On Definitional Processes in Knowledge Reconstruction Systems

1. Introduction

Defining concepts covering a certain domain usually yields a hierarchical taxonomy and a corresponding terminology. This has been the m.tn Import of semantic networks, frame languages, and even semantic data models [vtz. Hammer/ Mcleod 19811. Especlaly with the first kind of representation schemes, problems arise If the necessary distinctions are not observed. I wll look at frame oriented languages, In particular those claimed to support ·termlnologklll adequ.if In know­ ledge engineering. An example taken from Brachman et al. [19831 wll Illustrate some aspects of the problem. They define the concept of ·tamllj" as:

·a social structure with, among other things, a male parent who Is a man, a female parent who Is a woman, and some number of chHdren, an persons·

corresponclng to the KRYPTON expression:

(PrlmGeneric

(ConjGenertc

(NRGeneric

(VRGeneric socal-structure ma,._,_..,, man)

male-parent 1 1)

(NRGeneric

CVRGeneric soc'-1-structure fema»-,.,-ent woman)

female-parent 1 1)

CVRGeneric social-structure child person)))

a poor one, however. (2) leads to similar problems . In KRL [Bobrow/Winograd 19771, the subconcept "motor-vehicle" could be given as:

motor-vehlce UNIT SPECIALIZATION [(SELF (a vehicle)) (motor (XOR combustion steam electro))]

As a consequence, It Is undecidable (In general It should be semi-decidable):

• (a) whether a sibling concept, e.g. "muscle-driven-vehicle" with a slot "instrument" (pedals, oars etc.), is disjoint from "motor-vehicle", and, hence:
• (b) whether "motor-vehicle" can have an "Instrument" or not, and whether "muscle-driven-vehicle" can have a "motor" or not.

This has also been stated by HuSmann [19841, but has been overlooked by Hayes and Hendrix [Hayes/Hendrix 19801.

An adequate specialization can only be achieved by method (3) . Using predicate calculus, the above example becomes:

motor-vehicle (x) ::= vehicle (x) A propelling-force (x) = motor muscle-driven-vehicle (x) ::=

vehicle (x) A propelling-force (x) = muscle

A category such as "propelling-force" applying to all instances of the superconcept is restricted in some way yielding a new subconcept. By this method of taxonomic subcategorization It will be (seml-)decidable that both concepts defined above are mutually exclusive, and exhaustive, depending on whether the "values" of "propelling-force" are exhausted or not.

Subcategorlzatlon Is closely related to attribution. The latter raises a question mostly ignored In the literature:

Is the value of an attribute something ascribed to the object, tMrt of the object, or even an object in its own right (e.g. "motor" In the above example)?

I hold the first opinion. Attributes refer to a scale, a set of values representing criteria of observation (be it real or forman. A typical example is:

height (John) = 180cm

"180cm~ Is a value pertaining to a scale of height, an interval scale. Beyond this, there are ordinal and nominal scales. All scales are construed by comparing the object in question with one or more base objects. Thus, in the above example, "motor" Is a value of a nomln~ scale, i.e. it does not repre­ sent a part or thing In Its own right .

This fundamental issue Is obscured in representation languages such as KL-ONE, as "attributes" and "values" are not distinguished from "roles" in "systems", or "relations" to other "entitles".

Method (4), "role differentiation" as Included In KL-ONE, is not a basic method of taxonomic specialization (see below 3.2) .

3. System--agregatlon, role and collection concepts

In simple predicate logic and In most AI "representation languages", a universe of discourse is assumed to contain only atomic Individuals. A set of Interrelated entitles, however, must be modelled as a new complex entity. To account for this epistemological process I call system aggregation, extend­ Ing the predicate logic by simple aggregates otherwise called tuples will suffice. The "family" example then reads:

family (x) ::= 3 u v w: x = <u,v,w> A family-rei (u,v,wl

family-rei (u,v,wl ::=

man (u) A father-of (u,w) A ...

A children (w) A ... A married (u,vl A ...

or by extending KRYPTON:

(SystemGeneric family-rei father: man mother: woman chHdren: set-of-persons)

The "family-rei" may be as complex as necessary for a certain purpose. The "parts" of the system can now be adressed as genuine roles In a system (not as relationships or attributes). Role concepts become definable In there own right :

father (x) ::= 3 u: family (u) A u = <x,y,z>

or In KRYPTONic style:

(RoleGeneric family father)

Within a general definitional capability, this extension Is a necessary one to define genuine role concepts. The . "role sets" of KL -ONE do not pertain, because they mix up attribu­ tion and relationships, resulting, e.g. In the amazing "number restriction" for attribute values (on a scale, there can be only

one value) .

As system aggregation allows for viewing a set of Inter­ related entitles as a new entity, further speclaUzatlon be­ comes feasible. However, now It comes In different flavors, namely by way of (1) attribute value restriction, and (2) role filler restriction, e.g.

1. ·A poor family Is a family with Income < 1000·
2. ·A royal family Is a family where the father Is a king or the mother Is a queen·

The latter example us" the role concepts ·k~ng· and ·queen· (where the system II ·empire·) to restrict the flier of the father or mother role In ·family".

Subcategorlzatlon of role concepts also Is at least two­ fold:

• (a) attribute value or role flhr restriction

(b) role differentiation

The first pertains to the entity as a role flier, e.g.

tall-father (x) ::= father (x) A height (x) = tall

This specialization corresponding to a subrelatlon does not necessarily Imply a ·role differentiation", as Brachman et al. [op. clt.l appear to have In mind. It Is a subcategorlzatlon pertaining to an entity as a flier, not to the role Itself. A subrole should be differentiated from Its superrole by some function or relation In the system, not by an arbitrary at­ tribute or component restriction, say:

·A crown prince Is a subrole of a child In a royal family with inheritance claim where sex = male·

crown-prince ::=

3 u: royal-family (u) A u = cy,z,W> A x . E w A sex (x) = male A Inheritance-claim (x,y)

A system may be viewed simply u a collection, thus abstracting from the specific relationships, and an entity may . be viewed simply as a member of a system, thus abstracting from Its specific role:

family-collection (x) ::=

family-member (x) ::= 3 u: family (u) A x E u

We have to extend predicate logic by set-construction and membership (another conservative extension). The difficulty with the above example Is that all subsystems, ·children· In this case, have to be collected together. Otherwise, a more complex model of the family would be required, e.g. by modei­ Hng a variant number of chlld-subroles.

4. Analytic and a,nthetlc knowledge

For every simple concept, there Is a set (conjunction) of value-restrictions that uniquely determine Its position In a definitional lattice. This knowledge Is, of course, purely analytle. It provides with a certain set of propositions pertaining to a ·possible. Instance of the concept In question.

We have to make the necessary distinction of analytic and synthetic knowledge. Clearly, that the lack of a certain part or role results In ·the malfunction of the system as a whole, Is synthetic knowledge. The concept of "a chair with two legs· could be derived from the concept of a chair by role restriction. However, whether this makes sense can be doubted. Hence, a definitional component could be restricted by an assertlona/ one comprising the constraint, say:

·A chair must have at least three legs"

thus preventing a knowledge engineer to conceive of "Im­ possible concepts" . An "elephant with three legs" or a "truck with two wheels" Is not a concept derived by simple number restriction, but, based on a description of a possible con­ tingent state of a system, a new concept dealing with a possible maffunctlon of a system.

Analytic and synthetic knowledge, though tightly Inter~ twined, should be kept separatly requiring different modes of To model the correct deductions, I.e. to provide with epistemo­ logically adequate representations of knowledge, a semantics

reasoning, deduction and (probabilistic) abduction, respectively. Is required richer than the slot-and-fiNer-restriction scheme can account for.

rt.t.renca1

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