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Decision heuristics based on an Abstraction/Refinement model

Ofer Strichman

Roman Gershman

Technion

(HaifaSat)

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SAT solving “Naïve” point of view:

Searches in the decision tree, prunes subspaces. Creates “blocking clauses” that restrain the solver from

choosing the same bad path again.

This point of view fails to explain why 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

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

Model is an (over approximating) abstraction of M if:

A degenerated case: Formula is an (over-approximation) abstraction

of F if:

F !or simply:

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Abstraction of formulas

Now consider Binary Resolution:

(AÇ x) Æ (B Ç :x) ! (A Ç B)

over-approximates

F !

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Resolution Graph

O1 O2

O3O4

O6

O7

i1

c-1

i2

i3

O5

c-2

C-3

i4

Binary DAG with intermediate and conflict clauses.

Each node in the graph is an abstraction of its descendants

Collapsed DAG with multi-degree nodes

C-1

C-3

C-2

O1 O2 O3 O4O5 O6

O7

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Refinement of models and formulas

An intermediate model is a refinement of if:

An intermediate formula is a refinement of if:

F ! , ! or simply:

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Why all this theory? …

Because Conflict Clauses are derived through a process of resolution.

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

Push conflict clauses to a stack. Find the first unsatisfied clause and choose a

variable from this clause. If all conflict clauses are satisfied, choose a

variable according to the VSIDS (Zchaff) heuristic.

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Berkmin heuristic

tail-first conflict clause

A new conflict clause

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Let φ denote the original formula F abstracts φ (φ ! F ) is a refinement of F with respect to φ

(φ ! , ! F )

Berkmin heuristic

tail-first conflict clause F

Check of abstract assignment fails

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path.

Generally: hundreds of clauses can be between a clause and its resolving clauses.

Berkmin heuristic

C-1

C-3

C-2

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A General Heuristic for choosing the next clause

1. Mark all roots.

2. Choose an unresolved marked clause V (If there are none - exit)

3. Decide a variable from V until it is satisfied.

4. Mark V’s children

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The Clause-Move-To-Front (CMTF) heuristic

Is an instantiation of the general heuristic Does not need to store the whole graph. More focused than Berkmin.

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Progressing on the resolve graph

Progress with “Best-First” according to some criterion.

Must store the whole resolve graph in memory – this is frequently infeasible.

HaifaSat’s strategy: Do not store graph Be more abstraction-focused than Berkmin

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The CMTF heuristic

Position conflict clauses together with their resolving clauses in the end of a list.

Find the first unsatisfied clause and choose a variable from this clause.

If all conflict clauses are satisfied, choose a variable according to the VMTF (Siege) heuristic.

Gives us the ‘first-layer approximation’ of the graph.

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CMTF

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.

C-1

C-3

C-2

C-0

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Given a clause: choose a variable.

The Activity of a variable v: Activity score of a variable increases when it is a

resolution variable, but… only when the clause it helped resolving is currently

relevant, and… it happened recently

A recursive computation embedded in the First-UIP scheme.

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DecisionConflict

Decision Level

Time

work invested in refuting x=1

(some of it seems wasted)

Cx=1 Refutation of x=1

C1

C5

C4

C3

C2

Activity Score

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ConflictDecision

C1Decision Level

Time

x=1 Refutation of x=1

C5

C4

C3

C2

C

C5

C2

C4

Weight is given to variablesresolved-on in the process of resolving C

C

Activity Score

C0

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Results (sec., average)Benchmark (#) Berkmin+VSIDS CMTF+RBS

Hanoi (5) 530 130

IP (4) 395 203

Hanoi03 (4) 1342 426

Check-int (4) 3323 681

Bmc2 (6) 1030 1261

Fifo8 (4) 3944 1832

Fvp2 (22) 8638 1995

W08 (3) 5347 2680

Ibm02 (9) 9710 3875

01_rule (20) 33642 19171

11_rule_2 (20) 34006 22974

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(CMTF + RBS) Vs. Berkmin(both implemented inside HaifaSat)

Berkmin + VSIDS Vs. CMTF + RBS (HaifaSat)

0

500

1000

1500

2000

2500

3000

3500

0500100015002000250030003500

Berkmin + VSIDS

CM

TF

+ R

BS

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HaifaSat Vs. zChaff 2004

HaifaSat Vs. zChaff

0

500

1000

1500

2000

2500

3000

3500

0500100015002000250030003500

zChaff 2004

Hai

faS

at

h-c-rbs

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Results –SAT05 (Industrial)

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Results –SAT05 (Industrial)

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Competition...

Independently, very similar principles were discovered by Dershowitz, Hana 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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And now...

Two research directions Better refinement strategies. Hints.

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Recall the general framework:

1. Mark all roots.

2. Choose an unresolved marked clause V (If there are none - exit)

3. Decide a variable from V until it is satisfied.

4. Mark V’s children

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But,

HaifaSat does not really traverse the resolution graph.

The assumption is: the graph is too large to store in memory.

But, there are news: A new technique developed in IBM-Haifa allows to shrink

the graph stored in memory by two orders of magnitude. The search for new refinement strategies is now open...

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Refinement-driven Vs. Conflict-driven search.

O-1 O-2 O-3 O-4 O-5 O-6 O-7 O-8

W(c) = ci 2 antecedents(c) W(ci)

How should we balancebetween refinement-driven and conflict-driven strategies ?

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Hints

An (unpublished) idea by (Kroening, Yorav, Shacham)

Hints are constraints (clauses) that are conjectured to be true.

A separate BCP processes the set of conjectured clauses. An implied literal becomes the next decision. A conflict is ignored.

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Hints (cont.)

The original use of hints: high-level knowledge. We suggest: prune ‘seemingly hopeless

branches (SHB)’ Define a monotonically decreasing function

f: decision-level time-interval If time at decision level dl > f(dl) prune the branch.

This branch is seemingly hopeless.

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Hints: Example

A hint clause: (:l1 :l2 :l3) Perhaps a better idea:

Keep track on which subset S of l1 ... l3 were used in the SHB. The negation of literals in S is a better hint.

l1

l2

l3 : l3

SHB

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Hints vs. restarts

Not entirely orthogonal techniques. A restart is effective because of randomization

and/or learning. Hints are more directed: they push the solver

away from seemingly hopeless branches. Also: it is activated due to local consideration,

and not a global clock. Bart Selman: “You can not restart too much”

Perhaps now: “You can not hint too much”