Artificial Intelligence: A Modern Approach
3rd Edition
ISBN: 9780136042594
Author: Stuart Russell, Peter Norvig
Publisher: Prentice Hall
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Chapter 7, Problem 11E
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Sentence in CNF
- Each possible world can be written as a conjunction of literals.
- Asserting that a possible world is not the case can be written by negating that...
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A propositional logic expression is in full disjunctive normal form if it is a disjunction (i.e., a term that uses only the 'OR' operator) of one or more conjunctions (i.e., terms that uses only the 'AND'
operator), and each of the variables in the expression appears once (and only once) in each conjunction. Using this definition, what is the full disjunctive normal form of the expression below?
(¬((P V (¬R)) → (¬Q)))
Select one:
(¬P AQA¬R) V (PA¬Q A ¬R) V (PAQA¬R)
(¬P ^¬QA¬R) V (¬P ^ Q ^ ¬R) V (P ^ Q A R)
(¬P A¬QA ¬R) V (¬P AQA¬R) V (PAQA¬R)
O none of these options
(¬P AQA¬R) V (PAQA¬R) V (PAQA R)
(¬P AQA ¬R) V (¬P ^QA R) V (P AQA R)
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.
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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Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, computer-science and related others by exploring similar questions and additional content below.Similar questions
- How to Draw a first-order predicate logic proposition for complete sentences?arrow_forwardProve the predicate logic statement is valid: (∀ x)(P(x)) ∧ (∀ x)(Q(x)) → P(a) ∧ Q(b)arrow_forwardProve 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))arrow_forward
- For this question about predicate logic, please note that, even though the 'nonsense' words are only nouns and verbs, you do not need to know the meaning of the words being used in order to answer this question. Consider a universe of discourse that contains (among other things) all gudgeons. If the predicates B(x), C(x), and G(x) represented the assertions x brabbles, x corrades, and x groaks, respectively, then which of the following would be an accurate translation of the following assertion? some gudgeons do not brabble even though they corrade and groak Select one: (-B(=) ^ C(z) ^ G(x)) O none of these options (-B(2) v C(x) V G(=)) O Vz (-B(x) ^ C(2) V G(z) O I (¬B(x) V C(x) ^ G(x)) (-B(2) ^ C(x) V G(=)) (-B(=) ^ C(x) V G(=) O Frarrow_forwardFor this question about predicate logic, please note that, even though the 'nonsense' words are only nouns and verbs, you do not need to know the meaning of the words being used in order to answer this question. Consider a universe of discourse that contains (among other things) all gudgeons. If the predicates K(x), G(x), and J(x) represented the assertions x kenchs, x groaks, and x jargogles, respectively, then which of the following would be an accurate translation of the following assertion? if any gudgeon kenchs or that gudgeon does not groak then that gudgeon does not jargogle Select one: ((K(=) v G(2) → J(=) IE O (K(e)v -G(=) O Væ V → ¬J(x) (K(=) ^ -G(2)) → ¬J(x) TE O → ¬J(x) O none of these options (Ke)v -G(=) O Va V V ¬J(x) (K(2) V -G(2) v V ¬J(x)arrow_forwardExercise 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.arrow_forward
- Please Help with the question below: Suppose you are given some facts in First-Order Logics: a) Andi is a professor b) All professors are people. c) Ani is the dean. d) All Deans are professors. e) All professors consider the dean a friend or don’t know him. f) Everyone is a friend of someone. g) People only criticize people that are not their friends. h) Andi criticized Ani. Prove that: Ani is not Andi’s friend.arrow_forwardConvert each of the following argu- ments into formal statements, e.g., define sentences existentially and/or universally quantified statements. Then determine which rules of logic have been applied and explain whether or not they have been ap- plied correctly.arrow_forwardLogic's potential uses need to be taken into account (propositional and predicate).arrow_forward
- Question 3 VX(P(X) v Q(X))→ (VXP(X) V VXQ(X)) The above expression follows from the valid argument forms of logic and the rules for quantifiers. True False Question 4 Give an interpretation (in words) of the predicates in the previous question that shows you understand why your answer is correct.arrow_forwardConsider the predicates Martian (x): x is a MartianisGreen(x): x is green Use equivalence laws of first-order logic to identify the expression that is logically equivalent to ∀x (Martian(x) ˄ isGreen(x)) Group of answer choices ∃x (Martian(x)) ˅ ∃x (isGreen(x)) ∃x (¬Martian(x)) ˄ ∃x (¬isGreen(x)) ∀x (Martian(x)) ˄ ∀x (isGreen(x)) ∀x (¬Martian(x)) ˄ ∀x (¬isGreen(x))arrow_forwardConsider a First-Order Logic (FOL) language L with two unary predicate symbols P and Q, one unary function f, and two constants a and b. The following KB has been specified in this language (note that I use directly "for all" to stand for the logical quantifier of "for-all"). 1. (forall x)(Q(x) -> P(x)) (Note: the sentence is logically equivalent to (forall x) (~ Q(x) \/ P(x))).2. P(b)3. Q(b) 4. Q(f(a)) 5. P(f(b)) Consider an interpretation of <D, I>, where D={d1, d2, d3}, and d1 is denoted by a, d2 is denoted b. For the function f, I(f,d1)=d2, I(f,d2)=d3, I(f,d3)=d1. and I(P)={d1, d2}, I(P)={d1, d2}. Is this interpretation a model of L?arrow_forward
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