问题
How can I convert some regular language to its equivalent Context Free Grammar? Is it necessary to construct the DFA corresponding to that regular expression or is there some rule for such a conversion?
For example, consider the following regular expression
01+10(11)*
How can I describe the grammar corresponding to the above RE?
回答1:
Change A+B to grammar
G -> A G -> BChange A* to
G -> (empty) G -> A GChange AB to
G -> AB
and proceed recursively on A and B. Base cases are empty language (no productions) and a single symbol.
In your case
A -> 01
A -> 10B
B -> (empty)
B -> 11B
If the language is described by finite automaton:
- use states as nonterminal symbols
- use language as set of terminal symbols
- add a transition p -> aq for any transition p -> q on letter a in the original automaton
- use initial state as initial symbol in the grammar
回答2:
I guess you mean convert it to a formal grammar with rules of the form V->w, where V is a nonterminal and w is a string of terminals/nonterminals. To start, you can simply say (mixing CFG and regex syntax):
S -> 01+10(11)*
Where S is the start symbol. Now let's break it up a bit (and add whitespace for clarity):
S -> 0 A 1 0 B
A -> 1+
B -> (11)*
The key is to convert *es and +es to recursion. First, we'll convert the Kleene star to a plus by inserting an intermediate rule that accepts the empty string:
S -> 0 A 1 0 B
A -> 1+
B -> (empty)
B -> C
C -> (11)+
Finally, we'll convert + notation to recursion:
S -> 0 A 1 0 B
A -> 1
A -> A 1
B -> (empty)
B -> C
C -> 11
C -> C 11
To handle x?, simply split it into a rule producing empty and a rule producing x .
回答3:
Actually, different CFG grammars can produce the same language. So given a regular expression (regular language), its mapping back a CFG is not unique.
Definitely, you can construct a CFG that result in a given regular expression. The above answers shown some ways to achieve this.
Hope this gives you a high level idea.
来源：https://stackoverflow.com/questions/2639468/convert-regular-expression-to-cfg