Exercises, II part Forward Chaining: 12 Jul 2012 Exercises, II - - PDF document

Exercises, II part

Exercises, II part

Exercises, II part

Forward Chaining: 12 Jul 2012

Consider the following set of clauses in propositional logic: J = ⇒ Q A ∧ I = ⇒ J E ∧ F = ⇒ I B = ⇒ F A ∧ B = ⇒ E A B

1 Show and execution of the forward chaining algorithm on

this set of clauses for proving Q (show the and-or graph and the evolution of the count table).

2 What happens if we replace the first clause of the

knowledge base J = ⇒ Q with J ∧ G = ⇒ Q (and nothing else changes in the knowledge base)?

Exercises, II part

Forward Chaining: 27 Sep 2012

Consider the following set of clauses in propositional logic: H ∧ S = ⇒ Q M ∧ S = ⇒ H H ∧ A = ⇒ M M ∧ B = ⇒ S A ∧ B = ⇒ M A B

1 Show an execution of the forward chaining algorithm on

this set of clauses for proving Q (show the and-or graph and the evolution of the count table).

2 Briefly comment on the complexity and completeness of

the forward chaining approach.

Exercises, II part

Inference: 12 Jul 2012

Consider the following Joint Probability Table for the three binary ran- dom variables A, B, C. P(A, B, C) A B C 0.108 T T T 0.012 T T F 0.072 T F T 0.008 T F F 0.016 F T T 0.064 F T F 0.144 F F T 0.576 F T F Compute the following queries:

1 P(C|A=T,B=T) 2 P(C|A=T)

Exercises, II part

BN: 13 Feb. 2017

Consider the Bayesian Network in the Figure. Answer the following questions:

1

State whether P(A|B,C) = P(A|B,C,D). Motivate your answer.

2

State whether P(A|C) = P(A|B,C,D). Motivate your answer.

3

State how many numbers we need to represent the joint distribution for this network assuming variables are all boolean. Motivate your answer.

Exercises, II part

BN: 16 Giu. 2016

Consider the Bayesian Network in Figure, where every variable is by- nary. Answer the following ques- tions:

1

Is it true that P(B|A) = P(B|A,C) ? Motivate your answer.

2

Write down the equation to compute the query P(D|A=true,C=true) using the CPT associated with the network.

3

How many independent numbers must be stored to answer all the possible queries for this Bayesian Network ?

Exercises, II part

BN: 21 Sep. 2016

Consider the Bayesian Network in Figure, where every variable is by- nary. Answer the following ques- tions:

1

State how many parameters must be provided to compute the joint probability table of this BN.

2

Assume the CPT that defines P(E|B,C,D) is specified with a noisy-or, state how many parameters must be provided to compute the joint probability table of this BN.

3

State whether P(D|C,E) = P(D|A,B,C,E). Motivate your answer.

Exercises, II part

Minimax: 16 Giu. 2016

Consider the min-max tree in Figure. Is it possible to provide a set of values to replace the question marks so that an Aplha-Beta pruning algorithm would perform the specified cuts ? If so provide such values otherwise motivate your answer.

Exercises, II part

Minimax: 26 Sep 2013

Consider the min-max tree in the Figure. Show a trace of the execution

• f an Aplha-Beta pruning algorithm, indicating what will be the first move

selected by the max player and which cuts are performed by the algorithm.

Exercises, II part

DPOP: 22 Jun. 2015

Consider the following binary cost network: Variables, X = {x1, x2, x3, x4, x5}. Constraints Ch = { } and Cs = {F12(x1, x2), F13(x1, x3), F15(x1, x5), F23(x2, x3), F34(x3, x4), F45(x4, x5)}. Consider the DPOP algorithm: show the pseudo-tree obtained by using a DFS visit with the MCN heuristic. Start from node x1; give the dimension of the biggest message exchanged by the algorithm (in terms of values stored in the table); compute the total number of messages that DPOP would use to solve this problem. In general, is this number dependent on the pseudo-tree used ?

Exercises, II part

DPOP: 30 Apr. 2014 (partial test)

Consider the following binary cost network: Variables, X = {x1, x2, x3, x4, x5}, Domains, D1 = {G, B},D2 = D3 = D4 = D5 = {R, B}, Constraints Ch = { } and Cs = {F12(x1, x2), F13(x1, x3), F14(x1, x4), F15(x1, x5), F23(x2, x2), F35(x3, x5)}. Where each Fij has the following form Fij(xi, xj) =

• −1

if values are the same

• therwise

Consider the PseudoTrees that correspond to the following DFS visit

• rders o1 = {x5, x1, x4, x2, x3} and o2 = {x1, x4, x3, x2, x5}. Answer

the following questions: state which order is better considering message size and computation, motivate your answer. solve the problem by using the DPOP algorithm with a PseudoTree of your choice.