Lookahead Techniques

Lookahead Techniques

Marijn J.H. Heule

http://www.cs.cmu.edu/~mheule/15816-f19/

Automated Reasoning and Satisfiability, September 26, 2019

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DPLL Procedure

Look-ahead Architecture

Look-ahead Learning

Autarky Reasoning

Tree-based Look-ahead

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DPLL Procedure


Look-ahead Learning

Autarky Reasoning


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SAT Solving: DPLL

Davis Putnam Logemann Loveland [DP60,DLL62]

Recursive procedure that in each recursive call:Simplifies the formula (using unit propagation)

Splits the formula into two subformulas• Variable selection heuristics (which variable to split on)• Direction heuristics (which subformula to explore first)

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DPLL: Example

FDPLL := (x1 ∨ x2 ∨ x3) ∧ (x1 ∨ x2 ∨ x3) ∧(x1 ∨ x2 ∨ x3) ∧ (x1 ∨ x3) ∧ (x1 ∨ x3)

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DPLL: Example


x3

0 1

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DPLL: Example


x3

0 1

x2

x1 x3

0 1

0 1 1 0

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DPLL: Slightly Harder Example

Slightly Harder Example

Construct a DPLL tree for:

(a ∨ b ∨ c) ∧ (a ∨ b ∨ c) ∧(b ∨ c ∨ d) ∧ (b ∨ c ∨ d) ∧(a ∨ c ∨ d) ∧ (a ∨ c ∨ d) ∧(a ∨ b ∨ d)

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Look-ahead: Definition

DPLL with selection of (effective) decision variablesby look-aheads on variables

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Look-ahead:

Assign a variable to a truth value

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Look-ahead:


Simplify the formula

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Look-ahead:



Measure the reduction

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Look-ahead:




Learn if possible

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Look-ahead:




Learn if possible

Backtrack

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DPLL Procedure


Look-ahead Learning

Autarky Reasoning


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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)

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Look-ahead: Example


(x1 ∨ x6) ∧ (x4 ∨ x5 ∨ x6) ∧ (x5 ∨ x6)

ϕ = {x2=0}

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Look-ahead: Example


(x1 ∨ x6) ∧ (x4 ∨ x5 ∨ x6) ∧ (x5 ∨ x6)

ϕ = {x2=0, x1=0}

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Look-ahead: Example


(x1 ∨ x6) ∧ (x4 ∨ x5 ∨ x6) ∧ (x5 ∨ x6)

ϕ = {x2=0, x1=0, x6=0}

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Look-ahead: Example


(x1 ∨ x6) ∧ (x4 ∨ x5 ∨ x6) ∧ (x5 ∨ x6)

ϕ = {x2=0, x1=0, x6=0, x3=1}

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Look-ahead: Properties

Very expensive

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Very expensive

Effective compared to cheap heuristics

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Very expensive


Detection of failed literals (and more)

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Very expensive



Strong on random k-SAT formulae

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Very expensive



Strong on random k-SAT formulae

Examples: march, OKsolver, kcnfs

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DEMO

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Look-ahead: Reduction heuristics

Number of satisfied clauses

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Number of implied variables

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Number of implied variables

New (reduced, not satisfied) clauses

• Smaller clauses more important

• Weights based on occurring

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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 )

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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) then

6: return satisfiable

7: return DPLL (F(ldecision ← 0))

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DPLL Procedure


Look-ahead Learning

Autarky Reasoning


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Local Learning

Look-ahead solvers do not perform global learning,in contrast to contrast to conflict-driven clauselearning (CDCL) solvers

Instead, look-ahead solvers learn locally:Learn small (typically unit or binary) clauses that are validfor the current node and lower in the DPLL tree

Locally learnt clauses have to be removed duringbacktracking

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Failed Literals and Double Look-aheads

A literal l is called a failed literal if the look-ahead on l = 1results in a conflict:

failed literal l is forced to false followed by unit propagation

if both x and x are failed literals, then backtrack

Failed literals can be generalized by double lookahead: assigntwo literals and learn a binary clause in case of a conflict.

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(x1 ∨ x6) ∧ (x4 ∨ x5 ∨ x6) ∧ (x5 ∨ x6)

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(x1 ∨ x6) ∧ (x4 ∨ x5 ∨ x6) ∧ (x5 ∨ x6)

ϕ = {x4=0, x6=1}

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(x1 ∨ x6) ∧ (x4 ∨ x5 ∨ x6) ∧ (x5 ∨ x6)

ϕ = {x4=0, x6=1, x1=1}

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(x1 ∨ x6) ∧ (x4 ∨ x5 ∨ x6) ∧ (x5 ∨ x6)

ϕ = {x4=0, x6=1, x1=1, x2=1}

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(x1 ∨ x6) ∧ (x4 ∨ x5 ∨ x6) ∧ (x5 ∨ x6)

ϕ = {x4=0, x6=1, x1=1, x2=1, x3=1}

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Hyper Binary Resolution [Bacchus 2002]

Definition (Hyper Binary Resolution Rule)

(x ∨ x1 ∨ x2 ∨ · · · ∨ xn) (x̄1 ∨ x ′) (x̄2 ∨ x ′) . . . (x̄n ∨ x ′)

(x ∨ x ′)

binary edge

hyper edge

hyper binary edge

x̄ ′

x

x̄1

. . .x̄2

x̄n

x ′

x̄

Hyper Binary Resolution Rule:

combines multiple resolution steps into one

uses one n-ary clauses and multiple binary clauses

special case hyper unary resolution where x = x ′

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Look-ahead: Hyper Binary Resolvents


(x1 ∨ x6) ∧ (x4 ∨ x5 ∨ x6) ∧ (x5 ∨ x6)

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(x1 ∨ x6) ∧ (x4 ∨ x5 ∨ x6) ∧ (x5 ∨ x6)

ϕ = {x2=0}

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(x1 ∨ x6) ∧ (x4 ∨ x5 ∨ x6) ∧ (x5 ∨ x6)

ϕ = {x2=0, x1=0}

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(x1 ∨ x6) ∧ (x4 ∨ x5 ∨ x6) ∧ (x5 ∨ x6)

ϕ = {x2=0, x1=0, x6=0}

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(x1 ∨ x6) ∧ (x4 ∨ x5 ∨ x6) ∧ (x5 ∨ x6)

ϕ = {x2=0, x1=0, x6=0, x3=1}

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(x1 ∨ x6) ∧ (x4 ∨ x5 ∨ x6) ∧ (x5 ∨ x6)

ϕ = {x2=0, x1=0, x6=0, x3=1}

hyper binary resolvents:

(x2 ∨ x6) and (x2 ∨ x3)

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(x1 ∨ x6) ∧ (x4 ∨ x5 ∨ x6) ∧ (x5 ∨ x6)

ϕ = {x2=0, x1=0, x6=0, x3=1}

hyper binary resolvents:

(x2 ∨ x6) and (x2 ∨ x3)

Which one is more useful?

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Look-ahead: Necessary assignments


(x1 ∨ x6) ∧ (x4 ∨ x5 ∨ x6) ∧ (x5 ∨ x6)

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(x1 ∨ x6) ∧ (x4 ∨ x5 ∨ x6) ∧ (x5 ∨ x6)

ϕ = {x1=1}

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(x1 ∨ x6) ∧ (x4 ∨ x5 ∨ x6) ∧ (x5 ∨ x6)

ϕ = {x1=1, x2=1}

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(x1 ∨ x6) ∧ (x4 ∨ x5 ∨ x6) ∧ (x5 ∨ x6)

ϕ = {x1=1, x2=1, x3=1}

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(x1 ∨ x6) ∧ (x4 ∨ x5 ∨ x6) ∧ (x5 ∨ x6)

ϕ = {x1=1, x2=1, x3=1, x4=1}

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(x1 ∨ x6) ∧ (x4 ∨ x5 ∨ x6) ∧ (x5 ∨ x6)

ϕ = {x1=1, x2=1, x3=1, x4=1}


(x1 ∨ x6) ∧ (x4 ∨ x5 ∨ x6) ∧ (x5 ∨ x6)

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(x1 ∨ x6) ∧ (x4 ∨ x5 ∨ x6) ∧ (x5 ∨ x6)

ϕ = {x1=1, x2=1, x3=1, x4=1}


(x1 ∨ x6) ∧ (x4 ∨ x5 ∨ x6) ∧ (x5 ∨ x6)

ϕ = {x1=0}

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(x1 ∨ x6) ∧ (x4 ∨ x5 ∨ x6) ∧ (x5 ∨ x6)

ϕ = {x1=1, x2=1, x3=1, x4=1}


(x1 ∨ x6) ∧ (x4 ∨ x5 ∨ x6) ∧ (x5 ∨ x6)

ϕ = {x1=0, x6=0}

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(x1 ∨ x6) ∧ (x4 ∨ x5 ∨ x6) ∧ (x5 ∨ x6)

ϕ = {x1=1, x2=1, x3=1, x4=1}


(x1 ∨ x6) ∧ (x4 ∨ x5 ∨ x6) ∧ (x5 ∨ x6)

ϕ = {x1=0, x6=0, x3=1}

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St̊almarck’s Method

In short, St̊almarck’s Method is a procedure that generalizesthe concept of necessary assignments.

For each variable x , (Simplify(F |x) ∩ Simplify(F |x)) \ F isadded to F .

The above is repeated until fixpoint, i.e., until∀x : (Simplify(F |x) ∩ Simplify(F |x)) \ F = ∅

Afterwards the procedure is repeated using all pairs forvariables x and y : Add (Simplify(F |xy ) ∩ Simplify(F |xy) ∩Simplify(F |xy ∩ Simplify(F |xy)) \ F to F .

The second round is very expensive and can typically not befinished in reasonable time.

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DPLL Procedure


Look-ahead Learning

Autarky Reasoning


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Look-ahead: Autarky definition

An autarky is a partial assignment that satisfies all

clauses that are “touched” by the assignment

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clauses that are “touched” by the assignmenta pure literal is an autarky

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each satisfying assignment is an autarky

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the remaining formula is satisfiability equivalentto the original formula

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the remaining formula is satisfiability equivalentto the original formula

An 1-autarky is a partial assignment that

satisfies all touched clauses except one

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Look-ahead: Autarky detection


(x1 ∨ x6) ∧ (x4 ∨ x5 ∨ x6) ∧ (x5 ∨ x6)

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(x1 ∨ x6) ∧ (x4 ∨ x5 ∨ x6) ∧ (x5 ∨ x6)

ϕ = {x1=1}

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(x1 ∨ x6) ∧ (x4 ∨ x5 ∨ x6) ∧ (x5 ∨ x6)

ϕ = {x1=1, x2=1}

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(x1 ∨ x6) ∧ (x4 ∨ x5 ∨ x6) ∧ (x5 ∨ x6)

ϕ = {x1=1, x2=1, x3=1}

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(x1 ∨ x6) ∧ (x4 ∨ x5 ∨ x6) ∧ (x5 ∨ x6)

ϕ = {x1=1, x2=1, x3=1, x4=1}

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(x1 ∨ x6) ∧ (x4 ∨ x5 ∨ x6) ∧ (x5 ∨ x6)

ϕ = {x1=1, x2=1, x3=1, x4=1}

Flearning satisfiability equivalent to (x5 ∨ x6)

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(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

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Look-ahead: Autarky or Conflict on 2-SAT Formulae

Lookahead techniques can solve 2-SAT formulae inpolynomial time. Each lookahead on l results:

1. in an autarky: forcing l to be true

2. in a conflict: forcing l to be false

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Look-ahead: Autarky or Conflict on 2-SAT Formulae

Lookahead techniques can solve 2-SAT formulae inpolynomial 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

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http://www.cs.utexas.edu/~marijn/game/2SAT

Look-ahead: 1-Autarky learning


(x1 ∨ x6) ∧ (x4 ∨ x5 ∨ x6) ∧ (x5 ∨ x6)

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(x1 ∨ x6) ∧ (x4 ∨ x5 ∨ x6) ∧ (x5 ∨ x6)

ϕ = {x2=0}

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(x1 ∨ x6) ∧ (x4 ∨ x5 ∨ x6) ∧ (x5 ∨ x6)

ϕ = {x2=0, x1=0}

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(x1 ∨ x6) ∧ (x4 ∨ x5 ∨ x6) ∧ (x5 ∨ x6)

ϕ = {x2=0, x1=0, x6=0}

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(x1 ∨ x6) ∧ (x4 ∨ x5 ∨ x6) ∧ (x5 ∨ x6)

ϕ = {x2=0, x1=0, x6=0, x3=1}

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(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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DPLL Procedure


Look-ahead Learning

Autarky Reasoning


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Given a formula F which includes the clauses (a ∨ b) and(a ∨ c), tree-based look-ahead can reduce the look-ahead costs.

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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Lookahead Techniques

Marijn J.H. Heule


Automated Reasoning and Satisfiability, September 26, 2019

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Lookahead Techniques

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