Let C { x#y| x,y ∈ {0,1}* and x ≠ y} show that C is a context free Language...
To prove that C is a context-free language, we need to construct a context-free grammar (CFG) or a pushdown automaton (PDA) that generates or accepts the language C. I will provide a step-by-step explanation of how to construct a CFG for C:
Step 1: Define the variables:
- Start variable: S
- Non-terminal variables: X, Y, Z, W, and V
Step 2: Define the terminal symbols:
- Terminal symbols: 0 and 1
Step 3: Define the production rules:
- S -> XY
- X -> 0X1 | 1X0 | λ (lambda)
- Y -> 0Y1 | 1Y0 | λ
- Z -> 0Z0 | 1Z1 | W
- W -> 0W0 | 1W1 | XZ | λ
- V -> XZ | λ
Explanation of the production rules:
- The start variable S generates a string with an X followed by a Y.
- X generates strings that begin with a 0 or 1, followed by more X's and alternating between 0 and 1.
- Y generates strings that begin with a 0 or 1, followed by more Y's and alternating between 0 and 1.
- Z generates strings that alternate between 0 and 1 and must have at least one W in between.
- W generates strings that alternate between 0 and 1 and can have XZ or be empty (lambda).
- V generates strings that must have at least one XZ or be empty (lambda).
Step 4: Define the language:
- C is generated by the CFG if and only if it can be generated by S.
By following these steps, we have constructed a context-free grammar that generates the language C. Therefore, we have shown that C is a context-free language.
To show that a language is context-free, we need to demonstrate that it can be generated by a context-free grammar, or recognized by a pushdown automaton. In this case, we will construct a context-free grammar (CFG) for the language C.
Let's start by analyzing the language C. The language C contains all strings of the form "xy" where x and y are strings of 0s and 1s, and x is not equal to y. This condition implies that there must be at least one position in the string where the corresponding characters in x and y are different.
To construct a CFG for C, we need to define non-terminal symbols, production rules, and a start symbol. Let's denote the start symbol as S.
1. Define non-terminal symbols:
- S: This will be the start symbol of the grammar.
- X: This non-terminal will represent the common prefix of x and y.
- Y: This non-terminal will represent the point of difference between x and y.
- Z: This non-terminal will represent the common suffix of x and y.
2. Define production rules:
- S -> XY
- X -> 0X | 1X | ε (represents the common prefix)
- Y -> 0Z1 | 1Z0 (represents the point of difference)
- Z -> 0Z | 1Z | ε (represents the common suffix)
3. Define the start symbol:
- Our start symbol will be S.
This CFG can generate all strings in the language C. The derivation process follows these rules:
- S derives XY
- X derives 0X, 1X, or ε (the common prefix)
- Y derives 0Z1 or 1Z0 (the point of difference)
- Z derives 0Z, 1Z, or ε (the common suffix)
By applying these production rules, we can generate strings of the form "xy" where x and y belong to {0,1}* and x is not equal to y.
Hence, we have successfully constructed a CFG for the language C, demonstrating that C is a context-free language.