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Practical SAT Solving: Look-ahead Techniques
Marijn J.H. HeuleWarren A. Hunt Jr.
The University of Texas at Austin
Heule & Hunt (UT Austin) Look-ahead Techniques 1 / 1
Sat solving: DPLL
Davis Putnam Logemann Loveland [DP60,DLL62]
Simplify (Unit Propagation)
Split the formula
Variable Selection HeuristicsDirection heuristics
Heule & Hunt (UT Austin) Look-ahead Techniques 2 / 1
DPLL: Example
FDPLL := (x1 ∨ x2 ∨ ¬x3) ∧ (¬x1 ∨ x2 ∨ x3) ∧(¬x1 ∨ ¬x2 ∨ x3) ∧ (x1 ∨ x3) ∧ (¬x1 ∨ ¬x3)
Heule & Hunt (UT Austin) Look-ahead Techniques 3 / 1
DPLL: Example
FDPLL := (x1 ∨ x2 ∨ ¬x3) ∧ (¬x1 ∨ x2 ∨ x3) ∧(¬x1 ∨ ¬x2 ∨ x3) ∧ (x1 ∨ x3) ∧ (¬x1 ∨ ¬x3)
x3
1 0
Heule & Hunt (UT Austin) Look-ahead Techniques 3 / 1
DPLL: Example
FDPLL := (x1 ∨ x2 ∨ ¬x3) ∧ (¬x1 ∨ x2 ∨ x3) ∧(¬x1 ∨ ¬x2 ∨ x3) ∧ (x1 ∨ x3) ∧ (¬x1 ∨ ¬x3)
x3
1 0
x2
x1 x3
0 1
0 1 1 0
Heule & Hunt (UT Austin) Look-ahead Techniques 3 / 1
Look-ahead: Definition
DPLL with selection of (effective) decision variables by
look-aheads on variables
Heule & Hunt (UT Austin) Look-ahead Techniques 4 / 1
Look-ahead: Definition
DPLL with selection of (effective) decision variables by
look-aheads on variables
Look-ahead:
Assign a variable to a truth value
Heule & Hunt (UT Austin) Look-ahead Techniques 4 / 1
Look-ahead: Definition
DPLL with selection of (effective) decision variables by
look-aheads on variables
Look-ahead:
Assign a variable to a truth value
Simplify the formula
Heule & Hunt (UT Austin) Look-ahead Techniques 4 / 1
Look-ahead: Definition
DPLL with selection of (effective) decision variables by
look-aheads on variables
Look-ahead:
Assign a variable to a truth value
Simplify the formula
Measure the reduction
Heule & Hunt (UT Austin) Look-ahead Techniques 4 / 1
Look-ahead: Definition
DPLL with selection of (effective) decision variables by
look-aheads on variables
Look-ahead:
Assign a variable to a truth value
Simplify the formula
Measure the reduction
Learn if possible
Heule & Hunt (UT Austin) Look-ahead Techniques 4 / 1
Look-ahead: Definition
DPLL with selection of (effective) decision variables by
look-aheads on variables
Look-ahead:
Assign a variable to a truth value
Simplify the formula
Measure the reduction
Learn if possible
Backtrack
Heule & Hunt (UT Austin) Look-ahead Techniques 4 / 1
Look-ahead: Example
Flearning := (¬x1 ∨ ¬x3 ∨ x4) ∧ (¬x1 ∨ ¬x2 ∨ x3) ∧(¬x1 ∨ x2) ∧ (x1 ∨ x3 ∨ x6) ∧ (¬x1 ∨ x4 ∨ ¬x5) ∧
(x1 ∨ ¬x6) ∧ (x4 ∨ x5 ∨ x6) ∧ (x5 ∨ ¬x6)
Heule & Hunt (UT Austin) Look-ahead Techniques 5 / 1
Look-ahead: Example
Flearning := (¬x1 ∨ ¬x3 ∨ x4) ∧ (¬x1 ∨ ¬x2 ∨ x3) ∧(¬x1 ∨ x2) ∧ (x1 ∨ x3 ∨ x6) ∧ (¬x1 ∨ x4 ∨ ¬x5) ∧
(x1 ∨ ¬x6) ∧ (x4 ∨ x5 ∨ x6) ∧ (x5 ∨ ¬x6)
ϕ = {x2=0}
Heule & Hunt (UT Austin) Look-ahead Techniques 5 / 1
Look-ahead: Example
Flearning := (¬x1 ∨ ¬x3 ∨ x4) ∧ (¬x1 ∨ ¬x2 ∨ x3) ∧(¬x1 ∨ x2) ∧ (x1 ∨ x3 ∨ x6) ∧ (¬x1 ∨ x4 ∨ ¬x5) ∧
(x1 ∨ ¬x6) ∧ (x4 ∨ x5 ∨ x6) ∧ (x5 ∨ ¬x6)
ϕ = {x2=0, x1=0}
Heule & Hunt (UT Austin) Look-ahead Techniques 5 / 1
Look-ahead: Example
Flearning := (¬x1 ∨ ¬x3 ∨ x4) ∧ (¬x1 ∨ ¬x2 ∨ x3) ∧(¬x1 ∨ x2) ∧ (x1 ∨ x3 ∨ x6) ∧ (¬x1 ∨ x4 ∨ ¬x5) ∧
(x1 ∨ ¬x6) ∧ (x4 ∨ x5 ∨ x6) ∧ (x5 ∨ ¬x6)
ϕ = {x2=0, x1=0, x6=0}
Heule & Hunt (UT Austin) Look-ahead Techniques 5 / 1
Look-ahead: Example
Flearning := (¬x1 ∨ ¬x3 ∨ x4) ∧ (¬x1 ∨ ¬x2 ∨ x3) ∧(¬x1 ∨ x2) ∧ (x1 ∨ x3 ∨ x6) ∧ (¬x1 ∨ x4 ∨ ¬x5) ∧
(x1 ∨ ¬x6) ∧ (x4 ∨ x5 ∨ x6) ∧ (x5 ∨ ¬x6)
ϕ = {x2=0, x1=0, x6=0, x3=1}
Heule & Hunt (UT Austin) Look-ahead Techniques 5 / 1
Look-ahead: Properties
Very expensive
Heule & Hunt (UT Austin) Look-ahead Techniques 6 / 1
Look-ahead: Properties
Very expensive
Effective compared to cheap heuristics
Heule & Hunt (UT Austin) Look-ahead Techniques 6 / 1
Look-ahead: Properties
Very expensive
Effective compared to cheap heuristics
Detection of failed literals (and more)
Heule & Hunt (UT Austin) Look-ahead Techniques 6 / 1
Look-ahead: Properties
Very expensive
Effective compared to cheap heuristics
Detection of failed literals (and more)
Strong on random k-Sat formulae
Heule & Hunt (UT Austin) Look-ahead Techniques 6 / 1
Look-ahead: Properties
Very expensive
Effective compared to cheap heuristics
Detection of failed literals (and more)
Strong on random k-Sat formulae
Examples: march, OKsolver, kcnfs
Heule & Hunt (UT Austin) Look-ahead Techniques 6 / 1
Look-ahead: Reduction heuristics
Number of satisfied clauses
Heule & Hunt (UT Austin) Look-ahead Techniques 7 / 1
Look-ahead: Reduction heuristics
Number of satisfied clauses
Number of implied variables
Heule & Hunt (UT Austin) Look-ahead Techniques 7 / 1
Look-ahead: Reduction heuristics
Number of satisfied clauses
Number of implied variablesNew (reduced, not satisfied) clauses
Smaller clauses more importantWeights based on occurring
Heule & Hunt (UT Austin) Look-ahead Techniques 7 / 1
Look-ahead: Architecture
xa
xb xc
0 1
?
1 0 0
DPLL
x1 x2 x3 x4
FLA
0
9
1
9
0
13
1
6
0 1
7
0
10
1
8
LookAhead
H(xi )
Heule & Hunt (UT Austin) Look-ahead Techniques 8 / 1
Look-ahead: Pseudo-code
1: F := Simplify (F)2: If F is empty then return satisfiable
3: If ∅ ∈ F then return unsatisfiable
4: 〈F ; ldecision〉 := LookAhead( F )
5: if DPLL( F(ldecision ← 1)) = satisfiable then6: return satisfiable
7: end if
8: return DPLL( F(ldecision ← 0))
Heule & Hunt (UT Austin) Look-ahead Techniques 9 / 1
Look-ahead: Local learning
Flearning := (¬x1 ∨ ¬x3 ∨ x4) ∧ (¬x1 ∨ ¬x2 ∨ x3) ∧(¬x1 ∨ x2) ∧ (x1 ∨ x3 ∨ x6) ∧ (¬x1 ∨ x4 ∨ ¬x5) ∧
(x1 ∨ ¬x6) ∧ (x4 ∨ x5 ∨ x6) ∧ (x5 ∨ ¬x6)
Heule & Hunt (UT Austin) Look-ahead Techniques 10 / 1
Look-ahead: Local learning
Flearning := (¬x1 ∨ ¬x3 ∨ x4) ∧ (¬x1 ∨ ¬x2 ∨ x3) ∧(¬x1 ∨ x2) ∧ (x1 ∨ x3 ∨ x6) ∧ (¬x1 ∨ x4 ∨ ¬x5) ∧
(x1 ∨ ¬x6) ∧ (x4 ∨ x5 ∨ x6) ∧ (x5 ∨ ¬x6)
ϕ = {x2=0}
Heule & Hunt (UT Austin) Look-ahead Techniques 10 / 1
Look-ahead: Local learning
Flearning := (¬x1 ∨ ¬x3 ∨ x4) ∧ (¬x1 ∨ ¬x2 ∨ x3) ∧(¬x1 ∨ x2) ∧ (x1 ∨ x3 ∨ x6) ∧ (¬x1 ∨ x4 ∨ ¬x5) ∧
(x1 ∨ ¬x6) ∧ (x4 ∨ x5 ∨ x6) ∧ (x5 ∨ ¬x6)
ϕ = {x2=0, x1=0}
Heule & Hunt (UT Austin) Look-ahead Techniques 10 / 1
Look-ahead: Local learning
Flearning := (¬x1 ∨ ¬x3 ∨ x4) ∧ (¬x1 ∨ ¬x2 ∨ x3) ∧(¬x1 ∨ x2) ∧ (x1 ∨ x3 ∨ x6) ∧ (¬x1 ∨ x4 ∨ ¬x5) ∧
(x1 ∨ ¬x6) ∧ (x4 ∨ x5 ∨ x6) ∧ (x5 ∨ ¬x6)
ϕ = {x2=0, x1=0, x6=0}
Heule & Hunt (UT Austin) Look-ahead Techniques 10 / 1
Look-ahead: Local learning
Flearning := (¬x1 ∨ ¬x3 ∨ x4) ∧ (¬x1 ∨ ¬x2 ∨ x3) ∧(¬x1 ∨ x2) ∧ (x1 ∨ x3 ∨ x6) ∧ (¬x1 ∨ x4 ∨ ¬x5) ∧
(x1 ∨ ¬x6) ∧ (x4 ∨ x5 ∨ x6) ∧ (x5 ∨ ¬x6)
ϕ = {x2=0, x1=0, x6=0, x3=1}
Heule & Hunt (UT Austin) Look-ahead Techniques 10 / 1
Look-ahead: Local learning
Flearning := (¬x1 ∨ ¬x3 ∨ x4) ∧ (¬x1 ∨ ¬x2 ∨ x3) ∧(¬x1 ∨ x2) ∧ (x1 ∨ x3 ∨ x6) ∧ (¬x1 ∨ x4 ∨ ¬x5) ∧
(x1 ∨ ¬x6) ∧ (x4 ∨ x5 ∨ x6) ∧ (x5 ∨ ¬x6)
ϕ = {x2=0, x1=0, x6=0, x3=1}
(local) constraint resolventsa.k.a. hyper binary resolvents:
(x2 ∨ x3) and (x2 ∨ ¬x6)
Heule & Hunt (UT Austin) Look-ahead Techniques 10 / 1
Look-ahead: Necessary assignments
Flearning := (¬x1 ∨ ¬x3 ∨ x4) ∧ (¬x1 ∨ ¬x2 ∨ x3) ∧(¬x1 ∨ x2) ∧ (x1 ∨ x3 ∨ x6) ∧ (¬x1 ∨ x4 ∨ ¬x5) ∧
(x1 ∨ ¬x6) ∧ (x4 ∨ x5 ∨ x6) ∧ (x5 ∨ ¬x6)
Heule & Hunt (UT Austin) Look-ahead Techniques 11 / 1
Look-ahead: Necessary assignments
Flearning := (¬x1 ∨ ¬x3 ∨ x4) ∧ (¬x1 ∨ ¬x2 ∨ x3) ∧(¬x1 ∨ x2) ∧ (x1 ∨ x3 ∨ x6) ∧ (¬x1 ∨ x4 ∨ ¬x5) ∧
(x1 ∨ ¬x6) ∧ (x4 ∨ x5 ∨ x6) ∧ (x5 ∨ ¬x6)
ϕ = {x1=1}
Heule & Hunt (UT Austin) Look-ahead Techniques 11 / 1
Look-ahead: Necessary assignments
Flearning := (¬x1 ∨ ¬x3 ∨ x4) ∧ (¬x1 ∨ ¬x2 ∨ x3) ∧(¬x1 ∨ x2) ∧ (x1 ∨ x3 ∨ x6) ∧ (¬x1 ∨ x4 ∨ ¬x5) ∧
(x1 ∨ ¬x6) ∧ (x4 ∨ x5 ∨ x6) ∧ (x5 ∨ ¬x6)
ϕ = {x1=1, x2=1}
Heule & Hunt (UT Austin) Look-ahead Techniques 11 / 1
Look-ahead: Necessary assignments
Flearning := (¬x1 ∨ ¬x3 ∨ x4) ∧ (¬x1 ∨ ¬x2 ∨ x3) ∧(¬x1 ∨ x2) ∧ (x1 ∨ x3 ∨ x6) ∧ (¬x1 ∨ x4 ∨ ¬x5) ∧
(x1 ∨ ¬x6) ∧ (x4 ∨ x5 ∨ x6) ∧ (x5 ∨ ¬x6)
ϕ = {x1=1, x2=1, x3=1}
Heule & Hunt (UT Austin) Look-ahead Techniques 11 / 1
Look-ahead: Necessary assignments
Flearning := (¬x1 ∨ ¬x3 ∨ x4) ∧ (¬x1 ∨ ¬x2 ∨ x3) ∧(¬x1 ∨ x2) ∧ (x1 ∨ x3 ∨ x6) ∧ (¬x1 ∨ x4 ∨ ¬x5) ∧
(x1 ∨ ¬x6) ∧ (x4 ∨ x5 ∨ x6) ∧ (x5 ∨ ¬x6)
ϕ = {x1=1, x2=1, x3=1, x4=1}
Heule & Hunt (UT Austin) Look-ahead Techniques 11 / 1
Look-ahead: Necessary assignments
Flearning := (¬x1 ∨ ¬x3 ∨ x4) ∧ (¬x1 ∨ ¬x2 ∨ x3) ∧(¬x1 ∨ x2) ∧ (x1 ∨ x3 ∨ x6) ∧ (¬x1 ∨ x4 ∨ ¬x5) ∧
(x1 ∨ ¬x6) ∧ (x4 ∨ x5 ∨ x6) ∧ (x5 ∨ ¬x6)
ϕ = {x1=1, x2=1, x3=1, x4=1}
Flearning := (¬x1 ∨ ¬x3 ∨ x4) ∧ (¬x1 ∨ ¬x2 ∨ x3) ∧(¬x1 ∨ x2) ∧ (x1 ∨ x3 ∨ x6) ∧ (¬x1 ∨ x4 ∨ ¬x5) ∧
(x1 ∨ ¬x6) ∧ (x4 ∨ x5 ∨ x6) ∧ (x5 ∨ ¬x6)
Heule & Hunt (UT Austin) Look-ahead Techniques 11 / 1
Look-ahead: Necessary assignments
Flearning := (¬x1 ∨ ¬x3 ∨ x4) ∧ (¬x1 ∨ ¬x2 ∨ x3) ∧(¬x1 ∨ x2) ∧ (x1 ∨ x3 ∨ x6) ∧ (¬x1 ∨ x4 ∨ ¬x5) ∧
(x1 ∨ ¬x6) ∧ (x4 ∨ x5 ∨ x6) ∧ (x5 ∨ ¬x6)
ϕ = {x1=1, x2=1, x3=1, x4=1}
Flearning := (¬x1 ∨ ¬x3 ∨ x4) ∧ (¬x1 ∨ ¬x2 ∨ x3) ∧(¬x1 ∨ x2) ∧ (x1 ∨ x3 ∨ x6) ∧ (¬x1 ∨ x4 ∨ ¬x5) ∧
(x1 ∨ ¬x6) ∧ (x4 ∨ x5 ∨ x6) ∧ (x5 ∨ ¬x6)
ϕ = {x1=0}
Heule & Hunt (UT Austin) Look-ahead Techniques 11 / 1
Look-ahead: Necessary assignments
Flearning := (¬x1 ∨ ¬x3 ∨ x4) ∧ (¬x1 ∨ ¬x2 ∨ x3) ∧(¬x1 ∨ x2) ∧ (x1 ∨ x3 ∨ x6) ∧ (¬x1 ∨ x4 ∨ ¬x5) ∧
(x1 ∨ ¬x6) ∧ (x4 ∨ x5 ∨ x6) ∧ (x5 ∨ ¬x6)
ϕ = {x1=1, x2=1, x3=1, x4=1}
Flearning := (¬x1 ∨ ¬x3 ∨ x4) ∧ (¬x1 ∨ ¬x2 ∨ x3) ∧(¬x1 ∨ x2) ∧ (x1 ∨ x3 ∨ x6) ∧ (¬x1 ∨ x4 ∨ ¬x5) ∧
(x1 ∨ ¬x6) ∧ (x4 ∨ x5 ∨ x6) ∧ (x5 ∨ ¬x6)
ϕ = {x1=0, x6=0}
Heule & Hunt (UT Austin) Look-ahead Techniques 11 / 1
Look-ahead: Necessary assignments
Flearning := (¬x1 ∨ ¬x3 ∨ x4) ∧ (¬x1 ∨ ¬x2 ∨ x3) ∧(¬x1 ∨ x2) ∧ (x1 ∨ x3 ∨ x6) ∧ (¬x1 ∨ x4 ∨ ¬x5) ∧
(x1 ∨ ¬x6) ∧ (x4 ∨ x5 ∨ x6) ∧ (x5 ∨ ¬x6)
ϕ = {x1=1, x2=1, x3=1, x4=1}
Flearning := (¬x1 ∨ ¬x3 ∨ x4) ∧ (¬x1 ∨ ¬x2 ∨ x3) ∧(¬x1 ∨ x2) ∧ (x1 ∨ x3 ∨ x6) ∧ (¬x1 ∨ x4 ∨ ¬x5) ∧
(x1 ∨ ¬x6) ∧ (x4 ∨ x5 ∨ x6) ∧ (x5 ∨ ¬x6)
ϕ = {x1=0, x6=0, x3=1}
Heule & Hunt (UT Austin) Look-ahead Techniques 11 / 1
Look-ahead: Necessary assignments
Flearning := (¬x1 ∨ ¬x3 ∨ x4) ∧ (¬x1 ∨ ¬x2 ∨ x3) ∧(¬x1 ∨ x2) ∧ (x1 ∨ x3 ∨ x6) ∧ (¬x1 ∨ x4 ∨ ¬x5) ∧
(x1 ∨ ¬x6) ∧ (x4 ∨ x5 ∨ x6) ∧ (x5 ∨ ¬x6)
ϕ = {x1=1, x2=1, x3=1, x4=1}
Flearning := (¬x1 ∨ ¬x3 ∨ x4) ∧ (¬x1 ∨ ¬x2 ∨ x3) ∧(¬x1 ∨ x2) ∧ (x1 ∨ x3 ∨ x6) ∧ (¬x1 ∨ x4 ∨ ¬x5) ∧
(x1 ∨ ¬x6) ∧ (x4 ∨ x5 ∨ x6) ∧ (x5 ∨ ¬x6)
ϕ = {x1=0, x6=0, x3=1}
Heule & Hunt (UT Austin) Look-ahead Techniques 11 / 1
Look-ahead: Autarky definition
An autarky is a partial assignment that satisfies all clausesthat are “touched” by the assignment
Heule & Hunt (UT Austin) Look-ahead Techniques 12 / 1
Look-ahead: Autarky definition
An autarky is a partial assignment that satisfies all clausesthat are “touched” by the assignment
a pure literal is an autarky
Heule & Hunt (UT Austin) Look-ahead Techniques 12 / 1
Look-ahead: Autarky definition
An autarky is a partial assignment that satisfies all clausesthat are “touched” by the assignment
a pure literal is an autarky
each satisfying assignment is an autarky
Heule & Hunt (UT Austin) Look-ahead Techniques 12 / 1
Look-ahead: Autarky definition
An autarky is a partial assignment that satisfies all clausesthat are “touched” by the assignment
a pure literal is an autarky
each satisfying assignment is an autarky
the remaining formula is satisfiability equivalent
to the original formula
Heule & Hunt (UT Austin) Look-ahead Techniques 12 / 1
Look-ahead: Autarky definition
An autarky is a partial assignment that satisfies all clausesthat are “touched” by the assignment
a pure literal is an autarky
each satisfying assignment is an autarky
the remaining formula is satisfiability equivalent
to the original formula
An 1-autarky is a partial assignment thatsatisfies all touched clauses except one
Heule & Hunt (UT Austin) Look-ahead Techniques 12 / 1
Look-ahead: Autarky detection
Flearning := (¬x1 ∨ ¬x3 ∨ x4) ∧ (¬x1 ∨ ¬x2 ∨ x3) ∧(¬x1 ∨ x2) ∧ (x1 ∨ x3 ∨ x6) ∧ (¬x1 ∨ x4 ∨ ¬x5) ∧
(x1 ∨ ¬x6) ∧ (x4 ∨ x5 ∨ x6) ∧ (x5 ∨ ¬x6)
Heule & Hunt (UT Austin) Look-ahead Techniques 13 / 1
Look-ahead: Autarky detection
Flearning := (¬x1 ∨ ¬x3 ∨ x4) ∧ (¬x1 ∨ ¬x2 ∨ x3) ∧(¬x1 ∨ x2) ∧ (x1 ∨ x3 ∨ x6) ∧ (¬x1 ∨ x4 ∨ ¬x5) ∧
(x1 ∨ ¬x6) ∧ (x4 ∨ x5 ∨ x6) ∧ (x5 ∨ ¬x6)
ϕ = {x1=1}
Heule & Hunt (UT Austin) Look-ahead Techniques 13 / 1
Look-ahead: Autarky detection
Flearning := (¬x1 ∨ ¬x3 ∨ x4) ∧ (¬x1 ∨ ¬x2 ∨ x3) ∧(¬x1 ∨ x2) ∧ (x1 ∨ x3 ∨ x6) ∧ (¬x1 ∨ x4 ∨ ¬x5) ∧
(x1 ∨ ¬x6) ∧ (x4 ∨ x5 ∨ x6) ∧ (x5 ∨ ¬x6)
ϕ = {x1=1, x2=1}
Heule & Hunt (UT Austin) Look-ahead Techniques 13 / 1
Look-ahead: Autarky detection
Flearning := (¬x1 ∨ ¬x3 ∨ x4) ∧ (¬x1 ∨ ¬x2 ∨ x3) ∧(¬x1 ∨ x2) ∧ (x1 ∨ x3 ∨ x6) ∧ (¬x1 ∨ x4 ∨ ¬x5) ∧
(x1 ∨ ¬x6) ∧ (x4 ∨ x5 ∨ x6) ∧ (x5 ∨ ¬x6)
ϕ = {x1=1, x2=1, x3=1}
Heule & Hunt (UT Austin) Look-ahead Techniques 13 / 1
Look-ahead: Autarky detection
Flearning := (¬x1 ∨ ¬x3 ∨ x4) ∧ (¬x1 ∨ ¬x2 ∨ x3) ∧(¬x1 ∨ x2) ∧ (x1 ∨ x3 ∨ x6) ∧ (¬x1 ∨ x4 ∨ ¬x5) ∧
(x1 ∨ ¬x6) ∧ (x4 ∨ x5 ∨ x6) ∧ (x5 ∨ ¬x6)
ϕ = {x1=1, x2=1, x3=1, x4=1}
Heule & Hunt (UT Austin) Look-ahead Techniques 13 / 1
Look-ahead: Autarky detection
Flearning := (¬x1 ∨ ¬x3 ∨ x4) ∧ (¬x1 ∨ ¬x2 ∨ x3) ∧(¬x1 ∨ x2) ∧ (x1 ∨ x3 ∨ x6) ∧ (¬x1 ∨ x4 ∨ ¬x5) ∧
(x1 ∨ ¬x6) ∧ (x4 ∨ x5 ∨ x6) ∧ (x5 ∨ ¬x6)
ϕ = {x1=1, x2=1, x3=1, x4=1}
Flearning satisfiability equivalent to (x5 ∨ ¬x6)
Heule & Hunt (UT Austin) Look-ahead Techniques 13 / 1
Look-ahead: Autarky detection
Flearning := (¬x1 ∨ ¬x3 ∨ x4) ∧ (¬x1 ∨ ¬x2 ∨ x3) ∧(¬x1 ∨ x2) ∧ (x1 ∨ x3 ∨ x6) ∧ (¬x1 ∨ x4 ∨ ¬x5) ∧
(x1 ∨ ¬x6) ∧ (x4 ∨ x5 ∨ x6) ∧ (x5 ∨ ¬x6)
ϕ = {x1=1, x2=1, x3=1, x4=1}
Flearning satisfiability equivalent to (x5 ∨ ¬x6)
Could reduce computational cost on UNSAT
Heule & Hunt (UT Austin) Look-ahead Techniques 13 / 1
Look-ahead: Autarky or Conflict on 2-SAT Formulae
Lookahead techniques can solve 2-SAT formulae in
polynomial time. Each lookahead on l results:
1 in an autarky: forcing l to be true
2 in a conflict: forcing l to be false
Heule & Hunt (UT Austin) Look-ahead Techniques 14 / 1
Look-ahead: Autarky or Conflict on 2-SAT Formulae
Lookahead techniques can solve 2-SAT formulae in
polynomial time. Each lookahead on l results:
1 in an autarky: forcing l to be true
2 in a conflict: forcing l to be false
SAT gameby Olivier Roussel
http://www.cs.utexas.edu/~marijn/game/2SAT
Heule & Hunt (UT Austin) Look-ahead Techniques 14 / 1
Look-ahead: 1-Autarky learning
Flearning := (¬x1 ∨ ¬x3 ∨ x4) ∧ (¬x1 ∨ ¬x2 ∨ x3) ∧(¬x1 ∨ x2) ∧ (x1 ∨ x3 ∨ x6) ∧ (¬x1 ∨ x4 ∨ ¬x5) ∧
(x1 ∨ ¬x6) ∧ (x4 ∨ x5 ∨ x6) ∧ (x5 ∨ ¬x6)
Heule & Hunt (UT Austin) Look-ahead Techniques 15 / 1
Look-ahead: 1-Autarky learning
Flearning := (¬x1 ∨ ¬x3 ∨ x4) ∧ (¬x1 ∨ ¬x2 ∨ x3) ∧(¬x1 ∨ x2) ∧ (x1 ∨ x3 ∨ x6) ∧ (¬x1 ∨ x4 ∨ ¬x5) ∧
(x1 ∨ ¬x6) ∧ (x4 ∨ x5 ∨ x6) ∧ (x5 ∨ ¬x6)
ϕ = {x2=0}
Heule & Hunt (UT Austin) Look-ahead Techniques 15 / 1
Look-ahead: 1-Autarky learning
Flearning := (¬x1 ∨ ¬x3 ∨ x4) ∧ (¬x1 ∨ ¬x2 ∨ x3) ∧(¬x1 ∨ x2) ∧ (x1 ∨ x3 ∨ x6) ∧ (¬x1 ∨ x4 ∨ ¬x5) ∧
(x1 ∨ ¬x6) ∧ (x4 ∨ x5 ∨ x6) ∧ (x5 ∨ ¬x6)
ϕ = {x2=0, x1=0}
Heule & Hunt (UT Austin) Look-ahead Techniques 15 / 1
Look-ahead: 1-Autarky learning
Flearning := (¬x1 ∨ ¬x3 ∨ x4) ∧ (¬x1 ∨ ¬x2 ∨ x3) ∧(¬x1 ∨ x2) ∧ (x1 ∨ x3 ∨ x6) ∧ (¬x1 ∨ x4 ∨ ¬x5) ∧
(x1 ∨ ¬x6) ∧ (x4 ∨ x5 ∨ x6) ∧ (x5 ∨ ¬x6)
ϕ = {x2=0, x1=0, x6=0}
Heule & Hunt (UT Austin) Look-ahead Techniques 15 / 1
Look-ahead: 1-Autarky learning
Flearning := (¬x1 ∨ ¬x3 ∨ x4) ∧ (¬x1 ∨ ¬x2 ∨ x3) ∧(¬x1 ∨ x2) ∧ (x1 ∨ x3 ∨ x6) ∧ (¬x1 ∨ x4 ∨ ¬x5) ∧
(x1 ∨ ¬x6) ∧ (x4 ∨ x5 ∨ x6) ∧ (x5 ∨ ¬x6)
ϕ = {x2=0, x1=0, x6=0, x3=1}
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Look-ahead: 1-Autarky learning
Flearning := (¬x1 ∨ ¬x3 ∨ x4) ∧ (¬x1 ∨ ¬x2 ∨ x3) ∧(¬x1 ∨ x2) ∧ (x1 ∨ x3 ∨ x6) ∧ (¬x1 ∨ x4 ∨ ¬x5) ∧
(x1 ∨ ¬x6) ∧ (x4 ∨ x5 ∨ x6) ∧ (x5 ∨ ¬x6)
ϕ = {x2=0, x1=0, x6=0, x3=1}
(local) 1-autarky resolvents:(¬x2 ∨ ¬x4) and (¬x2 ∨ ¬x5)
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Tree-based Look-ahead
Given a formula F which includes the clauses (a ∨ b̄) and (a ∨ c̄),tree-based look-ahead can reduce the costs of look-aheads.
F
a
b c
2
3 4
5
1 6
implication
action
1 propagate a
2 propagate b
3 backtrack b
4 propagate c
5 backtrack c
6 backtrack a
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Homework puzzle
Populate the matrix with the following numbers:
10, 40, 50, 12, 21, 31, 35, 43, 46, 56, 74, 75, 83, 87, 89, 98such that no digit repeats in any row, column or diagonal.
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Practical SAT Solving: Look-ahead Techniques
Marijn J.H. HeuleWarren A. Hunt Jr.
The University of Texas at Austin
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