Question

What is the CFG? L3 = {w ∈ {0, 1, #} ∗ | w = x # y, x, y ∈ {0, 1}∗ and |x| = |y|}

Note: # is an ordinary symbol in this alphabet

Answer 1

A student is asking about a Context-Free Grammar for a language with strings containing two equal-length halves separated by a '#'. The appropriate CFG ensures that every character added to one half has a corresponding character in the other half. An example CFG for this language is S → 0S0 | 1S1 | #.

Step-by-step explanation:

The student is asking about a Context-Free Grammar (CFG) for the language L3 which consists of strings w, where w can be divided into two halves x and y separated by a '#' symbol, and x and y contain any combination of the symbols '0' or '1' with the condition that the lengths of x and y are equal. To construct a CFG for L3, we need to ensure that for every '0' or '1' added to x, a corresponding '0' or '1' is added to y.

A possible CFG for L3 is as follows:

• S → 0S0 | 1S1 | #

This grammar states that the string starts and ends with the same character ('0' or '1'), and in between, it can recursively have the same structure or just the '#' symbol. This ensures that for every terminal symbol on the left part (x), there is a matching terminal symbol on the right part (y), both in character and count, therefore preserving the condition |x| = |y|.