a signment
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A signment. tack. SAT’10 Conference Edinburgh, Scotland, UK July 11, 2010 . hrinking. Alexander Nadel, Intel, Israel Vadim Ryvchin, Intel&Technion , Israel. Agenda. Introduction Algorithm Description Heuristics Experimental Results Conclusions. What is Shrinking?. - PowerPoint PPT PresentationTRANSCRIPT
hrinking
SAT’10 ConferenceEdinburgh, Scotland, UK
July 11, 2010
A signment tack
Alexander Nadel, Intel, Israel
Vadim Ryvchin, Intel&Technion, Israel
AgendaIntroductionAlgorithm DescriptionHeuristicsExperimental ResultsConclusions
What is Shrinking?Shrinking is an algorithm that
boosts modern DPLL-based SAT solvers on difficult industrial benchmarks
by
dynamically filtering out irrelevant assignments
The HistoryShrinking was
◦introduced in the Jerusat solver (Nadel’02).
◦enhanced in the Chaff’04 solver (Mahajan&Fu&Malik’04) .
Jerusat and Chaff’04 are the SAT’04 competition winners in industrial categories.
Goals of this WorkEmpirically demonstrate that
shrinking is critical for solving difficult industrial families in modern solvers.
Provide a thorough description of the shrinking algorithm.
We also discuss shrinking heuristics and introduce some new ones.
Explanation: a SAT Solver Run Snapshot after a Conflict
Assignedvariables
First assigned variable
Parent resolution: a resolution proof for the
validity of the flip
(Each rectangle is a clause)
Variable to be flipped
Explanation: a SAT Solver Run Snapshot after a Conflict
Parent resolution: a resolution proof for the
validity of the flip
(Each rectangle is a clause)
Yellow variables are the asserting clause variables
Explanation: a SAT Solver Run Snapshot after a Conflict
•The assignments to all the blue variables were irrelevant:• Unnecessary for generating
the parent resolution.
•The blue variables might had been used for resolution proof that was subsequently pruned.
Explanation: a SAT Solver Run Snapshot after a Conflict
• Irrelevant assignments are bad:• They slow down BCP.• They violate the locality
principle: proof should be dependent on as few assignments as possible.
• Shrinking: remove irrelevant assignments before flipping.
Example: Standard Backtracking and Flipping w/o Shrinking
Example: Standard Backtracking and Flipping w/o Shrinking
Non-chronological backtracking will remove this assignment (if it defines a new decision level).
Example: Standard Backtracking and Flipping w/o Shrinking
Backtrack!
Non-chronological backtracking removed an assignment.
Example: Standard Backtracking and Flipping w/o Shrinking
Flip
Example: Shrinking in Action
Shrink on this area. Shrinking will remove all the irrelevant assignments here.
Example: Shrinking in Action
Shrink and backtrack!
Shrinking removed many irrelevant assignments.
Example: Shrinking in Action
Flip
The Shrinking AlgorithmAfter a conflict:
1. Record a conflict clause as usual!2. If the shrinking condition holds,
shrink as follows:1. Backtrack to the shrinking backtrack
level.2. Assign the unassigned literals of the
conflict clause to false. Propagate each assignment with BCP.
The order of variable assignments is defined by the sorting scheme.
The Three Elephants that Support Clever Shrinking
The shrinking condition
The sorting scheme
The shrinking backtrack level
Intuition behind Shrinking HeuristicsIntuition behind the shrinking condition:
◦Shrink when the search halts to be useful.Usefulness is estimated by looking at
the asserting clause:◦Short means useful
The most intuitive widely used measure: short clauses should prune the search space better
◦Few decision levels means useful Variables of the same decision level are
interrelated: if one is assigned, the others have good chances to be assigned soon
Locality principle rules
Intuition behind Shrinking HeuristicsIntuition behind the sorting
scheme:◦It is advantageous to locally reshuffle
the VSIDS scores from time to time.Intuition behind the shrinking
backtrack level determination◦Quite straightforward: subject for
future work
Shrinking in Jerusat1. Jerusat’s shrinking condition
◦The conflict clause contains no more than one variable from each decision level.◦ A decision-level-based scheme
2. Jerusat’s sorting scheme◦Same order as before shrinking.
3. Jerusat’s shrinking backtrack level◦Highest possible decision level where all
the literals of the conflict clause become unassigned.
Shrinking in Chaff’041. Chaff’s shrinking condition
◦ The conflict clause length exceeds x. x is initialized with 95 and is changed dynamically.
The goal is to be close to the mean length of the conflict clauses.◦ Secondary condition
2. Chaff’s sorting scheme◦ Reverses the order.
Empirically better than using the same order. Dynamic reshuffling to VSIDS scores.
3. Chaff’s shrinking backtrack level◦ Lowest possible decision level in the conflict clause
that is less than the next higher decision level by at least 2. If no such level exists, shrinking is not performed.
New Heuristics for ShrinkingDecision-level-based shrinking
condition◦Generalization of Jerusat’s shrinking
condition in the spirit of Glucose’s clause deletion scheme (Audemard&Simon’09)
◦The algorithm d the number of decision levels in the
asserting clause. Shrink if d > x’
x’ is similar to Chaff’s threshold on clause length, but it is applied to the number of decision levels in the clauses instead of to clause length.
New Heuristics for ShrinkingActivity ordering sorting scheme
◦Sort the variables of the conflict clause by VSIDS’s scores, from highest to lowest. Should work well, if no corrections to
VSIDS scores are required.
Experimental ResultsWe implemented shrinking in Eureka
and an enhanced version of Minisat.Time threshold was 3 hours.We used 8 benchmark families:
◦4 of Miroslav Velev’s families.◦2 families from the SAT’04 competition:
Goldberg; Maris◦2 SAT-Race’06 families.
166 instances overall
Experimental Results: the Algorithms1. No shrinking2. Base shrinking
◦ Eureka’s version: similar to Chaff with two minor changes
3. Base with the decision-level-based shrinking condition
4. Base with the activity ordering sorting scheme
vliw_un
sat_4.
0
vliw_un
sat_2.
0
fvp-un
sat.3.0 mari
s
goldb
erg TS1 TS2
fvp-sa
t.3.0
0
5
10
15
20
25
30
35
40
45
50
No Shrinking Dec. Level-based Cond.Base Activity Sorting
Eureka: Solved Instances per Family
vliw_un
sat_4.
0
vliw_un
sat_2.
0
fvp-un
sat.3.0 mari
s
goldb
erg TS1 TS2
fvp-sa
t.3.0
0
5
10
15
20
25
30
35
40
45
50
No Shrinking Dec. Level-based Cond.Base Activity Sorting
Eureka: “Base“ Beats “No Shrinking”!
Minisat: Solved Instances per Family
maris
fvp-un
sat.3.0
vliw_un
sat_4.
0
vliw_un
sat_2.
0
goldb
erg
fvp-sa
t.3.0 TS1 TS2
0
5
10
15
20
25
30
35
40
45
50
No Shrinking Dec. Level-based Cond.Base Activity Sorting
Minisat: “Dec. Level-based“ Beats “No Shrinking”!
maris
fvp-un
sat.3.0
vliw_un
sat_4.
0
vliw_un
sat_2.
0
goldb
erg
fvp-sa
t.3.0 TS1 TS2
0
5
10
15
20
25
30
35
40
45
50
No Shrinking Dec. Level-based Cond.Base Activity Sorting
Performance Summary: Solved Instances Overall (out of 166)
Minisat EurekaNo Shrinking 117 126
Base 137 151
Decision Level-Based
140 150
Activity Ordering 133 149
Impact of the Decision Level-Based Shrinking Condition Empirical recommendations:
◦ Recommended as the default for Minisat Solves more instances overall and faster for
6/8 families◦ Very useful for particular families for both
solvers. Resulting intuition:
◦ The number of decision levels in the clauses appears to be a good way of measuring usefulness.
But using the clause size measure is sometimes advantageous.
Experimental Results: Impact of the Activity Ordering Sorting Scheme Empirical recommendations:
◦ Not recommended as the default Causes a serious deterioration of Minisat
performance on two Velev families. Yet, improves the performance for SAT Race families.
Not helpful for most of the families and overall for Eureka.
Resulting intuition◦ A local reshuffling of VSIDS scores is
usually helpful. Reminder: the base scheme reverses the
order
Future WorkStudy the interplay between
restarts and shrinking.
Improve the heuristics further
ConclusionsShrinking is a dedicated technique for
filtering out irrelevant assignments in modern DPLL-based SAT solvers
Shrinking is empirically critical◦Many families couldn’t be solved without
shrinking (within Minisat and Eureka).◦Shrinking improves the performance for
almost all the considered families for both solvers.
Thanks for Your
Attention!
Backup Slides
Shrinking vs. Modern TechniquesConsider the following three
techniques: ◦Frequent restarts ◦A clause-based heuristic◦RSAT’s phase saving
Is the combined effect of these techniques similar to shrinking?◦No! Empirically, shrinking is highly
beneficial in Eureka, which applies all these techniques.
Why isn't the Effect of Shrinking Achieved by Other TechniquesBecause of the simultaneous effect of
the following:1. The shrinking condition differs from the
restart conditions.2. Shrinking restarts the search only partially.3. Unlike clause-based heuristics, shrinking
continues selecting variables from the top-most conflict clause, even if it was satisfied.
4. Shrinking uses a dedicated sorting (re-ordering) of the variables in the top-most conflict clause.
Calculation of the Threshold on Conflict Clause Length in Chaff x is initialized with 95 Do the following each 800 conflicts
◦ (mean, stdev) := mean and standard deviation of last 800 learned clause lengths
◦ center := mean + 0.5 * stdev; ◦ ulimit := mean + stdev◦ if x ≥ center then
x := x − 5◦ if x < center then
x := x + 5◦ if x > ulimit then
x := ulimit◦ if x < 5 then
x := 5 return x
Performance SummaryMinisat Eureka
No Shr. Solved 117 126
No Shr. Time 547709 479498
Base Shr. Solved 137 151
Base Shr. Time 349096 207061
Dec. Level Shr. Solved
140 150
Dec. Level Shr. Time 322304 223239
Act. Order Shr. Solved
133 149
Act. Order Shr. Time 381340 220437
Another Type of Shrinking