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COMP9414: Arti cial Intelligence

Tutorial 4: Propositional Logic

1. Translate the following sentences into Propositional Logic.

(i) If Jane and John are not in town we will play tennis [do both of them have to be away?] (ii) It will either rain today or it will be dry today [is“dry”the same as“not raining”?]

(iii) You will not pass this course unless you study [this means“if you do not study”] To do the translation

(a) Identify a scheme of abbreviation for basic sentences

(b) Identify logical connectives between variables

2. Convert the following formulae into Conjunctive Normal Form (CNF).

(i) P → Q

(ii)  (P → ¬Q) → R

(iii) ¬(P ∧ ¬Q) → (¬R ∨ ¬Q)

3. Show using the truth table method that the corresponding inferences are valid.

(i) P → Q, ¬Q |= ¬P

(ii) P → Q |= ¬Q → ¬P

(iii) P → Q, Q → R |= P → R

Check your answers using the Python program tableau prover .py.

4. Repeat Question 3 using resolution. In this case, show

(i) P → Q, ¬Q ⊢ ¬P    (ii) P → Q ⊢ ¬Q → ¬P

(iii) P → Q, Q → R ⊢ P → R

5. Determine whether the following sentences are valid (i.e. tautologies) using truth tables.

(i)  ((P ∨ Q) ∧ ¬P) → Q

(ii)  ((P → Q) ∧ ¬(P → R)) → (P → Q)

(iii) ¬(¬P ∧ P) ∧ P

(iv)  (P ∨ Q) → ¬(¬P ∧ ¬Q)

Check your answers using the Python program tableau prover .py.

6. Repeat Question 5 using resolution. In this case, try to show

(i) ⊢ ((P ∨ Q) ∧ ¬P) → Q

(ii) ⊢ ((P → Q) ∧ ¬(P → R)) → (P → Q)

(iii) ⊢ ¬(¬P ∧ P) ∧ P

(iv) ⊢ (P ∨ Q) → ¬(¬P ∧ ¬Q)