Artificial Intelligence: A Modern Approach
3rd Edition
ISBN: 9780136042594
Author: Stuart Russell, Peter Norvig
Publisher: Prentice Hall
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Textbook Question
Chapter 7, Problem 19E
A sentence is in disjunctive normal form (DNF) if it is the disjunction of conjunctions of literals. For example, the sentence (A ∧ B ∧ ￢C) ∨ (￢A ∧ C) ∨ (B ∧￢C) is in DNF.
- a. Any propositional logic sentence is logically equivalent to the assertion that some possible world in which it would be true is in fact the case. From this observation, prove that any sentence can be written in DNF.
- b. Construct an
algorithm that converts any sentence in propositional logic into DNF. (Hint: The algorithm is similar to the algorithm for conversion to CNF given in Section 7.5.2.) - c. Construct a simple algorithm that takes as input a sentence in DNF and returns a satisfying assignment if one exists, or reports that no satisfying assignment exists.
- d. Apply the algorithms in (b) and (c) to the following set of sentences:
A ⇒ B
B ⇒ C
C ⇒ ￢A.
- e. Since the algorithm in (b) is very similar to the algorithm for conversion to CNF, and since the algorithm in (c) is much simpler than any algorithm for solving a set of sentences in CNF, why is this technique not used in automated reasoning?
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Exercise 1 Let B be a set of Boolean variables and P be a propositional logic formula over B. If P always evaluates to 1, no matter than assignments to each b ∈ B, then P is called a tautology. Formulate this definition as an expression in first order predicate logic.
Prove the following predicate logic statements valid or invalid:
A) ∀x(A(x) → B(x)) ∧ ∀x(A(x) ∨ ¬C(x)) ∧ ∃x(¬B(x)) → ¬∃x(¬A(x) ∧ ¬B(x) ∧ ¬C(x))
B) ∀x(A(x) → B(x)) ∧ ∃x(A(x) ∨ B(x)) → ∃x(A(x) ∧ B(x))
1. Teachers in the Middle Ages supposedly tested the real-time propositional logic ability of a student via a technique known as an obligato game. In an obligato game, a number of rounds is set and in each round the teacher gives the student successive assertions that the student must either accept or reject as they are given. When the student accepts an assertion, it is added as a commitment; when the student rejects an assertion its negation is added as a commitment. The student passes the test if the consistency of all commitments is maintained throughout the test.
a.) Suppose that in a three-round obligato game, the teacher first gives the student the proposition p → q, then the proposition ¬(p ∨ r) ∨ q, and finally the proposition q. For which of the eight possible sequences of three answers will the student pass the test?
b.) Explain why every obligato game has a winning strategy.
Chapter 7 Solutions
Artificial Intelligence: A Modern Approach
Ch. 7 - Suppose the agent has progressed to the point...Ch. 7 - (Adapted from Barwise and Etchemendy (1993).)...Ch. 7 - Prob. 3ECh. 7 - Which of the following are correct? a. False |=...Ch. 7 - Prob. 5ECh. 7 - Prob. 6ECh. 7 - Prob. 7ECh. 7 - We have defined four binary logical connectives....Ch. 7 - Prob. 9ECh. 7 - Prob. 10E
Ch. 7 - Prob. 11ECh. 7 - Prob. 12ECh. 7 - Prob. 13ECh. 7 - Prob. 14ECh. 7 - Prob. 15ECh. 7 - Prob. 16ECh. 7 - Prob. 17ECh. 7 - Prob. 18ECh. 7 - A sentence is in disjunctive normal form (DNF) if...Ch. 7 - Prob. 20ECh. 7 - Prob. 21ECh. 7 - Prob. 23ECh. 7 - Prob. 24ECh. 7 - Prob. 25ECh. 7 - Prob. 26ECh. 7 - Prob. 27E
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