Show that language over unary alphabet is context free iff is regular.

In summary, a language over a unary alphabet is context-free if and only if it is regular. This can be proven by showing that a context-free grammar can be used to generate the language, and a finite automaton can be used to recognize the language.
  • #1
smith92
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Let L be language over unary alphabet {0}. Show that L is context free iff L is regular.
Could someone give me a hint how to solve this problem?

I solved similar problem: Show that L* is regular, but I still don't have idea how to solve previous.

Second problem could be solved in this way (e - empty word):

If L is empty or has only empty word, then L* has only empty word, so it is regular.
If L is not empty then exists smallest k that 0^k is in L.

We can define L = { w in L* | w mod k = i } and show that each L is regular language. L* is sum of L[0] ... L[k-1], so it is finite sum of regular language - it is regular language.
 
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  • #2


For the first problem, we can use the fact that a language is context-free if and only if it can be generated by a context-free grammar. A context-free grammar is a set of production rules that can be applied to a starting symbol to generate strings in the language.

For a unary alphabet {0}, we can define a context-free grammar as follows:

S -> 0S | e

Where S is the starting symbol and e is the empty word. This grammar generates all strings in L by repeatedly adding a 0 to the beginning of the string or by generating the empty word.

Now, to show that L is regular if and only if it is context-free, we can use the fact that a language is regular if and only if it can be recognized by a finite automaton. A finite automaton is a machine with a finite number of states that reads input symbols and transitions between states based on those symbols.

For a unary alphabet {0}, we can define a finite automaton as follows:

Q = {q0, q1} - states
Σ = {0} - input alphabet
q0 - starting state
q1 - accepting state
δ(q0, 0) = q1 - transition from q0 to q1 on input 0
δ(q1, 0) = q1 - transition from q1 to q1 on input 0

This finite automaton recognizes all strings in L by starting in the state q0 and transitioning to q1 after reading a 0. The automaton stays in q1 for any additional 0s in the string, and accepts the string if it ends in q1.

Therefore, we have shown that L is context-free if and only if it is regular.
 

Related to Show that language over unary alphabet is context free iff is regular.

1. What is a unary alphabet?

A unary alphabet is a finite set of symbols or characters that contains only one element. This means that all words or strings in the language are composed of the same symbol or character repeated multiple times.

2. What does it mean for a language to be context-free?

A context-free language is a type of formal language that can be generated by a context-free grammar, which consists of a set of rules for combining symbols to form strings. These languages are recognized by a non-deterministic pushdown automaton, which is a type of theoretical machine used in computer science.

3. How does a language over a unary alphabet differ from a language over other alphabets?

Languages over unary alphabets are simpler than languages over other alphabets because they have only one symbol. This means that there are no rules for combining different symbols, and all words in the language are simply repetitions of the same symbol.

4. What is the relationship between regular and context-free languages over a unary alphabet?

A language over a unary alphabet is regular if and only if it is context-free. This means that a language over a unary alphabet can be recognized by a finite state automaton (regular language) if and only if it can be generated by a context-free grammar (context-free language).

5. What is the significance of proving that a language over a unary alphabet is context-free if and only if it is regular?

This proof demonstrates the close relationship between regular and context-free languages. It also shows that languages over unary alphabets are simpler and more limited than languages over other alphabets, as they can be recognized by simpler theoretical machines. This proof is also useful in the study of formal languages and their applications in computer science.

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