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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a) Let Γ be a set of sentences of L1 (the language ofpropositional logic) and let φ be a sentence of L1.Consider:If the argument from Γ to φ is not valid, theargument from Γ to ¬φ is valid.Is this claim correct? Why or why not?
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.
Part 1: Proof-Theoretic Concepts Show that each of the following pairs of sentences are provably equivalent in SL
1. P → R, ¬R → ¬P
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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