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Assignment 1: Proofs and Simple Programs: Solutions
1. If E is the set of even numbers (0, 2, 4, 6.), and N the set of natural
numbers (0, 1, 2, 3.), give examples of:
(a) A 1-1 function from N to E,
(b) An onto function from E to N .
Be sure to prove tha
Assignment 5: Recursion Theorem, Strings, Turing Machines
1. (a) Use the Recusion theorem to show that there is a partially computable function f that satises the equations
f (x, 0) = x + 2
f (x, 1) = 2f (x, 2x)
f (x, 2t + 2) = 3f (x, 2t)
f (x, 2t + 3) =
Assignment 4: Parameter Theorem, R.E. Sets,
Reducibility, Rice-Shapiro Theorem
This assignment is due Friday, March 12th, at the beginning of class (9:00am).
1. Suppose that f : N N is a strictly increasing function: in other words,
f (n + 1) > f (n) for
Assignment 2:
More Programs and Primitive Recursive Functions: Solutions
1. Let P (x) be a computable predicate. If f is dened by
f (x1 , x2 ) =
x1 + x2
if P (x1 + x2 );
otherwise.
show that f is partially computable.
The following program computes f :
Y
CPSC 513 Winter 2010 Midterm: Solutions
1. (a) Dene what it means for a function to be computable.
A function f : N m N is computable if it is total, and there exists
a program P such that P (x1 . . . xm ) = f (x1 . . . xm ).
(b) Show that the function
f
Assignment 3: Numbering and Universal Programs: Solutions
1. The Fibonacci sequence is given by
F (0) = 0,
F (1) = 1,
F (n + 2) = F (n + 1) + F (n).
Show that F is primitive recursive. (Hint: use the pairing function).
We cannot directly give recursion
Assignment 6: Reducibility and Oracles: Solutions
1. Recall that the strengthened Parameter theorem says that for n, m > 0,
n
n
Sm (u1 , . . . un , y) = Sm (u , . . . u , y)
1
n
implies that u1 = u , . . . , un = u .
1
n
(a) Show, by giving a counter-exam