# Non-deterministic Turing machine that halts on at least one branches of computation

I'm looking at my textbook here from Michael Sipser and he says that a nondeterministic Turing machine is a decider if all its computation branches halt on all inputs. I think I recall seeing somewhere what you'd call a nondeterministic Turing machine that halts on at least one branch for all inputs, but may loop on others. Is there a name for such a thing? I see later in this chapter the word verifier, but that doesn't seem to fit... I think that refers to an algorithm.

A verifier for a language $A$ is an algorithm $V$, where $$A=\{w\mid V\text{ accepts }\langle w,c\rangle\text{ for some string c}\}.$$ We measure the time of a verifier only in terms of the length of $w$, so a polynomial time verifier runs in polynomial time in the length of $w$. A language $A$ is polynomially verifiable if it has a polynomial time verifier.

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Perhaps just in the definition of the language recognized by a NTM? An NTM accepts a string $w$ if there exists at least one computation path that ends in the accepting state ... but not necessarily this happens for all input strings (otherwise L(NTM) = \Sigma^* ) –  Vor Feb 12 at 16:46
I believe that you would say that the machine "accepts" the language. –  user6845 Feb 12 at 19:03