MATH 516301
MATH-516301
This question paper consists of 4 printed pages, each of which is identied by the reference MATH 516301 No calculators allowed
c UNIVERSITY OF LEEDS Examination for the Module MATH 5163 (January 2008) ADVANCED COMPUTABILITY AND UNS
University of Leeds
Department of Pure Mathematics
MATH 5164M
ADVANCED COMPUTABILITY AND UNSOLVABILITY
Problems 7
1) There is nothing useful one can say about incomputability discuss.
Write an essay answering the above question, covering not more than thr
University of Leeds
Department of Pure Mathematics
MATH 3163/5164M
COMPUTABILITY AND UNSOLVABILITY
Problems 6
Read CT, pages 139154, 161166, and 287296
1) Show that T is an equivalence relation.
2) Show that is a partial ordering on D .
3) (a) Show that f
University of Leeds Department of Pure Mathematics MATH 3163 COMPUTABILITY AND UNSOLVABILITY Problems 7 1) Write an essay, covering not more than three pages, describing the background to, and consequences of, Alan Turings discovery of the existence of a
University of Leeds
Department of Pure Mathematics
MATH 3163/5164M
COMPUTABILITY AND UNSOLVABILITY
Problems 4
1) A function f is increasing if f (x) < f (x + 1) for all x.
Show that an innite set A is computable if and only if it is the range of an increa
University of Leeds
Department of Pure Mathematics
MATH 3163/5164M
COMPUTABILITY AND UNSOLVABILITY
Problems 5
Read CT, pages 101115
1) (a) Show that if A m B and B is computable, then A is computable.
(b) Show that if A m B and B is computably enumerable,
University of Leeds
Department of Pure Mathematics
MATH 3163/5164M
COMPUTABILITY AND UNSOLVABILITY
Problems 3
Read CT, pages 8799 and 6668
1) Show that: A computable A c.e.
2) Show that the following sets are all c.e.
(i) K1 = cfw_x | Wx = ,
(ii) Range e
University of Leeds
Department of Pure Mathematics
MATH 3163/5164M
COMPUTABILITY AND UNSOLVABILITY
Problems 1
Read CT, pages 325
1) Show that the following functions are primitive recursive:
(i) sg(n) =
1
0
if n = 0
otherwise ,
(ii) n! = factorial n ,
(ii
University of Leeds
Department of Pure Mathematics
MATH 3163/5164M
COMPUTABILITY AND UNSOLVABILITY
Problems 2
Read CT, pages 3334, 6166 and 6981
1) Write Turing programs for the functions:
(b) g(x, y) = xy .
(a) f (x, y) = x + y,
2)D Say which function f
1. In my opinion I agree with Paul Singer that ethically animals have an equal
concern with humans. Animals are usually also affected by things humans
do. It is non-humans who bear the most direct burden of human
interference with nature. For example, obt