I see that * is claimed to be decidable in many documents, but I have never seen an example or easy demostration that it is decidable.What is the proof that * is decidable?
Prove * is decidable
computability
Theorem: The set $\Sigma^{*}$ of all words is decidable.Proof. According to the definition of decidability, we must provide a computable function $d$ which takes a word $w$ and outputs $1$ if $w \in \Sigma^{*}$, and outputs $0$ if $w \not\in \Sigma^{*}$. Such a function is very easily constructed, it is$$d(w) = 1,$$That is, because every word is in $\Sigma^{*}$, the decision function always says yes. QED.