Write a context-free grammar for the following language. Start with an English statement of the language, and give five good strings.
x1#x2# .. #xk, k ≥ 1, for some i and j: xi = xj^R, Σ = {a, b, #}
(notes: xi’s contain a’s and b’s only,
#’s appear only between xi substrings)
The given language consists of strings that are formed by concatenating substrings xi, separated by the symbol "#". Each substring xi is a mirror image of another substring xj^R, where ^R denotes the reverse of xj. The alphabet Σ consists of the symbols "a", "b", and "#".
Here is the context-free grammar for the given language:
S -> X
X -> X#X | A
A -> B | C
B -> aB | bB | ε
C -> aC | bC | a | b
Here are five good strings in the language:
1. a#b#ba
2. a#ab#b#baa
3. aa#ba#b#abaa
4. b#a#ab#b#ab#a#
5. aba#b#ba#
Note: ε represents the empty string.
The language can be described as follows in English: It consists of strings in the form of x1 # x2 # ... # xk, where k is greater than or equal to 1. Each xi is a string formed by a sequence of 'a' and 'b' characters only. Additionally, there must exist an i and a j (i ≠ j) such that xi is the reverse of xj.
Now, let's create a context-free grammar (CFG) to generate this language:
1. S -> X '#' | X '#' S
2. X -> A | B
3. A -> 'a' A | 'a'
4. B -> 'b' B | 'b'
Explanation:
- Rule 1: This rule represents the overall structure of the language. It states that a string can either start with an X followed by '#', or an X followed by '#' and then followed by another part that adheres to the S rule.
- Rule 2: This rule introduces the non-terminal X, which represents the xi substrings. It can be either an 'A' or a 'B'.
- Rule 3: This rule generates the 'A' non-terminal, which represents a sequence of 'a' characters. It can either be a single 'a' or 'a' followed by another 'A'.
- Rule 4: This rule generates the 'B' non-terminal, which represents a sequence of 'b' characters. It can either be a single 'b' or 'b' followed by another 'B'.
Here are five good strings that can be generated by this grammar:
1. "a#b#a"
2. "a#ba#b#ba"
3. "b#aa#b"
4. "aa#aaa#aaaa"
5. "bb#b#"
To generate more strings, you can start with S and derive them using the given rules of the grammar.