Bai giai tri tue nhan tao tut4

TRƯỜNG ĐẠI HỌC BÁCH KHOA TP.HCM Khoa Khoa học & Kỹ thuật Máy tính TUTORIAL SESSION 4 PROPOSITIONAL LOGIC 1.. Getting an A on the final and doing every exercise in the book is sufficien

TRƯỜNG ĐẠI HỌC BÁCH KHOA TP.HCM Khoa Khoa học & Kỹ thuật Máy tính

TUTORIAL SESSION 4 PROPOSITIONAL LOGIC

1 Let p, q, and r be the following propositions:

p: You get an A on the final exam

q: You do every exercise in the book

r: You get an A in this class

Write the following formulas using p, q, and r and logical connectives

a You get an A in this class, but you do not do every exercise in the book

b To get an A in this class, it is necessary for you to get an A on the final

c Getting an A on the final and doing every exercise in the book is sufficient for getting an A in this class

Solution:

a r ˄ ¬q

b r => p

c p ˄ q => r

2 Is the sentences valid, unsatisfiable, or neither?

a Smoke => Smoke

b Smoke => Fire

c ((Smoke  Heat) => Fire)  ((Smoke => Fire) ν (Heat => Fire))

Solution:

a valid

b neither

c neither

3 Express sentences in propositional logic:

a You can stay in dormitory only if you are an honor student or you are not a

graduate student

b You cannot pass the driver license exam if your knowledge exam’s score is under

60 unless you are older than 45 years old

c If the photo is digital or in black and white, then it is a portrait, else not

d If it is a portrait, then it is a picture of my friend

Solution:

a Symbolize the related predicates as follows:

p: you can stay in dormitory

q: you are a honor student

r: you are a graduate student

Propositional logic representation: p => q ˅ ¬r

b Symbolize the related predicates as follows:

p: you can pass the driver license exam

q: your knowledge exam’s score is under 60

r: you are older than 45 years old

Propositional logic representation: ¬r => (q => ¬p) (or (p ˄ q) => r)

c Symbolize the related predicates as follows:

p: the photo is digital

q: the photo is in black and white

r: the photo is a portrait

t: the photo is a picture of my friend

Propositional logic representation: p ˅ q  r

d Symbolize the related predicates as question 3c:

Propositional logic representation: r => t

4 Assuming that we have three propositional-logic-based KBs as follows:

a If the temperature and the pressure are constant then it does not rain

The temperature remained constant

It rained

Using resolution to answer the question: Did the pressure remain constant or not?

b Men eat when they are hungry

John always wears his best suit to eat

At this moment John is not hungry

Using resolution to answer the question: Is John wearing his best suit or not?

Solution:

a T: the temperature is constant

P: the pressure is constant

R: it’s rain

KB = {T ∧ P ⇒ ¬R, T, R} = {¬T ∨ ¬P ∨ ¬R, T, R}

Assume the pressure didn’t remain constant: α = ¬P

KB ∪ {¬α} = {¬T ∨ ¬P ∨ ¬R, T, R, P}

Resolution:

¬T ∨ ¬P ∨ ¬R T => ¬P ∨ ¬R

¬P ∨ ¬R R => ¬P

¬P P => []

Thus, the pressure didn’t remain constant

b E: John eats

H: John is hungry

S: John wears his best suit

KB = {H ⇒ E, E ⇒ S, ¬H} = {¬H ∨ E, ¬E ∨ S, ¬H}

Assume John is wearing his best suit: α = S

KB ∪ {¬α} = {¬H ∨ E, ¬E ∨ S, ¬H, ¬S}

Resolution:

¬H ∨ E ¬E ∨ S => ¬H ∨ S

¬H ∨ S ¬S => ¬H

Assume John is not wearing his best suit: α = ¬S

KB ∪ {¬α} = {¬H ∨ E, ¬E ∨ S, ¬H, S}

Resolution:

¬H ∨ E ¬E ∨ S => ¬H ∨ S

Thus, we can’t conclude John is wearing his best suit or not