13th International Conference on Membrane Computing - MTA Sztaki

A. Alhazov, R. Freund, H. Heikenwälder, M. Oswald, Yu. Rogozhin, S. Verlan
2.5 Grammars with Regular Control and Time-Varying Grammars
Another possibility to capture the idea of controlling the derivation in a grammar
as with a control graph is to consider the sequence of rules applied during a
computation and to require this sequence to be an element of a regular language:
A grammar with regular control and appearance checking is a construct
G C = (G, H C , L, F )
where G = (O, O T , w, P, =⇒ G ) is a grammar of type X and L is a regular
language over H C , where H C is the set of labels identifying the subsets of productions
from P in a one-to-one manner (H C is a bijective function on 2 P ), and
F ⊆ H C . The language generated by G C consists of all terminal objects z such
that there exist a string H C (P 1 ) · · · H C (P n ) ∈ L as well as objects w i ∈ O,
1 ≤ i ≤ n + 1, such that w = w 1 , z = w n+1 , and, for all 1 ≤ i ≤ n, either
– w i =⇒ G w i+1 by some production from P i or
– w i = w i+1 , no production from P i is applicable to w i , and H C (P i ) ∈ F .
It is rather easy to see that the model of grammars with regular control is
closely related with the model of graph-controlled grammars in the sense that
the control graph corresponds to the deterministic finite automaton accepting L.
Hence, we may also speak of a grammar with regular control and without appearance
checking if F = ∅, and if F = H C then G C is said to be a grammar with regular
control and unconditional transfer. The corresponding families of languages
are denoted by L (X-C (REG) ac
), L (X-C (REG)), and L (X-C (REG) ut
).
Obviously, the control languages can also be taken from another family of languages
Y , e.g., L (CF ), thus yielding the families L (X-C (Y ) ac
), etc., but in this
paper we shall restrict ourselves to the cases Y = REG and Y = REG 1∗ (k, p).
For Y = REG 1∗ (1, p), these grammars are also known as (periodically) timevarying
grammars, as a control language {H C (P 1 ) · · · H C (P p )} ∗ means that the
set of productions available at a time t in a derivation is P i if t = kp + i, k ≥ 0;
p is called the period of the time-varying system. The corresponding families
of languages generated by time-varying grammars with appearance checking,
without appearance checking, with unconditional transfer and with period p are
denoted by L (X-T V ac (p)), L (X-T V (p)), and L (X-T V ut (p)), respectively; if p
may be arbitrarily large, p is replaced by ∗ in these notions.
In many cases it is not necessary to insist that the control string
H C (P 1 ) · · · H C (P n ) of a derivation is in L, it usually also is sufficient that
H C (P 1 ) · · · H C (P n ) is a prefix of some string in L. We call this control weak
and replace C by wC and T V by wT V in the notions of the families of languages.
We should like to mention that in the case of wT V the control words
are just prefices of the ω-word (H C (P 1 ) · · · H C (P p )) ω .
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