I have found two interesting questions regarding the quotient of languages, described as:

$A/B=\{w \mid \exists z\in B\land wz\in A\}$

The first one is:

Let $A$ and $B$ be regular languages, prove that $A/B$ is decidable By using the proof from this other question, it can be proved that $A/B$ is regular if $A$ is regular too.

Then, since any regular language is decidable, $A/B$ will be regular (and decidable) too.

The second one is:

Let $A$ and $B$ be semi-decidable languages, prove that $A/B$ is semi-decidable


Let us keep this in mind: $w\in A/B \iff \exists z\in B: wz\in A$

Since we are working with semi-decidable machines, then we have no problem iterating over all $z$'s until we find a good one!

Therefore, the idea to solve it will be the following: Let $M_A$ be the TM for $A$, and let $M_B$ be the TM for $B$.

We want to create a new TM for $A/B$ like so:(when given $w$ as input)

  • Iterate through all $z\in \Sigma^*$
  • For each such $z$, if both $M_B(z)$ and $M_A(wz)$ accepted, then accept.

Notice, that there is still an important detail left! What happens if $M_B(z)$ never stops? We will never continue to search the other $z$'s!

To solve that, we will modify the TM as follows:

  • Let $T=\{\}$ be an empty set representing the set of all running "threads" we have created
  • For $z\in \Sigma^*$:
    • Create two new threads that will compute $M_B(z)$ and $M_A(wz)$ respectively. Add those two threads into $T$.
    • Do one step in every thread in $T$.
  • If at any point, there is some $z$ such that both threads $M_B(z)$ and $M_A(wz)$ have finished executing and both accepted, then accept as well.

This solution effectively runs the machine $M_B(z_0)$ (for a specific $z_0$) only one step for each new $z$ that we discover - instead of trying to run the machine until it halts. This allows us to continue to discover more new $z$'s even if the execution of those machines never ends!

I will leave it to you to verify that this machine really accepts $A/B$ :)


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