如果我们获得了语言L的语法，该如何找到L *的常规语法？

Question

Is there any general method to do so? For example, we have a general method to find grammar for L1 U L2 by adding a production S-> S1 | S2 where S1 and S2 are start symbols for grammars of L1 and L2 respectively. Thanks in advance..

Answer 1

In general, given a grammar G such that L(G) = L' , there is no algorithm which always produces a regular grammar G' such that L(G') = (L')* . For starters, (L')* may not be a regular language. Even if you allow the procedure to recognize this case and print "not a regular language" in such a case, this cannot be generally possible since it would allow us to determine whether arbitrary unrestricted grammars generate particular strings (the construction is not too hard but I won't provide it unless desired). This is an undecidable problem, so we can't recognize regular languages in unrestricted grammars.

Perhaps your question is whether there is a neat construction to do this if initially given a regular grammar. In that case, the answer is a definite and clear, "yes!" Here is one easily described (though possibly inefficient in practice) procedure for doing just that:

1. Convert the regular grammar into a nondeterministic finite automaton using the typical construction for doing so. There are easy constructions for left-regular and right-regular grammars.
2. Construct a regular expression from the nondeterministic finite automaton using any known construction. One such construction is typically used in proving equivalence.
3. Construct a new regular expression which is the Kleene closure of the one from the last step.
4. Construct a nondeterministic finite automaton from the regular expression from the last step, using a standard construction.
5. Construct a regular grammar from the nondeterministic finite automaton from the last step. There are known constructions for this.

Thus, we can mechanically go from regular grammar for L to regular grammar for L* .

If you just want ANY grammar for L* , the simplest would probably be to introduce a new start state S' and productions S' := S'S' | S S' := S'S' | S where S is the start symbol of your input grammar. This obviously does not give a regular grammar, however - if the input grammar generates a regular language, this one will do so as well.

Example: given the regular grammar

S := 0S | 1T
T := 0S | 1T | 1

A construction gives us this nondeterministic finite automaton:

q    s    q'
-    -    -
S    0    S
S    1    T
T    0    S
T    1    T
T    1    (H)

A construction gives us the regular expression:

(0*1)(0*1)*1

The Kleene closure of this is:

((0*1)(0*1)*1)*

We recognize from the standard construction that this automaton is equivalent:

q    s    q'
-    -    -
(I)  -    S
S    0    S
S    1    T
T    0    S
T    1    T
T    1    H
H    -    (I)

Whence the following regular grammar:

I := S | -
S := 0S | 1T
T := 0S | 1T | H
H := I