Computability, Unsolvability, Randomness
Math 497A: Homework #1
Stephen G. Simpson
Due Tuesday, September 4, 2007
1. Exhibit a register machine program which computes the exponential
function, exp(x, y)=xy. Remember that the variables xand yrange
over N, the set of natural numbers. Note that x0=1forallx,evenfor
x=0.
2. Exhibit a register machine program which computes the function
Rem(y, x) = the remainder of yon division by x.
For example, Rem(17,5)=2.
3. Assume that P(x, y, z) is a 3-place predicate which is computable. Con-
sider the 2-place partial function ψ(x, y) defined as follows:
ψ(x, y)'the least zsuch that P(x, y, z) holds, if such a z
exists. If such a zdoes not exist, then ψ(x, y ) is undefined.
Use register machine programs to prove that the function ψis com-
putable. (More precisely, ψis partial recursive.)
Hint: Since Pis a computable predicate, we may assume that we have
a program Pwhich computes the characteristic function of P. Show
how to embed P(or perhaps a variant of P) into a larger program, call
it Q, such that Qcomputes ψ.TheideaofQis that, given xand y,
Q(x, y) searches sequentially through the integers z=0,z=1,z=2,
..., to find the first zsuch that P(x, y , z) holds. In particular, Qwill
have the property that, for all xand y,Q(x, y) eventually halts if and
only if P(x, y, z )holdsforsomez.
4. Use the results of Problems 2 and 3 to prove that the 2-place function
LCM(x, y) = the least common multiple of xand yis computable.
Deduce that the 2-place function GCD(x, y) = the greatest common
divisor of xand y, is also computable.
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