Let be the set of letters. For a language over , we define as follows.
Here, denotes the length of the string . For example, if , then .
Answer the following questions.
(1) Let . Express using a regular expression.
(2) Let . Give a context-free grammar that generates .
(3) Let be a deterministic finite automaton, and let be the language accepted by . Here, , , , , and are the set of states, the transition function, the initial state, and the set of final states of , respectively. You may assume that the transition function is total. Give a finite automaton that accepts , with a brief explanation.
(4) If the proposition given below is true, then give how to construct a context-free grammar that generates , from a context-free grammar that generates (you may use push-down automata instead of context-free grammars), with a brief explanation. Otherwise, give a counterexample, with a brief explanation.
Proposition: "For every context-free language , is a context-free language."
Let . The language consists of strings of even length formed by repeating the substring "ab".
Solution:
To find , consider what does. For each string , there exists a string of the same length such that .
Given that consists of strings of the form , the possible strings must have lengths that can be paired with some such that their concatenation results in a string from .
The key observation is that can be any prefix of strings from . This gives us the possible regular expression:
This is because for any string formed from letters and , there is a corresponding such that as long as the length condition is satisfied.
Let . This language consists of strings where the numbers of s and s are matched in pairs.
Solution:
To generate , consider the strings that can form the beginning of valid strings in . Specifically, includes any prefix of that can be completed to a string in . Therefore, includes strings of the form where , , , and .
A context-free grammar generating can be defined as follows:
Variables:
Alphabet:
Rules:
Start Symbol:
This grammar generates strings in the form of prefixes of the strings in .
Proposition: "For every context-free language , is a context-free language."
Solution:
The proposition is false. A counterexample can be constructed as follows:
Consider the context-free language , which is known to be context-free. However, would include strings like where do not necessarily follow the strict condition . The language can be shown to be non-context-free because it requires balancing different parts of the string in a way that a context-free grammar cannot generally handle.
This counterexample demonstrates that is not necessarily context-free even if is.