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Suppose o is a map on strings to strings. For every language R, we let $o(R) := \{o(x) : x \in R\}$. If o(R) is a regular language for every regular language R, then prove or disprove that the language $\{xc o(x) :x \in A\}$ is context-free, where A is a regular language and c is a symbol not in the alphabet of A, $\Sigma$.

The claim seems to hold when o is the reverse of a string, or the bitwise negation of it. I think a useful idea is to modify the rules for the CFG for A that are of the form $X\rightarrow aY$ for some alphabet element a and variables X and Y. So it might be useful to define Q variables in the new context free grammar, one for each state in a DFA recognizing A.

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    $\begingroup$ Sounds good. Did you try it? Maybe try using the DFA for $R^R$ on the right-hand side $\endgroup$
    – rici
    Jun 17 at 14:34
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    $\begingroup$ Please indicate where you encountered this task and credit the source. We require you to credit the source of all copied material (as I have explained before): cs.stackexchange.com/help/referencing. $\endgroup$
    – D.W.
    Jun 17 at 19:32

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Your operation is very general, and in its generality your conjecture is not true.

For instance take for $o$ the identity, then we get the simple example $o(\{a,b\}^*) = \{wcw\mid w\in \{a,b\}^*\}$ which is not context-free.

Your examples show that a better conjecture would be that instead $\{ x c\, o(x^R)\mid x\in A\}$ is context-free, where the second copy $x$ is reversed. In that case for the identity, $\{ wcw^R \mid \dots\}$ is nicely context-free for regular $A$.

But your statement is too general, as in that case we can take $o$ itself to be the mirror image of a string, and we are in trouble again.

The following general statement holds: if $o$ is performed by a finite state transducer, a finite state automaton with output, then $\{ x c\, o(x^R)\mid x\in A\}$ is context-free, for regular $A$.

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