Pat, (01)
>... conclusion, however it was arrived at, is *recorded*
>in a way that is intended to be used by machinery - if only in a very
>simple and straightforward way, such as substituting one name for
>another - that the notation or encoding method that is used to record
>it in be provided with a precise, mathematically described,
>'logic-style' semantic theory. Not in order to impose a tyrannical
>mainstream logical cultural hegemony, and not to subtly trick users
>into using alien notations, and not to require that all users have a
>graduate degree in logic. Rather, the point of having such a
>semantics for the formalism is purely pragmatic: it provides the only
>secure, non-procedural basis for interoperability between all the
>different formalisms. No one formal notation is going to be the
>single final form that everyone uses. But as long as all the
>formalisms have a common semantic base, there is at least the
>possibility of making translators between them. That is what I have
>devoted the last several years to achieving: giving a variety of
>formalisms a common semantic base. So far we have done it for Common
>Logic, IKL, RDF, RDFS and OWL. (It looks as though OWL 1.1 will
>deviate from this is some subtle ways that may not be very important;
>more seriously, the long-awaited Rule Language (RIF) may be even less
>semantically aligned with the RDF/CL vision. Oh well, I did my best.) (02)
That's exactly what I'm aiming for with probability. (Inspired to no
small degree by what you've been doing with logic, I might add.) The
right semantics will interoperate cleanly with logic, and will extend
logic in a coherent way to allow us to reason coherently about the
likelihood of propositions that can be neither proven nor disproven
from the logical axioms of a theory. (03)
A logical theory gives you a set of possible worlds, but can say
absolutely nothing --- zilch!!! --- about a world except whether it
is implied by the axioms, inconsistent with the axioms, or neither.
For any sentence in the language, if there are possible worlds in
which it is true and others in which it is false, then its
truth-value is indeterminate. Period. No matter WHAT we know about
it. (04)
The vast majority propositions of practical interest can neither be
proven nor disproven. Therefore, logic alone can tell us NOTHING
about their truth-values! We often constrain our logical theories
to rule out wildly improbable propositions -- in effect pretending
they are impossible. This is highly dangerous, as any reactor safety
engineer or medical doctor will tell you. (05)
Probability theory allows us to say something about propositions we
can neither prove nor disprove. This is extremely useful. (06)
People tried for years to do AI without probability. They invented
certainty factors, which turned out to be mathematically equivalent
to a very restricted family of probability models. Embracing
probability theory enables you to analyze the kinds of situations
where certainty factors will work well and where they will get you
into trouble. It also allows you to build more sophisticated
probabilistic theories to handle situations where certainty factors
won't work. People invented neural networks which turned out to be a
kind of semi-parametric regression model (strictly speaking this
applies to feedforward backpropagation neural networks, but there are
statistical models for other kinds of neural networks as well).
People claimed it was a bad idea to use probability because it was
intractable, and then invented hacks that turned out to work or not
work in a given context, depending on whether they were or were not
close approximations to what you would get if you applied probability
sensibly in that context. Formulating a problem using probability
theory and seeking tractable approximations to the correct solution
turns out to be an excellent heuristic for finding tractable and
high-performing solutions. Duh! (07)
Wheel-reinvention is not necessarily a bad thing, by the way. It
often opens the minds of researchers in the original-wheel-inventing
field to whole classes of problems they had never considered before,
and when they turn their theoretical arsenals on those new problems,
they invent wonderful new theory for the field of the
wheel-reinventors. The problem isn't wheel-reinvention and the
resultant cross-pollination, which actually is a very good thing.
The problem is the mindset that the wheel-reinventors are stupid for
not having realized our field already invented this, or that the
people in that esoteric original-wheel-inventing field cannot
possibly have something coherent to say about my field. (08)
Anyway, when the dust settled, AI realized it needed probability and
decision theory, and researchers in AI now study advanced statistics,
to the betterment of all concerned. (09)
My research for the past several years has focused on developing a
formal semantics for probability theory that interoperates cleanly
with logic. I find it fascinating how difficult it is for many people
to appreciate what I'm aiming for, and the inability among many who
view themselves as "practical applications" people to appreciate the
value it brings. For example, in a recent discussion on whether we
could agree on a common semantics for various formalisms for
combining probability and logic, there were some well known and
highly respected researchers in the room who vociferously argued
against a possible worlds semantics on the grounds that, for example,
in a system for doing diagnostic reasoning, the statements are not
about possible worlds -- they are about symptoms and failures and
states of systems. We couldn't get past that blockage, and so we
ended up not coming to agreement on common semantics. (010)
But -- to take my own advice -- my inability to make inroads with
such people probably has less to do with any wrongheadedness on
their part, and more to do with my inability to articulate clearly
why a common semantics is both possible and a good thing to do. I've
been working on developing a clearer articulation of what the
business folks would call my "value proposition." Stay tuned. (011)
Kathy (012)
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