Transcript sspv2006 6628

Decision heuristics based on an
Abstraction/Refinement model
(HaifaSat)
Ofer Strichman
Roman Gershman
An earlier version was presented in IBM’s verification
conference (Haifa, Oct. 05).
Technion
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SAT solving
Technion

“Naïve” point of view:
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
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Searches in the decision tree, prunes subspaces.
Creates “blocking clauses” that do not allow the solver
to choose the same path again.
This point of view fails to explain why
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
We can solve many formulas with 105 variables,
We cannot solve other formulas with 103 variables
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A different point of view
Technion
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
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Modern solvers act as proof engines based on
resolution, rather than as search engines, with
structured problems.
Evidence: adding the shortest conflict clauses is
not the best strategy [R04].
Furthermore: certain strategies resemble a proof
by abstraction-refinement.
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Abstraction of models and formulas
Technion

Model
of M if:
is an (over approximating) abstraction

A QF formula is an (over-approximating)
abstraction of F if:
or simply:
F!
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Refinement of models and formulas
Technion
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Model

Formula
refines
if:
refines
if:
or simply:
F! ,
!
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Abstraction of formulas
Technion


F!
Now consider Binary Resolution:
(A _ x) ^ (B _ :x) !
(A _ B)
over-approximates

Conflict Clauses are derived through a process
of Binary Resolution.
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Resolution Graph
Technion
Binary DAG with intermediate
and conflict clauses.
Collapsed DAG with multi-degree nodes
C-3
C-3
i4
C-2
C-1
i3
i2 O
5
i1
O3
O4
O7
C-2
C-1
O6
O1
O2 O
3
O4
O5
O6
O7
O1 O2
Each node in the graph is an abstraction of its antecedents
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Conflict clauses
Technion

Several modern Decision Heuristics are guided
by the Conflict Clauses (e.g. Berkmin)

Hence, we can analyze them with the
Abstraction/Refinement model.
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Berkmin’s heuristic
Technion
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Push conflict clauses to a ‘stack’.
Find the first unsatisfied clause and choose a
variable from this clause.
Secondary heuristic: VSIDS (Zchaff).
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Berkmin heuristic
Technion
tailfirst conflict clause
A new
conflict
clause
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Berkmin heuristic
Technion
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Let F denote the original formula
abstracts F
(F ! )
refines
(F ! , !
)
tailfirst conflict clause
Check of abstract
assignment fails
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Berkmin heuristic

Does not focus on a specific Abstraction/Refinement
path.
Technion
C-3
C-2
C-1
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Generally: hundreds of clauses can be between a clause
and its resolving clauses.
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Choosing the next clause by traversing
the resolution graph: a general scheme
Technion
1.
2.
Mark all sinks.
Choose an unresolved marked clause C
(If there are none - exit)
3.
4.
process C
// attempt to satisfy C.
Mark C ’s antecedents.
Berkmin chooses a clause only if all its
descendents are already satisfied.
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Progressing on the resolve graph
Technion
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Progress with “Best-First” according to some
criterion.
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Must store the whole resolve graph in memory –
this is frequently infeasible.
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HaifaSat’s strategy:
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Do not store graph
Be more abstraction-focused than Berkmin
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The CMTF heuristic
Technion
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Position conflict clauses together with their resolving
clauses in the end of a list.
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Find the first unsatisfied clause and choose a variable
from this clause.
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secondary strategy: the VMTF (Siege) heuristic.
Gives us the ‘first-layer approximation’ of the graph.
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CMTF
Technion
C-3
C-2
C-1
C-0
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When C-3 is created, C-0, C-1 are moved to the head
of the list together with C-3.
C-2 is left in place.
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News
Technion
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A new technique developed in IBM-Haifa allows to
shrink the graph stored in memory by ~ an order
of magnitude.
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So, assuming the graph is in memory, the search
for new strategies is now open…
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A contention between conflict-driven and
abstraction-driven.
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Activity Score
T2(C)
T1(C)
time dedicated for refuting x=1
Technion
Time C is
asserting
time dedicated for creating C
C
x=1
Decision
Level
C5
C2
C1
C4
C3
Time
Decision
Conflict
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Given a clause: choose a variable.
The activity score.
Technion
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The Activity of a variable v:
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Let V be the set of variables that were resolved on in
T1(C) on the resolution path of C.
Add +1 for each v2V in the interval T2(C)
Occasionally, divide all scores.
A cheap recursive computation embedded in the
First-UIP scheme.
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Results
Technion
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165 industrial hard benchmarks from previous SAT
competitions.
All heuristics (Berkmin, VSIDS) implemented inside
HaifaSat for fair comparison.
Comparing…
Time
Fails
CMTF
Berkmin
10%
12-25%
RBS
VSIDS
23%
20-30%
Berkmin
a little better than zChaff 2004
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(CMTF + RBS) Vs. Berkmin
(both implemented inside HaifaSat)
Technion
Berkmin + VSIDS Vs. CMTF + RBS (HaifaSat)
3500
3000
2000
1500
CMTF + RBS
2500
1000
500
0
3500
3000
2500
2000
1500
1000
500
0
Berkmin + VSIDS
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HaifaSat Vs. zChaff 2004
Technion
HaifaSat Vs. zChaff
3500
3000
2000
h-c-rbs
1500
HaifaSat
2500
1000
500
0
3500
3000
2500
2000
1500
1000
500
0
zChaff 2004
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Results –SAT05 (Industrial)
Technion
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Results –SAT05 (Industrial)
Technion
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Competition...
Technion
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Independently, very similar principles were
discovered by Dershowitz, Hanna and Nadel
[SAT’05]
Reached very similar conclusions
Their ‘black-box’ Eureka SAT solver took several
first and second places in last year’s
competition.
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What we explore now
Technion

We are currently investigating the possibility to
use Hints for dealing with hard instances

Hints: a yet-unpublished work by
Kroening, Yorav, Shacham.
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The original use of hints: using presumed highlevel knowledge to speed SAT.
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Hints
Technion
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Hints are constraints (clauses) that are
conjectured to be true.
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A separate BCP processes the set of
conjectured clauses.
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An implied literal becomes the next decision.
A conflict is only used for deriving more hints.
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Hints (cont.)
Technion
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Use hints to prune ‘seemingly hopeless
branches (SHB)’
Define a monotonically decreasing function
f: decision-level  time-interval
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If time at decision level dl > f(dl) move to
another branch. It is seemingly hopeless.
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Actively: force backtracking
Passively: wait for a restart
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Hints: Example
l1
l2
l3
:l3
Technion
SHB
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A hint clause: (:l1 :l2 :l3).
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Option #1: hint = negation of partial assignment
Option #2: hint = projection of #1 to literals that were
used in the SHB.
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Combining Hints and restarts
Technion
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Some heuristics to be tried:
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Change restart policy, using f.
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Local restarts: only from a certain decision level
…
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Hints and restarts
Technion
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A restart is effective because of randomization
and/or learning. Hints will add new conjectures.
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Bart Selman: “You can not restart too much”
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Perhaps now: “You can not hint too much”
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