A Turing machine has a finite control, an unbounded tape divided into cells over a finite tape alphabet including a blank symbol, and a head that reads one cell at a time. For a deterministic Turing machine, the current state and scanned symbol determine the replacement symbol, a head move one cell left or right, and the next state. The input occupies finitely many cells initially; all other cells are blank. Distinguished halting states report acceptance or rejection.
A formal language is a decidable language if some Turing machine halts on every input string, accepting exactly the members and rejecting all nonmembers:
(1)
Halting on the nonmembers is part of the requirement; a machine that merely runs forever on some nonmembers need not decide the formal language.

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