Question

Find context-free grammars for the following languages (with n ≥0, m20).L (a"b":n≤m+3).

Answer 1

The context-free grammar for the language L consisting of strings where the number of 'a's is less than or equal to the number of 'b's plus 3 is described by the production rules S → aSb | T and T → aT | ε.

Step-by-step explanation:

The question asks to find a context-free grammar (CFG) for the language L defined as {anbm : n ≤ m + 3} where n and m are non-negative integers (with n ≥ 0, m ≥ 0). This language includes strings where the number of 'a's is less than or equal to the number of 'b's plus 3.

A CFG for the language L can be:

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• S → aSb | T
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• T → aT | ε

This CFG operates under the rules where:

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• S generates an equal number of 'a's and 'b's, starting with an 'a' and ending with a 'b' to ensure n ≤ m.
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• T allows for up to three additional 'a's, representing the up to 'm + 3' part of the language definition.
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• ε represents the empty string allowing for the case when n = 0.