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C00002 00002	halper[e85,jmc]		Abstract for Halpern's knowledge conference
C00008 00003	Common sense inertia compared to inertia in physics
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halper[e85,jmc]		Abstract for Halpern's knowledge conference
∂05-Sep-85  1616	VAL  	Abstract for Halpern's conference 

	Circumscription As A Solution to the Frame Problem

   The frame problem consists in describing how the properties of objects change
across events. To solve it, we should formalize the Common Sense Law of Inertia:
"Generally, things remain as they were".

   The first idea is to state this in terms of minimization:
"The difference between the values of a fluent in two situations separated by an
event is minimal, subject to the given properties of the event".

   What do we mean by "difference" here? If the fluent is predicate-valued, like
"on", then the difference is the extension of the exclusive OR applied to the two
values of the fluent. Thus we minimize the extensions of predicates, so
circumscription is relevant.

   To give a simple example of how this works, consider the blocks world in
which a block can be in one of only two places: on the table or on the floor.
The language will be more limited than that of situation calculus: we have
variables for blocks only, not for events or situations. Two unary predicates
ontable and ontable', will be used to talk about the positions of blocks in
two situations separated by an event. Axioms:

	Bi≠Bj (1≤i<j≤6); ontable Bi (1≤i≤3); ¬ontable Bj (4≤i≤6).

Let the event consist in putting B4 on the table. That is expressed by ontable' B4.
The frame problem consists in deciding which of ontable' Bi are true for i≠4. Add
the axiom

		¬ab x ⊃ (ontable x ≡ ontable' x),

and circumscribe ab with ontable' varied. The result is: ab x ≡ x≠B4, which is
exactly what we needed.

   We want to do this sort of things for more complex versions of the blocks world,
and for the language with variables for events and situations. In situation
calculus, a fluent is represented by a predicate p(x,s), where x is a tuple
of arguments, and the law of inertia for p can be expressed by

	-ab(x,e,s) ⊃ (p(x,s) ≡ p(x,result(e,s))).

Unfortunately, this doesn't quite work, at least in the blocks world in which
one can move blocks on top of one another, because there may be a conflict
between minimizing ab(x,e,s) for different situations s, and we get some
minimum values of ab which don't correspond to the picture we want to
formalize. (I'll check whether describing the simple version of the blocks
world above in the language of situation calculus already causes problems, or
they arise only for more complex versions).

   There are two ways out: (i) state that the action is impossible when the
preconditions aren't satisfied; (ii) state that abnormality now should be
minimized at a higher priority than abnormality later. Only (i) has been
explored so far, and it seems to work.  (I hope I'll have something to say
about (ii) by the time the full paper is due.) In this solution, since
"result" is a total function, some situation terms fail to denote possible
situations. We view the universe as containing the concepts of situations
rather than situations themselves, and introduce the predicate "exists" to
talk about the possibility of actions. For this formalization, we can prove
that the minimal point is unique and satisfies some conditions which show
that it is what we would intuitively expect.


Common sense inertia compared to inertia in physics

	The two principles resemble each other but are not identical.
The physics principle makes more precise predictions within its domain
of applicability.  Various authors have made much of certain errors
people make, and these errors can be regarded as supposing that
inertia in physics is merely an example of common sense inertia.

	The simplest example occurs when a subject is asked to predict
what will happen to a ball constrained to roll along a circular arc
when it comes to the end of the arc.  Physics tells us that it will
continue from the end of the arc in a straight line in the direction
it was going at the end of the arc.  A naive application of common
sense inertia tells us that it will continue doing what it was doing
before.  If what it was doing is characterized as moving in a circle,
then one may predict that it will continue to move in a circle -
contrary to fact.

	Notice that the common sense inertial prediction is language
dependent.  We can get the physics prediction from the common sense
law simply by characterizing what the ball was doing when the constraint
was removed as moving in a certain direction with a certain speed.
It may also be remarked that when a driver turns the wheel to a certain
position and holds it the car will continue in a circle.