Artificial Intelligence: A Modern Approach
3rd Edition
ISBN: 9780136042594
Author: Stuart Russell, Peter Norvig
Publisher: Prentice Hall
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Expert Solution & Answer
Chapter 7, Problem 17E
a.
Explanation of Solution
Resolution
- The negated goal is ￢G.
- The last two clauses is resolved to produce ￢C and ￢D...
b.
Explanation of Solution
Clauses
- First, each 2-CNF clause has two places to put literals...
c.
Explanation of Solution
Propositional resolution
- Resolving two 2-CNF clauses cannot increase the clause size...
d.
Explanation of Solution
Argument
- First, the number of 3-CNF clauses is O(n3), so for nonpolynomial complexity on the basis of the number of different clauses...
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Students have asked these similar questions
Consider the wffs:φ1 ≡ p1 → (p2 → (p3 → p4))φ2 ≡ (p1 ∧ p2 ∧ p3) → p4(a) Technically speaking, neither φ1 nor φ2 is well-formed since neither is allowed by the formal syntaxof propositional logic. Correct them. Note, however, that we will freely make such trivial ’errors’throughout this semester (as do most such courses).(b) Use truth tables (in the form defined in this course) to show that φ1 ↔ φ2.(c) After internalizing an intuitive understanding of this equality, propose an extension of it to natoms.(d) State the number of rows in a truth table for proving the extension.
Given a CFG G over {a, b}, G: S -> SAB | a | B | A -> a | ABaB-> b | ab(a) Prove that G is ambiguous.
Will the following strings be generated by G?If yes, show a derivation sequence. If no, do NOT need to explain.(b) baba
(c) aaaa
Suppose P and Q are (possibly molecular) propositional statements. Prove that P and Q are logically equivalent if any only if P ↔ Q is a tautology. Explain.
Hint: What do these concepts mean in terms of truth tables?
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
Knowledge Booster
Similar questions
- Given a boolean formula in conjunctive normal form with M clauses and N literals, find a satisfying assignment (if one exists), such that each clause has precisely two literals. With 2N vertices (one for each literal and its negation), create an implication digraph. For each phrase x + y, take into account the edges from y' to x and x' to y. In order to fulfil the condition x + y, both (i) x must be true if y is false and (ii) y must be true if x is false. If and only if no variable x is in the same strong component as its negation x', the formula is true. Additionally, a topological sort of the kernel DAG (which reduces each strong component to a single vertex) results in an assignment that is adequate.arrow_forwardSuppose propositional sentences and knowledge-base here. We have a knowledge-base called KB. KBPrime is defined as an union of (1) KB and (2) the negation of a propositional predicate P. Answer true/false to the following questions. Note that "|=" is used to denote entailment, and "|-" is used to denote derivation through resolution. a) If KB |= P, then KB |- P. That is, if KB entails P, then KB derives (using resolution) P. b) If KB |- P, then KB |= P. c) If KBPrime |- [], then KBPrime is not satisfiable. d) If KBPrime is not satisfiable, then KBPrime |- [].arrow_forwardGiven four propositions A, B, C, and D, how many models are there for the sentence “(C or D) and not A”?arrow_forward
- Translate the given sentence into a logical expression involving quantifiers, predicates, and logical connectives. Specify a simple domain (ex. "People") and the English statement that each predicate represents. 1. Everyone who lives in the United States has a relative who lives in another country.arrow_forwardExercise 1.4.1: Proving tautologies and contradictions. About Show whether each logical expression is a tautology, contradiction or neither. (a) (p ∨ q) ∨ (q → p) (b) (p → q) ↔ (p ∧ ¬q) (c) (p → q) ↔ p (d) (p → q) ∨ p (e) (¬p ∨ q) ↔ (p ∧ ¬q) (f) (¬p ∨ q) ↔ (¬p ∧ q)arrow_forwardA propositional knowledge-base KB consisting of five sentences is given below (note that "/\" is used to denote "logical and", and "\/" is used to denote "logical or", and "~" is used to denote "logical negation"): 1. (~P /\ ~Q ) -> R2. R -> S3. ~S4. P -> ~U5. U a) Explain in English (in Steps!) that KB |= Q. b) Show resolution steps that leads to KB |- Q. You should first convert the sentences in the KB into clauses.arrow_forward
- Consider the following simple Probabilistic Context-Free Grammar (PCFG), which we will use to parse the sentence “student types program with keyboard”. . Find each of the possible parses of the sentence using the grammar, and draw the parse tree for each of these parses. Show the probability at each node. b. For each parse, calculate its probability, which will be the product of the probabilities for all the rules used in generating it. Which one is the most probable and the most likely parse for a human reader?arrow_forwardA propositional knowledge-base KB consisting of five sentences is given below (note that "/\" is used to denote "logical and", and "\/" is used to denote "logical or", and "~" is used to denote "logical negation"): 1. (~P /\ ~Q ) -> R2. R -> S3. ~S4. P -> ~U5. U (a). Explain in English that KB |= Q.(b). Show resolution steps that leads to KB |- Q. You should first convert the sentences in the KB into clauses.arrow_forwardIf there are n propositional variables in a propositional expression, what is a function f(n) that yields the number of rows that we need in a truth table to fully examine all possible outcomes of that expression? Why?arrow_forward
- Consider 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_forwardSimplify using the Logical Equivalence Laws or Algebra of Propositions. Do not simplify by indicating the binaries. 22. (~P -> (P -> Q)) -> (Q -> (P -> P))arrow_forwardFind a satisfactory assignment (if one exists) given a boolean formula in conjunctive normal form with M clauses and N literals, such that each clause contains precisely two literals. Make an implication digraph with 2N vertices (one for each literal and its negation). Include edges from y' to x and x' to y for each clause x + y. To satisfy the condition x + y, (i) if y is false, x is true, and (ii) if x is false, y is true. The formula is satisfying if and only if no variable x is in the same strong component as its negation x'. Furthermore, a topological sort of the kernel DAG (contracting each strong component to a single vertex) produces a satisfactory assignment.arrow_forward
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