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Artificial Intelligence - Practice Homework 2 | CPSC 420, Assignments of Computer Science

Texas A&M University (A&M)Computer Science
Prof. Y. Choe

Material Type: Assignment; Professor: Choe; Class: ARTIFICIAL INTELLIGENCE; Subject: COMPUTER SCIENCE; University: Texas A&M University; Term: Unknown 1989;

Typology: Assignments

Pre 2010

Uploaded on 02/10/2009

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CPSC 420-500 Homework #2
Due 11/05/07 (Monday), 3pm
Handwritten or printed hardcopy must be submitted to the TA
Total: 130 pts
Yoonsuck Choe
October 29, 2007
1 First-Order Logic
Important: In this section, assume that w, x, y, z are variables; A, B, C, D are constants; and f(·), g(·), h(·)
are functions; and P(·), Q(·), R(·)are predicates.
1.1 Standard Forms
To do automatic theorem proving in first-order logic, you need to go through three steps to convert your
initial first-order logic expression into a standard form. These are:
1. Prenex normal form,
2. Conjunctive normal form, and
3. Skolemization.
Question 1 (12 pts): Convert to prenex normal form (4 points each):
1. ∀x, ¬(∃y, ¬P(x, y))
2. ¬∀x(P(x)∨ ¬(∃y, ¬Q(x, y)))
3. ¬∀x(∃y, Q(x, y)→ ¬P(x))
Question 2 (20 pts): Skolemize the expressions (4 points each):
1. ∃xP (x)
2. ∀x∃yP (x, y)
3. ∃x, ∃y, ∀zP (x, y )∧Q(y, z)
4. ∀x, ∃y, ∃P(x, y)∧Q(y , z)
5. ∀x, ∀y, ∃zP (x, y )∧Q(y, z)
Question 3 (9 pts): Convert the following into a standard form:
∀x, [¬P(x)→ ¬(∃y, Q(x, y))]
1
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Download Artificial Intelligence - Practice Homework 2 | CPSC 420 and more Assignments Computer Science in PDF only on Docsity!

CPSC 420-500 Homework

Due 11/05/07 (Monday), 3pm

Handwritten or printed hardcopy must be submitted to the TA Total: 130 pts

Yoonsuck Choe

October 29, 2007

1 First-Order Logic

Important: In this section, assume that w, x, y, z are variables; A, B, C, D are constants; and f (·), g(·), h(·) are functions; and P (·), Q(·), R(·) are predicates.

1.1 Standard Forms

To do automatic theorem proving in first-order logic, you need to go through three steps to convert your initial first-order logic expression into a standard form. These are:

  1. Prenex normal form,
  2. Conjunctive normal form, and
  3. Skolemization.

Question 1 (12 pts): Convert to prenex normal form (4 points each):

  1. ∀x, ¬ (∃y, ¬P (x, y))
  2. ¬∀x (P (x) ∨ ¬(∃y, ¬Q(x, y)))
  3. ¬∀x (∃y, Q(x, y) → ¬P (x))

Question 2 (20 pts): Skolemize the expressions (4 points each):

  1. ∃xP (x)
  2. ∀x∃yP (x, y)
  3. ∃x, ∃y, ∀zP (x, y) ∧ Q(y, z)
  4. ∀x, ∃y, ∃P (x, y) ∧ Q(y, z)
  5. ∀x, ∀y, ∃zP (x, y) ∧ Q(y, z)

Question 3 (9 pts): Convert the following into a standard form:

∀x, [¬P (x) → ¬(∃y, Q(x, y))]

1.2 Substitution and Unification

Question 1 (9 pts): Apply the following substitutions to the expressions (3 point each);

  1. Apply {x/f (A)} to P (x, y) ∨ Q(x).
  2. Apply {x/A, y/f (z)} to P (x, y) ∨ Q(x).
  3. Apply {y/x} to P (x, y) ∨ Q(x).

Question 2 (16 pts): For each of the following, (1) find the unifier, and (2) show the unified expression. For example, given P (A) and P (x), the unifier would be {x/A}, and the unified expression P (A). If the pair of expressions is not unifiable, indicate so. (4 points each):

  1. P (x, f (B)) and P (A, f (y))
  2. P (x, f (A)) and P (y, y)
  3. P (x, f (y), y) and P (A, f (g(w)), g(A))
  4. P (A, f (y), y, A) and P (x, f (g(x)), g(B), w)

Question 3 (20 pts): Show that R(A) is a logical consequence of the following. Use resolution. Turn into a normal form as necessary.

  1. ∀x, ∀y, (¬P (x) → (Q(x, y) ∨ R(y)))
  2. ∃x, ¬P (x)
  3. ∀w, ∀z, (¬Q(w, z) ∨ R(w))

2 Uncertainty and Probabilistic Reasoning

B T^ T^ F^  F^ 

E T  F T  F

P(A) . . .

.

P(B) .

P(E)

Alarm



Earthquake

JohnCalls MaryCalls



Burglary

A P(J) T^ F^  . .

A P(M) T^ F  .70.

.

Figure 1: Belief Network. See problem 1.

Question 1 (10 pts): Given the Belief network as shown in figure 1, calculate the two joint probability values and answer the question. Note that in this section P (·) denotes the probability of the event. (5 points each):