KR with SAT

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• KR with SAT
• Representation, Algorithms, Translations
• My Thanks to Toby Walsh

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Propositional Logic

• Propositional logic is concerned with the truth
  or falsehood of statements (propositions) like
• the valve is closed
• five plus four equals nine
• Connectives and ?
• or ?
• not ?
• implies ?
• equivalent ?

(X ? (Y ? Z)) ? ((X ? Y) ? (X ? Z)) X implies Y
and Z is the same as X implies Y and X implies Z
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Inference rules

• Logical inference is used to create new
  sentences that logically follow from a given set
  of predicate calculus sentences (KB).
• An inference rule is sound if every sentence X
  produced by an inference rule operating on a KB
  logically follows from the KB. (That is, the
  inference rule does not create any
  contradictions)
• An inference rule is complete if it is able to
  produce every expression that logically follows
  from (is entailed by) the KB. (Note the analogy
  to complete search algorithms.)

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Sound rules of inference

• Here are some examples of sound rules of
  inference.
• Each can be shown to be sound using a truth
  table A rule is sound if its conclusion is true
  whenever the premise is true.
• RULE PREMISE CONCLUSION
• Modus Ponens A, A gt B B
• And Introduction A, B A B
• And Elimination A B A
• Double Negation A A
• Unit Resolution A v B, B A
• Resolution A v B, B v C A v C

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Conjunctive Normal Form

• Every sentence in propositional logic can be
  transformed into conjunctive normal form
• i.e. a conjunction of disjunctions
• Algorithm
• Eliminate ? using the rule that (p ?q) is
  equivalent to (?p ? q)
• Use de Morgans laws so that negation applies to
  literals only
• Distribute ? and ? to write the result as a
  conjunction of disjunctions

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Conjunctive Normal Form - Example

• ?(p ?q) ? (r ?p)
• Eliminate implication signs
• ?(?p ? q) ? (?r ? p)
• Apply de Morgans laws
• (p ? ?q) ? (?r ? p)
• Apply associative and distributive laws
• (p ? ?r ? p) ? (?q ? ?r ? p)
• (p ? ?r) ? (?q ? ?r ? p)

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SAT

• The problem of finding a model for a formula in
  propositional logic is known as the propositional
  satisfiability problem (PSAT)
• When the formula is in CNF it is usually called
  SAT
• 1st problem shown to be NP-complete Cook 71
• Each disjunction of literals is called a clause
• Many real world problems can be formulated as SAT
  problems
• including CSPs
• 2SAT is a special case that can be solved in
  polynomial time

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Representing knowledge using SAT - Example

• Embassy ball (a diplomatic problem)
• King wants to invite PERU or exclude QATAR
• Queen wants to invite QATAR or ROMANIA
• Kings wants to snub ROMANIA or PERU
• Who can we invite?

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Representing knowledge using SAT - Example

• Embassy ball (a diplomatic problem)
• King wants to invite PERU or exclude QATAR
• Queen wants to invite QATAR or ROMANIA
• Kings wants to snub ROMANIA or PERU
• (P ? ?Q) ? (Q ? R) ? (?R ? ?P)
• is satisfied by Ptrue, Qtrue, Rfalse
• and by Pfalse, Qfalse, Rtrue

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Other applications of SAT

• Hardware verification

S Cin ? (P ? Q),
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Other applications of SAT

• Scheduling jobs in a factory
• Set of jobs, resource constraints, start and due
  dates,
• Just need to sequence jobs
• Xij true iff Job i occurs before Job j
• Design theory
• Experimental design, Latin squares with special
  properties
• Open problems solved by encoding problems into
  SAT

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Fiber-optic routing
Nodes connect point to point fiber optic links
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Fibre-optic routing
Input Ports
Output Ports
1
1
2
2
3
3
4
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Routing Node
How can we achieve conflict-free routing in each
node of the network?
Dynamic wavelength routing is a NP-hard problem.
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Routing Latin square problem

• Each channel cannot be repeated on input ports
• Each channel cannot be repeated on output ports

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Latin square

• Also called quasigroup
• Each colour occurs once in each row and column

Quasigroup or Latin Square (Order 4)
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Latin squares

• Can we complete a partially colored Latin square?
• NP-complete problem
• Applications in telecommunications

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SAT model for Latin square

• What are the variables?
• Xijk is true if square (i,j) has the colour k
• What are the constraints
• Row constraints
• (?Xijk ? ?Xi(j1)k), .
• Column constraints
• (?Xijk ? ?X(i1)jk),
• Xij1 ? ? Xij4

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SAT vs. CSP

• SAT CSP
• Binary domains Non-binary domains
• Non-binary constraints Binary constraints

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Encoding CSP as SAT

• A propositional variable Xij for each value j in
  the domain of the CSP variable Xi
• Xij is true if Xi takes value j
• Clauses that ensure each CSP variable is given a
  value
• for each i Xi1 ? ? Xim
• Clauses that ensure no CSP variable takes more
  than one values
• for each i,j,k, with j?k, ?Xij ? ?Xik
• (Binary) clauses that rule out any (binary)
  nogoods
• e.g. (if X15, X24) is not allowed we have a
  clause ?X15 ? ?X24

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Algorithms for SAT

• How do we test if a problem is SAT or not?
• Complete methods
• Return Yes if SATisfiable
• Return No if UNSATisfiable
• Incomplete methods
• If return Yes, problem is SATisfiable
• Otherwise timeout/run forever, problem can be SAT
  or UNSAT

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Complete SAT algorithms

• Truth tables
• Exponential time to construct
• Exponential space

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Complete SAT algorithms

• DLL algorithm
• Due to Davis, Logemann and Loveland from 1962
• Original algorithm by Davis Putnam in 1960
  could use exponential space
• DLL traded time for space to give linear space
  algorithm

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Complete SAT algorithms

• DLL algorithm
• Consider
• (X ? Y) ? (?X ? Z) ? (?Y ? Z) ?
• Basic idea
• Try Xtrue
• Remove clauses which must be satisfied

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Complete SAT algorithms

• DLL algorithm
• Consider
• (?X ? Z) ? (?Y ? Z) ?
• Basic idea
• Try Xtrue
• Remove clauses which must be satisfied
• Simplify clauses containing ?X

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Complete SAT algorithms

• DLL algorithm
• Consider
• ( Z) ? (?Y ? 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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Complete SAT algorithms

• DLL algorithm
• Branching heuristics
• Which variable to assign next?
• Which value (true/false) to try first?
• Other refinements
• Clause learning
• Intelligence backtracking
• Implementation tricks
• Worst-case complexity
• O(1.696n)

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Incomplete SAT algorithms

• Most fit a very simple schema
• Randomly guess truth assignment
• Repeat
• Pick a variable
• Flip its truth value
• Heuristics to pick variable
• Any var in UNSAT clause
• Var that maximizes number of SAT clauses if
  flipped
• GSAT
• Var in UNSAT clause flipped longest time ago

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k-SAT

• Subclass of SAT problem
• Exactly k variables in each clause
• (P ? ?Q) ? (Q ? R) ? (?R ? ?P) is in 2-SAT
• 3-SAT is NP-complete
• SAT can be reduced to 3-SAT
• 2-SAT is in P
• Exists linear time algorithm!

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Encoding CSP as SAT

• There are various encodings of SAT into CSP
• Can you think of any?

• SAT into CSP
• Dual Encoding
• Hidden Variable Encoding
• Literal Encoding
• Non-binary Encoding

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Dual Encoding

• A dual variable Vi associated with each clause Ci
• The domain of Vi consists of those tuples of
  truth values which satisfy the clause
• For example the clause x1 ? x2 is associated
  with a dual variable having domain (T,F), (F,T),
  (T,T)
• Binary constraints are posted between dual
  variables which are associated with clauses that
  share propositional variables
• For example there is a binary constraint between
  variables V1 and V2 , where V1 corresponds to
  clause x1 ? x2 and V2 corresponds to clause
  x1??x2

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Hidden Variable Encoding

• A hidden variable Vi associated with each clause
  Ci
• The domain of Vi consists of those tuples of
  truth values which satisfy the clause
• For example the clause x1 ? x2 is associated
  with a hidden variable having domain (T,F),
  (F,T), (T,T)
• We also have the original propositional
  variables with domains T,F
• There is a binary constraint between a hidden
  variable and a propositional variable if the
  clause represented by the hidden variable
  mentions the propositional variable
• For example there is a binary constraint between
  hidden variable V1 ,which corresponds to clause
  x1 ? x2 ,and variable x1

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Literal Encoding

• A variable Vi associated with each clause Ci
• The domain of Vi consists of those literals
  which satisfy the clause
• For example the clause x1 ? x2 is associated
  with a variable V1 having domain x1 , x2 and
  the clause x3 ? ?x2 is associated with a variable
  V2 having domain x3 , ?x2
• There is a binary constraint between two
  variables Vi, Vj if the associated clause Ci
  contains a literal whose complement is contained
  in the associated clause Cj
• For example there is a constraint between V1 and
  V2
• the constraint allows tuples (x1,x3), (x1,?x2),
  (x2,x3)
• domain size of variables is O(k), where k the
  length of the clauses

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Non-binary Encoding

• A variable Vi associated with each literal
• The domain of Vi is T,F
• A non-binary constraint is posted on variables
  appearing in the same clause
• This constraint disalllows partial assignments
  that do not satisfy the clause
• For example, associated with the clause x1 ? x2
  ? x3 is a non-binary constraint on variables x1 ,
  x2 , x3 which allows all tuples except (F,F,F)

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Experiments with 3-SAT

• Where are the hard 3-SAT problems?
• Sample randomly generated 3-SAT
• Fix number of clauses, l
• Number of variables, n
• By definition, each clause has 3 variables
• Generate all possible clauses with uniform
  probability

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Experiments with random 3-SAT

• Which are the hard instances?
• around l/n 4.3
• What happens with larger problems?
• Why are some dots red and others blue?
• This is a so-called phase transition

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Experiments with random 3-SAT

• Varying problem size, n
• Complexity peak appears to be largely invariant
  of algorithm
• complete algorithms like Davis-Putnam
• Incomplete methods like local search
• Whats so special about 4.3?

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Experiments with random 3-SAT

• Complexity peak coincides with satisfiability
  transition
• l/n lt 4.3 problems under-constrained and SAT
• l/n gt 4.3 problems over-constrained and UNSAT
• l/n4.3, problems on knife-edge between SAT and
  UNSAT

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Conclusions

• SAT is a useful problem class
• Theoretical importance
• prototypical NP-complete problem
• NP-completeness is usually proved by reduction
  to SAT
• Practical value
• can be used to represent and reason with
  knowledge in many domains
• SAT problems can be translated into CSPs and vice
  versa
• as to which representation is better It depends
  on the specific problem