proposed definition of covers

Def: Covers

The RIF covers (a portion of) a rule formalism if it can simulate that
formalism (portion).  

More precisely:

Let L be a set of rule-sets, i.e., (a portion of) the syntax of a rule
formalism, F.  Let D be a set of data-sets, i.e., (a portion of) the syntax
of the input data for F.  Let C be a set of conclusion-sets, i.e., (a
portion of) the syntax of the conclusions of F.  [Note:  x-sets need not be
"sets", they could be sequences, etc., etc.]

Let E be the "entailment" relationship in F, i.e, running-rule set l on
data-set d under F results in E(l,d).

Let M be a mapping from L into RIF rule-sets, from D into RIF data-sets,
and from C into RIF conclusion-sets, such that "essentially" different F
conclusion-sets are mapped into different RIF conclusion-sets.  

[Yes, this is open to abuse, but I think that this is the best that can be
done, because there may be "inessential" features of F.  This also depends
on there being no "inessential" differences among RIF conclusion-sets, but
this could be fixed by using a notion of equivalence on RIF conclusion-sets
and pushing back the equivalence into the condition on M.  I should be able
to come up with a definition that takes into account all these quibbles, if
needed.]

Let the RIF entailment relationship be R.

RIF covers (the portion of) F precisely when

	for all l in L and d in D such that E(l,d) is in C
		R(M(l),M(d)) = M(E(l,d))

[To handle "inessential" differences in RIF conclusion sets, change
equality to a suitable version of equivalence, or even some weaker "similar
to" notion.]

peter

Received on Friday, 9 June 2006 12:23:06 UTC