Propositional Satisfiability (SAT)

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Propositional Satisfiability(SAT)

• Toby Walsh
• Cork Constraint Computation Centre
• University College Cork
• Ireland
• 4c.ucc.ie/tw/sat/

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Every morning

• I cycle across the River Lee in Cork
• And see this rather drab house

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Every morning

• I read the plaque on the wall of this house
• Dedicated to the memory of George Boole
• Professor of Mathematics at Queens College (now
  University College Cork)

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George Boole (1815-1864)

• Boolean algebra
• The Mathematical Analysis of Logic, Cambridge,
  1847
• The Calculus of Logic, Cambridge and Dublin
  Mathematical journal, vol. 3, 1848
• Reduced propositional logic to algebraic
  manipulations

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George Boole (1815-1864)

• Boolean algebra
• The Mathematical Analysis of Logic, Cambridge,
  1847
• The Calculus of Logic, Cambridge and Dublin
  Mathematical journal, vol. 3, 1848
• Reduced propositional logic to algebraic
  manipulations

Moon crater named after him close to the Babbage
crater
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Outline

• What is SAT?
• How do we solve SAT?
• Why is SAT important?

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Propositional logic

• Long history
• Aristotle
• Leibnitz
• Boole
• Foundation for formalizing deductive reasoning
• Used extensively to automate such reasoning in
  Artificial Intelligence

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A diplomatic problem

• As chief of staff, you are to sent out
  invitations to the embassy ball.
• The ambassador instructs you to invite Peru or
  exclude Qatar.
• The vice-ambassador wants you to invite Qatar or
  Romania or both.
• A recent diplomatic incidents means that you
  cannot invite both Romania and Peru
• Who do you invite?

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A diplomatic problem

• As chief of staff, you are to sent out
  invitations to the embassy ball.
• The ambassador instructs you to invite Peru or
  exclude Qatar.
• The vice-ambassador wants you to invite Qatar or
  Romania or both.
• A recent diplomatic incidents means that you
  cannot invite both Romania and Peru
• Who do you invite?

• P v -Q
• Q v R
• -(R P)

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A diplomatic problem

• As chief of staff, you are to sent out
  invitations to the embassy ball.
• The ambassador instructs you to invite Peru or
  exclude Qatar.
• The vice-ambassador wants you to invite Qatar or
  Romania or both.
• A recent diplomatic incidents means that you
  cannot invite both Romania and Peru
• Who do you invite?

• P v -Q
• Q v R
• -(R P)
• P,Q,-R or -P,-Q,R

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Propositional satisfiability (SAT)

• Given a propositional formula, does it have a
  model (satisfying assignment)?
• 1st decision problem shown to be NP-complete
• Usually focus on formulae in clausal normal form
  (CNF)

P iff Q, Q iff R, -(P iff R)
P -gt Q, -Q, P
P v Q, P -Q
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Clausal Normal Form

• Formula is a conjunction of clauses
• C1 C2
• Each clause is a disjunction of literals
• L1 v L2 v L3,
• Empty clause contains no literals (False)
• Unit clauses contains single literal
• Each literal is variable or its negation
• P, -P, Q, -Q,

P v Q, -P v -Q
P v -Q, Q v -R, P v -R
P v Q v R, -P v -R
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Clausal Normal Form

• k-CNF
• Each clause has k literals
• 3-CNF
• NP-complete
• Best current complete methods are exponential
• 2-CNF
• Polynomial (indeed, linear time)

P v Q, -P v -Q
P v -Q, Q v -R, P v -R
P v Q v R, -P v -R
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Converting to CNF

• Eliminate iff and implies
• P iff Q gt P implies Q, Q implies P
• P implies Q gt -P v Q
• Push negation down
• -(P Q) gt -P v -Q
• -(P v Q) gt -P -Q

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Converting to CNF

• Clausify using De Morgans laws
• E.g. P v (Q R) gt (P v Q) (P v R)
• In worst case, formula may grow exponentially in
  size
• Can introduce new literals to prevent this blow up

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How do we solve SAT?

• Systematic methods
• Truth tables
• Davis Putnam procedure
• Local search methods
• GSAT
• WalkSAT
• Tabu search, SA,
• Exotic methods
• DNA, quantum computing,

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Truth tables

• Enumerate all possible truth assignments
• P Q R C1P v Q C2-Q v -R C1C2
• T T T T F
  F
• T T F T T
  T
• T F T T T
  T
• T F F

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Davis Putnam procedure

• Introduced by Davis Putnam in 1960
• Resolution rule required exponential space
• Modified by Davis, Logemann and Loveland in 1962
• Resolution rule replaced by splitting rule
• Trades space for time
• Modified algorithm usually inaccurately called
  Davis Putnam procedure

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Davis Putnam procedure

• Consider
• (X or Y) (-X or Z) (-Y or Z)
• Basic idea
• Try Xtrue

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Davis Putnam procedure

• Consider
• (X or Y) (-X or Z) (-Y or Z)
• Basic idea
• Try Xtrue
• Remove clauses which must be satisfied

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Davis Putnam procedure

• Consider
• (-X or Z) (-Y or Z)
• Basic idea
• Try Xtrue
• Remove clauses which must be satisfied
• Simplify clauses containing -X

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Davis Putnam procedure

• Consider
• ( Z) (-Y or Z)
• Basic idea
• Try Xtrue
• Remove clauses which must be satisfied
• Simplify clauses containing -X
• Can now deduce that Z must be true

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Davis Putnam procedure

• Consider
• ( Z) (-Y or Z)
• Basic idea
• Try Xtrue
• Remove clauses which must be satisfied
• Simplify clauses containing -X
• Can now deduce that Z must be true
• At any point, may have to backtrack and try
  Xfalse instead

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Procedure DPLL(C)

• (SAT) if C then SAT
• (Empty) if empty clause in C then UNSAT
• (Unit) if unit clause, l then
• DPLL(Cl/True)
• (Split) if DPLL(Cl/True) then SAT else
• DPLL(Cl/False)

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Procedure DPLL(C)

• (Pure) if l is pure in C then
• DPLL(Cl/True)
• (Taut) if l v -l in C then
• DPLL(C - l v -l)
• Neither rules are needed for completeness and do
  not tend to improve solving efficiency

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Procedure DPLL(C)

• (SAT) if C then SAT
• (Empty) if empty clause in C then UNSAT
• (Unit) if unit clause, l then
• DPLL(Cl/True)
• (Split) if DPLL(Cl/True) then SAT else
• DPLL(Cl/False)
• Unit rule not needed for completeness but
  improves efficiency greatly

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Procedure DPLL(C)

• Non-deterministic
• Which literal do we branch upon?
• Good choice can reduce solving time
  significantly
• Popular heuristics include
• MOMs (most occurrences in minimal size clauses)
• Literal that gives most unit propagation

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Procedure DPLL(C)

• Other refinements
• Non-chronological backtracking (conflict
  directed backjumping, )
• Clause learning (at end of branch, identify
  cause for failure and add this as an additional
  clause)
• Fast data structures (two literal watching for
  fast unit propagation)

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Procedure DPLL(C)

• Space complexity
• O(n)
• Time complexity
• O(1.618n)
• Average and best case often much better than this
  worst case (see next lecture)

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Brief History of DP

• 1st generation (1960s)
• DP, DLL
• 2nd generation (1980s/90s)
• POSIT, Tableau,
• 3rd generation (mid 1990s)
• SATO, satz, grasp,
• 4th generation (2000s)
• Chaff, BerkMin,
• 5th generation?

Actual Japanese 5th Generation Computer (from FGC
Museum archive)
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Brief History of DP

• 1st generation (1960s)
• DP, DLL
• 2nd generation (1980s/90s)
• POSIT, Tableau,
• 3rd generation (mid 1990s)
• SATO, satz, grasp,
• 4th generation (2000s)
• Chaff, BerkMin,
• 5th generation?
• Will it need a paradigm shift?

Actual Japanese 5th Generation Computer (from FGC
Museum archive)
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Moores Law

• Are we keeping up with Moores law?
• Number of transistors doubles every 18 months
• Number of variables reported in random 3SAT
  experiments doubles every 3 or 4 years
• Were falling behind each year!
• Even though were getting better performance due
  to Moores law!

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Local search for SAT

• Repair based methods
• Instead of building up a solution, take complete
  assignment and flip variable to satisfy more
  clauses
• Cannot prove UNSATisfiability
• Often will solve MAX-SAT problem

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Papadimitrous procedure

• T random truth assignment
• Repeat until T satisfies all clauses
• v variable in UNSAT clause
• T T with vs value flipped
• Semi-decision procedure
• Solves 2-SAT in expected O(n2) time

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GSAT Selman, Levesque, Mitchell AAAI
92

• Repeat MAX-TRIES times or until clauses satisfied
• T random truth assignment
• Repeat MAX-FLIPS times or until clauses satisfied
• v variable which flipping maximizes number of
  SAT clauses
• T T with vs value flipped
• Adds restarts and greediness
• Sideways flips important (large plateaus to
  explore)

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WalkSAT Selman, Kautz, Cohen AAAI 94

• Repeat MAX-TRIES times or until clauses satisfied
• T random truth assignment
• Repeat MAX-FLIPS times or until clauses satisfied
• c unsat clause chosen at random
• v var in c chosen either greedily or at random
• T T with vs value flipped
• Focuses on UNSAT clauses

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Novelty McAllester, Selman, Kautz AAAI
97

• Repeat MAX-TRIES times or until clauses satisfied
• T random truth assignment
• Repeat MAX-FLIPS times or until clauses satisfied
• c unsat clause chosen at random
• v var in c chosen greedily
• If v was last var in clause flipped and rand(1)ltp
  then
• v var in c with 2nd highest score
• T T with vs value flipped
• Encourages diversity (like Tabu search)

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Other local search methods

• Simulated annealing
• Tabu search
• Genetic algorithms
• Hybrid methods
• Combine best features of systematic methods (eg
  unit propagation) and local search (eg quick
  recovery from failure)

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Exotic methods

• Quantum computing
• DNA computing
• Ant computing

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Beyond the propositional

• Linear 0/1 inequalities
• Quantified Boolean satisfiability
• Stochastic satisfiability
• Modal satisfiability

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Linear 0/1 inequalities

• Constraints of the form
• Sum ai . Xi gt c
• SAT can easily be expressed as linear 0/1 problem
• X v -Y v Z gt X (1-Y) Z gt 1

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Linear 0/1 inequalities

• Often good for describing problems involving
  arithmetic
• Counting constraints
• Objective functions
• Solution methods
• Complete Davis-Putnam like methods
• Local search methods like WalkSAT(PB)

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Quantified Boolean formulae

• Propositional SAT can be expressed as
• Exists P, Q, R . (P v Q) (-Q v R)
• Can add universal quantifiers
• Forall P . Exists Q, R . (P v Q) (-Q v R) ..
• Useful in formal methods, conditional planning

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Quantified Boolean formulae

• SAT of QBF is PSPACE complete
• Can be seen as game
• Existential quantifiers trying to make it SAT
• Universal quantifiers trying to make it UNSAT
• Solution methods
• Extensions of the DPLL procedure
• Local search methods just starting to appear

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Stochastic satisfiability

• Randomized quantifier in addition to the
  existential
• With prob p, Q is True
• With prob (1-p), Q is False
• Can we satisfy formula in some fraction of
  possible worlds?
• Useful for modelling uncertainty present in real
  world (eg planning under uncertainty)

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Conclusions

• Defined propositional SAT
• Complete methods
• Davis Putnam, DPLL,
• Local search procedures
• GSAT, WalkSAT, Novelty,
• Extensions