LIF
Computational Linguistics Research Group
Albert-Ludwigs-Universitat Freiburg im Breisgau Germany
A TERMINOLOGICAL QUALIFICATION CALCULUS FOR PREFERENTIAL REASONING UNDER UNCERTAINTY Klemens Schnattinger & Udo Hahn
1996 LIF
REPORT 5/96
A TERMINOLOGICAL QUALIFICATION CALCULUS FOR PREFERENTIAL REASONING UNDER UNCERTAINTY Klemens Schnattinger & Udo Hahn LIF
Computational Linguistics Research Group Albert-Ludwigs-Universitat Freiburg Werthmannplatz 1 79085 Freiburg, Germany
http://www.coling.uni-freiburg.de fschnattinger,
[email protected]
Abstract
We introduce a qualitative model of uncertain reasoning and illustrate its application in the framework of a natural language understanding task. Considering uncertain reasoning as a preferential choice problem between alternative hypotheses, the model we provide assigns quality labels to single evidences for or against a hypothesis, combines the generated labels in terms of the overall credibility of a single hypothesis, and, nally, computes a preference order on the entire set of competing hypotheses. This model of quality-based uncertain reasoning is entirely embedded in a terminological logic framework.
Appeared in: G. Gorz, S. Holldobler (Eds.),KI'96 - Advances in Arti cial Intelligence. Proceedings of the 20th Annual German Conference on Arti cial Intelligence, Dresden, Germany, September 17-19, 1996. Berlin etc.: Springer, 1996, 349-362.
In: G. G{{accent 127 o}}rz, S. H{{accent 127 o}}lldobler (Eds.),KI’96- Advances in Artificial Intelligence. Proceedings of the 20th AnnualGerman Confe
A Terminological Quali cation Calculus for Preferential Reasoning under Uncertainty Klemens Schnattinger & Udo Hahn Text Knowledge Engineering Lab Universitat Freiburg, Europaplatz 1, D-79085 Freiburg, Germany LIF
fschnattinger,
[email protected]
Abstract. We introduce a qualitative model of uncertain reasoning and
illustrate its application in the framework of a natural language understanding task. Considering uncertain reasoning as a preferential choice problem between alternative hypotheses, the model we provide assigns quality labels to single evidences for or against a hypothesis, combines the generated labels in terms of the overall credibility of a single hypothesis, and, nally, computes a preference order on the entire set of competing hypotheses. This model of quality-based uncertain reasoning is entirely embedded in a terminological logic framework.
1 Introduction In this paper, we develop a qualitative, preference-based model of uncertain reasoning. Decision-making under uncertainty is here considered as the choice between several alternatives (or hypotheses). The quali cation calculus we introduce serves as a system of preference computations that treats the problem of choosing from among several alternatives as a quality-based decision task and decomposes it into three constituent parts: the continuous generation of quality labels for single hypotheses, the estimation of the overall credibility of single hypotheses, and the computation of a preference order for the entire set of competing hypotheses. The key notion of quality labels captures di erent types of evidences for or against single alternatives and their speci c statuses, i.e., their signi cance, reliability or strength. This approach and its complete embedding in a terminological reasoning framework are motivated by requirements which emerged from our work in the overlapping elds of natural language parsing [7, 16] and learning from texts [5]. In order to cope with lexical and conceptual underspeci cation of the relevant knowledge sources in a constructive way the parsing process is interwoven with concept learning tasks. The common basis for text understanding as well as concept learning from texts are terminological knowledge representation structures. Both tasks are also characterized by the common need to evaluate alternative representation structures, either re ecting parsing ambiguities or multiple concept hypotheses. In order to deal with the emerging indeterminacy, e.g., in the learning task, two types of evidences are considered. The rst one re ects structural linguistic properties of phrasal patterns or discourse contexts unknown words occur in (assuming that the type of grammatical construction exercises a
particular interpretative force on the lexical item to be learned). The second one relates to conceptual properties of particular concept hypotheses as they are generated and continuously re ned by the ongoing text understanding process (e.g., consistency relative to already given knowledge, independent justi cation from several sources). Each of these grammatical, discourse or conceptual indicators is assigned a particular quality label. The application of quality macro operators, taken from the quali cation calculus, to these atomic quality labels nally determines which out of several alternative hypotheses are actually preferred.
2 Formal Foundations We consider the problem of uncertain reasoning from a new methodological perspective, viz. one based on metareasoning about statements expressed in a terminological representation language. Terminological assertions are rei ed, contexts are used for the encapsulation of qualifying reasoning processes on rei ed terminological assertions, and truth-preserving translation rules mediate between di erent contexts. Hence, we exploit the full classi cation power from standard terminological systems for metareasoning. A detailed discussion of the underlying architecture is given in [19]. Terminological Logics. We use a concept description language with a standard set-theoretical semantics (the interpretation function I ). It has several constructors combining atomic concepts, roles and individuals (see Tables 1 and 3). By means of terminological axioms a symbolic name can be de ned for each concept and role term; concepts and roles are associated with concrete individuals by assertional axioms (see Tables 2 and 4). A survey of the major properties of terminological languages is given by [22]. Syntax
Semantics
Catom CuD CtD :C 9R:C 8R:C
I j Catom is atomicg fd 2 Catom C I \ DI C I [ DI I n C I fd 2 I j RI (d) \ C I =6 ;g fd 2 I j RI (d) C I g
Syntax
Semantics f(d; e) 2 RIatom j Ratom is atomic g
Axiom Semantics Terminological Axioms A =: C AI = C I
AvC
AI C I
Assertional Axioms a:C aI 2 C I Table 1. Syntax/Semantics for Concept Constructors Table 2. Axioms for Concept Constructors
Ratom RuS C jR RjC R?1 C D (R1 ; ::; Rn )
RI \ S I f(d; d0 ) 2 RI j d 2 C I g f(d; d0 ) 2 RI j d0 2 C I g f(d; d0 ) 2 I I j (d0 ; d) 2 RI g C I DI RI1 :: RIn
Table 3. Syntax/Semantics for Role Constructors
Axiom Semantics Terminological Axioms Q =: R QI = RI
QvR
Q I RI
Assertional Axioms aRb (aI ; bI ) 2 RI Table 4. Axioms for Role Constructors
Reif
=: 8binary-rel:Roles u 8domain:Things u
8range:Things u 8hypo-rel:Hypo binary-rel v Roles domain v Roles range v Roles hypo-rel v Roles
Table 5. General Data Structure for Rei cation < (a : C ) r : Reif u r binary-rel inst-of u r domain a u r range C u r hypo-rel h < (a R b) r : Reif u r binary-rel R u r domain a u r range b u r hypo-rel h Table 6. Rei cation Function < Rei cation in Terminological Logics. We have chosen a particular \data structure", itself expressed in terminological logics to make the rei cation format explicit (see Table 5). It provides the common ground for expressing qualitative assertions about the plausibility or credibility of various alternatives (see the translation rule schema below). Roles is the concept for all roles including the relations inst-of and isa, Things is the (meta)concept for all concepts and instances and Hypo is the concept denoting all hypothesis spaces. The symbol Reif denotes the concept for all rei cators, i.e., the anchoring terms introduced by the rei cation, and binary-rel, domain, range and hypo-rel denote its associated roles. With these conventions, we are able to de ne the bijective rei cation function < : Axiom ! Rex, where Axiom is the set of assertional axioms and Rex is the corresponding set of rei ed expressions (see Table 6)1. Hence,
