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The predicate holding of tape symbols is defined explicitly as a primitive recursive predicate (union format):
Movement. Only the following movement may be performed by the heads of Turing machines:
the head stays at the same square - this is represented by the number ,
the head moves one square left - this is represented by the number ,
the head moves one square right - this is represented by the number .
The predicate which holds if is one of the above possible movements is defined explicitly as a primitive recursive predicate (union format):
Solution. The Turing machine computing will be defined if we determine the following objects:
a set of symbols represented here by the predicate ,
a set of states represented here by the predicate , and
a set of movements represented here by the predicate ,
a binary transition function denoted here by .
Note that the tape alphabet and the set of movements is already fixed. The predicates and the transition function must satisfy:
finiteness of the set of states:
finiteness of the transition function:
existence of two distinguished states and called respectively the initial state and the final state:
property of being a transition function:
The Turing machine computes the partial function . Its contraction is explicitly defined as a partial recursive function: