Problem 5. Define the language{p: $ is a 3CNF formula over n variables and has a satisfying assignmentin which every clause has at least one FALSE literal.}NOTALL3SAT =In this problem you will prove that NOTALL3SAT is NP-Complete. If any truth assignment that satisfies phas the property that at least one literal in every clause is FALSE, then we say that the truth assignmentis a NOTALL3-satisfying assignment.A. Show that if all the variables in a NOTALL3-satisfying assignment are negated, then the result isalso a NOTALL3-satisfying assignment for p.B. Suppose that o has m clauses. Now, transform $(x1, ... , xn) into another 3CNF formulaY(x1, ... , Xn, yo, Y1, .. , Ym) as follows:Convert the ith clause (z¡1 V zi2 V Zt3) of o into (z¡1 V Zi2 V Yi) ^ (ỹ, V zi3 V yo)Note that p has 2m clauses, and that yo appears in m of them.Show that if p has a NOTALL3-satisfying assignment, then p is satisfiable.C. Show that if o is satisfiable then p has a NOTALL3-satisfying assignment.D. Prove that NOTALL3SAT is NP-Complete.

Solution A) In an NOTALL3SAT assignment each clause has at least one literal assigned 1 and at least…

Answered: NOTALL3SAT = {p: $ is a 3CNF formula…

NOTALL3SAT = {p: $ is a 3CNF formula over n variables and has a satisf ying assignment in which every clause has at least one FALSE literal.}

Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

Problem 1PE

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Question

Problem 5. Define the language
{p: $ is a 3CNF formula over n variables and has a satisfying assignment
in which every clause has at least one FALSE literal.}
NOTALL3SAT =
In this problem you will prove that NOTALL3SAT is NP-Complete. If any truth assignment that satisfies p
has the property that at least one literal in every clause is FALSE, then we say that the truth assignment
is a NOTALL3-satisfying assignment.
A. Show that if all the variables in a NOTALL3-satisfying assignment are negated, then the result is
also a NOTALL3-satisfying assignment for p.
B. Suppose that o has m clauses. Now, transform $(x1, ... , xn) into another 3CNF formula
Y(x1, ... , Xn, yo, Y1, .. , Ym) as follows:
Convert the ith clause (z¡1 V zi2 V Zt3) of o into (z¡1 V Zi2 V Yi) ^ (ỹ, V zi3 V yo)
Note that p has 2m clauses, and that yo appears in m of them.
Show that if p has a NOTALL3-satisfying assignment, then p is satisfiable.
C. Show that if o is satisfiable then p has a NOTALL3-satisfying assignment.
D. Prove that NOTALL3SAT is NP-Complete.

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