MINIMAL MODELS (research topic)


	The idea is to provide a method for AI programs to jump
to conclusions that what they know about is all there is.  It
seems that this form of reasoning is common with humans and
while risky, it often leads to the right answer.

	The example is the %2missionary and cannibals%1 problem.
When we hear the statement of the problem we conclude that these
are all the relevant facts and therefore there is no bridge across
the river and the boat is normal (i.e. is usable = has nothing
wrong with it).

	We can try to introduce a corresponding notion in the
predicate calculus by introducing the notion of a %2minimal
model%1 of a set ⊗S of sentences.

	Let ⊗A and ⊗A' be two sets with %2A ⊂ A'%1, and let
⊗M and ⊗M' be two models of a set ⊗S of sentences with domains
⊗A and ⊗A' respectively and such that for every predicate
letter ⊗P and every %2a, b, ... , c ε A%1, we have

	%2true("P(a,b,...,c)",M') ≡ true("P(a,b,...,c)",M)%1.

We then call⊗M' an extension of ⊗M. 
⊗M is called a minimal model if it is not a proper extension of another.
(For the notion of minimal model to work properly, it may be
necessary to allow null domains in predicate calculus).

	A sentence ⊗p is minimally entailed by a set ⊗S of sentences,
if ⊗p is true in all minimal models of ⊗S. 

	Any sentence that is entailed in the ordinary sense is
minimally entailed, but the converse is not true.  In some
sense, the minimal consequences of a set of sentences are
those obtained by assuming that no objects exist other than
those that can be shown to exist.

	If we take Peano's axioms leaving out induction, the
minimal models are just the standard model, and the minimal
consequences are the sentences true in the standard model.
The AI applications will be less fancy, i.e. we will merely
use minimal models to jump to the conclusion that there is
no bridge.

	It is not clear to me that minimal models are the right
way to jump to this kind oπ conclusion.  The first step in
proceeding further with the idea is to construct some examples.