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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

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

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

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

……

(x0)∧

(-x0∨x1)∧

(-x2∨-x3∨-x4)∧

……

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

/* 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();

}

}

}

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

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

activity description from Lintao Zhang and Sharadmalik

“The Quest for Efficient Boolean Satisfiability Solvers”

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

VSIDS is proposed by chaff first.

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

VSIDS Decision HeuristicInitial 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

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

- Conflict analysis adds a new learnt clause to DB

(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

Periodically, the scores are divided by a constant factor

- In this example, a constant factor is 2

(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

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 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

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

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 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 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

DLevel=1

Conflict Clause Analysise=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

DLevel=1

Conflict Clause Analysise=F

a=T

-b∨-c∨-d

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

DLevel=1

Conflict Clause Analysise=F

a=T

-b∨-c∨-d

-b∨-c∨h

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

DLevel=1

Conflict Clause Analysise=F

a=T

-b∨-c∨-d

-b∨-c∨h

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

DLevel=1

Conflict Clause Analysise=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 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

- 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

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

Example

- Right-down shows trail, that is an assignment stack

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

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

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

Example

- Finally we find a model, that is, satisfying assignment

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

by NiklasEen and NiklasSörensson

in SAT 2003

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

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