Exercise 1 (Effective operations on computable functions).
Show that there exists a total computable function k such that, given indices
e and e0 of computable functions f and g, delivers an index k(e, e0 ) of the
composite function f ◦ g.1
Exercise 2 (Kleene predicates).
Use Kleene’s S predicate to show that, if f is an injective computable function
(but not necessarily total or surjective) then f −1 is computable, where:
the x such that f (x) = y if y ∈ Ran(f )
−1
f (y) =
undefined
if y 6∈ Ran(f )
1 Recall
that the composite function is defined by f ◦ g(x) = f (g(x)).
1