00100	MINIMAL MODELS (research topic)
00200	
00300	
00400		The idea is to provide a method for AI programs to jump
00500	to conclusions that what they know about is all there is.  It
00600	seems that this form of reasoning is common with humans and
00700	while risky, it often leads to the right answer.
00800	
00900		The example is the %2missionary and cannibals%1 problem.
01000	When we hear the statement of the problem we conclude that these
01100	are all the relevant facts and therefore there is no bridge across
01200	the river and the boat is normal (i.e. is usable = has nothing
01300	wrong with it).
01400	
01500		We can try to introduce a corresponding notion in the
01600	predicate calculus by introducing the notion of a %2minimal
01700	model%1 of a set ⊗S of sentences.
01800	
01900		Let ⊗A and ⊗A' be two sets with %2A ⊂ A'%1, and let
02000	⊗M and ⊗M' be two models of a set ⊗S of sentences with domains
02100	⊗A and ⊗A' respectively and such that for every predicate
02200	letter ⊗P and every %2a, b, ... , c ε A%1, we have
02300	
02400		%2true("P(a,b,...,c)",M') ≡ true("P(a,b,...,c)",M)%1.
02500	
02600	We then call⊗M' an extension of ⊗M. 
02650	⊗M is called a minimal model if it is not a proper extension of another.
02700	(For the notion of minimal model to work properly, it may be
02800	necessary to allow null domains in predicate calculus).
02900	
03000		A sentence ⊗p is minimally entailed by a set ⊗S of sentences,
03100	if ⊗p is true in all minimal models of ⊗S. 
03200	
03300		Any sentence that is entailed in the ordinary sense is
03400	minimally entailed, but the converse is not true.  In some
03500	sense, the minimal consequences of a set of sentences are
03600	those obtained by assuming that no objects exist other than
03700	those that can be shown to exist.
03800	
03900		If we take Peano's axioms leaving out induction, the
04000	minimal models are just the standard model, and the minimal
04100	consequences are the sentences true in the standard model.
04200	The AI applications will be less fancy, i.e. we will merely
04300	use minimal models to jump to the conclusion that there is
04400	no bridge.
04500	
04600		It is not clear to me that minimal models are the right
04700	way to jump to this kind oπ conclusion.  The first step in
04800	proceeding further with the idea is to construct some examples.