What happened with types?
fritz@rodin.wustl.edu (Fritz Lehmann)
Date: Mon, 17 May 93 14:53:27 CDT
From: fritz@rodin.wustl.edu (Fritz Lehmann)
Message-id: <9305171953.AA05848@rodin.wustl.edu>
To: cg@cs.umn.edu, interlingua@ISI.EDU
Subject: What happened with types?
MacGregor, Hayes and Sowa discussed earlier whether
KIF should accomodate typed variables. I believe it
should, despite the fact that many are unable (so far)
to formalize the distinctions between types and monadic
predicates which they nontheless observe and use.
Among the eight main research families of semantic
networks and "logics" surveyed in my 1992 collection,
seven use some kind of type hierarchy. A typed logic is
simply very convenient, regardless of whether it can be
"dispensed with in principle". A lot of KIF, e.g.
functions, can be dispensed with in principle --- this
should not be a criterion for exclusion from KIF. Using
a hierarchy of ordered sorts (types) speeds automatic
theorem-proving dramatically, as shown in 1985 by
Walther and by Cohn. If KIF has no types, this cannot
be exploited without some painful process of "type
discovery" or specification.
A distinguishing feature of Knowledge
Representation is that predicates (and relations) do
form hierarchies. This is not a feature of first-order
logic, in which predicates might form only an
unstructured jumble. The poset of (explicit or
implicit) IS-A links is a formal mathematical
structuring of the second-order theory. Since it only
concerns "a theory" which is "second-order", it does not
interest the first-order logician, for whom all
predicates are essentially unrelated.
If a hierarchy exists, it is an order-theoretic
factorization of the predicates; any remaining logic
(left in the ABOX, in KRYPTONite terms) is a kind of
quotient.
We test drugs as "safe and effective". Genesereth &
Hayes evidently questioned whether types are "effective"
-- do they also contend that types are not "safe"? So
far, I see no pernicious harm caused by including in KIF
a distinction which about 95% of AI practitioners use
and which apparently causes no first-order trouble.
Yours truly, Fritz Lehmann
[P.S. Incidentally, some people profess to see a real-
world kind/quality distinction among predicates, not
just a difference in the uses to which a predicate is
put. David Israel wrote (on inheritance systems):
"There is to be one tree for kinds of things and another
for qualities of things. Kinds must be distinguished
>From qualities: being a cat must be distinguished (in
kind, no noubt) from being red." A number of formal
theories contend. I believe in this real-world
distinction but only as a ranking difference in a
second-order theory, not as a first-order distinction.
In Rudolf Wille's theory of Concept Lattices, the
powerset of individuals has a Galois connection with the
dual powerset of monadic predicates and the respective
Galois closures form a complete lattice of concepts.
Typically, the more predicates true of (inherited by) a
concept in the lattice, the fewer the individuals in the
set of described individuals. A high concept describes
many individuals and inherits few predicates, and a low
concept does the opposite. "Cat" inherits an enormous
bundle of predicates which are true of something in
virtue of its being a cat. "Red" inherits virtually
nothing. For this reason, "Red" is not useful as a type
but "Cat" is. Lattice-theoretically, "red" is a "meet-
irreducible" node of the concept lattice, meaning that
it is not a specific combination of more general
qualities. The "intent" (set of inherited qualities) is
the filter or upper ideal of a concept in the lattice,
and the "extent" (set of covered individuals) is the
(lower) order ideal. This view allows intermediate
concepts like "Bohemian" which can be viewed both ways.
So I myself do no not formalize any sharp distinction
except for the extremes (meet- and join- irreducibles).]
25 Seton, Irvine, CA 92715 714-733-0566 fritz@rodin.wustl.edu
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