13th International Conference on Membrane Computing - MTA Sztaki

S. Verlan, J. Quiros
For a regular grammar G the system (1) becomes linear. By considering a
finite automaton A = (V, Q, q 0 , Q f , δ) equivalent to G we obtain that system (1)
corresponds to the following system (recall that x is considered as a constant)
Q = xMQ + F (2)
where
– Q = [q 1 . . . q n ] t , q i ∈ Q, 1 ≤ i ≤ n is the vector containing all states,
– F = [a 0 . . . a n ] t , is the final state characteristic vector, i.e., a i = 1 if q i is a
final state and 0 otherwise.
– M is the transfer matrix of the automaton A, i.e., the incidence matrix of
the graph represented by A with negative values replaced by zero.
We remark that in the case of a regular language it is also possible to count
the number of words of length n by summing the columns corresponding to
the final states of the n-th power of the transfer matrix of the corresponding
automaton:
f i (n) = ∑
q j∈Q f
(M n ) i,j
It is known that the generating series f for a regular language is rational.
That implies that there exists a finite recurrence f(n) = ∑ k
j=1 a jf(n−j), k > 0,
a j ∈ Z which holds for large n.
Example 1. Consider the regular language L I recognized by the following automaton
q 0
0
8 888888
q 2
❆ ❆❆❆❆❆❆❆
1
8 1
0
q 1
0
q 3
1
1
q 4
Then the final state characteristic vector F of this automaton is defined by
F = [0, 1, 0, 1, 0] t and the transfer matrix M by
⎛ ⎞
0 1 1 0 0
0 0 0 1 0
M =
⎜0 1 0 0 0
⎟
⎝0 1 0 0 1⎠
0 1 0 0 0
436