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Consider the function $f$ that maps strings over $\{a, b, c\}$ to strings over $\{0, 1\}$ by replacing each $a$ by 0, each $b$ by 0, and each $c$ by 1. For example $f(cabbc) = 10001$. The function $f$ extends naturally to languages: if $L$ is a language over $\{a, b, c\}$, then $f(L)$ is the language over $\{0, 1\}$ defined as $\{f(w) \mid w\text{ is in }L\}$.

Prove that if a language $L$ is decidable, then so is $f(L)$.

I am thinking that since $L$ is decidable it must be regular so there exists a DFA for $L$. $f(L)$ is closed -- dk how to prove though. Then $f(L)$ is also regular. Therefore is also decidable.

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    $\begingroup$ What have you tried? $\endgroup$
    – Yamar69
    Oct 21 '19 at 18:40
  • $\begingroup$ I am thinking that since L is decidable it must be regular so there exists a dfa for L. f(L) is closed -- dk how to prove though. Then f(L) is also regular. therefore is also decidable $\endgroup$ Oct 21 '19 at 21:39
  • $\begingroup$ Not all decidable languages are regular. For example, the language $\{a^nb^n \mid n \geq 0 \}$ is decidable but not regular. $\endgroup$
    – siracusa
    Oct 22 '19 at 4:21
  • $\begingroup$ can you please explain reason why do you think that $L$ is decidable implies $L$ is regular. Or are you talking about some specific $L$ like $0^+$ or something like that? $\endgroup$ Oct 22 '19 at 12:31
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Let $f$ be any computable mapping such that $|f(x)| \geq c|x|$ for some $c > 0$. For every language $L$, if $L$ is computable then so is $f(L)$.

To see this, let us be given an input $y$. We go over all strings $x$ of length at most $|y|/c$. For each of them, we check whether $x \in L$ and $f(x) = y$. If we found such a string $x$, then $y$ is in the language, otherwise it isn't.

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Ok , if L is decidable , then some TM machine decides L

Let M be a TM that decides L

Now we construct a TM N that decides f(L) :

N on input w : "

Generate all possible mapped (original) strings x from w by :

  1. Replacing all 1s in w by c

  2. Replacing all 0s in w by all possible combinations of as & bs

Run M on every x if M accepts any x accept , else reject "

So for ex , w =100 , N generates all possible mapped strings , caa , cab , cba , cbb , if M accepts any of these , then there is a string x , such that f(x) = w and x is in L , if none of them is accepted then we reject

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