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Literature.
[1] Downloadable lecture notes available through the web page of the course.
[2] J. Komara and P. J. Voda. Lecture Notes in Theory of Computability. 2001.
[3] P. J. Voda. Theory of Recursive Functions & Computability. 2000.
[4] I. Korec. Uvod do teorie algoritmov. MFF UK. Bratislava. 1983.
Arithmetization of Turing machines. The predicate holds if the number is the code of a Turing machine ; that is, if the following conditions hold:
The number is a sequence of quintuples:
Uniqueness condition. The numbers , and in the quintuple of are uniquelly determined by the numbers and :
Sort condition. If the quintuple is an element of then is an internal state, and are tape symbols, and represents a movement operation:
Arithmetization of the transition function. The ternary function is arithmetization of the transition function. The function satisfies
Arithmetization of one-step computation. The binary function takes a configuration of a Turing machine , where is the code of , and yields the next configuration obtained from by one computational step of . Otherwise, the application yields .
Exercise. (bonus 3 points) Show that there is a primitive recursive function and a primitive recursive predicate such that
Exercise. Find a well-formed Turing index for the successor function , i.e. find the number such that
Exercise. Find a well-formed Turing index for the predecessor function , i.e. find the number such that
Exercise. (1 point) Find a well-formed Turing index for the zero function , i.e. find the number such that
Exercise. Find a well-formed Turing index for the identity function , i.e. find the number such that
Exercise. Find a well-formed Turing index for the identity function , i.e. find the number such that
Exercise. Find a well-formed Turing index for the addition function , i.e. find the number such that