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Just did a test about the subject that had the following question:

I know it seems trivial and my first reaction was "well of course its true" but the epslilon kinda threw me off.

$L_2$={ab,$\epsilon$} $L_1$={a} , is there a computable reduction from $L_1\leq L_2$ ,: True Or False

I assumed by contradiction that its true and did the following :

My idea was that if we always reject epsilon , then the starting state would always reject on the empty string , therefor no reduction would exists.

, if $\epsilon\notin L_1$ then the computable function f must uphold: $f(\epsilon)\notin L_2\iff \epsilon\notin L_1$ Therefor the turning machine will always stop on the empty string , so she'll also stop on $\epsilon\cdot a=a$ therefor $f(\epsilon a)=f(a)\notin L_1$ in contradiction to the assumption

Now the answer was true apparently but I can't really understand why , I'm also not 100% sold on my solution , but since I managed to prove it I just went with it , so I assume that I have some mistake in the part of $f(\epsilon a)=f(a)$.

Thanks,

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  • $\begingroup$ $\varepsilon$ is not in $L_1$, so your "if and only if" simplifies to $f(\varepsilon) \not\in L_2$. I'm not sure what the argument about the Turing machine stopping is. The reduction is a computable total function $f$, therefore any Turing machine computing $f$ will halt for all possible input strings $x$ regardless of whether $x \in L_1$ or $x \not\in L_1$. $\endgroup$
    – Steven
    Jul 12 at 17:49
  • $\begingroup$ Thanks , I thought that since she isn't accepting $f(\epsilon)$ then she must always reject any input which I'm still not sure which its true or not , but the answer below gave a reduction which finds a reduction and overcomes this problem. $\endgroup$ Jul 12 at 17:59
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Here a possible function $f$ that provides the reduction from $L_1$ to $L_2$.

$$ f(x) = \begin{cases} \varepsilon & \mbox{if $x=a$} \\ b & \mbox{otherwise} \end{cases}. $$

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