Hi Mitch:

we are in agreement. You basically consider one s-t path in the strategy
and we do that too: we consider all s-t paths one by one and each such path
is of your form [to t1 to t2 to t3 ...].

I like your definition of POSS because it is a formalization of
the notion of "while there is still a possibility of success".
And POSS is formulated in terms of strategy paths and objects,
and not referring to a PathSet concept for class graphs.

I adapt the definition slightly:

POSS(c1, t, o1) = those edges e outgoing from o1
s.t. there is an object graph O (consistent
with the class graph C), containing the object o1 of class c1, 
an object o2 of type
(not class) t, and a path in O from o1 to o2 such that the first edge
in the path is e.

The goal of POSS is to offer a simpler semantics. You also offer 
an algorithm to compute POSS doing transitive closures. I think
the traversal graph construction we already have is more efficient
because it deals with all s-t paths in one sweep. There could
be exponentially many s-t paths but the traversal graph construction
is still polynomial.

Mitch writes:

>The object graph slice starting with o1 is the slice built by following
>the edges POSS(class(o),t) starting with o = o1 and continuing until
>every path terminates at an object of type t.

I would like to add to this:

>The object graph slice starting with o1 is the slice built by following
>the edges POSS(class(o),t) starting with o = o1 and continuing until
every path terminates (at an object of type t
or it terminates prematurely).

I think that we arrive at a much more straightforward definition 
of traversals that can be better understood by
programmers.

-- Karl