If we have an interactive system (quadruple):

   (S,Phi) : (Sigma S : Set)S -> Fam(Fam S),

where 

   Phi = \s->(C(s),\c->(R(s,c),\r->s[c/r])), 

then we define two relations a <= s and s < a between states s : S and
countable ordinals a:

          0 <= s   ==   N1
      (a+1) <= s   ==   (Sigma c : C(s), Pi r : R(c,s)) a <= s[c/r] 
(sup_n a_n) <= s   ==   (Pi n : Nat) a_n <= s

           0 > s   ==   N0
       (a+1) > s   ==   (Pi c : C(s), Sigma r : R(c,s)) a > s[c/r] 
 (sup_n a_n) > s   ==   (Sigma n : Nat) a_n > s

Intended meaning : 
  a <= s  :  the `interactive potential' of s can be at least a
  s < a   :  the `interactive potential' of s can be bounded by a
(If `interactive potential' means anything .. !) 

***************

Perhaps there is some (pseudo-)quantifiable notion of `potential
difference' between states?  (The work done in getting from one to the
other.)  If one thinks of the `work' as simulating a (high-level)
state sa using a (low-level) state sc, then it seems not unreasonable
that

  Sim(sa,sc) = ( Pi c : C(sa), Sigma p : C^*(sc), 
                 Pi t : R^*(sc,p), Sigma r : R(sa,c) )Sim(sa[c/r],sc[p/t]) 

where Sim(sa,sc) is the set of simulations of sa using sc, and `^*'
refers to the closure of the interactive system. (So p : C^*(sc) is a
`tree with leaves', t : R^*(sc,p) is a path to a leaf, and sc[t/p] is
the leaf state.)

Such a simulation has the form: 

  \c->(p(c),\t->(r(c,t),m(c,t))) 
  where m(c,t) : Sim(sa[c/r(c,t)],sc[p(c)/t]) 

There is a natural reflexive and transitive order between
simulations, which seems to capture the idea that one 
simulation is at least as "efficient" as another: 

  (p,r,m) <= (p',r',m')  
  ==  (for all c : C(sa)) 
         the tree p'(c) is a prefix of the tree p(c), and
         (for all paths t : R^*(sc,p(c))) 
             m(c,t) <= m'(c,t'), where t' is the prefix of t in p'(c)

(The formal definition is quite long.) 

Perhaps there is some way to `bound the complexity' of a simulation
(above and below) by an ordinal?