Presentacin de PowerPoint

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Local Consistency in Weighted CSP and Inference
in Max-SAT
Student Federico Heras Advisor Javier Larrosa
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Overview

• Max-SAT is an optimization problem. We propose
• A new Max-SAT framework.
• An extension of DPLL.
• An extension of the resolution Rule.
• As result we obtain
• Algorithms with a logic flavour.
• Nice connections with Weighted CSP.

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(Weighted) Max-SAT

• Weighted clause (C,w)
• C clause, w weight (cost of violation)
• WCNF formula F(C,w),.. set of weighted
  clauses
• Truth assignment cost sum of costs of
  unsatisfied clauses
• Max-SAT instance lt F, ? gt
• Model truth assignment with cost lt ?
• Goal minimum cost model
• Fl instantiate set of clauses F with literal
  l
• if l?C then remove C from F
• if l?C then remove l from C

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(Weighted) Max-SAT

• (?,w) is an obvious lower bound of the optimum
• ? is an upper bound
• if wgt ? then (C,w)(C, ?)
• (C, ?) is a mandatory clause
• If (?,?)?F then there is a contradiction
• sum of costs v?w minvw, ?
• subtraction of costs

a b a ? T T a T
a b
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Max-DPLL
Function Max-DPLL(F, T) nat FUP(F) if
(?, T)? F then ret T if FØ then ret 0 if
F(?, w) then ret w lSelectLiteral(F) vMax
-DPLL(Fl, T) vMax-DPLL(Fl,v) ret v
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Max-DPLL

• UP(Unit Propagation) Basic Rules UCR.
• Notation
• O?T replace O by T.
• x, y, z, boolean variables
• A, B, C clauses (set of literals)
• Basic Rules
• (A,0) ?
• (x ? x ? A,w) ?
• (A,w),(A,u) ? (A,w?u)
• if (w?u ?) then (A,w),(A ? B,u) ? (A,w),(A ?
  B, ?)
• UCR (Unit clause Reduction). Selects a mandatory
  clause (l,T) and instantiates the corresponding
  value in accordance to the literal in that clause.

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Basic Rules Example (? 5)
(?,1), (x,3), (x,1), (x,1), (y,3), (y,2), (x ?
y ? z,1)
(?,1), (x,4), (x,1), (y,3), (y,2), (x ? y ?
z,1)
(?,1), (x, T), (x,1), (y,3), (y,2), (x ? y ?
z,1)
(?,2), (y,3), (y,2), (y ? z,1)
(?,2), (y, T), (y,2), (y ? z,1)
(?,4), (z,1)
(?,4), (z, T)
(?,4)
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Max-SAT Resolution
(A ? B,m),(x ? A,u-m), (x ? B, w-m), (x ? A ?
B,m), (x ? A ? B,m)
(x ? A,u), (x ? B,w) ?
where mminu,w
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Max-SAT Resolution
(A v B,m),(x v A,u-m),(x v B, w-m), (x v A v
B,m),(x v A v B,m)
(x v A,u), (x v B,w)
x A B
uu00w0w0
m00 0 m000
??? ? ? ? ? ?
u-mu-m0 0 0 0 00
0000w-m0w-m0
??? ? ? ? ? ?
??? ? ? ? ? ?
0m000000
??? ? ? ? ? ?
000000m0
f f ff f tf t ff t tt f ft f tt t ft t t
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Neighborhood Resolution
NRES Resolution with AB
(A,m),(x ? A,u-m), (x ? A, w-m)
(x ? A,u), (x ? A,w) ?
(where mminu,w)
If A0 (NRES0)
(?,m),(x,u-m), (x,w-m)
(x,u), (x,w)
(where mminu,w)

• NRES0 is equivalent to NC for WCSP.
• NRES1 is equivalent to AC for WCSP.

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Modus Ponens
(Binary)MP (Modus Ponens) Resolution with Ay
and Bfalse.
(x ? y,u-m),(x,w-m), (y, m), (x ? y, m)
(x ? y,u), (x,w) ?
(where mminu,w)
(Binary)MPE (Modus ponens Empty) Resolution with
Ay and Bfalse and an additional unary clause.
(x ? y,u-m),(x,w-m), (y, v-m), (x ? y,
m) (?,m)
(where mminu,w,v)
(x ? y,u), (x,w),(y,v) ?
Equilance to local consistencies??
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Inference Example (? 5)
(x,2),(x v y,2),(x v y v z,1), (x v y v z,1)
, (y,1)
NRES2
(x,2),(x v y,2),(x v y,1) , (y,1)
NRES1
(x,2),(x v y,1),(x,1), (y,1)
NRES0
(x,1),(?,1),(x v y,1), (y,1)
MPE
MP
NRES0
(?,2),(x v y,1)
(x v y,1),(?,1),(y,1),(y,1)
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Future Work

• Extend our weigthed logic framework.
• Extend Modus Ponens to n-ary clauses (not only
  binary)..
• Apply n-ary rules to generic weighted CSP.

Federico Heras Viaga fheras_at_lsi.upc.edu Javier
Larrosa Bondia larrosa_at_lsi.upc.edu Universitat
Politècnica de Cataunya, UPC, Spain