Factoring Out Assumptions to Speed Up MUS Extraction Jean-Marie - - PowerPoint PPT Presentation

Factoring Out Assumptions to Speed Up MUS Extraction

Jean-Marie Lagniez1 Armin Biere2 11 July 2013

1 Univ. Lille-Nord de France – CRIL/CNRS UMR8188 – Lens, F-62307, France 2 Institute for Formal Models and Verification Johannes Kepler University, Linz, Austria

Minimal Unsatisfiable Set (MUS)

x ∨ y ∨ z x ∨ ¬y x ∨ ¬z ¬x ∨ y ∨ z x ∨ w w ∨ z ∨ ¬y ¬x ∨ ¬y ¬x ∨ ¬z w ∨ ¬x ∨ ¬z

UNSAT

The formula is unsatisfiable : why? Subset of constraints minimally unsatisfiable Two approaches:

→ constructive → destructive

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Minimal Unsatisfiable Set (MUS)

x ∨ y ∨ z x ∨ ¬y x ∨ ¬z ¬x ∨ y ∨ z x ∨ w w ∨ z ∨ ¬y ¬x ∨ ¬y ¬x ∨ ¬z w ∨ ¬x ∨ ¬z

The formula is unsatisfiable : why? Subset of constraints minimally unsatisfiable Two approaches:

→ constructive → destructive

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Minimal Unsatisfiable Set (MUS)

x ∨ y ∨ z x ∨ ¬y x ∨ ¬z ¬x ∨ y ∨ z x ∨ w w ∨ z ∨ ¬y ¬x ∨ ¬y ¬x ∨ ¬z w ∨ ¬x ∨ ¬z

The formula is unsatisfiable : why? Subset of constraints minimally unsatisfiable Two approaches:

→ constructive → destructive

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Minimal Unsatisfiable Set (MUS)

x ∨ y ∨ z x ∨ ¬y x ∨ ¬z ¬x ∨ y ∨ z x ∨ w w ∨ z ∨ ¬y ¬x ∨ ¬y ¬x ∨ ¬z w ∨ ¬x ∨ ¬z

SAT

The formula is unsatisfiable : why? Subset of constraints minimally unsatisfiable Two approaches:

→ constructive → destructive

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Minimal Unsatisfiable Set (MUS)

x ∨ y ∨ z x ∨ ¬y x ∨ ¬z ¬x ∨ y ∨ z x ∨ w w ∨ z ∨ ¬y ¬x ∨ ¬y ¬x ∨ ¬z w ∨ ¬x ∨ ¬z

The formula is unsatisfiable : why? Subset of constraints minimally unsatisfiable Two approaches:

→ constructive → destructive

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Minimal Unsatisfiable Set (MUS)

x ∨ y ∨ z x ∨ ¬y x ∨ ¬z ¬x ∨ y ∨ z x ∨ w w ∨ z ∨ ¬y ¬x ∨ ¬y ¬x ∨ ¬z w ∨ ¬x ∨ ¬z

UNSAT

The formula is unsatisfiable : why? Subset of constraints minimally unsatisfiable Two approaches:

→ constructive → destructive

2 / 16

Minimal Unsatisfiable Set (MUS)

x ∨ y ∨ z x ∨ ¬y x ∨ ¬z ¬x ∨ y ∨ z x ∨ w w ∨ z ∨ ¬y ¬x ∨ ¬y ¬x ∨ ¬z w ∨ ¬x ∨ ¬z

The formula is unsatisfiable : why? Subset of constraints minimally unsatisfiable Two approaches:

→ constructive → destructive

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Minimal Unsatisfiable Set (MUS)

x ∨ y ∨ z x ∨ ¬y x ∨ ¬z ¬x ∨ y ∨ z x ∨ w w ∨ z ∨ ¬y ¬x ∨ ¬y ¬x ∨ ¬z w ∨ ¬x ∨ ¬z

MUS!

The formula is unsatisfiable : why? Subset of constraints minimally unsatisfiable Two approaches:

→ constructive → destructive

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Minimal Unsatisfiable Set (MUS)

x ∨ y ∨ z x ∨ ¬y x ∨ ¬z ¬x ∨ y ∨ z x ∨ w w ∨ z ∨ ¬y ¬x ∨ ¬y ¬x ∨ ¬z w ∨ ¬x ∨ ¬z

MUS!

The formula is unsatisfiable : why? Subset of constraints minimally unsatisfiable Two approaches:

→ constructive → destructive SAT Incremental

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From SAT to Incremental SAT

Solving the SAT problem

Modern SAT solvers are based on the CDCL paradigm Dynamic heuristics:

→ VSIDS, polarity, cleaning learned clauses and restart

Solving incrementally SAT

Successive calls of a SAT solver Keeping a lot of information between the different runs

→ VSIDS, polarity, cleaning learned clauses and restart

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From SAT to Incremental SAT

Solving the SAT problem

Modern SAT solvers are based on the CDCL paradigm Dynamic heuristics:

→ VSIDS, polarity, cleaning learned clauses and restart

Solving incrementally SAT

Successive calls of a SAT solver Keeping a lot of information between the different runs

→ VSIDS, polarity, cleaning learned clauses and restart → learned clauses

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From SAT to Incremental SAT

Solving the SAT problem

Modern SAT solvers are based on the CDCL paradigm Dynamic heuristics:

→ VSIDS, polarity, cleaning learned clauses and restart

Solving incrementally SAT

Successive calls of a SAT solver Keeping a lot of information between the different runs

→ VSIDS, polarity, cleaning learned clauses and restart → learned clauses

Adding selectors

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Selectors

a1 ∨ x ∨ y ∨ z a2 ∨ x ∨ ¬y a3 ∨ x ∨ ¬z a4 ∨ ¬x ∨ y ∨ z a5 ∨ x ∨ w a6 ∨ w ∨ z ∨ ¬y a7 ∨ ¬x ∨ ¬y a8 ∨ ¬x ∨ ¬z a9 ∨ w ∨ ¬x ∨ ¬z

To activate/deactivate the ith clause :

→ assign ai to false to activate the clause → assign ai to true to deactivate the clause

Used to know which initial clauses participating to the creation of each learned clause

a1 ∨ x ∨ y ∨ z a2 ∨ x ∨ ¬y a1 ∨ a2 ∨ x ∨ z

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Selectors

a1 ∨ x ∨ y ∨ z a2 ∨ x ∨ ¬y a3 ∨ x ∨ ¬z a4 ∨ ¬x ∨ y ∨ z a5 ∨ x ∨ w a6 ∨ w ∨ z ∨ ¬y a7 ∨ ¬x ∨ ¬y a8 ∨ ¬x ∨ ¬z a9 ∨ w ∨ ¬x ∨ ¬z

To activate/deactivate the ith clause :

→ assign ai to false to activate the clause → assign ai to true to deactivate the clause

Used to know which initial clauses participating to the creation of each learned clause

a1 ∨ x ∨ y ∨ z a2 ∨ x ∨ ¬y a1 ∨ a2 ∨ x ∨ z Selectors impact on the size of the clauses

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Factoring-out Assumptions

Introducing abbreviations to factor out assumptions

The replaced part consists of all assumptions and previously added abbreviations Connections between the abbreviations and the replaced literals is stored in a definition map

(p1 ∨ · · · ∨ pn ∨ a1 ∨ · · · ∨ am) is factored out into (p1 ∨ · · · ∨ pn ∨ ℓ) and ℓ → a1 ∨ · · · ∨ am

• G[ℓ]

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Definition Map

Under assumptions {¬a1, ¬a2, ¬a3, ¬a4, ¬a5, ¬a6, . . .}

learned clauses antecedents factoring factored clauses

G

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Definition Map

Under assumptions {¬a1, ¬a2, ¬a3, ¬a4, ¬a5, ¬a6, . . .}

α1 : p2 ∨ p7 ∨ a1 ∨ a2 ∨ a4 {. . .} learned clauses antecedents factoring factored clauses

G

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Definition Map

Under assumptions {¬a1, ¬a2, ¬a3, ¬a4, ¬a5, ¬a6, . . .}

α1 : p2 ∨ p7 ∨ a1 ∨ a2 ∨ a4 {. . .} learned clauses antecedents factoring α′

1 : p2 ∨ p7 ∨ ℓ1

factored clauses

G

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Definition Map

Under assumptions {¬a1, ¬a2, ¬a3, ¬a4, ¬a5, ¬a6, . . .}

α1 : p2 ∨ p7 ∨ a1 ∨ a2 ∨ a4 {. . .} learned clauses antecedents factoring α′

1 : p2 ∨ p7 ∨ ℓ1

factored clauses

G

a2 a1 a4 ℓ1

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Definition Map

Under assumptions {¬a1, ¬a2, ¬a3, ¬a4, ¬a5, ¬a6, . . .}

α1 : p2 ∨ p7 ∨ a1 ∨ a2 ∨ a4 α2 : p2 ∨ a2 ∨ a3 {. . .} {. . .} learned clauses antecedents factoring α′

1 : p2 ∨ p7 ∨ ℓ1

α′

2 : p2 ∨ ℓ2

factored clauses

G

a2 a1 a4 ℓ1

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Definition Map

Under assumptions {¬a1, ¬a2, ¬a3, ¬a4, ¬a5, ¬a6, . . .}

α1 : p2 ∨ p7 ∨ a1 ∨ a2 ∨ a4 α2 : p2 ∨ a2 ∨ a3 {. . .} {. . .} learned clauses antecedents factoring α′

1 : p2 ∨ p7 ∨ ℓ1

α′

2 : p2 ∨ ℓ2

factored clauses

G

a3 a2 a1 a4 ℓ1 ℓ2

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Definition Map

Under assumptions {¬a1, ¬a2, ¬a3, ¬a4, ¬a5, ¬a6, . . .}

α1 : p2 ∨ p7 ∨ a1 ∨ a2 ∨ a4 α2 : p2 ∨ a2 ∨ a3 α3 : p7 ∨ p4 ∨ p6 ∨ ℓ1 ∨ a4 {. . .} {. . .} {α1, . . .} learned clauses antecedents factoring α′

1 : p2 ∨ p7 ∨ ℓ1

α′

2 : p2 ∨ ℓ2

α′

3 : p7 ∨ p4 ∨ p6 ∨ ℓ3

factored clauses

G

a3 a2 a1 a4 ℓ1 ℓ2

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Definition Map

Under assumptions {¬a1, ¬a2, ¬a3, ¬a4, ¬a5, ¬a6, . . .}

α1 : p2 ∨ p7 ∨ a1 ∨ a2 ∨ a4 α2 : p2 ∨ a2 ∨ a3 α3 : p7 ∨ p4 ∨ p6 ∨ ℓ1 ∨ a4 {. . .} {. . .} {α1, . . .} learned clauses antecedents factoring α′

1 : p2 ∨ p7 ∨ ℓ1

α′

2 : p2 ∨ ℓ2

α′

3 : p7 ∨ p4 ∨ p6 ∨ ℓ3

factored clauses

G

a3 a2 a1 a4 ℓ1 ℓ2 ℓ3

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Definition Map

Under assumptions {¬a1, ¬a2, ¬a3, ¬a4, ¬a5, ¬a6, . . .}

α1 : p2 ∨ p7 ∨ a1 ∨ a2 ∨ a4 α2 : p2 ∨ a2 ∨ a3 α3 : p7 ∨ p4 ∨ p6 ∨ ℓ1 ∨ a4 α4 : p6 ∨ p8 ∨ ℓ2 ∨ a5 {. . .} {. . .} {α1, . . .} {α2, . . .} learned clauses antecedents factoring α′

1 : p2 ∨ p7 ∨ ℓ1

α′

2 : p2 ∨ ℓ2

α′

3 : p7 ∨ p4 ∨ p6 ∨ ℓ3

α′

4 : p6 ∨ p8 ∨ ℓ4

factored clauses

G

a3 a2 a1 a4 ℓ1 ℓ2 ℓ3

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Definition Map

Under assumptions {¬a1, ¬a2, ¬a3, ¬a4, ¬a5, ¬a6, . . .}

α1 : p2 ∨ p7 ∨ a1 ∨ a2 ∨ a4 α2 : p2 ∨ a2 ∨ a3 α3 : p7 ∨ p4 ∨ p6 ∨ ℓ1 ∨ a4 α4 : p6 ∨ p8 ∨ ℓ2 ∨ a5 {. . .} {. . .} {α1, . . .} {α2, . . .} learned clauses antecedents factoring α′

1 : p2 ∨ p7 ∨ ℓ1

α′

2 : p2 ∨ ℓ2

α′

3 : p7 ∨ p4 ∨ p6 ∨ ℓ3

α′

4 : p6 ∨ p8 ∨ ℓ4

factored clauses

G

a5 a3 a2 a1 a4 ℓ1 ℓ2 ℓ3 ℓ4

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Definition Map

Under assumptions {¬a1, ¬a2, ¬a3, ¬a4, ¬a5, ¬a6, . . .}

α1 : p2 ∨ p7 ∨ a1 ∨ a2 ∨ a4 α2 : p2 ∨ a2 ∨ a3 α3 : p7 ∨ p4 ∨ p6 ∨ ℓ1 ∨ a4 α4 : p6 ∨ p8 ∨ ℓ2 ∨ a5 α5 : p2 ∨ p5 ∨ a2 {. . .} {. . .} {α1, . . .} {α2, . . .} {. . .} learned clauses antecedents factoring α′

1 : p2 ∨ p7 ∨ ℓ1

α′

2 : p2 ∨ ℓ2

α′

3 : p7 ∨ p4 ∨ p6 ∨ ℓ3

α′

4 : p6 ∨ p8 ∨ ℓ4

α′

5 : p2 ∨ p5 ∨ a2

factored clauses

G

a5 a3 a2 a1 a4 ℓ1 ℓ2 ℓ3 ℓ4

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Definition Map

Under assumptions {¬a1, ¬a2, ¬a3, ¬a4, ¬a5, ¬a6, . . .}

α1 : p2 ∨ p7 ∨ a1 ∨ a2 ∨ a4 α2 : p2 ∨ a2 ∨ a3 α3 : p7 ∨ p4 ∨ p6 ∨ ℓ1 ∨ a4 α4 : p6 ∨ p8 ∨ ℓ2 ∨ a5 α5 : p2 ∨ p5 ∨ a2 α6 : p7 ∨ p4 ∨ ℓ3 ∨ ℓ4 {. . .} {. . .} {α1, . . .} {α2, . . .} {. . .} {α3, α4, . . .} learned clauses antecedents factoring α′

1 : p2 ∨ p7 ∨ ℓ1

α′

2 : p2 ∨ ℓ2

α′

3 : p7 ∨ p4 ∨ p6 ∨ ℓ3

α′

4 : p6 ∨ p8 ∨ ℓ4

α′

5 : p2 ∨ p5 ∨ a2

α′

6 : p7 ∨ p4 ∨ ℓ5

factored clauses

G

a5 a3 a2 a1 a4 ℓ1 ℓ2 ℓ3 ℓ4

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Definition Map

Under assumptions {¬a1, ¬a2, ¬a3, ¬a4, ¬a5, ¬a6, . . .}

α1 : p2 ∨ p7 ∨ a1 ∨ a2 ∨ a4 α2 : p2 ∨ a2 ∨ a3 α3 : p7 ∨ p4 ∨ p6 ∨ ℓ1 ∨ a4 α4 : p6 ∨ p8 ∨ ℓ2 ∨ a5 α5 : p2 ∨ p5 ∨ a2 α6 : p7 ∨ p4 ∨ ℓ3 ∨ ℓ4 {. . .} {. . .} {α1, . . .} {α2, . . .} {. . .} {α3, α4, . . .} learned clauses antecedents factoring α′

1 : p2 ∨ p7 ∨ ℓ1

α′

2 : p2 ∨ ℓ2

α′

3 : p7 ∨ p4 ∨ p6 ∨ ℓ3

α′

4 : p6 ∨ p8 ∨ ℓ4

α′

5 : p2 ∨ p5 ∨ a2

α′

6 : p7 ∨ p4 ∨ ℓ5

factored clauses

G

a5 a3 a2 a1 a4 ℓ1 ℓ2 ℓ3 ℓ4 ℓ5

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Initialisation

The definition map G can be interpreted as a non-cyclic circuit Abbreviations can be computed after all assumptions have been assigned In the MUS behaviour, the set of assumptions equals to the set of entries and it remains the same over all incremental calls

Example: under assumptions {a1, ¬a2, ¬a3, ¬a4, ¬a5, ¬a6, . . .} G

a5 a3 a2 a1 a4 ℓ1 ℓ2 ℓ3 ℓ4 ℓ5

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Initialisation

The definition map G can be interpreted as a non-cyclic circuit Abbreviations can be computed after all assumptions have been assigned In the MUS behaviour, the set of assumptions equals to the set of entries and it remains the same over all incremental calls

Example: under assumptions {a1, ¬a2, ¬a3, ¬a4, ¬a5, ¬a6, . . .} G

a5 a3 a2 a1 a4 ℓ1 ℓ2 ℓ3 ℓ4 ℓ5

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Initialisation

The definition map G can be interpreted as a non-cyclic circuit Abbreviations can be computed after all assumptions have been assigned In the MUS behaviour, the set of assumptions equals to the set of entries and it remains the same over all incremental calls

Example: under assumptions {a1, ¬a2, ¬a3, ¬a4, ¬a5, ¬a6, . . .} G

a5 a3 a2 a1 a4 ℓ1 ℓ2 ℓ3 ℓ4 ℓ5

7 / 16

Assumptions Core Analysis

Under assumptions {¬a1, ¬a2, ¬a3, ¬a4, ¬a5, ¬a6, . . .} α′

1 : p2 ∨ p7 ∨ ℓ1

α′

2 : p2 ∨ ℓ2

α′

3 : p7 ∨ p4 ∨ p6 ∨ ℓ3

α′

4 : p6 ∨ p8 ∨ ℓ4

α′

5 : p2 ∨ p5 ∨ a2

α′

6 : p7 ∨ p4 ∨ ℓ5

α7 : p2 ∨ ℓ1 factored clauses G

a5 a3 a2 a1 a4 ℓ1 ℓ2 ℓ3 ℓ4 ℓ5

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Assumptions Core Analysis

Under assumptions {¬a1, ¬a2, ¬a3, ¬a4, ¬a5, ¬a6, . . .} α′

1 : p2 ∨ p7 ∨ ℓ1

α′

2 : p2 ∨ ℓ2

α′

3 : p7 ∨ p4 ∨ p6 ∨ ℓ3

α′

4 : p6 ∨ p8 ∨ ℓ4

α′

5 : p2 ∨ p5 ∨ a2

α′

6 : p7 ∨ p4 ∨ ℓ5

α7 : p2 ∨ ℓ1 factored clauses ⊥ G

a5 a3 a2 a1 a4 ℓ1 ℓ2 ℓ3 ℓ4 ℓ5

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Assumptions Core Analysis

Under assumptions {¬a1, ¬a2, ¬a3, ¬a4, ¬a5, ¬a6, . . .} α′

1 : p2 ∨ p7 ∨ ℓ1

α′

2 : p2 ∨ ℓ2

α′

3 : p7 ∨ p4 ∨ p6 ∨ ℓ3

α′

4 : p6 ∨ p8 ∨ ℓ4

α′

5 : p2 ∨ p5 ∨ a2

α′

6 : p7 ∨ p4 ∨ ℓ5

α7 : p2 ∨ ℓ1 factored clauses ⊥ p2 p2 G

a5 a3 a2 a1 a4 ℓ1 ℓ2 ℓ3 ℓ4 ℓ5

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Assumptions Core Analysis

Under assumptions {¬a1, ¬a2, ¬a3, ¬a4, ¬a5, ¬a6, . . .} α′

1 : p2 ∨ p7 ∨ ℓ1

α′

2 : p2 ∨ ℓ2

α′

3 : p7 ∨ p4 ∨ p6 ∨ ℓ3

α′

4 : p6 ∨ p8 ∨ ℓ4

α′

5 : p2 ∨ p5 ∨ a2

α′

6 : p7 ∨ p4 ∨ ℓ5

α7 : p2 ∨ ℓ1 factored clauses ⊥ p2 p2 ℓ1 G

a5 a3 a2 a1 a4 ℓ1 ℓ2 ℓ3 ℓ4 ℓ5

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Assumptions Core Analysis

Under assumptions {¬a1, ¬a2, ¬a3, ¬a4, ¬a5, ¬a6, . . .} α′

1 : p2 ∨ p7 ∨ ℓ1

α′

2 : p2 ∨ ℓ2

α′

3 : p7 ∨ p4 ∨ p6 ∨ ℓ3

α′

4 : p6 ∨ p8 ∨ ℓ4

α′

5 : p2 ∨ p5 ∨ a2

α′

6 : p7 ∨ p4 ∨ ℓ5

α7 : p2 ∨ ℓ1 factored clauses ⊥ p2 p2 ℓ1 ℓ2 G

a5 a3 a2 a1 a4 ℓ1 ℓ2 ℓ3 ℓ4 ℓ5

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Assumptions Core Analysis

Under assumptions {¬a1, ¬a2, ¬a3, ¬a4, ¬a5, ¬a6, . . .} α′

1 : p2 ∨ p7 ∨ ℓ1

α′

2 : p2 ∨ ℓ2

α′

3 : p7 ∨ p4 ∨ p6 ∨ ℓ3

α′

4 : p6 ∨ p8 ∨ ℓ4

α′

5 : p2 ∨ p5 ∨ a2

α′

6 : p7 ∨ p4 ∨ ℓ5

α7 : p2 ∨ ℓ1 factored clauses ⊥ p2 p2 ℓ1 ℓ2 G

a5 a3 a2 a1 a4 ℓ1 ℓ2 ℓ3 ℓ4 ℓ5

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Assumptions Core Analysis

Under assumptions {¬a1, ¬a2, ¬a3, ¬a4, ¬a5, ¬a6, . . .} α′

1 : p2 ∨ p7 ∨ ℓ1

α′

2 : p2 ∨ ℓ2

α′

3 : p7 ∨ p4 ∨ p6 ∨ ℓ3

α′

4 : p6 ∨ p8 ∨ ℓ4

α′

5 : p2 ∨ p5 ∨ a2

α′

6 : p7 ∨ p4 ∨ ℓ5

α7 : p2 ∨ ℓ1 factored clauses ⊥ p2 p2 ℓ1 a3 a2 G

a5 a3 a2 a1 a4 ℓ1 ℓ2 ℓ3 ℓ4 ℓ5

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Assumptions Core Analysis

Under assumptions {¬a1, ¬a2, ¬a3, ¬a4, ¬a5, ¬a6, . . .} α′

1 : p2 ∨ p7 ∨ ℓ1

α′

2 : p2 ∨ ℓ2

α′

3 : p7 ∨ p4 ∨ p6 ∨ ℓ3

α′

4 : p6 ∨ p8 ∨ ℓ4

α′

5 : p2 ∨ p5 ∨ a2

α′

6 : p7 ∨ p4 ∨ ℓ5

α7 : p2 ∨ ℓ1 factored clauses ⊥ p2 p2 ℓ1 a3 a2 G

a5 a3 a2 a1 a4 ℓ1 ℓ2 ℓ3 ℓ4 ℓ5

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Assumptions Core Analysis

Under assumptions {¬a1, ¬a2, ¬a3, ¬a4, ¬a5, ¬a6, . . .} α′

1 : p2 ∨ p7 ∨ ℓ1

α′

2 : p2 ∨ ℓ2

α′

3 : p7 ∨ p4 ∨ p6 ∨ ℓ3

α′

4 : p6 ∨ p8 ∨ ℓ4

α′

5 : p2 ∨ p5 ∨ a2

α′

6 : p7 ∨ p4 ∨ ℓ5

α7 : p2 ∨ ℓ1 factored clauses ⊥ p2 p2 a3 a2 a1 a4 G

a5 a3 a2 a1 a4 ℓ1 ℓ2 ℓ3 ℓ4 ℓ5

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Experiments: MUS Competition

300 instances from the MUS competition 2011 Timeout limited to 1800 seconds Memory limited to 7800 Mo Use of the MUS extractor MUSer.2

→ default options (destructive + model rotation) → use of MINISAT solver

Plug our approach MINISAT+abr to MUSer.2 Intel R CoreTM2 Quad Processor Q9550 with 2.83 GHz CPU frequency with 8 GB memory and running Ubuntu 12.04

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Experiments: Factoring Out Assumptions

MINISAT (261) MINISAT+abr (272)

1 10 100 1000 1 10 100 1000

Figure: Running time

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Reduce learned clause database

Keeping all learned clauses slows down the solver Determining which learned clauses to keep is essential What are the necessary clauses to prove the inconsistancy?

→ use abbreviation information to refine the approximation

G

a5 a3 a2 a1 a4 ℓ1 ℓ2 ℓ3 ℓ4 ℓ5

α′

1 : p2 ∨ p7 ∨ ℓ1

α′

2 : p2 ∨ ℓ2

α′

3 : p7 ∨ p4 ∨ p6 ∨ ℓ3

α′

4 : p6 ∨ p8 ∨ ℓ4

α′

5 : p2 ∨ p5 ∨ a2

α′

6 : p7 ∨ p4 ∨ ℓ5

α7 : p2 ∨ ℓ1

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Reduce learned clause database

Keeping all learned clauses slows down the solver Determining which learned clauses to keep is essential What are the necessary clauses to prove the inconsistancy?

→ use abbreviation information to refine the approximation

G

a5 a3 a2 a1 a4 ℓ1 ℓ2 ℓ3 ℓ4 ℓ5

α′

1 : p2 ∨ p7 ∨ ℓ1

α′

2 : p2 ∨ ℓ2

α′

3 : p7 ∨ p4 ∨ p6 ∨ ℓ3

α′

4 : p6 ∨ p8 ∨ ℓ4

α′

5 : p2 ∨ p5 ∨ a2

α′

6 : p7 ∨ p4 ∨ ℓ5

α7 : p2 ∨ ℓ1

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Reduce learned clause database

Keeping all learned clauses slows down the solver Determining which learned clauses to keep is essential What are the necessary clauses to prove the inconsistancy?

→ use abbreviation information to refine the approximation

G

a3 a2 a1 a4 ℓ1 ℓ2 ℓ3

α′

1 : p2 ∨ p7 ∨ ℓ1

α′

2 : p2 ∨ ℓ2

α′

3 : p7 ∨ p4 ∨ p6 ∨ ℓ3

α′

4 : p6 ∨ p8 ∨ ℓ4

α′

5 : p2 ∨ p5 ∨ a2

α′

6 : p7 ∨ p4 ∨ ℓ5

α7 : p2 ∨ ℓ1

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Reduce learned clause database

Keeping all learned clauses slows down the solver Determining which learned clauses to keep is essential What are the necessary clauses to prove the inconsistancy?

→ use abbreviation information to refine the approximation

G

a3 a2 a1 a4 ℓ1 ℓ2

α′

1 : p2 ∨ p7 ∨ ℓ1

α′

2 : p2 ∨ ℓ2

α′

3 : p7 ∨ p4 ∨ p6 ∨ ℓ3

α′

4 : p6 ∨ p8 ∨ ℓ4

α′

5 : p2 ∨ p5 ∨ a2

α′

6 : p7 ∨ p4 ∨ ℓ5

α7 : p2 ∨ ℓ1

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Experiments

MINISAT+abr (272) MINISAT+abr+g (275)

1 10 100 1000 1 10 100 1000

Figure: Running time

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Minimization of the learned clauses

Learned clauses can be minimized: recursive minimization Clause minimization usually improves SAT solver performance With many assumptions, clause minimization is not effective

→ assumptions are not obtained by unit propagation → non-assumption literals are often blocked by assumptions → the number of deleted literals is rather small

Ignoring assumptions during the minimization step

→ the resulting “minimized” clause might even increase in size → no more non-assumption literals than the original clause

MINISAT MINISAT+abr MINISAT+abr+g #solved(MO) #solved(MO) #solved(MO) without 259(15) 272(3) 273(3) classic 261(13) 272(3) 275(1) full 238(25) 276(0) 281(0)

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Conclusion and perspectives

Introduction of the factoring out assumptions in the context of incremental SAT solving under assumptions: MINISAT+abr

→ techniques that work well for a large number of assumptions → improve the speed of the BCP procedure

Additionnal information collected from the definition map to reduce the learned clause database Application of new form of clause minimization Good results when our approach is combined with MUSer.2 Combine our techniques with more recent results on MUS preprocessing (inprocessing) Apply our approach to high-level MUS extraction Improve the data structure used to save the definition map

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MINISAT MINISAT+abr

1 10 100 1000 10000 1 10 100 1000 10000

Figure: Average size of learned clauses

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MINISAT MINISAT+abr

1 10 100 1 10 100

Figure: Average number of traversed literals

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MINISAT+abr MINISAT+abr+g

1 10 100 1000 1 10 100 1000

Figure: Memory used (in Mega Bytes)

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10 100 1000 50 100 150 200 250 memory used (in Mega Bytes) number of instances solved MINISAT(full) MINISAT(classic) MINISAT(without) MINISAT+abr+g(without) MINISAT+abr+g(classic) MINISAT+abr+g(full)

Figure: Memory usage of MUSer

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