ECE 421 Homework Problems – Tutorial #5 Theme: Multi-Layer Perceptrons solution

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Question 1 (Exercise 7.1 from LFD)
Consider a target function f (top row) composed of three perceptron components h1, h2 and h3.
The + and − regions of h1, h2 and h3 are illustrated (from left to right in bottom row).
x2 +

x1

+

+
x2 +

x1
x2 +
x1

x2

x1
+
h1 h2 h3
(a) Show that
f = h1h2h3 + h1h2h3 + h1h2h3. (1)
(b) Is there a systematic way of going from a target function composed of perceptrons to a Boolean
formula like in the part (a)? [Hint: consider only the + regions of the target function and use the
disjunctive normal form (OR and ANDs).]
1
ECE421 Tutorial 5 Exercises Due: Sunday February 14 11:59 PM
Question 2 (Exercise 7.2 from LFD)
(a) Extend the boolean OR and AND to more than two inputs, i.e., OR(x1, . . . , xM) = +1 if any
one of the M inputs is +1; and AND(x1, . . . , xM) = +1 only if all the inputs are +1. Give
graph representations of OR(x1, . . . , xM) and AND(x1, . . . , xM).
(b) Give the graph representations of the perceptron h(x) = sign(w
>x).
(c) Give the graph representation of OR(x1, x2, x3).
Question 3 (Problem 3, Midterm 2019)
Assume two logical inputs (that can either be 0 or 1)
x1 0 1 0 1
x2 0 0 1 1
and the following single-layer model.
+
x1
x2
ϕ
Activation
ν
y = ϕ(ν) = (
1 ν ≥ 2
0 ν < 2
w1 = 1
w2 = 1
(a) Given the 4 different sets of inputs of x1 and x2, calculate the output y. What function can
be represented by this model?
(b) How can the following function be implemented by changing only the threshold value (ν)
(weights are same as before).
x1 0 1 0 1
x2 0 0 1 1
g(x1, x2) 0 1 1 1
(c) Can the following function be implemented with the given single-layer model (one set of inputs
and an activation function)? If no explain why, if yes give an example.
x1 0 1 0 1
x2 0 0 1 1
z(x1, x2) 0 1 1 0