u{D) ~ min (u(~h, Q.E.D.
1)
(by tho 1.2). By tho 1.2 again, this implies our claim.
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