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If a language L is regular then the given operation sort(L) is context free
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Problem
I was asked in one of my homework tasks this next question and I'm not quite sure how to handle it:
Sort(w) is an operation which takes a string (w) from a given language and sorts the characters in an ascending order. Sort(L) is the outcome language of the operation Sort(w) on each and every string (w) in the language L.
I was asked to prove or disprove the following claim:
If L is a regular language over the alphabet {0,1} where 0<1 then sort(L) is context free.
I would appreciate any hints or clues on how to approach this type of question.
Sort(w) is an operation which takes a string (w) from a given language and sorts the characters in an ascending order. Sort(L) is the outcome language of the operation Sort(w) on each and every string (w) in the language L.
I was asked to prove or disprove the following claim:
If L is a regular language over the alphabet {0,1} where 0<1 then sort(L) is context free.
I would appreciate any hints or clues on how to approach this type of question.
Solution
The way to solve this sort of question is using your problem-solving skills. Here are three ways to approach this question:
-
Take some DFA (or NFA) for $L$. Now construct a PDA for $\mathsf{sort}(L)$ along the following lines. The PDA starts by reading $0$s, pushing them into the stack. At some point it switches to a second phase in which it only accepts $1$s. At each step it first pops an arbitrary number of $0$s from the stack and feeds them to the DFA, and then reads the next $1$ if any, feeding it to the DFA. It accepts if after processing all the $0$s in the stack and all the $1$s in the input, the DFA is in an accepting state.
-
Take a left regular grammar for $L$ (so the rules are of the form $A\to\epsilon$, $A\to a$, and $A\to a B$), and replace $A\to 1B$ with $A\to B1$. The result is a context-free grammar for $\mathsf{sort}(L)$.
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Parikh's theorem states that $\mathsf{sort}(L)$ is of the form
$$
\bigcup_{i=0}^m \{ 0^{x_i+n_1 y_{i,1}+\cdots+n_{\ell_i} y_{i,\ell_i}} 1^{z_i + n_1 w_{i,1} + \cdots + n_{\ell_i} w_{i,\ell_i}} : n_1,\ldots,n_{\ell_i} \geq 0 \},
$$
where $x_i,y_{i,j},z_i,w_{i,j} \geq 0$. It's not hard to check that all languages of this form are context-free, for example by constructing a context-free grammar.
Note how all of these approaches rely on the alphabet being binary. The language $(012)^$ shows that this assumption is necessary. (Similarly, the language $(01)^$ shows that $\mathsf{sort}(L)$ need not be regular.)
Parikh's theorem implies that for languages over the binary alphabet, $\mathsf{sort}(L)$ is context-free whenever $L$ is context-free (that is, we don't need to assume that $L$ is regular, it suffices to assume that it is context-free). You can try proving this fact directly.
-
Take some DFA (or NFA) for $L$. Now construct a PDA for $\mathsf{sort}(L)$ along the following lines. The PDA starts by reading $0$s, pushing them into the stack. At some point it switches to a second phase in which it only accepts $1$s. At each step it first pops an arbitrary number of $0$s from the stack and feeds them to the DFA, and then reads the next $1$ if any, feeding it to the DFA. It accepts if after processing all the $0$s in the stack and all the $1$s in the input, the DFA is in an accepting state.
-
Take a left regular grammar for $L$ (so the rules are of the form $A\to\epsilon$, $A\to a$, and $A\to a B$), and replace $A\to 1B$ with $A\to B1$. The result is a context-free grammar for $\mathsf{sort}(L)$.
-
Parikh's theorem states that $\mathsf{sort}(L)$ is of the form
$$
\bigcup_{i=0}^m \{ 0^{x_i+n_1 y_{i,1}+\cdots+n_{\ell_i} y_{i,\ell_i}} 1^{z_i + n_1 w_{i,1} + \cdots + n_{\ell_i} w_{i,\ell_i}} : n_1,\ldots,n_{\ell_i} \geq 0 \},
$$
where $x_i,y_{i,j},z_i,w_{i,j} \geq 0$. It's not hard to check that all languages of this form are context-free, for example by constructing a context-free grammar.
Note how all of these approaches rely on the alphabet being binary. The language $(012)^$ shows that this assumption is necessary. (Similarly, the language $(01)^$ shows that $\mathsf{sort}(L)$ need not be regular.)
Parikh's theorem implies that for languages over the binary alphabet, $\mathsf{sort}(L)$ is context-free whenever $L$ is context-free (that is, we don't need to assume that $L$ is regular, it suffices to assume that it is context-free). You can try proving this fact directly.
Context
StackExchange Computer Science Q#75627, answer score: 4
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