Four. Let u(n) be the nth wide variety so as of length that is the sum of
squares. Show that u(n) is primitive recursive.
5. Let R(x, t) be a primitive recursive predicate. Let
g(x,y) =max R(x,t),
1:5.Y
i.E., g(x, y) is the biggest price of-t :::;; y for which R(x, t) is actual; if
there may be none, g(x, y) = zero. Prove that g(x, y) is primitive recursive.
6. Let gcd(x, y) be the best common divisor of x and y. Show that
gcd(x, y) is primitive recursive.
7. Let lcm(x, y) be the least common a couple of of x and y. Show that
lcm(x, y) is primitive recursive.
Eight. Pairing Functions and G6del Numbers fifty nine
8. Give a computable predicate P(x1 , ••• , xn, y) such that the function
min Y P(x 1 , ••• , x n, y) isn't always computable.
Nine.* A characteristic is essential if it can be obtained from the functions s, n,
u), + , ..:... By a finite collection of packages of composition, bounded
summation, and bounded product. (By software of bounded summation we suggest obtaining the characteristic r.R-o f(t, x 1 , ••• , xn) from
f(t, x1 , ••• , xn), and in addition for bounded product.)
(a) Show that every simple characteristic is primitive recursive.
(b) Show that x · y, xY, and x! Are fundamental.
(c) Show that if n + 1-ary predicates P and Q are primary, then
so are - P, P V Q, P & Q, ('Vt), YP(t, x 1 , ••• , xn),
(3t),; YP(t, x, ' ... ' xn), and min,,; YP(t, x,' ... ' xn).
(d) Show that Prime(x) is essential.
(e) Let the binary function exp/x) be described
exp0 (x) = x
expy+ ,(x) = 2exp,(x).
Show that for every basic feature f(x1 , ••• , xn), there may be a
consistent k such that f(x1 , ••• , xn) ~ expk(maxx1 , ••• , xn). [Hint:
Show that for every n there is an m ~ n such that x · expn(x) ~
expm(x) for all x.]
(f) Show that exp/x) is not fundamental. Conclude that the elegance of
basic features is a right subset of the class of primitive
recursive capabilities.

You work for a massive business enterprise in which there are many meetings, each of internal workforce
and among workforce and external customers. Meetings are recorded in formal minutes.
The employer's documents of mins are big, and the cloth must be saved for lots
years considering it could be necessary to test back on decisions taken early in massive
initiatives.
You are requested to layout a retrieval machine in order that organisation group of workers can discover mins
on a selected topic. Because of the legal implications that past discussions and
decisions may additionally have, the employer is particularly worried that the new retrieval
gadget might be reliable and effective.
Outline the design of your system, indicating the particular capabilities it'll have that
are supposed to satisfy the agency's necessities (you could anticipate that minutes
are usually definitely dated and have specific lists of participants). [10 marks]
The employer is willing to permit the installation of a pilot gadget so your approach
can be evaluated below sensible situations.
Describe, in element, your layout for the assessment: what statistics, operational
situations and factors of your machine could you take into account, and why? What
overall performance measures could you apply, and why? [10 marks]
five Security
"Robert Morris Senior became liable for Unix protection, Robert Morris Junior for
the Internet malicious program. The father did much extra damage to Internet security than
the son" (Whitfield Diffie). Discuss. [20 marks]
four
CST.2000.7.Five
6 Neural Computing
Explain the key ideas of a Hopfield synthetic neural network for content material-addressable,
associative reminiscence. In explaining how memories are stored and retrieved, make certain
to outline the notions of:
• configuration area
• connectivity matrix
• solid attractor
• basin of enchantment
• network ability, and its dependence at the range of "neurones"
[10 marks]
Marshall as many strains of proof as you can to help the view that in human
imaginative and prescient "what you notice is your personal 'images', instead of the retinal picture as
faithfully recorded by using photoreceptors in the eye". Explain the importance of this
commentary for vision concept and for machine imaginative and prescient. [8 marks]
Suppose you had been trying to design a system imaginative and prescient device based totally as carefully as
feasible upon human imaginative and prescient. Would you purpose to design within the visible illusions that
almost all of us "see" as nicely? (These include the distortions of geometrical form,
attitude and relative length illusions, and many others.) If such residences emerged as accidental
outcomes of your imaginative and prescient design, could you recollect them to be functions, or
bugs?

CS 301 Data Structures I. Suppose that by special measurement it has been discovered that the following functions describe the precise action of some algorithms. For each function, describe the asymptotic behavior in terms of big-Oh notation. 14-3n+7n a) b) (5 * log n) * (3 * n) II. Consider the following program segment. Describe the asymptotic execution time as a function of n using big-Oh notation. In each case, you can assume that the remaining portion of the loop will require constant execution time. a) b) Homework 1 (exam study guide) c) for (int i = 0; i