halt; other models require all entries to be filled. Every part of the machine (i.e. its state, symbol-collections, and used tape at any given time) and its actions (such as printing, erasing and tape motion) is finite, discrete and distinguishable; it is the unlimited amount of tape and runtime that gives it an unbounded amount of storage space. Formal definition Following , a (one-tape) Turing machine can be formally defined as a 7-tuple where is a finite, non-empty set of tape alphabet symbols; is the blank symbol (the only symbol allowed to occur on the tape infinitely often at any step during the computation); is the set of input symbols, that is, the set of symbols allowed to appear in the initial tape contents; is a finite, non-empty set of states; is the initial state; is the set of final states or accepting states. The initial tape contents is said to be accepted by if it eventually halts in a state from . is a partial function called the transition function, where L is left shift, R is right shift. If is not defined on the current state and the current tape symbol, then the machine halts; intuitively, the transition function specifies the next state transited from the current state, which symbol to overwrite the current symbol pointed by the head, and the next head movement. In addition, the Turing machine can also have a reject state to make rejection more explicit. In that case there are three possibilities: accepting, rejecting, and running forever. Another possibility is to regard the final values on the tape as the output. However, if the only output is the final state the machine ends up in (or never halting), the machine can still effectively output a longer string by taking in an integer that tells it which bit of the string to output. A relatively uncommon variant allows "no shift", say N, as a third element of the set of directions . The 7-tuple for the 3-state busy beaver looks like this (see more about this busy beaver at Turing machine examples): (states); (tape alphabet symbols); (blank symbol); (input symbols); (initial state); (final states); see state-table below (transition function). Initially all tape cells are marked with . Additional details required to visualize or implement Turing machines In the words of van Emde Boas (1990), p. 6: "The set-theoretical object [his formal seven-tuple description similar