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Introduction to MiniSat v1.14. Presented by Yunho Kim Provable Software Lab, KAIST. Overview VSIDS Decision Heuristic Conflict Clause Analysis Example. Contents. MiniSat is a fast SAT solver developed by Niklas Eén and Niklas Sörensson

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introduction to minisat v1 14

Introduction to MiniSat v1.14

Presented by Yunho Kim

Provable Software Lab, KAIST

contents

Overview

VSIDS Decision Heuristic

Conflict Clause Analysis

Example

Contents

MiniSat v1.14, Yunho Kim, Provable Software Lab, KAIST

overview

MiniSat is a fast SAT solver developed by NiklasEén and NiklasSörensson

    • MiniSatwon all industrial categories in SAT 2005 competition
    • MiniSatwon SAT-Race 2006
  • MiniSat is simple and well-documented
    • Well-defined interface for general use
    • Helpful implementation documents and comments
    • Minimal but efficient heuristic
Overview

MiniSat v1.14, Yunho Kim, Provable Software Lab, KAIST

overview1

CNF is a logical AND of one or more clauses

Clause is a logical OR of one or more literals

Literal is either the positive or the negativevariable

Overview

(x2∨x4∨x5∨x6) ∧ (x0∨-x1∨x6)∧ (-x2∨-x3∨-x4)

(x2∨x4∨x5∨x6), (x0∨-x1∨x6), (-x2∨-x3∨-x4)

Variables: x0, x1, x2, x3, x4, x5, x6

Literals: x0, -x1, x2, -x2, -x3, x4, -x4, x5, x6

Definition from Lintao Zhang and Sharadmalik

“The Quest for Efficient Boolean Satisfiability Solvers”

MiniSat v1.14, Yunho Kim, Provable Software Lab, KAIST

overview2

Unit clause is a clause in which all but one of literals is assigned to False

  • Unit literal is the unassigned literal in a unit clause
    • (x0) is a unit clause and ‘x0’ is a unit literal
    • (-x0∨x1) is a unit clause since x0 has to be True
    • (-x2∨-x3∨-x4) can be a unit clause if the current assignment is that x3 and x4 are True
  • Boolean Constrain Propagation(BCP) is the process of assigning the True value to all unit literals
Overview

……

(x0)∧

(-x0∨x1)∧

(-x2∨-x3∨-x4)∧

……

MiniSat v1.14, Yunho Kim, Provable Software Lab, KAIST

overview3

/* overall structure of Minisat solve procedure */

Solve(){

while(true){

boolean_constraint_propagation();

if(no_conflict){

if(no_unassigned_variable) return SAT;

decision_level++;

make_decision();

}else{

if (no_dicisions_made) return UNSAT;

analyze_conflict();

undo_assignments();

add_conflict_clause();

}

}

}

Overview

MiniSat v1.14, Yunho Kim, Provable Software Lab, KAIST

vsids decision heuristic

Variable State Independent Decaying Sum(VSIDS)

    • decision heuristic to determine what variable will be assigned next
    • decision is independent from the current assignment of each variable
  • VSIDS makes decisions based on activity
    • Activity is a literal occurrence count with higher weight on the more recently added clauses
VSIDS Decision Heuristic

activity description from Lintao Zhang and Sharadmalik

“The Quest for Efficient Boolean Satisfiability Solvers”

MiniSat v1.14, Yunho Kim, Provable Software Lab, KAIST

vsids decision heuristic1

VSIDS is proposed by chaff first.

Initially, the score for each literal is the occurrence on the given input CNF instance

VSIDS Decision Heuristic

Initial constraints

(x1∨x4) ∧

(x1∨-x3∨-x5) ∧

(x1∨x5∨x7) ∧

(x2∨x8) ∧

(-x7∨-x3∨x6) ∧

(-x7∨x5∨-x6) ∧

(x7∨x5∨-x9)

MiniSat v1.14, Yunho Kim, Provable Software Lab, KAIST

vsids decision heuristic2

When a clause is added to DB, the related counters are incremented

    • Conflict analysis adds a new learnt clause to DB
VSIDS Decision Heuristic

(x1∨x4) ∧

(x1∨-x3∨-x5) ∧

(x1∨x5∨x7) ∧

(x2∨x8) ∧

(-x7∨-x3∨x6) ∧

(-x7∨x5∨-x6) ∧

(x7∨x5∨-x9)∧

(x7∨x6∨-x9)

MiniSat v1.14, Yunho Kim, Provable Software Lab, KAIST

vsids decision heuristic3

Periodically, the scores are divided by a constant factor

    • In this example, a constant factor is 2
VSIDS Decision Heuristic

(x1∨x4) ∧

(x1∨-x3∨-x5) ∧

(x1∨x5∨x7) ∧

(x2∨x8) ∧

(-x7∨-x3∨x6) ∧

(-x7∨x5∨-x6) ∧

(x7∨x5∨-x9)∧

(x7∨x6∨-x9)

MiniSat v1.14, Yunho Kim, Provable Software Lab, KAIST

vsids decision heuristic4

Literals on more recently added clause have higher weighted scores

VSIDS Decision Heuristic

(x1∨x4) ∧

(x1∨-x3∨-x5) ∧

(x1∨x5∨x7) ∧

(x2∨x8) ∧

(-x7∨-x3∨x6) ∧

(-x7∨x5∨-x6) ∧

(x7∨x5∨-x9)∧

(x7∨x6∨-x9) ∧

(-x9∨x8)

MiniSat v1.14, Yunho Kim, Provable Software Lab, KAIST

vsids decision heuristic5

VSIDS in MiniSat is slightly different from chaff

    • MiniSat does not consider polarity
    • Before adding thefirst learnt clause, all the scores are 0
    • All the scores are decaying by 5% when a conflict occurs
    • The activities of all variables occurring in some clause used in the conflict analysis are increased
VSIDS Decision Heuristic

MiniSat v1.14, Yunho Kim, Provable Software Lab, KAIST

conflict clause analysis
Conflict Clause Analysis
  • A conflict happens when one clause is falsified by unit propagation
  • Analyze the conflicting clause to infer a clause
    • (-x3∨-x2∨-x1) is conflicting clause
  • The inferred clause is a new knowledge
    • A new learnt clause is added to constraints

Assume x4 is False

(x1∨x4) ∧

(-x1∨x2) ∧

(-x2∨x3) ∧

(-x3∨-x2∨-x1) Falsified!

Omitted clauses

MiniSat v1.14, Yunho Kim, Provable Software Lab, KAIST

conflict clause analysis1
Conflict Clause Analysis
  • Learnt clauses are inferred by conflict analysis
  • They help prune future parts of the search space
    • Assigning False to x4 is the casual of conflict
    • Adding (x4) to constraints prohibit conflict from –x4
  • Learnt clauses actually drive backtracking

(x1∨x4) ∧

(-x1∨x2) ∧

(-x2∨x3) ∧

(-x3∨-x2∨-x1) ∧

omitted clauses ∧

(x4) learnt clause

MiniSat v1.14, Yunho Kim, Provable Software Lab, KAIST

conflict clause analysis2
Conflict Clause Analysis

/* conflict analysis algorithm */

Analyze_conflict(){

cl = find_confclicting_clause();

/* Loop until cl is falsified and zero or one propagated literals in current decision level are remained */

While(!stop_criterion_met(cl)){

lit = choose_literal(cl); /* select the last propagated literal */

Var = variable_of_literal(lit);

ante = antecedent(var);

cl = resolve(cl, ante, var);

}

add_clause_to_database(cl);

/* backtrack level is the lowest decision level for which the learnt clause is unit clause */

back_dl = clause_asserting_level(cl);

return back_dl;

}

Algorithm from Lintao Zhang and Sharadmalik

“The Quest for Efficient Boolean Satisfiability Solvers”

MiniSat v1.14, Yunho Kim, Provable Software Lab, KAIST

conflict clause analysis3

DLevel=2

DLevel=1

Conflict Clause Analysis

e=F

a=T

Conflict

-b∨-c∨-d

Example slides are from CMU 15-414 course ppt

MiniSat v1.14, Yunho Kim, Provable Software Lab, KAIST

conflict clause analysis4

DLevel=2

DLevel=1

Conflict Clause Analysis

e=F

a=T

-b∨-c∨-d

MiniSat v1.14, Yunho Kim, Provable Software Lab, KAIST

conflict clause analysis5

DLevel=2

DLevel=1

Conflict Clause Analysis

e=F

a=T

-b∨-c∨-d

-b∨-c∨h

MiniSat v1.14, Yunho Kim, Provable Software Lab, KAIST

conflict clause analysis6

DLevel=2

DLevel=1

Conflict Clause Analysis

e=F

a=T

-b∨-c∨-d

-b∨-c∨h

MiniSat v1.14, Yunho Kim, Provable Software Lab, KAIST

conflict clause analysis7

DLevel=2

DLevel=1

Conflict Clause Analysis

e=F

a=T

-b∨-c∨-d

-b∨-c∨h

-b∨e∨f∨hlearnt clause

MiniSat v1.14, Yunho Kim, Provable Software Lab, KAIST

conflict clause analysis8

DLevel=1

Conflict Clause Analysis

e=F

a=T

b=F

-b∨-c∨-d

-b∨-c∨h

-b∨e∨f∨h

MiniSat v1.14, Yunho Kim, Provable Software Lab, KAIST

example
Example
  • Left side shows status of clauses
  • Green literals are True, reds are False and yellows are Undefined

MiniSat v1.14, Yunho Kim, Provable Software Lab, KAIST

example1
Example
  • Right-up side shows variable data
  • Reason indicates what clause the variable propagated from
  • Level is a decision level

MiniSat v1.14, Yunho Kim, Provable Software Lab, KAIST

example2
Example
  • Right-down shows trail, that is an assignment stack

MiniSat v1.14, Yunho Kim, Provable Software Lab, KAIST

example3
Example
  • First assign False to x7

MiniSat v1.14, Yunho Kim, Provable Software Lab, KAIST

example4
Example
  • BCP from –x7

MiniSat v1.14, Yunho Kim, Provable Software Lab, KAIST

example5
Example
  • Second assign False to x3

MiniSat v1.14, Yunho Kim, Provable Software Lab, KAIST

example6
Example
  • C0 is a conflicting clause. Analyze it

MiniSat v1.14, Yunho Kim, Provable Software Lab, KAIST

example7
Example

Activity is changed

  • First resolve last propagated literal
  • c0 and c2 are resolved w.r.t x0

MiniSat v1.14, Yunho Kim, Provable Software Lab, KAIST

example8
Example
  • A new learnt clause is (-x2∨x4∨x6∨x7)

MiniSat v1.14, Yunho Kim, Provable Software Lab, KAIST

example9
Example

Activities are decayed for next added clause

  • Adds a learnt clause to DB and go on search

MiniSat v1.14, Yunho Kim, Provable Software Lab, KAIST

example10
Example
  • Finally we find a model, that is, satisfying assignment

MiniSat v1.14, Yunho Kim, Provable Software Lab, KAIST

references

An Extensible SAT-solver

by NiklasEen and NiklasSörensson

in SAT 2003

References

MiniSat v1.14, Yunho Kim, Provable Software Lab, KAIST

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