E fficient SAT S olving u nder A ssumptions

Efficient SAT Solving Under AssumptionsAlexander Nadel1 and Vadim Ryvchin1,2

1 – Intel, Haifa, Israel2 – Technion, Haifa, Israel

SAT’12, Trento, Italy

Motivation: Real-Life Experience at Intel

Critical non-incremental property unsolved in 48 hours by the SAT solver during FV of Intel’s latest design

The default flow: Minisat-like incremental SAT solving under assumptions • The negation of the property is the only assumption in our case

Solution: model the property as a unit clause solved in 30 minutes!• Propagation of the single assumption by SatELite was extremely helpful:

resulted in “chain reaction”

Our paper generalizes the case-study to incremental SAT solving under assumptions

Basic DefinitionsInput: {}

– formula in CNF form = {} – set of assumptions

• – a literal (unit clause)

Invocation decides the satisfiability of:

Recall: Clause Database Simplification

1. Propagation of unit clauses2. Elimination of satisfied clauses 3. Removal of falsified literals from clauses

Used in leading SAT solvers Compliant with incremental solving

Recall: SatELite Preprocessing1. Variable Elimination2. Subsumption3. Self-subsuming resolution

Used in leading SAT solvers Non-Compliant with incremental solving

Approaches to Incremental SAT Solving under Assumption Literal-based Single instance (LS):

• One incremental solver instance • Assumptions are chosen as first decisions

Clause-based Multiple instances (CM): • Multiple solver instances • Assumptions are provided as temporary unit clauses

• LS is the current state-of-the-art

Literal-based Single instance (LS) Create instance For each incremental call do

1. Run solver over

The same instance of the solver is reused

Clause-based Multiple instances (CM) For each incremental call do

1. Create new instance o Where the set Ai comprise the temporary clauses

2. Add pervasive conflict clauses to the instanceo Pervasive clauses: conflict clauses generated during previous

invocation that do not depend on • Solve

A new solver instance is created each time

Temporary vs. Pervasive

C1 C2 C3 C4 C5 C6 C7 C8

C10 C11 C12 C13 C14 C15

C20C19 C21C17 C18

C22 C23=()

C9

C16

Input clausesLegend: Derived clauses Temporary input clauses


C1 C2 C3 C4 C5 C6 C7 C8

C10 C11 C12 C13 C14 C15

C20C19 C21C17 C18

C22 C23=()

C9

C16

Input clausesLegend: Derived clauses Temporary input clauses

Temporary conflict clauses


C1 C2 C3 C4 C5 C6 C7 C8

C10 C11 C12 C13 C14 C15

C20C19 C21C17 C18

C22 C23=()

C9

C16

Input clausesLegend: Pervasive conflict clauses

Temporary input clauses

Temporary conflict clauses

LS vs. CM LS: Efficient Learning

All conflict clauses are pervasiveHeuristics take advantage of incrementalityAssumptions are not propagated by simplification

CM: Efficient Assumption PropagationSome conflict clauses are temporaryHeuristics start from zero at every invocationAssumptions are propagated by simplification

The problem: How to Propagate Assumptions with SatELite in Incremental SAT? Our experience showed that propagating assumptions with SatELite

is vital

SatELite could not be used to propagate assumptions for either LS or CM in incremental settings:

LS: • Incremental SatELite was not well-defined (before our other paper to be

presented next )• It is still unknown how to propagate assumption using SatELite, even if SatELite

is applied incrementally

CM:• It was undefined how to distinguish between temporary and pervasive conflict

clauses after applying SatELite

Outline of Our Work1. Enabling assumption propagation with SatELite for

incremental SAT by making SatELite and CM compliant• Much simpler than doing the same for LS

2. Mitigate the advantages of LS over CM by:• Transforming temporary clauses to pervasive• Solving related incremental chunks using one SAT instance (if

limited look-ahead information is available)

Our algorithms outperform LS on Intel instances of incremental SAT under assumption.

Make SatELite Compliant with CM SatELite over temporary clauses:

• Variable elimination / Self-subsuming resolution : o new clause is marked as temporary iff or is temporary

• Subsumption:o no change

Improve Learning Efficiency The problem:

• Our algorithm propagates assumptions with SatELite, but:• Conflict clause learning is still more efficient for LS, since:

o All the conflict clauses are pervasive for LSo Some conflict clauses are temporary for CM

Solution: transform temporary conflict clauses to pervasive after CM’s invocation

Transform Temporary to Pervasive – T2P For every temporary clause : transform to pervasive by

adding to every assumption depends on• By analyzing the resolution derivation

Problem: Conflict clauses might become much longer Solution: Limit the size

• The threshold is controlled by a parameter: T2P Threshold

Transform Temporary to Pervasive – T2P

α1=a

Legend:

α2=b α3=¬a v c v d α4=¬b v ¬d α5=¬c v e α6=¬c v ¬e

α7=c v d α8=¬d α9=¬c

α10=c

α11=

Input clauses

Pervasive conflict clauses

Assumptions Temporary conflict clauses


α1=a

Legend:


α7=c v d α8=¬d α9=¬c

α10=c

α11=

Input clauses



¬a


Legend:


α7=¬a v c v d α8=¬d α9=¬c

α10=c

α11=

Input clauses



¬b


Legend:

α3=¬a v c v d α4=¬b v ¬d α5=¬c v e α6=¬c v ¬e

α7=¬a v c v d α8=¬b v ¬d α9=¬c

α10=c

α11=

Input clauses



¬a ¬b


Legend:



α10=¬a v ¬b v c

α11=

Input clauses



¬a v ¬ b


Legend:



α10=¬a v ¬b v c

α11=

Input clauses




Legend:


α9=¬c

α10=¬a v ¬b v c

α11=

Input clauses



Improve the Efficiency of Heuristics The problem:

• Our algorithm propagates assumptions with SatELite, and• We know how to make all the clauses pervasive, but:• Heuristics are still incremental for LS, while our algorithm needs to

collect information from scratch for each invocation

Solution: use a single SAT solver instance for multiple calls, if step look-ahead information is available

Step Look-AheadF1 A1F2 A2…Fk Ak

Fj*k+1 Aj*k+1Fj*k+2 Aj*k+2…Fj*k+k Aj*k+k

Fk+1 Ak+1Fk+2 Ak+2…Fk+k Ak+k

Essential conditions for applying step look-ahead:

Step (window) 1:– available at invocation 1

- equisatisfiability

Step (window) i>1: similar

LS Using Step Look-Ahead Adjustment of LS to take advantage of step look-ahead

• Proposed recently in the context of BMC by Khasidashvili&Nadel, HVC’11

Single instance Literal-based with Step look-ahead (LSS):• All step (window) clauses are added at once• Assumptions are chosen as first decisions• Advantage over LS: has a wider view of the problem• The same drawback as LS: no simplification over assumptions; no

preprocessing

LSS – Window 2F1 A1F2 A2F3 A3


F4 A4F5 A5F6 A6

1. 2. For j=4..6

1. Solve(, )

Our Algorithm: CLMSF1 A1F2 A2…Fk Ak


Fk+1 Ak+1Fk+2 Ak+2…Fk+k Ak+k

CLMS: Multiple instances Clause/Literal-based with Step look-aheadCLMS invokes the SAT solver for each window k times over a single SAT instance

Solve(, , ): Solve formula Using temporal clauses Under assumptions

Our Algorithm: CLMS – Window 1F1 A1F2 A2F3 A3


F4 A4F5 A5F6 A6

1. Create Instance 2. (temporary clauses)

3. Optionally:1. Freeze 2. Apply SatELite to

4. For j=1..3 do1. Solve(, , )

5. Optionally: Transform temporary to pervasive

6. Store pervasive in 7. Delete Instance

Our Algorithm: CLMS – Window 2F1 A1F2 A2F3 A3


F4 A4F5 A5F6 A6

1. Create Instance 2. (temporary clauses)

3. Optionally:1. Freeze 2. Apply SatELite to

4. For j=4..6 do1. Solve(, , )

5. Optionally: Transform temporary to pervasive

6. Store pervasive in 7. Delete Instance

Experimental Results Benchmark Set:

• Instances generated by incremental BMC under assumptionso Generated by an incremental model checker

• May be invoked multiple times with different assumptions and properties• Essential to reduce the debug loop time for validation engineers

• 3 satisfiable families – 128 instances• 4 unsatisfiable families – 81 instances • Algorithm Implementation in Intel’s internal Eureka SAT Solver• Timeout: 3600sec

Machines:• Intel® Xeon® 4Ghz 32Gb of memory

Variables To Assumptions Ratio

Unsatisfiable Instances

State-of-the-art LS


State-of-the-art CM


2nd: SatELite with temporary clauses + CLMS step 10 + T2P Thr. 100

Unsatisfiable InstancesWinner: SatELite with temporary clauses + CLMS step 50; No T2P

Unsatisfiable Instances: Summary CM outperforms LS, since:

• The average clause size is higher for LS by 1-2 orders of magnitude

• Simplification removes 1-2 orders of magnitude more clauses for CM

Unsurprisingly, both CLMS (the step) and SatELite are helpful

T2P is not helpful, since:• SatELite is slowed down significantly due to T2P

Satisfiable Instances

State-of-the-art CM


SatELite with temporary clauses + CLMS step 50; No T2P


2nd: State-of-the-art LS

Satisfiable InstancesWinner: SatELite with temporary clauses + CLMS step 10 + T2P Thr. 100

Satisfiable Instances: Summary LS is second best:

• Incrementality for heuristics proves to be essential for relatively easy satisfiable instances

The combination of our algorithms proves to be the best• SatELite with temporary clauses• CLMS with step 10 • T2P with threshold 100

Experimental Results: Summary The following algorithm proves to be the best overall (2nd

on unsatisfiable and 1st on satisfiable):• SatELite with temporary clauses• CLMS with step 10 • T2P with threshold 100

The gap between our algorithms and LS is especially significant on hard unsatisfiable instances.

Thank You!

E fficient SAT S olving u nder A ssumptions

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