LIF
Computational Linguistics Research Group
Albert-Ludwigs-Universit¨at Freiburg im Breisgau Germany
A SKETCH OF A QUALIFICATION CALCULUS Klemens Schnattinger& Udo Hahn
1996
LIF
REPORT 3/96
A SKETCH OF A QUALIFICATION CALCULUS Klemens Schnattinger& Udo Hahn
LIF
Computational Linguistics Research Group Albert-Ludwigs-Universit¨at Freiburg Werthmannplatz 1 79085 Freiburg, Germany
http://www.coling.uni-freiburg.de fschnattinger,
[email protected]
Abstract We introduce a symbolic, qualitative model of uncertain reasoning and apply it to a concept acquisition problem in the framework of natural language text understanding. Considering uncertain reasoning as a choice problem between different alternatives (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, finally, computes a preference order for the entire set of competing hypotheses. This model of quality-based uncertain reasoning is entirely embedded in a description logic framework.
Appeared in: FLAIRS' 96 - Proceedings of the 9th Florida Artificial Intelligence Research Symposium, Key West, Florida, May 20-22, 1996, Florida AI Research Society, 1996, pp.198-203.
A Sketch of a Qualification Calculus Klemens Schnattinger
Udo Hahn
Freiburg University Computational Linguistics Lab Europaplatz 1, D-79085 Freiburg, Germany schnattinger,hahn @coling.uni-freiburg.de LIF
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In: FLAIRS’96 - Proceedings of the 9th Florida Artificial Intelligence Research Symposium, Key West, Florida, May 20-22, 1996,
Abstract We introduce a symbolic, qualitative model of uncertain reasoning and apply it to a concept acquisition problem in the framework of natural language text understanding. Considering uncertain reasoning as a choice problem between different alternatives (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, finally, computes a preference order for the entire set of competing hypotheses. This model of quality-based uncertain reasoning is entirely embedded in a description logic framework.
1 INTRODUCTION In this paper, we develop a symbolic, qualitative model of uncertain reasoning, one which is entirely embedded in a terminological reasoning framework. Decision-making under uncertainty can be considered as choice between several alternatives or hypotheses. The qualification calculus we propose can be considered 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 can be related to different types of evidences for or against single alternatives. Depending on the availability of observational instances that can be related to single evidences, the calculus we provide characterizes different types of evidences by certain quality labels which
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indicate the specific status, i.e., the significance, reliability and strength of the evidence under consideration. This approach is motivated by requirements which emerged from our work in the overlapping fields of natural language parsing and learning from texts. Both tasks are characterized by the common need to evaluate alternative representation structures, either reflecting parsing ambiguities or multiple concept hypotheses. For instance, in the course of learning from texts, various and often conflicting concept hypotheses for a single item are formed as the learning environment usually provides only inconclusive evidence for exhaustively determining the properties of the concept to be learned. Moreover, in “real-world” natural language understanding systems, processing large text corpora, the underdetermination of results can often not only be attributed to incomplete knowledge provided for that concept in the source texts, but it may also be due to imperfect parsing results (originating from lacking linguistic or conceptual specifications, or ungrammatical input). Therefore, competing hypotheses at different levels of validity and reliability are the rule rather than the exception and, thus, require appropriate formal treatment. In order to deal with the emerging indeterminacy, e.g., in the learning task, two types of evidences are considered. The first one reflects structural linguistic properties of phrasal patterns or discourse contexts in which unknown words occur (assuming that the type of grammatical construction exercises a particular interpretative force on the lexical item to be learned). The second one reflects conceptual properties of particular concept hypotheses as they are generated and continuously refined by the ongoing text understanding process (e.g., consistency relative to already given knowledge, independent justification 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 qualification calculus, to these atomic quality labels finally determines which out of several alternative hypotheses actually hold(s).
Syntax Semantics
Terminological A. Axiom Semantics
I j Catom fd 2 Catom
C uD C tD :C 9R:C 8R:C
Catom is atomicg A =: C AI = C I I I C \D A v C AI C I C I [ DI Assertional A. I n CII I I Axiom Semantics fd 2 j R (d) \ C 6= ;g I 2 CI I I I a : C a fd 2 j R (d) C g
Table 1: Syntax and Semantics for Table 2: Axioms for Concept Constructors
Concept Constructors Syntax
Semantics
f(d; e) 2 RIatom j
Ratom
Terminological A. Axiom Semantics
Ratom is atomicg Q =: R QI = RI RuS RI \ S I I I f(d; d0 ) 2 RI j d 2 C I g Q v R Q R C jR RjC f(d; d00 ) 2 RII j d0 2I C I g Assertional A. ? 1 R f(d; d ) 2 j Axiom Semantics (d0 ; d) 2 RI g a R b (aI ; bI ) C D C I DI I I 2 RI (R1 ; ::;Rn ) R1 :: Rn
Table 3: Syntax and Semantics for
Table 4: Axioms for
Role Constructors
Role Constructors
2 TECHNICAL FOUNDATIONS Technically, 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. Based on the reification of terminological assertions, we use contexts for the encapsulation of qualifying reasoning processes and truth-preserving translation rules for the mediation between those contexts. Hence, we gain the full classification power from standard terminological systems for our metareasoning approach. Description Logics. We use a standard concept description language, referred to as CDL, which has several constructors combining atomic concepts, roles and individuals to define the terminological theory of a domain (for a subset, see Tables 1 and 3; cf. Woods & Schmolze (1992) for a survey of terminological languages). Concepts are unary predicates, roles are binary predicates over a domain , with individuals being the elements of . We assume a common set-theoretical semantics for CDL — an interpretation I is a function that assigns to each concept symbol (the set A) a subset of the domain , I A ! , to each role symbol (the set P) a binary relation of , I P ! , and to each individual symbol (the set I) an element of , I I ! . Concept terms and role terms are defined inductively. Tables 1 and 3 state corresponding constructors for concepts and roles, together with their semantics. C and D denote concept terms, while R and S denote roles. RI d represents the set of role fillers of the individual d, i.e., the set of individuals e with d; e 2 RI . jjRI d jj denotes the number of role fillers.
:
2 :
2
:
()
( )
()
R EIF
=: 8BINARY-REL:ROLES u 8DOMAIN:T HINGS u
8RANGE:T HINGS u 8HYPO -REL:H YPO v ROLES DOMAIN v RANGE v ROLES HYPO -REL v BINARY- REL
ROLES ROLES Table 5: General Data Structure for Reification
< (a : C ) = r : R EIF u r BINARY-REL INST-OF u r DOMAIN a u r RANGE C u r HYPO -REL h < (a R b) = r : R EIF u r BINARY-REL R u r DOMAIN a u r RANGE b u r HYPO -REL h Table 6: A Sketch of the Reification Functions < By means of terminological axioms (for subsets, cf. Tables 2 and 4) a symbolic name can be defined for each concept and role term. We may supply necessary and suf: ficient constraints (using ) or only necessary constraints (using v) for concepts and roles. A finite set of such axioms is called the terminology or TBox. Concepts and roles are associated with concrete individuals by assertional axioms (see Table 2 and 4; a; b denote individuals). A finite set of such axioms is called the world description or ABox. An interpretation I is a model of an ABox with regard to a TBox, iff I satisfies the assertional and terminological axioms.
=
Reification. We here restrict ourselves to the reification of the assertional axioms. The remaining constructors can be reified in a straightforward way based on the scheme outlined below (cf. Schnattinger et al. (1995)). We have chosen a particular “data structure”, itself expressed in CDL (see Table 5) to make the reification format explicit. It provides the common ground for expressing qualitative assertions at various degrees of plausibility or credibility. R OLES is the concept for all roles including the relations INST- OF and ISA (it thus represents the set Pext = P [ finst-of; isag), T HINGS is the (meta)concept for all concepts and instances (it represents the set A [ I) and HYPO is the concept denoting all hypothesis spaces. The symbol R EIF denotes a concept and BINARYREL, DOMAIN, RANGE and HYPO - REL denote its associated roles. With these conventions, we are able to define the (bijective) reification function < t:termh r:term, where t:termh is a terminological term known to be true in hypothesis space h (i.e., h H YPO ) and r:term is its corresponding reified term (a fragmentary definition is given in Table 6). By analogy, we may also define the inverse function
