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Consider the following sentence in 3-CNF with m=8 clauses. (A v B v D) A (B vč v D) ^ (c v Ā V D) A (A v B V D) (c vā v D) (A v B v D) S = A V B V В у (B v C v D) A (c vĀ v D ^ (A V B V C) ^ (A v B V C) V Determine whether this sentence is satisfiable. Answer either YES or NO.

Consider the following sentence in 3-CNF with m=8 clauses. (A v B v D) A (B vč v D) ^ (c v Ā V D) A (A v B V D) (c vā v D) (A v B v D) S = A V B V В у (B v C v D) A (c vĀ v D ^ (A V B V C) ^ (A v B V C) V Determine whether this sentence is satisfiable. Answer either YES or NO.

Computer Networking: A Top-Down Approach (7th Edition)
Computer Networking: A Top-Down Approach (7th Edition)
7th Edition
ISBN:9780133594140
Author:James Kurose, Keith Ross
Publisher:James Kurose, Keith Ross
Chapter1: Computer Networks And The Internet
Section: Chapter Questions
Problem R1RQ: What is the difference between a host and an end system? List several different types of end...
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In Boolean logic, a sentence is in 3-Conjunctive Normal Form (abbreviated 3-CNF) if it of the form C¡ ^ C2 ^ C3 ^ .…. A Cm, where S is the conjunction of m clauses, with each clause being a disjunction of S three literals. For example, the following sentence is in 3-CNF, with m=4 clauses. (A v B v č) ^ (A v c v D)^ (B v vc v D)A (a v E v D) S = A V B V C A V C D) A B V В For example, the first clause is equivalent to this in English: "either A or B or Not C". As long as one of these three literals is TRUE then this clause evaluates to TRUE. If none of these literals is TRUE (i.e., A=FALSE, B=FALSE, C=TRUE), then this clause evaluates to FALSE. We say that S is satisfiable if there exists an assignment of TRUE/FALSE values to each variable (A, B, C, D) so that the entire sentence S evaluates to TRUE. What this means is that each of the m clauses must evaluate to TRUE. The above sentence indeed is satisfiable; one possible assignment is A=TRUE, B=FALSE, C=FALSE, D=TRUE. The first two clauses are TRUE since A=TRUE, while the last two clauses are TRUE since B=FALSE. Since all four clauses evaluate to TRUE, S is indeed satisfiable. Consider the following sentence in 3-CNF with m=8 clauses. (A v E v D) ^ (B v č v D) ^ (c v Ā v D) ^ (B v č v D) ^ (c vĀ V ) A (ī v B v č) S (A V B V V C v A v D) ^ (A V B V C) ^ A V Determine whether this sentence is satisfiable. Answer either YES or NO.
Transcribed Image Text:In Boolean logic, a sentence is in 3-Conjunctive Normal Form (abbreviated 3-CNF) if it of the form C¡ ^ C2 ^ C3 ^ .…. A Cm, where S is the conjunction of m clauses, with each clause being a disjunction of S three literals. For example, the following sentence is in 3-CNF, with m=4 clauses. (A v B v č) ^ (A v c v D)^ (B v vc v D)A (a v E v D) S = A V B V C A V C D) A B V В For example, the first clause is equivalent to this in English: "either A or B or Not C". As long as one of these three literals is TRUE then this clause evaluates to TRUE. If none of these literals is TRUE (i.e., A=FALSE, B=FALSE, C=TRUE), then this clause evaluates to FALSE. We say that S is satisfiable if there exists an assignment of TRUE/FALSE values to each variable (A, B, C, D) so that the entire sentence S evaluates to TRUE. What this means is that each of the m clauses must evaluate to TRUE. The above sentence indeed is satisfiable; one possible assignment is A=TRUE, B=FALSE, C=FALSE, D=TRUE. The first two clauses are TRUE since A=TRUE, while the last two clauses are TRUE since B=FALSE. Since all four clauses evaluate to TRUE, S is indeed satisfiable. Consider the following sentence in 3-CNF with m=8 clauses. (A v E v D) ^ (B v č v D) ^ (c v Ā v D) ^ (B v č v D) ^ (c vĀ V ) A (ī v B v č) S (A V B V V C v A v D) ^ (A V B V C) ^ A V Determine whether this sentence is satisfiable. Answer either YES or NO.
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