Chapter%2034:%20Satisfiability

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Chapter 34 Satisfiability

• Dan Hand
• COT 4810
• January 27, 2003

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Presentation Organization

• Overview
• Typical problem applications
• Boolean expressions
• Logical network
• Solution Methods
• Exhaustive search
• Davis-Putnum Algorithm
• Relation class NP
• Applications to less obvious problems

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Overview

• The Satisfiability Problem in its most general
  form is determining whether it is possible to
  satisfy a given set of constraints
• Could take many forms
• Boolean expressions logic circuits
• Many other applications as well
• Was the original NP-Complete Problem
• Shown in 1970 by Stephen Cook

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CNF-SAT Problem

• CNF Acronym for Conjunctive Normal Form which
  is a boolean expression given as a product of
  sums
• Let us consider the example
• (A ? B) ? (B ? C) ? (A ? B)
• Could easily use an exhaustive search in the form
  of a truth table to solve this problem

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Circuit-SAT Problem

• Given a logic network, is there a sequence of
  inputs such that the network returns true as
  output?
• Let the the logical network be represented by the
  equation
• (A B) (B C) (A B)
• We can do several things to determine whether
  this network can be satisfied
• One of the first things we might try is to
  simplify the network to a sum of products if
  possible and look for solutions

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Davis-Putnam Algorithm

• Recursive Algorithm that creates a search tree by
  making assignments to the remaining variables at
  each stage of the algorithm
• Performs better on average than an exhaustive
  search
• Effectively prunes failed branches from the
  search tree

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Davis-Putnam Algorithm Continued

• Lets apply the Davis-Putnam Algorithm to our
  example
• (A ? B) ? (B ? C) ? (A ? B)

Davis-Putnam Algorithm
procedure split(E) if E has an empty clause,
then return if E has no clauses, then exit with
current partial assignment select next
unassigned variable, xi in E split(E(xi0)) spli
t(E(xi1))
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Davis-Putnam Algorithm Continued
Application to (A ? B) ? (B ? C) ? (A ? B)
(A ? B) ? (B ? C) ? (A ? B)
A0
A1
B ? (B ? C) ? B
B ? C
B0
B1
B0
B1
(?)
(?)
C
?
C0
C1
(?) Failed Partial Assignment ? Successful
Partial Assignment
?
(?)
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Davis-Putnam Algorithm Continued

• Unfortunately in the worse case the Davis-Putnam
  Algorithm is still exponential
• Currently there is no known solution that
  provides a worse case performance better than
  exponential

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Other Applications?

• So far we have seen instances of the
  Satisfiability problem to pure logic applications
• However it is possible to apply to many other
  types of problems as well
• In fact the problem of Satisfiability is a
  central problem and has limitless applications

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Complexity Classes

• Review
• P Problems which can be solved in polynomial
  time
• NP Problems which can be solved in
  non-deterministic polynomial time
• All problems in the NP class can be reduced (in
  polynomial time) to problems which are said to be
  NP-Complete
• P NP??? Could be shown if one could find a
  polynomial time solution for a NP-Complete
  problem such as the Satisfiability problem.

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Complexity Classes Continued
Intractable Problems
Class NP
Can be visualized in the form of a set chart with
problem hardness increasing radially
Class P
Increasing Hardness
NP-Complete Problems exist at very edge of Class
NP
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Transformation Of Other NP Problems Into SAT
Problems
Provides a method to transform any problem in NP
class into an instance of the Satisfiability
problem. Therefore, we can say that the
Satisfiability problem is NP-Complete.
Instance of the Satisfiability Problem
Any Problem In Complexity Class NP
Transformation
Some Examples Include Hamiltonian cycle
traveling salesperson, 3-graph coloring,
partition problem, etc
Transform the problem to be solved from an
optimization to a decision. This can be done in
polynomial time
Output is a series of true or false
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Implications

• Since the Satisfiability problem is NP-Complete,
  if a polynomial time solution for the
  Satisfiability problem was known then we could
  solve all the problems in the NP complexity class
  in polynomial time.
• This would essentially collapse the complexity
  class NP into P

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Example Transform Traveling Salesperson Into Sat
Problem

• Original Problem Statement What is the shortest
  round trip path that can be taken to visit each
  city in a set exactly once?
• Output is a path (assuming one exists)
• Transformed Problem Statement Is there a round
  trip path that can be taken to visit each city in
  a set exactly once less than some finite number?
• Output here is either yes or no
• Original problem can be answered by repeating the
  transformed problem statement with different
  values

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Example Is A Graph 3-Colorable?

• Given a graph we would like to know if the
  graphs vertices can be colored with only three
  colors such that no vertices with an edge
  connecting them share the same color.
• Homework