Hybrid SAT Solving using Reference Points

Stephan KottlerUniversität Tübingen

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Motivation

o DPLL / CDCLo Learning, backjumping and lazy data-

structures huge instances tractableo Early decisions are crucialo Many restarts to escape from bad parts in

the search tree

o When restartingo Keep learnt clauses, activities, phase savingo How close is partial assign. to a model? (sat)o Analysis of partial assignment (sat/unsat)

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Modificationloop{ if ( all vars assigned ) return SAT choose free var and decide polarity bcp while ( conflict ) {

if ( level = 0 ) return UNSATanalyse conflict, backjump and learn lemmabcp

} if ( reached restart limit )

restart}

Hold onTake closer look at partial assign.

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Continue on partial assignments

o Extend to full assignmento Random values to free varso Consider subset of clauses for bcp

oExtend to fulfill all binary clauses

o Modify full assignmento Use local search (incomplete)o More reasoning: DMRP

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Decision Making with Reference Point (DMRP)

clauses

(¬a v ¬b) ∧(¬a v ¬c) ∧(¬b v ¬c) ∧(a v ¬b v d) ∧(a v b v c v ¬d)

∧ …1110

reference point

0110

0111

dcba

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Solving with reference points

Maintain assignment A to all variableso some clauses U are not satisfied by Ao try to modify A to satisfy more (all)

clausesomodification of A must keep previously

satisfied clauses (S) satisfied !!!okeep track of clauses of S that are temporarily unsatisfied

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Modification of reference point

clauses

(¬a v ¬b) ∧(¬a v ¬c) ∧(¬b v ¬c) ∧(a v ¬b v d) ∧(a v b v c v ¬d) ∧(b v e) ∧(¬e v a) ∧ …

edcba

reference point

00111

00110

00100

10100

10101

01110

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Profound decisions need more information

o Keep track of o reference point / unsatisfied clauseso temporarily modified ref. pointo temporarily unsatisfied clauses

o Functionalityo backtracking on reference pointso compute clauses that get unsat by

ref. p.o still handle learning from conflicts

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clauses

(¬a v ¬b) ∧(¬a v ¬c) ∧(¬b v ¬c) ∧(a v ¬b v d) ∧(a v b v c v ¬d) ∧(b v e) ∧(¬e v a) ∧ …

Trying to modify the ref. point

occurrence

lists

a b cedref point watchers

edcbareference point

00111

00110

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Temporarily unsat clauses

Lev 0 C01 C02 C03 C04

L12Lev 1 C11 C12 C13

Lev 2 C21 C22 C23 C24

Lev 3 C31 C32

L02L01 L04L03

L11 L13

L22L21 L24L23

L32L31

o Backtracking very fasto One Operation per level o No maintenance for ref. point watchers!

o Find temporarily unsat clauseso No occurrence list of literals!

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Making decisionsChoose a temporarily unsat clauseChoose Literal witho least BreakCount

o least sat clauses become unsat

o highest activity valueo highest MakeCount

o most unsat clauses are fulfilled

Further adaptions of data-structure

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Design of data structures

o Lazy data structures o little effort to maintain datao ref. point watchers (watched lits)o computation of MakeCount

o Consequenceso frequently called functions still cheap

(propagation)o less frequent calls require more computationo garbage collection

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Setup with Luby restartso Idea: Grant different restart strategies

equal computation time1 1 2 1 1 2 4 1 1 2 1 1 2 4 8 1 ... * fact

o Shortest runs use DMRPo Extend partial assig. by performing bcp on

binary clauseso Increase conflict limit if DMRP is successful o Different strategies to choose next unsat

clause and next temporarily unsat clause

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Results

o Tests on 576 instances of last four sat competitions / races

o Timeout 1200 sec.

217,85 sec204,41 sec161,98 secavg time

325396386all

193221210unsat

132175176sat

Hybrid comp 09

Hybrid part. assign

CDCL (Luby)

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Hardware Verification

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Past

o Submitted 2 solver versions to sat-competition 2009o (CDCL, hybrid with reference points)

o Differences to former hybrid version:o DMRP initialised with partial assignmentso DMRP used for (many) short runs. Previous

version run DMRP with very high conflict limits

o Less memory consumption

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Present and Future

o Use DMRP in combination with CDCL in multi-threaded solver

o Better heuristic to determine conflict limit for DMRP runs

o More criteria to analyse partial assignments (sat instances)

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Thank you for your attention