3.3 KiB
Instructions:
12 FRQ, written answers 3 MCQ, multiple choice 1 Matching question, algorithms (role of algorithms) 15 total questions (?)
1 A4 size double sided hand written notes allowed (Important!!!) 2 hour exam Partial marks allowed for partially correct answers Bring a calculator (Important!!!)
Part 1 Important Questions
Horn form for logic? Why are these conditions not solvable without a truth table?
Part 2 Important Questions
1
Arithmetic assertions can be written in first order logic with the predicate symbol <, the function symbols + and x, and the constant symbols 0 and 1. Additional predicates can also be defined with bi-conditionals
a) Represent the property "x is and even number"
Ax Even(x) <=> Ey x=y+y
b) Represent the property "x is prime"
Ax Prime(x) <=> Ey,z x=y * z => y = 1 V z = 1
c) Goldbach's conjecture is the conjecture (unproven as of yet) that "every even number is equal to the sum of two primes". Represent this conjecture as a logical sentence.
Ax Even(x)=> Ey,z Prime(y) /\ Prime(z) /\ x=y+z
2
Find the values for the probabilities a and b in joint probability table below so that the binary variables X and Y are independent
| X | Y | P(X, Y) |
|---|---|---|
| t | t | 3/5 |
| t | f | 1/5 |
| f | t | a |
| f | f | b |
| Due to probability being max 1, we know that a + b must be 1/5 | ||
| P(Yt)/P(Yf) = a/b = 3 | ||
| b = 1/20 | ||
| a = 3/20 |
3
idk where R comes from, look into slides about bayes theorem Show the three forms of independence in Equation (12.11) are equivalent P(a|b) = P(a) or P(b|a) = P(b) or P(a /\ b) = P(a) * P(b) / R(?)
First two are logically the same, just inverted
From bayes theorem P(a | b) * P(b) = P(a) * P(b) / R(?)
P(a /\ b) = P(a | b) * P(b)
4
Consider the following propability distrobutions:
| A | P(A) |
|---|---|
| t | 0.8 |
| f | 0.2 |
| A | B | P(B|A) |
|---|---|---|
| t | t | 0.9 |
| t | f | 0.1 |
| f | t | 0.6 |
| f | f | 0.4 |
| B | C | P(C|B) |
|---|---|---|
| t | t | 0.8 |
| t | f | 0.2 |
| f | t | 0.8 |
| f | f | 0.2 |
| Given these tables and no other assumptions, calculate the following probabilities. | ||
| a. P(a, ~b) | ||
| = P(a) * P(~b | a) | |
| = 0.8 * 0.1 | ||
| = 0.08 | ||
| b. P(b) | ||
| = P(bt | a) * P(a) + P(bt | ~a) * P(~a) |
| = 0.9 * 0.8 + 0.6 * 0.2 | ||
| = 0.84 |
5
Let A and B be Boolean Random variables. You are given the following probabilities P(A=true) = 0.5 P(B=true |A=true) = 1 P(B=true) = 0.75
What is P(B=true|A=false)?
6
Consider the XOR function of three binary input attributes, which produces the value 1 if and only if an odd number of the three input attributes has value 1.
Draw a minimal sized decision tree for the three input XOR function.
Three layer decision three, A > B > C. Output of tree would be 0 1 1 0 1 0 0 1 if on the left of the decision is always 0 and 1 is right
7
Consider the problem of separating N data points into +ve and -ve examples using a linear separator. Clearly this can always be done for N=2 points on a line of dimension d=1, regardless of how many points are labeled or where they are located (unless the points are in the same place)
a) Show that it can always be done for N=3 points on a plane of dimension d=2 unless they are co-linear.
b) Show that it cannot (or can we?) always be done for N=4 points on a plane of dimension d=2