Algebraic Foundation For Formal Language Theory IV

From: Mark Hopkins (whopkins_at_alpha2.csd.uwm.edu)
Date: 05/20/04 

Date: 20 May 2004 17:51:04 GMT

Part IV: Algebraic Chomsky-Schuetzenberger Theorem
         2nd version. "Vacuum expectation values".

>From May 19:
 
There one also one other note I forgot to add. In the context I
described of Maxwell-Boltzmann Fock spaces, the identity Dir_2:
             |0><0| + ... + |n-1><n-1| = 1
in fact, does NOT hold. In general, given the relations
                  Dir_1: <i||j> = delta^i_j
the most you can say is that |0><0| + ... + |n-1><n-1| = P is a projection
operator. In fact, in the Maxwell-Boltzmann Fock space example it will
be P = 1 - W, where W is projects the space H* onto the "0-particle"
subspace (i.e., W is the "vacuum state" projection). So, one has:
             |0><0| + ... + |n-1><n-1| + W = 1
with
             W^2 = W; <i| W = 0 = W |i>.
This also holds for other two Fock space examples I cited.
 
So, the general theorem may, in fact, only need to use P_2 instead of C_2
and there may, in fact, not be a real need for the relation set Dir_2.
 
In the context of PDA's the above relation set is actually the more
natural one to consider with the interpretations:
            <i| = push stack symbol i
            |i> = test for stack symbol i & pop
            W = test for empty stack.
 
Then, in place of the original theorem, one should also be able to state
and prove a 2nd version:
            C(M) W = W (P_n x R(M)) W, for all n > 1;
with a similar generalization for all Kleene algebras:
            U(C(K)) = W (P_n x K) W, for all n > 1.
 
The operation W e W in analogous physical contexts is:
                      W e W = <e> W
where
                      <e> = "vacuum expectation value" of e.