The standard definition of computability, for a sequence $s\in\{0,1\}^\omega$, is that there is a Turing machine outputting $s[i]$ on input $i$.

I'm looking for strengthenings of this notion; for example, in the above definition it's not decidable whether there is a $1$ in $s$; or, given $i$, whether there is a $1$ in $s$ at position $\ge i$. I would be happy to be shown a "natural" definition of computability that makes these predicates computable.

To the above: if there were an algorithm that, from the Turing machine producing $s$, tells us whether $s$ contains a $1$ then I could do the following: from any Turing machine $M$, program a Turing machine outputting $s[i]=1$ if $M$ stops after $\le i$ steps. This sequence $s$ is obviously computable --- I said how to compute it --- but an algorithm determining if the sequence contains a $1$ would solve the Halting problem.

A search through the literature didn't show anything, so links and references are most welcome!