SAT

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SAT

• J. Gonzalez

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Summary

• Introduction
• Algorithms for SAT
• Splitting and Resolution
• Local Search
• Integer Programming Method
• Global Optimization
• SAT Solvers Features
• Final Thoughts

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SAT

• Decission Problem
• Assing true and false values to make the sentence
  true
• NP-Complete

. . .
Literal
Literal
?
Clauses
Clauses
?
Sentences In CNF

• Literals An atom or its negation
• Clauses Disjunction of Literals
• Sentence in CNF Conjunction of Clauses

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CNF and DNF

• CNF Conjunctive normal form
• DNF Disjunctive normal form

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CSP (Constraint Satisfaction Problems)

• Variables WA, NT, Q, NSW, V, SA, T
• Domains Di red,green,blue
• Constraints
• WA ? NT,
• WA ? SA
• NT ? SA
• NT ? Q
• SA ? Q
• SA ? NSW
• SA ? V
• Q ? NSW
• NSW ? V
• V ? T

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CSP (Constraint Satisfaction Problems)

• Variables WA, NT, Q, NSW, V, SA, T
• Domains Di red,green,blue
• Constraints
• WA ? NT,
• NT ? SA
• NT ? Q
• SA ? Q
• SA ? NSW
• SA ? V
• Q ? NSW
• NSW ? V
• V ? T

XWA,red ? XWA,green ? XWA,blue XNT,red ?
XNT,green ? XNT,blue XWA,red ?
XNT,red XWA,green ? XNT,green XWA,blue ?
XNT,blue
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CSP (Constraint Satisfaction Problems)

• Variables WA, NT, Q, NSW, V, SA, T
• Domains Di red,green,blue
• Constraints
• WA ? NT,
• NT ? SA
• NT ? Q
• SA ? Q
• SA ? NSW
• SA ? V
• Q ? NSW
• NSW ? V
• V ? T

XWA,red ? XWA,green ? XWA,blue XNT,red ?
XNT,green ? XNT,blue XWA,red ?
XNT,red XWA,green ? XNT,green XWA,blue ?
XNT,blue
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CSP (Constraint Satisfaction Problems)

• Variables WA, NT, Q, NSW, V, SA, T
• Domains Di red,green,blue
• Constraints
• WA ? NT,
• NT ? SA
• NT ? Q
• SA ? Q
• SA ? NSW
• SA ? V
• Q ? NSW
• NSW ? V
• V ? T

(XWA,red ? XWA,green ? XWA,blue) ? (XNT,red ?
XNT,green ? XNT,blue) ? (XWA,red ? XNT,red)
? (XWA,gree ? XNT,gree) ? (XWA,blue ?
XNT,blue)
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CSP (Constraint Satisfaction Problems)

• SAT is an special case of CSP.
• CSP to SAT
• CSP
• Discrete CSP
• N-queen problem
• Graph coloring problem
• Scheduling problem
• Binary CSP
• SAT problem
• Max-SAT problem

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Summary

• Introduction
• Algorithms for SAT
• Splitting and Resolution
• Local Search
• Integer Programming Method
• Global Optimization
• SAT Solvers Features
• Final Thoughts

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Algorithms for SAT

• Discrete Constrained Algorithms
• Discrete, satisfies all clauses
• Goal satisfy all clauses (constraints )
• Usually uses Splitting (search) and/or Resolution
• Discrete Unconstrained Algorithms
• Discrete, minimize a function
• Minimize the number of unsatisfied clauses
• Usually uses Local Search.
• Constrained Programming Algorithms
• Non discrete, satisfies all clauses
• From CNF to IP (Integer Programming).
• Usually uses Linear Programming relaxations
• Unconstrained Global Optimization Algorithms
• Non discrete, minimize a function (constrains
  included in such function)
• From a formula on Boolean Space (a decision
  problem) to an unconstrained problem on real
  Space (unconstrained global optimization
  problem).
• Usually uses many global optimization methods.

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Summary

• Introduction
• Algorithms for SAT
• Splitting and Resolution
• Local Search
• Integer Programming Method
• Global Optimization
• SAT Solvers Features
• Final Thoughts

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Splitting and Resolution

• Replace one formula by one or more other
  equivalent formulas.
• Splitting
• A variable v is replaced by True and False.
• Two sub-formulas are generated.
• The original formula has a satisfying truth
  assignment iff either sub-formula has one
  satisfying truth assignment .

Ptrue
PFalse
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Splitting and Resolution

• Stop recursion
• Any formula with no variables is True
• All subformulas are false and there is no
  formulas with variables

Satisfiable
Unsatisfiable
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Splitting and Resolution

• Resolution Using a resolvent to create new
  clauses.
• Resolvent clauses transformation based on a
  given variable v.

Resolvent
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Splitting and Resolution

• Resolution Using a resolvent to create new
  clauses.
• Resolvent clauses transformation based on a
  given variable v.

Resolvent
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Splitting and Resolution

• Stop recursion
• Formulas with an empty clause have no solution
• No more resolvent could be created

Unsatisfiable
Satisfiable
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Splitting and Resolution

• First methods
• DP (Davis-Putnam) Resolution
• DPL (Davis-Putnam-Loveland) Splitting (
  depth-first search avoids memory explosion)

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Summary

• Introduction
• Algorithms for SAT
• Splitting and Resolution
• Local Search
• Integer Programming Method
• Global Optimization
• SAT Solvers Features
• Final Thoughts

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Local Search

• Begins with an initial vector y0
• F(y0) Number of unsatisfiable formulas
• We define neighbourhood function N(yi)
• N(y0)y1
• F(yi1)gtF(y0) strategies are applied to help
  escape from the local minima.

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True(C), False(A,B,D)
F2
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True(C), False(A,B,D)
F2
True(A,C), False(B,D)
F1
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True(C), False(A,B,D)
F2
True(A,C), False(B,D)
F1
True(A,B,C), False(D)
F1
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True(C), False(A,B,D)
F2
True(A,C), False(B,D)
F1
True(A,B,C), False(D)
True(A,C,D), False(B)
F1
F1
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True(C), False(A,D,B)
F2
True(C,D), False(A,B)
F1
True(C,D,A), False(B)
True(C,D,B), False(A)
F1
F0
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Summary

• Introduction
• Algorithms for SAT
• Splitting and Resolution
• Local Search
• Integer Programming Method
• Global Optimization
• SAT Solvers Features
• Final Thoughts

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Integer Programming Method (IP)
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Integer Programming Method (IP)
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Integer Programming Method (IP)
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Integer Programming Method (IP)
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Integer Programming Method (IP)
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Integer Programming Method (IP)

• Solving
• Linear Programs (LP) solved by the Simplex
• Integer Programs (IP) no fast technique.
• Integer Linear Programs (ILP) Is a LP
  constrained by integrality restrictions
• Method
• Solve a LP
• If solution is integer gt end
• If solution is non integer, round off such values
• If it is a solution gt end
• Else adds a new constraint and try again

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Summary

• Introduction
• Algorithms for SAT
• Splitting and Resolution
• Local Search
• Integer Programming Method
• Global Optimization
• SAT Solvers Features
• Final Thoughts

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Global optimization

• Continuous Unconstrained Formulation
• From discrete to continuous (UniSAT)
• Try to minimize a function instead of trying to
  verify concrete restrictions.

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Summary

• Introduction
• Algorithms for SAT
• Splitting and Resolution
• Local Search
• Integer Programming Method
• Global Optimization
• SAT Solvers Features
• Final Thoughts

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Which is the best method?

• Local search
• Faster for satisfiable CNF
• Can not prove unsatisfiability
• DP algorithm (split, resolution)
• Slower for satisfiable CNF
• Can prove unsatisfiability

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Algorithm categories

• Complete
• Find the unique hard solution
• Determine whether or not a solution exits
• Give the variable settings for one solution
• Find all solutions or an optimal solution
• Prove that there is no solution
• Incomplete
• Verify that there is no solution but could not
  find one.
• Can not optimize solution quality

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Performance evaluation

• Performance evaluation
• Experimentally
• Works for random formulas
• Does not work for worst-case formulas (too many
  formulas for every given size)
• Sometimes inconclusive
• Analytically
• Works for random formulas
• Works for worst-case formulas
• Does not work for typical formulas (which is his
  mathematical structure)
• Need simplified algorithms

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P-Subclasses of SAT

• Solved in polynomial time
• Determine if a formula (or a portion) is
  polynomial-time solvable
• The study of this subclasses reveals the nature
  of these easy SAT formulas
• Subclasses
• 2-SAT CNF formula with clauses of one or two
  literals only
• Horn Formulas CNF formula where every formula
  has at most one positive literal.
• Extended Horn Formulas Rounding the results of a
  Linear Programming method we obtain the real
  solutions.
• Balanced formulas When a Linear Programming
  method can be used to obtain integer solutions.

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Applications

• Mathematics
• Computer Science and AI
• Machine Vision
• Robotics
• Computer-aided manufacturing
• Database systems
• Text processing
• Computer graphics
• Integrated circuit design automation
• Computer architecture design
• High-speed networking
• Communications
• Security

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Summary

• Introduction
• Algorithms for SAT
• Splitting and Resolution
• Local Search
• Integer Programming Method
• Global Optimization
• SAT Solvers Features
• Final Thoughts

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A few names

• SatELite CNF minimizer (MiniSAT)
• MiniSAT the SAT solver properly
• Authors Niklas Eén and Niklas Sörensson
• Vallst by Daniel Vallstrom
• Kcnfs is a generalization of cnfs. Olivier
  Dubois and Gilles Dequen
• Ranov by by LC Researcher Anbulagan and Nghia
  Duc Pham
• Zchaff Boolean Satisfiability Research Group at
  Princeton University

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Some last ideas

• Produce each solution gt exponential in the worst
  case (whether or not PNP)
• List an exponential number of solutions
• Solutions in compressed form
• Cylinders of solutions variables not listed
  could have any value.
• BDD Binary Decision Diagrams
• SAT solvers can be faster than other non-NP
  systems under some circumstances.

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Bibliography

• Algorithms for the Satisfiability (SAT) Problem
  A Survey.
• Jun Gu, Paul W. Purdom, John Franco, Benjamin W.
  Wah
• SAT 2004 http//www.satisfiability.org/SAT04/
• SAT 2005 http//www.satisfiability.org/SAT05/
• SAT 2006 http//www.easychair.org/FLoC-06/SAT.htm
  l