问题
I've been reading: "Tips for creating Context free grammar" post for learning purposes and I nearly understand the concept, but I don't quite understand the following.
If we have:
L = {am bn | m >= n}.
I understand this:
S --> B
B --> aBb
A --> aA
But what I don't understand is the concept of adding to the end of these particular values, such as:
S --> B | ^
B --> aBb | A
A --> aA | a
Why do we add ^ (null), A, and a to the end of these lines? What do they do and why do we need them?
All help is greatly appreciated.
回答1:
You need them to be able to construct the strings that are in the language L.
B --> aBb | A means that if you have a nonterminal B it can be replaced by either aBb or A. (Capital letters represent nonterminals and the lower case letters represent terminals).
Let's take a look at the grammar:
S --> B | ^
B --> aBb | A
A --> aA | a
Why do we have a | ^? You need it to be able to generate the empty string. The empty string is clearly part of the language L since it contain an equal amount of as and bs.
Why do we have | A? To be able to use the rule for A. Now you can replace B with A so you can insert either aA or a. You need this rule to be able to generate strings that have more as than bs.
Why do we have | a? To be able to replace A with a without adding a new nonterminal that need to be replaced.
When I look at this grammar I would say that you need to change A --> aA | a to A --> aA | a | ^ to be able to generate strings with an equal amount of as and bs. (So you can replace A with the empty string (or null) instead of having to add an extra a)
Let's say you want to generate the string aaabb:
S //You start with S
B //Rule: S --> B
aBb //Rule: B --> aBb
aaBbb //Rule: B --> aBb
aaAbb //Rule: B --> A
aaabb //Rule: A --> a
回答2:
The need for the extra ^,A and a is to complete the grammar set.
Thus A --> aA takes care of [aa, aaa, aaaa, ...]
A --> aA | a takes care of [a, aa, aaa, aaaa, ...]
B --> aBb takes care of [ab, aabb, aaabbb, ...]
So, B --> aBb | A should take care of [a, ab, aa, aab, aaa, aaab, aabb, ...]
The only one missing is ^
So now complete the grammar with S --> B | ^ [^, a, aa, ab, aaa, aab, aaaa, aaab, aabb, ...]
As you can see, it is not possible to generate the 2nd half from the first half, but it is also a subset of the grammar. So, in order to complete the grammar, that 2nd half is crucial as well. Look at the wikipedia article for clearer examples.
http://en.wikipedia.org/wiki/Context-free_grammar
来源：https://stackoverflow.com/questions/15983477/why-s-and-a-a-in-context-free-grammars