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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] J. Komara and P. J. Voda. Metamathematics of Computer Programming. 2001.
Definition. Predicate is recursively enumerable in increasing order if there exists an increasing recursive function enumerating that predicate, that is for all we have
Note that in this case the predicate is infinite, i.e. it holds for infinitely many numbers.
Exercise. (1 point) Prove the following theorem:
For an infinite predicate , the predicate is recursive iff it is recursively enumerable in increasing order.
Remark. Prove the claim for the special cases in the next two theories.
Exercise. Suppose that is an increasing recursive function. Show that the predicate satisfying
is a recursive predicate.
Exercise. Suppose that is an infinite recursive predicate. Show that there is an increasing recursive function such that
Definition. A predicate is said to be
recursively semidecidable, or simply semirecursive, if it is the domain of a partial recursive function,
recursively enumerable if it is empty or the range of a recursive function.
Exercise. (1 point) Prove the following theorem:
a predicate is semirecursive iff it is a recursively enumerable predicate.
Reamrk. Prove the claim for the special cases in the next two theories.
Exercise. Suppose that is a partial recursive function such that
Let be an index of . Suppose further that holds for the number . Define a recursive function such that
Exercise. Suppose that is a recursive function such that
Define a partial recursive function such that
Exercise. (1 point) Prove Post's theorem for semirecursive predicates:
a predicate is recursive iff both and are semirecursive predicates.
Remark. Prove the claim for the special cases in the next two theories.
Exercise. Suppose that is a recursive predicate. Define a partial recursive function such that