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My assignment says:

"Determine if the following statement is correct: If $A$ and $A \cup B$ are decidable, then $B$ is decidable."

The solution says:

"Incorrect. If $B = H_0 \subseteq \{0,1\}^*$ is the halting problem with input $\epsilon$ and $A = \{0,1\}^*$, then $A$ and $A \cup B$ are decidable and $B$ is undecidable.

My question is:

How can $A \cup B$ be decidable if $B$ is undecidable? If for example $b \in B$ then $b \in A \cup B$ but a Turing Machine may not halt on $b$ because $B$ is undecidable, then how can $A \cup B$ still be decidable with $b \in A \cup B$ ?

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    $\begingroup$ Take $A = \{0,1\}^*$. $\endgroup$ – Yuval Filmus Sep 16 at 10:10
  • $\begingroup$ If it’s not clear enough (since it was mentioned in the solution you were given): in this case A union B is decidable no matter what B is, because A = A union B = all possible strings. The Turing machine can halt immediately without caring about B at all. $\endgroup$ – gnasher729 Sep 16 at 10:37
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If $A=\{0,1\}^*$ then $A\cup B=\{0,1\}^*$, regardless of what $B$ is. $\{0,1\}^*$ is decidable, and choosing to express it as something involving undecidable things doesn't change that fact.

The question "Is $w$ in $A\cup B$?" is equivalent to "Is at least one of the following statements true? $w$ is in $A$; $w$ is in $B$; $w$ is in both $A$ and $B$?" In the case $A=\{0,1\}^*$, we can answer "Yes, $w$ is in $A$" and the rest of the questions are irrelevant. It doesn't even matter if we can't answer "Is $w$ in $B$?", because we don't need to.

Here's a more practical example that might help. Suppose I give you a number $x$ and ask you "Is $x\in\text{Integers}\cup\text{Primes}$?" Figuring out if a large number is prime is quite difficult, but you can easily answer my question by checking if $x$ is an integer. You can ignore the "prime" part completely.

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