Re: Good and Bad IS-A hierarchies (Fritz Lehmann)
Date: Mon, 24 Jul 95 19:47:46 CDT
From: (Fritz Lehmann)
Message-id: <>
Subject: Re: Good and Bad IS-A hierarchies
Precedence: bulk
     Pat Hayes wrote:
---begin quote---
[stuff on graph-theoretic "bundles" of axioms]
None of this is usefully
reflected in any kind of isa heirarchy: its not a choice between different
concepts, but different ways to axiomatise the same collection of concepts.
Moreover, there is no way to organize the concepts, or even the axioms,
into neat little packets so that the various alternatives can be assembled
by choosing some and ignoring others.
There just are genuine alternatives, and one has to make committments in
selecting a temporal theory to work with.

I seem to detect, in the object-oriented flavor which informs so much
current work in ontologies, a residue of the old bias against the use of
axioms. But theres no way around it: if you want to answer questions, you
have to be able to draw conclusions, and conclusions involve making
connections between things. The basic unit of meaning is not a concept but
a theory, ie a collection of axioms.
Pat Hayes
---end quote---

     IS-A hierarchies do accomplish (some of) this
bundling.  IS-A with strict inheritance is a terse second-order
specification of a large class of first-order constraints;
constraints are in turn a particular kind of axioms.
It is exactly the "bundling" effect of hierarchies that makes them

     Anytime you can factor out a "BOX" (in the KL-ONE sense)
in a knowledge base you have accomplished something.  Divide and Conquer.
Admittedly the particular axiom-set determined by the hierarchy is only one
kind among the many kinds of axiom-sets needed, but it's a practically
important one.  Aside from conceptual clarity, it gives you the benefits
described by Walther and by Cohn for "order-sorted-logics" (= semantic nets
with nestable-negation and an IS-A/type-lattice): dramatic theorem-proving

     We still need other ("ABOX") axioms, of course, that don't fit in
any of the other "BOX"-factors, even if there are multiple, different
subsumption hierarchies ("BOXES") having nodes combined in one proposition.

     For "vivid" models (existential, without nestable negations) Gerard Ellis'
and my ICCS-94 paper (which you saw/suffered through?) gives the
mathematics of the "fret product" which combines all the "BOXES" into the big
hierarchy of all possible descriptions, ordered by subsumption. ("Exploiting
the Induced Order on Type Labeled Graphs for Fast Knowledge Retrieval", in
Conceptual Structures: Current Practice, Tepfenhart et al., eds., Lect.
Notes on A.I. No. 835, Springer, Berlin, 1994.)

     IS-A isn't everything, but it's still good.

                          Yours truly,   Fritz Lehmann
GRANDAI Software, 4282 Sandburg Way, Irvine, CA 92715, U.S.A.
Tel:(714)-856-0671               email: