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We define an operation three on strings as three(c1c2c3c4c5c6...) = c3c6... then the above-described definition is extended to languages. Prove that the class of regular languages is closed under this operation.

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    $\begingroup$ If $R$ is regular, there is a DFA which recognises it. Given a DFA $D$, how can you make a finite automaton which recognises $\mathit{three}(L(D))$? ($D$ doesn't really have to be deterministic. But it might help your thinking.) $\endgroup$
    – rici
    Oct 24 '21 at 3:37
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Let $Q$ be a $DFA$ recognizing $L$. To show that $\mathrm{three}(L)$ is regular, you only need to construct a $NFA$ deciding $\mathrm{three}(L)$ (this suffices, since any $NFA$ can be transformed to an equivalent $DFA$). Now, the key idea is that if the $NFA$ is reading, say, the first character $c_3$, then it can guess what two characters $c_1,c_2$ occurred before it.

More formally, suppose that $Q = (\Sigma,S,s_0,\delta,F)$. Given a string $\overline{x} \in \Sigma^*$ and states $s,s' \in S$ we will use $s,\overline{x} \to s'$ to denote the fact that $Q$, starting from state $s$ while reading the first character of $\overline{x}$, ends up in state $s'$ after reading the entire string $\overline{x}$. Consider now the following non-deterministic finite automata $Q' = (\Sigma,S\cup \{s_0'\},s_0',\delta',F)$, where $s_0' \notin S$ and the transition function $\delta$ is defined as follows:

  • For every $s\in S$ and $z\in \Sigma$ we define $$\delta(s,z) := \{s' \in S \mid \exists x,y \in \Sigma : s,xyz \to s'\}$$
  • For every $z\in \Sigma$ we define $$\delta(s_0',z) := \{s'\in S \mid \exists x,y \in \Sigma:s_0,xyz \to s'\}$$

It is straightforward to verify that $L(Q') = \mathrm{three}(L)$.

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You might use closure properties of the regular languages: (inverse) homomorphisms and intersection (by regular languages).

Assuming the language is over the alphabet $\Sigma$, make two copies $\Sigma_i = \{ \sigma_i\mid \sigma\in\Sigma \}$ of that alphabet, $i=0,1$.

The homomorfism $h$ maps elements from $\Sigma_0\cup\Sigma_1$ onto their original $h:\sigma_i\mapsto\sigma$. The inverse morphism $h^{-1}: \Sigma^* \to (\Sigma_0\cup\Sigma_1)^*$ nondeterministically chooses a $0$ or $1$ copy for each symbol.

Now let $g$ be the homomorfism on $\Sigma_0\cup\Sigma_1$ that deletes symbols from $\Sigma_0$ and maps symbols from $\Sigma_1$ onto their original.

Now the following composition keeps every third symbol, while deleting the other symbols $w\mapsto g(\;h^{-1}(w) \cap (\Sigma_0\Sigma_0\Sigma_1)^*(\Sigma_0^0\cup\Sigma_0^1\cup\Sigma_0^2)\;)$.

This method has the advantage that it also works for context-free languages.

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