Solving model-based diagnosis problems with max-SAT solvers and vice versa

Solving Model-Based Diagnosis Problems with

Max-SAT Solvers and Vie Versa

Alexander Feldman

1 , 4

, Gregory Provan

2

, Johan de Kleer

3

,

Stephan Robert

4

, and Arjan van Gemund

1

1

DelftUniversityofTehnology,Mekelweg4,2628 CD,Delft,The Netherlands

{a.b.feldman,a.j..vangemund}tudelft.nl

2

UniversityCollegeCork,CollegeRoad,Cork,Ireland

g.provans.u.ie

3

PaloAltoResearh Center,3333Coyote HillRoad, PaloAlto,CA94304,USA

dekleerpar.om

4

HauteEoled'IngénierieetdeGestiondu CantondeVaud

Route deCheseaux1,CH-1401Yverdon-les-Bains,Switzerland

stephan.robertheig-vd.h

ABSTRACT

Inthispaperwebringloseromputationof

onsisteny-basedardinality-minimaldiag-

nosisandsolvingMax-SAT.Weproposetwo

algorithms for translating between those:

(1)Diorama (DIagnOsis-based algoRithm

for mAx-sat optiMizAtion) for translating

ardinality-minimalonsistenybaseddiag-

nosistoMax-SATand(2)Meridian(Max-

sat-basEdalgoRIthmforDIAgNosis)forthe

other way around. While the former ap-

proah hasbeen studied,solving Max-SAT

instaneswithadiagnostisolveris, tothe

bestofourknowledge, novel. We ongure

MeridianwiththeStohastiLoalSearh

(SLS)solversfromtheUBCSATsuite,per-

form extensive experimentation on fault-

modelsofthe

74XXX

^/

ISCAS85

^iruitsand

omparethe resulting optimality (interms

of approximation error inthe minimal-ar-

dinality)to theone ofthe MBD algorithm

Safari. Theresultsshowthattheoptimal-

ityofSafariisuptoseveral-orders-of- mag-

nitude better than that of the SLS-based

Max-SAT solvers. We ongure Diorama

with Safari and experiment on instanes

from the Max-SAT ompetitions. While

the performane of Diorama/Safari on

raftedMax-SATproblemsisslightlyworse

ompared to UBCSAT, Diorama/Safari

outperforms at leastseveral-orders-of- mag-

nitudeallUBCSATalgorithmsonsmallin-

dustrialMax-SATinstanes.

1 INTRODUCTION

Model-Based Diagnosis (MBD) inferene algo-

rithmsforpropositionaldiagnosismodelshave,in

This is an open-aess artile distributed under the

terms of the Creative Commons Attribution 3.0

UnitedStatesLiense,whihpermitsunrestriteduse,

distribution, and reprodution in any medium, pro-

videdtheoriginalauthorandsoureareredited.

general, been reated speially for the diagno-

sis problem, e.g., GDE (de

Kleer and Williams,

1987),

andmorereentlySafari (F

eldman etal.,

2010).

Thesealgorithmshavebeenimprovedover

theyearsbytakingadvantageofpropertiesofthe

diagnosis problem, e.g., by fousing on themost

likely diagnoses, orusing a notion of ontinuity 1

ofthediagnosisspae (F

eldmanet al., 2010).

With regardto domain-independent problems,

signiant progress hasbeen made in developing

powerfulsolversforthesatisability (SAT)prob-

lem, e.g.,

(Gomes

et al., 2007).

This suesshas

prompted the use of SAT-solversfor many other

problems,suhasplanning

(Castellini

etal., 2003)

andiruittest-asegeneration (Iyer

etal., 2003).

Using aSAT-solverfor another(non-SAT)prob-

lem

P

^{entailsrewriting}

P

^{inSATformat;although}

this rewritingproessaninreasethesize ofthe

problem, the high eieny of SAT solversoften

makestherewritingproessworthwhile.

MBD is a version of propositional abdution,

whih is a more omplex problem than SAT.

Hene, although one an make alls to a SAT

solverduring the proessof omputingdiagnoses

(e.g.,SafariusesaninompleteSATsolver),one

annotuse astandardSAT solverdiretly for di-

agnostiinferene. Thisartileshowshowonean

useanextensionoftheSATproblem,alledMax-

SAT,to solveMBDproblems.

Max-SATis anoptimizationextensionof SAT.

Given a formula

Φ

^{in Conjuntive Normal Form}

(CNF), aMax-SAT (

Hoosand Stützle,2004 )

so-

lutionisavariableassignmentthatmaximizesthe

numberofsatisedlausesin

Φ

^{(inmostasesof}

interest

Φ

^isunsatisable, otherwiseanyvariable assignmentwhihsatises

Φ

^{isalsoaMax-SATso-}

lution). InpartialMax-SAT,someof thelauses

1

Aontinuoussubspaeislooselydenedasasub-

setofthetrue-pointsofaBooleanfuntion,suhthat

foreahtrue-pointitinludesanothertrue-point,dif-

ferentinonebitexatly.

Workshop on the Principles of Diagnosis, 13-16 October

2010, Portland, Oregon, USA, which should be cited to refer

to this work.

in

Φ

^{are designatedashard,theothersaresoft.}

AsolutiontothepartialMax-SATproblemshould

satisfyallhardlausesandmaximizethenumber

of satised soft lauses. Similarly, in weighted

Max-SATaweightisassignedtoeahlausein

Φ

and asolutionmaximizesthe sumofthe weights

ofthesatisedlauses.

The ontributions of this paperare as follows.

(1) We are the rst to ast the Max-SAT prob-

lem as an MBD problem, for whih we propose

an algorithm alled Diorama (DIagnOsis-based

algoRithm for mAx-sat optiMizAtion). (2) We

showthatDiorama/Safarioutperformsthetra-

ditional UBCSAT

(Tompkins

and Hoos, 2005)

Max-SAT algorithms by at least two-orders-of-

magnitudeonalassofindustrialMax-SATprob-

lems,eventhoughitperformsslightlyworsethan

Stohasti Loal Searh (SLS) Max-SAT algo-

rithmsonraftedMax-SATompetitionproblems.

(3) We propose an algorithm, alled Meridian

(Max-sat-basEd algoRIthm for DIAgNosis), that

translatesanMBDproblemto aMax-SATprob-

lem. (4) We empirially show that Meridian

ongured withtraditionalMax-SATislessopti-

malthanspeializedMBDsolverssuhasSafari

therebyrevealingalargelassof Max-SATprob-

lems that exposeontinuous properties amenable

togreedyalgorithmslikeDiorama/Safari.

2 RELATED WORK

MBD has resemblane to Max-SAT (Hoos

and

Stützle, 2004)

and we have onduted extensive

experimentation with both omplete Max-SAT

(partial and weighted) and Max-SAT based on

StohastiLoalSearh(SLS).Empirialevidene

shows that although Max-SAT an ompute di-

agnoses in many of the ases, the performane

ofMax-SATdegradeswheninreasingtheiruit

sizeortheardinalityoftheinjetedfaults.

Fu and Malik (2006)

onstrut apartial Max-

SAT algorithm that uses UNSAT oresprovided

by SAT solvers. The algorithm of Fu and Malik

iterativelyrelaxesUNSAToresuntiltheCNFin-

putbeomessatisable. Thedierenefromtheir

approah and Diorama is that they do not ex-

pliitlyuseadiagnostialgorithmtondasingle

minimalunsatisableore.

Aninterestingapproahto solvingMax-SATis

proposed by de Givry, et al.

(2003)

where they

astaMax-SATproblemasaweightedonstraint

satisfation problem. On the other side, solving

thediagnosis problemas aCOP(ConstraintOp-

timization Problem)iswell-knownfromWilliams

andRagno (

2007 )

.

OnthesideofsolvingdiagnosiswithMax-SAT,

Kutsunaetal.

(

2009 )

useapartialMax-SATalgo-

rithmtosolveseveraldiagnostiautomotiveprob-

lems. Similarly, Chen et al.

(2009)

use partial

Max-SATtosolveproblemsofdebuggingsequen-

tialiruits. All theseapproahesdierfromours

in that theysolvespei diagnostiproblems as

opposed to empirially studying the general per-

formaneharateristisofMax-SATanddiagnos-

tialgorithms.

3 TECHNICALBACKGROUND

Amodel ofanartifatisrepresentedasaproposi-

tionalformulaoversomesetofvariables. Wedis-

ern subsets of these variables asassumable and

observable.

Denition 1 (DiagnostiSystem). Adiagnosti

system

DS

^{is dened as the triple}

DS = hSD

^,

COMPS

^,

OBSi

^,where

SD

^{is a}propositionalthe- ory over a set of variables

V

^,

COMPS ⊆ V

^,

OBS ⊆ V

^,

COMPS

^{is theset of} assumables,and

OBS

^{is theset of}observables.

Throughout this paper we assume that

OBS ∩ COMPS = ∅

^and

SD 6| =⊥

^.

Not allpropositionaltheoriesare of interestto

MBD.TraditionallyMBDusespropositionalthe-

oriesthatdesribenominaland,optionally,faulty

behaviorofaninteronnetedset of omponents.

Werstdeneweak-faultmodels,i.e.,modelsthat

desribenominalbehavioronly,aoneptreferred

alsotoasignoraneofabnormalbehavior.

Denition 2 (Weak-Fault Model). Adiagnosti

system

DS = hSD, COMPS, OBSi

^{belongs to the}

lass

WFM

ifor

COMPS = {h ₁ , h ₂ , . . . , h n }

^,

SD

isequivalentto

(h ₁ ⇒ F ₁ )∧(h ₂ ⇒ F ₂ )∧. . .∧(h n ⇒ F n )

^and

COMPS∩V ^′ = ∅

^,where

V ^′

^{isthesetofall}

variablesappearingin the propositionalformulae

F ₁ , F ₂ , . . . , F n

^.

Modelingoffaultsmakestheproblemofomput-

ingdiagnosesmoreomplex (de

Kleeretal., 1992),

butaninreasethepreisionofadiagnostialgo-

rithm. Modelsthat haveknowledge of faults are

formalizedbelow.

Denition3(Strong-FaultModel). Adiagnosti

system

DS = hSD, COMPS, OBSi

^{belongs to the}

lass

SFM

i

SD

^{is equivalentto}

(h ₁ ⇒ F ₁ , 1 ) ∧ (¬h ₁ ⇒ F ₁ , 2 ) ∧ . . . ∧ (h n ⇒ F n, 1 ) ∧ (¬h n ⇒ F n, 2 )

suhthat

1 ≤ i, j ≤ n, k ∈ {1, 2}

^,

{h i } ⊆ COMPS

^,

F {j,k}

^{is a} propositional formula, and none of

h i

appearsin

F j,k

^.

In this paper, in addition to

WFM

, we experi- mentwithstuk-at-zero(S-A-0)andstuk-at-one

(S-A-1) models. S-A-0 and S-A-1 are sublasses

of

SFM

inwhih theoutputof amalfuntioning omponentisassumedeither

⊥

^or

⊤

^.

Denition4(Diagnosis). Givenadiagnostisys-

tem

DS = hSD, COMPS, OBSi

^{, an}observation

α

oversomevariablesin

OBS

^{,andaonjuntionof}

literals

ω

^,

ω

^{isadiagnosisi}

SD ∧ α ∧ ω 6| =⊥

^.

Givenaonjuntionofliterals

ω

^{wedenotetheset}

ofnegativeliteralsin

ω

^as

Lit ⁻ (ω)

^.

Denition 5 (Subset-MinimalDiagnosis). A di-

agnosis

ω ^⊆

^{is dened as} subset-minimal, if no other diagnosis

ω ˜ ^⊆

^,

ω ˜ ^⊆ 6= ω ^⊆

^{, exists suh that}

Lit ⁻ (˜ ω ^⊆ ) ⊂ Lit ⁻ (ω ^⊆ )

^.

In the MBD literature, a range of types of pre-

ferred diagnosis has been proposed. In the fol-

lowing denition weonsider theimportant from

thepratialperspetiveardinalityordering.

Denition6(Cardinality-MinimalDiagnosis). A

diagnosis

ω ˜ ^≤

^{is dened as} ardinality-minimal if nootherdiagnosis

ω ˜ ^≤

^,

ω ˜ ^≤ 6= ˜ ω ^≤

^{,existssuhthat}

Lit ⁻ (˜ ω ^≤ ) < Lit ⁻ (ω ^≤ )

^.

OntheMax-SATside weleaveoutalldenitions

as there is an agreement in the literature. Note

that most SAT and Max-SAT solvers(as well as

manydiagnostisolvers) aept

SD

^inCNFonly.

Any propositional formula an be onverted to

CNFtaking into onsiderationanumberof om-

plexityandother issues

(Feldman

etal., 2010).

4 MBDFRAMEDAS MAX-SAT

InthissetionwedemonstratetheuseofMax-SAT

forsolvingMBDproblems.

4.1 A Max-SAT-Based MBDAlgorithm

Weproposeanalgorithm,alledMeridian(Max-

sat-basEd algoRIthm for DIAgNosis), for om-

putingardinality-minimaldiagnoses(seeDef.6).

MeridianusestheapproahofSangetal.

(2007)

for enoding Most ProbableExplanation (MPE)

asMax-SAT.ComputingMPEisidentialtoom-

puting a most-probable diagnosis in amore gen-

eral framework. Algorithm 1omputesdiagnoses

byallingaMax-SATorale.

Note that the diagnosti problems we solve in

thispaperanbetranslatedtomultipleoptimiza-

tionproblemswhihanbesolvedwithSAT-based

methods

(Giunhiglia

and Maratea, 2006).

The

Maximum Satisable Subset (MSS) problem, for

example,isdualtotheMinimalUnsatisableSub-

set problem (

Bailey and Stukey, 2005 )

and the

twoanbesolvedwithMax-SATandMin-UNSAT

solvers,respetively (

LitonandSakallah,2005 )

.

From those, we have found preferene in the

researh ommunity towards Max-SAT, and for

pratialreasonswethereforeompareSafarito

Max-SAT.

Algorithm 1 adds a unit lause with weight

1

^{for eah assumable (line 6). Note that these}

weight/lausepairsareaddedto

W

^{whih isrst}

set (lines 2 4) to ontainall the lauses of the

model

SD

^{. Asaresult,inline8,}

W

^ontainsboth

the lausesof the original systemdesription

SD

and theunitlauses addedfor eah assumablein

line 57. Theweight ofeahinput lauseis set

to avaluegreater thanthe numberof allassum-

ables(line 3). The loopin lines 811omputes

adiagnosiswith aalltoMax-SATandifadiag-

nosisexists,itisaddedtotheresult(line10),and

itsnegationisadded totheoriginalsetoflauses

(line9)topreventsubsequentomputationofthe

samediagnosis. Notethat thenegationofaterm

is,onveniently,alause.

DependingontheimplementationoftheMax-

SAT allin line 8 of Alg. 1 we have afamily of

Max-SATalgorithms fordiagnosis: (1) if Max-

SATisapartialMax-SATsolver,Alg.1omputes

diagnosesorderedbyardinality;(2)ifMax-SAT

is a weighted Max-SAT solver, Alg. 1 omputes

diagnosesorderedbyprobability;and(3)ifMax-

SAT is based on SLS, not everyiteration of the

Algorithm1Meridian: analgorithmforMBD

basedonweightedMax-SAT

1: funtion Meridian(

DS, α

^{) returnsaset of}

diagnoses

inputs:

DS

^{,diagnostisystem}

DS

⁼

hSD, COMPS, OBSi α

^,term,observation

loalvariables:

W

^{,set ofweightand}

lause pairs

Ω

^{,setofdiagnoses}

ω

^{,diagnosisterm}

c i

^,lause

h i

^,variable

2: for all

c i ∈

^Clauses

(SD)

^do

3:

W ← W ∪ h∞, c i i

4: endfor

5: for all

h i ∈ COMPS

^do

6:

W ← W ∪ h1, h i i

7: endfor

8: while

ω ←

^Max-SAT

(W )

^do

9:

W ← W ∪ h∞, ¬ωi

10:

Ω ← Ω ∪ ω

11: endwhile

12: return

Ω

13: endfuntion

mainloopyields adiagnosis. Wehaverun exten-

siveexperimentswithallthreeMax-SATvariants,

whihwedesribeinthefollowingsub-setions.

4.2 ExperimentalResultswith SLS

Max-SAT

Intheexperimentsthatfollowweomparetheop-

timalityof Meridian(intermsofapproximation

errorin theminimal-ardinality)ongured with

anumberofSLS-basedMax-SATalgorithmsfrom

the UBCSAT suite (

Tompkins and Hoos, 2005 )

.

WeomparetheresultstoSafari,astate-of-the-

art stohasti MBD algorithm (

Feldman et al.,

2010 )

.

There are two issues that ompliate the use

of SLSMax-SATin diagnostialgorithms. First,

there is no simple termination riterion in diag-

nostialgorithmsbasedonSLSMax-SAT,i.e.,we

keeptheloal diagnosis and restartSafari after

anumber of suessive unsuessful ips, while

there isnonotionof unsuessful ip(fromthe

viewpoint of diagnosis) in Max-SAT. As we will

see from our experimentation, ipping avariable

whih dereases the weight (or number) of ur-

rentlysatisedlausesmaybeneessarytoesape

plateaus and/orloal optima,henethe aumu-

lation of suh ips annot beused as atermina-

tionriterion. Seond,diagnostiMax-SATprob-

lemshavetwotypesofonstraints: hardandsoft.

Thehard onstraintsarethelauses of theorigi-

nal (nominal) model, while the soft onstraints

are the unit lauses reeivedfrom theassumable

variables. An SLS Max-SAT algorithm doesnot

distinguishbetweenthosehardandsoftlauses;if

suh an algorithm guaranteed the satisfation of

the hard-onstraintsit would be lassiedashy-

bridand notstohasti. These reasons makethe

use of algorithms based on SLS Max-SAT prob-

lemati in pratial diagnosis. Despite that, we

have onduted extensive experimentation with

UBCSATinordertoevaluatethepotentialofSLS

Max-SATinMBD.

TooverometheterminationproblemswithSLS

Max-SAT,forthefollowingexperiments,wehave

hosenobservationsleadingtoknownsinglefaults.

For eah

WFM

wehavehosen

50

observations.

We haveongured the SLS Max-SAT searh to

terminateafter

100 000

^{variableipsandwehave}

modiedAlg.1toterminateafter

10

^allstoMax-

SAT, this

10

^{runs. The resulting optimality of}

algorithmsbasedon SLSMax-SATin omputing

singlefaultdiagnosesisshowninTable1. Wehave

run experimentswithall algorithmsoralgorithm

variantsimplementedbytheUBCSATsuite,ov-

eringthealgorithms RGSAT,Shöening,CDRW,

throughGSAT.

The data in Table 1 show best ases. From

eah of the

500

^{Max-SAT invoations per algo-}

rithm/iruit(

50

^{singlefaults,}

10

^{runsperexperi-}

ment)wehave(1)ignoredallresultswhihdonot

satisfyallhard onstraints,(2) reordedthebest

diagnosti ardinality ahieved in the hill limb-

ing (reall that these are single-faults hene the

best result is

1

^{) and (3) reorded the number of}

steps(bitips) in whih this best diagnostiar-

dinalitywasahieved(the numberof bit-ipsare

giveninparenthesesbelowtheoptimalitynumber

inTable1).

Table 1 showsthe generally poorperformane

ofSLSMax-SATalgorithms. Inmostoftheases

the algorithmould either neversatisfyall hard-

onstraints or ahieved inreasingly worse ardi-

nalitywith thegrowthofthe iruit. Exeptions

arethetwovariantsofSAPS

(Hutter

etal., 2002)

andweattributethisrelativelygoodoptimalityof

SAPS to itsmehanismfor assigning andupdat-

ingweightsto lausesbasedonthelauselength.

Reall that in our diagnostiproblems lauses of

assumable literals have unit weights while hard-

onstraintshaveweightsgreaterthanthenumber

of assumable literals. Despite that, in the best

asefor

c7552

^,SAPSneeded

77 264

^bitipstond

theoptimal single-fault diagnosis. Inomparison

Safari performed

11

^{bit ips, and although an}

LTMS/SATonsistenyhekofSafariisstritly

more expensivethan the onsisteny heking of

SLSMax-SAT(the formerisworst-aseNP-hard

whilethelatterisinP),Safariisomputationally

moreeientonaverage.

Figure 1 illustrates the progress of two SLS

Max-SAT invoations. The Conit-Direted

Random Walk (CDRW) (

Papadimitriou, 1991 )

startswitharandomvariableassignmentandips

the most protable (for inreasing the satised

weight)variable. Thisoftenleadstoviolatedhard-

onstraints(duetoippingofnon-assumablevari-

ables), and the restartswhih are needed for es-

apingthosesituationsleadtotherelativelynoisy

asentofCDRW.OtherSLSMax-SATalgorithms

like HSAT (Gent

and Walsh, 1993)

avoid down-

wardips(ipswhihdereasetheurrentlysatis-

edweight),quiklyinreasingthesatisedweight

but ultimatelygetstukin loal optima. A lose

inspetion of Fig. 1 reveals that HSAT osillates

forevershort ofsatisfyingallhardonstraints.

5 MAX-SATFRAMED AS MBD

In what follows we disuss the use of MBD for

solvingMax-SATproblems.

5.1 An MBD-Based Max-SAT Algorithm

Algorithm 2, alled Diorama (DIagnOsis-based

algoRithm for mAx-sat optiMizAtion), shows a

verysimpletranslationfromaMax-SATproblem

inCNFtoadiagnostiproblem.

Algorithm 2Diorama: analgorithm for Max-

SAToptimizationbasedonMBD

1: funtionDiorama(

Φ

^{)returnsaterm}

inputs:

Φ

^,setoflauses

loalvariables:

DS = hSD, COMPS, OBSi

^,

diagnostisystem

c i

^,lause

h i

^,variable

2: for all

c i ∈ Φ

^do

3:

SD ← SD ∧ {h i ⇒ c i }

4:

COMPS ← COMPS ∪ h i

5: endfor

6: returnMBD

(DS, ⊤)

7: endfuntion

The loopin lines 2- 4 of Alg.2 modies eah

lause in the input problem

Φ

^{. Note that line 3}

addsexatlyoneliteraltoeahinputlause

c i

^as,

given a lause

c = x ₁ ∨ x ₂ ∨ · · · ∨ x n

^{, we have}

h ⇒ (x ₁ ∨ x ₂ ∨ · · · ∨ x n ) ≡ ¬h ∨ x ₁ ∨ x ₂ ∨ · · · ∨ x n

and theright-handside of thelast equivalene is

alsoalause. Line4addsatotalof

|Φ|

^assumable

variablesto

|COMPS|

^where

|Φ|

^{isthenumberof}

lausesin

Φ

^.

Algorithm2alwaysreatesasystemdesription

SD ∈ WFM

(f. Def.2). NoteaswellthatAlg.2 invokesthe MBDorale in line 6 with anempty

observation (for any propositional formula

Φ

^we

have

Φ ∧ ⊤ ≡ Φ

^).

In a striter paper one an formally show the

orretnessof Diorama, i.e., one anprovethat

Alg.2alwaysomputesanoptimalMax-SAT so-

lutionifitisonguredwithanMBDoralethat

omputes at least one ardinality-minimal diag-

nosis. Theomplexityof Dioramaisdominated

by the omplexity of the Max-SAT solver. The

omplexityofAlg.2is

O(|Φ|) + Ψ

^where

Ψ

^isthe

omplexityoftheMBDorale. Wewill,however,

leavethisdisussionshortinordertoprovidemore

extensiveempirial evideneon theoptimality of

Diorama.

5.2 ExperimentalResultswith a

Stohasti MBDOrale

InourrstseriesofMax-SATexperimentswehave

onguredAlg.2withthestohastiMBDorale

Table1:OptimalityofSLS-basedMax-SATMeridianandSafarion

7 4 X X X

/

IS C A S 8 5 W F M

sandthenumberofsteps(inparentheses)inwhih thisoptimalityhasbeenahieved Name RGSAT Shöening CDRW URW IRoTS RoTS G2WSAT SAPS

a b

SAPS

Adaptive Novelty

+

Novelty

+

Novelty WalkSAT/TABU

WalkSAT HSAT GWSAT GSAT Safari

74182

1 1 1 9 1 1 1 1 1 1 1 1 1 1 1 1 1 1 (4 3 3 1 ) (1 6 3 ) (5 0 1 ) (2 9 8 9 6 ) (1 2 6 5 ) (1 9 ) (4 0 ) (4 0 ) (7 6 ) (4 2 2 6 0) (2 8 ) (5 6 ) (2 1 ) (3 7 ) (2 6 ) (1 8 6 ) (3 8 ) (2 )

74L85

1 1 1 − 1 3 1 1 1 1 1 1 5 1 1 1 1 1 (6 5 6 2 3 ) (2 7 6 ) (3 5 3 ) ( − ) (7 2 3 6 ) (2 7 ) (3 9 ) (1 9 9 ) (1 4 3 ) (8 0 7 8 2 ) (9 8 1 5 ) (1 2 6 7 ) (4 7 ) (1 16 ) (3 4 ) (6 2 9 ) (3 4 ) (4 )

74283

1 1 1 − 1 1 1 1 1 1 1 1 4 1 1 1 1 1 (6 3 4 2 9 ) (1 1 3 4 ) (5 7 3 ) ( − ) (3 6 4 4 ) (2 6 ) (2 1 8 0 6 ) (1 6 6 ) (1 1 2 ) (2 5 0 0 3 ) (3 1 9 0 0 ) (4 3 5 ) (3 5 ) (1 0 3 4 ) (2 9 ) (1 8 7 ) (3 2 ) (2 )

74181

4 1 1 − 1 1 3 1 1 1 1 1 1 1 0 1 3 1 3 1 (4 8 3 9 5 ) (4 5 2 5 ) (6 5 4 2 ) ( − ) (4 9 9 6 7 ) (1 4 9 8 ) (1 8 1 4 ) (1 4 6 ) (4 5 3 ) (1 0 4 5 ) (8 1 8 8 3 ) (2 0 6 0 ) (7 5 ) (2 1 2 3 ) (6 7 ) (1 1 3 9 ) (5 4 ) (3 )

432

2 8 1 1 − 4 3 6 4 1 1 7 1 8 − 1 1 3 1 1 6 1 (4 3 1 7 6 ) (1 8 6 6 ) (7 4 5 2 ) ( − ) (8 2 4 9 6 ) (1 0 1 ) (1 0 2 6 2 ) (8 0 6 ) (5 6 4 ) (2 0 2 1 1 ) (7 5 4 8 8 ) (1 6 3 9 3 ) ( − ) (9 0 5 ) (1 3 3 ) (2 4 1 6 4 ) (1 6 7 ) (2 )

499

− 1 1 − 2 9 − 6 1 1 7 5 1 3 − 1 − 1 − 1 ( − ) (2 4 2 0 3 ) (9 5 2 4 2 ) ( − ) (9 9 9 9 9 ) ( − ) (9 8 6 8 3 ) (4 2 5 3 4 ) (6 6 5 1 9 ) (2 1 1 2 ) (8 7 5 8 9 ) (4 9 5 1 6 ) ( − ) (4 7 1 0 2 ) ( − ) (4 5 4 6 5 ) ( − ) (1 )

880

− 1 1 − − − 2 6 1 1 3 5 2 3 3 0 − 1 − 1 4 6 1 ( − ) (3 6 2 6 9 ) (8 3 8 8 ) ( − ) ( − ) ( − ) (9 5 6 1 ) (4 2 4 4 ) (1 5 5 1 ) (5 6 8 0 ) (8 0 9 2 6 ) (3 4 9 8 9 ) ( − ) (9 8 2 5 ) ( − ) (2 7 0 2 6 ) (3 0 4 ) (5 )

1355

− 2 1 − − − 4 4 1 1 5 2 5 1 5 3 − 1 − 1 − 1 ( − ) (9 3 5 4 9 ) (3 4 2 3 5 ) ( − ) ( − ) ( − ) (8 3 9 4 4 ) (9 6 7 0 ) (4 9 7 5 ) (8 4 7 7 7 ) (6 7 4 5 4 ) (8 5 3 6 5 ) ( − ) (3 4 7 8 6 ) ( − ) (9 3 6 5 6 ) ( − ) (5 )

1908

2 4 0 − − − − − 9 1 1 1 9 4 9 8 1 0 1 − 1 1 2 5 1 1 3 9 1 (6 8 6 4 9 ) ( − ) ( − ) ( − ) ( − ) ( − ) (9 7 5 6 7 ) (9 5 9 4 2 ) (1 9 6 6 4 ) (5 1 4 2 ) (7 9 4 4 2 ) (4 3 5 4 1 ) ( − ) (5 0 6 6 0 ) (6 5 4 ) (9 4 6 9 5 ) (6 4 3 ) (6 )

2670

− − − − − − 1 2 2 1 1 1 2 6 1 3 5 1 4 4 − 1 1 7 3 1 1 7 7 1 ( − ) ( − ) ( − ) ( − ) ( − ) ( − ) (9 3 7 9 9 ) (9 5 4 7 ) (6 6 3 5 ) (1 9 0 0 0 ) (6 7 0 4 2 ) (6 5 3 2 ) ( − ) (8 3 2 9 2 ) (1 0 0 8 ) (8 5 5 8 1 ) (9 7 6 ) (5 )

3540

− − − − − − 1 6 7 1 1 1 7 0 1 7 7 2 0 3 − 1 − 1 6 − 1 ( − ) ( − ) ( − ) ( − ) ( − ) ( − ) (7 9 1 8 8 ) (1 6 3 7 6 ) (1 7 7 7 6 ) (8 8 0 5 2 ) (6 8 3 4 2 ) (6 8 6 6 3 ) ( − ) (9 9 2 0 9 ) ( − ) (9 3 6 3 9 ) ( − ) (9 )

5315

− − − − − − 2 6 1 1 1 2 7 0 2 7 5 3 2 9 − 1 4 − 7 0 − 1 ( − ) ( − ) ( − ) ( − ) ( − ) ( − ) (7 8 0 9 3 ) (7 9 4 3 8 ) (4 8 3 8 8 ) (6 4 0 7 9 ) (9 6 9 7 3 ) (4 8 4 9 7 ) ( − ) (9 8 5 5 7 ) ( − ) (8 9 4 6 2 ) ( − ) (9 )

6288

− − − − − − 3 2 9 4 8 3 6 0 3 7 2 − − 5 8 − − − 1 ( − ) ( − ) ( − ) ( − ) ( − ) ( − ) (2 3 1 9 1 ) (7 9 8 0 7 ) (7 3 7 5 2 ) (3 6 3 7 3 ) (9 1 6 3 6 ) ( − ) ( − ) (9 4 6 7 3 ) ( − ) ( − ) ( − ) (3 )

7552

− − − − − − 4 0 3 1 1 4 1 2 4 4 0 4 4 9 − 1 1 5 4 9 2 1 8 1 5 0 7 1 ( − ) ( − ) ( − ) ( − ) ( − ) ( − ) (1 2 9 0 5 ) (7 7 2 6 4 ) (4 2 2 8 4 ) (3 9 0 4 6 ) (9 0 4 8 5 ) (3 2 7 3 2 ) ( − ) (9 0 4 4 9 ) (2 5 4 3 ) (9 9 3 4 0 ) (2 5 5 2 ) (1 1 )

a Clausepenaltiesareinitializedtothelauseweightsandsmoothedbaktotheirinitialvalues. bClausepenaltiesareinitializedtothelauseweights.

0 500 1000 1500 2000 2500 8

8.2 8.4 8.6 8.8 9 9.2 x 10 ⁴

step

satisfied weight

Conflict−Directed Random Walk

0 100 200 300 400

8.5 8.6 8.7 8.8 8.9 9

x 10 ⁴ HSAT

step

satisfied weight

satisfied weight global optimum hard constraints

satisfied weight global optimum hard constraints

Figure1:ProgressoftwoSLSMax-SATalgorithmsinaweak-faultmodelof432,singlefaultobservation

Safari (F

eldman et al., 2010).

Safari is anap-

proximation-basedalgorithmand wehaveong-

uredittoomputeguaranteedsubset-minimaldi-

agnoses(itannotbeonguredtoomputeguar-

anteedardinality-minimaldiagnoses). Thesesub-

set-minimaldiagnosesare used asanapproxima-

tiontoardinality-minimaldiagnoses. Theresult-

ingalgorithmDiorama/SafariissimilartoSLS-

basedMax-SATalgorithmsliketheonedisussed

inSe.4.2.

Table 2 ompares the optimality of Dio-

rama/Safari to the algorithms from the UBC-

SATsuite. Theexperimentsare ontheproblems

from theSeondMax-SATEvaluation 2007. The

majority of those problems (

680

^{out of a total}

of

815

^{)arerandom 2-SAT2 and 3-SAT.Wehave}

ongured UBCSAT to terminate after

100 000

stepsandwehaverunit

10

^{timesforeahexper-}

iment. We ansee inTable2that theoptimality

of Diorama/Safari is slightly worse but om-

parable to the optimality of the UBCSAT algo-

rithms. Ingeneral,theoptimality,ofallUBCSAT

algorithmsandDiorama/Safariissimilarwhih

means that there are either (1) ontinuous diag-

nostisubspaesintheMax-SATinstanes 3

or(2)

the Max-SAT algorithms and Diorama/Safari

annotlimbaftertheinitialvariableassignment.

Table 3 shows the optimality of the UBCSAT

Max-SAT algorithms and Diorama/Safari on

theeightsmallestinstanesoftheMax-SAT2009

industrial benhmark. The 3, 5315, 6288,

7552, mot_omb1, mot_omb2, mot_omb3,

ands15850olumnsin Table3orrespondtothe

3_DD_s3_f1_e1_v1-bug-oneve-gate-0,5315-

bug-gate-0, 6288-bug-gate-0, 7552-bug-gate-0,

mot_omb1._red-gate-0,mot_omb2._red-gate-

0, mot_omb3._red-gate-0, and s15850-bug-

oneve-gate-0 instanes in the Max-SAT benh-

mark. WeanseethatDiorama/Safarioutper-

formsthetraditionalSLS-basedalgorithmsbytwo

tothree orders-of-magnitude. Thisisnotsurpris-

ingasthe5315,6288,7552instanesomefrom

the

ISCAS85

^{benhmark and we have seen the}

2

Reallthatalthoughthe2-SATdeisionproblem

is easy, the optimization Max-2-SAT problem is al-

ready

NP

^-hard.

3

SeeFeldman,etal. 2010 fordeningontinuity.

goodperformaneof Safariontheseinstanesin

Se.4. What ismoreinterestingisthat these re-

sultsholdforotherbenhmarkinstanesfromfor-

malveriation. s15850,forexample,omesfrom

ISCAS89

^andhas

534

^{D-typeip-ops. Notethat}

all these problem instanes result in solutions of

verysmallardinality.

6 CONCLUSION

This paperoersextensive empirialresearh on

theuse ofonsisteny-baseddiagnosisfor solving

Max-SATproblemsandvie-versa. Themainon-

tribution of this paper is solving more than

800

Max-SAT instanes with Diorama/Safari and

UBCSAT and more than

700 74XXX

^/

ISCAS85

problems withMeridian/UBCSAT and Safari.

We have experimented with

20

^{algorithms for}

SLS-based Max-SAT. The good result of Dio-

rama/Safarionsmallindustrial instanes show

that many Max-SAT problems of real-world im-

portane an be optimally and eiently solved

withgreedystohastialgorithms.

REFERENCES

(Bailey

andStukey, 2005)

James Baileyand Pe-

ter J. Stukey. Disoveryof minimal unsatis-

able subsets of onstraints using hitting set

dualization. InPro.PADL'05,pages174186,

2005.

(Castellini

etal., 2003)

ClaudioCastellini,Enrio

Giunhiglia, and Armando Tahella. SAT-

based planning in omplex domains: Conur-

reny,onstraintsandnondeterminism. Arti-

ialIntelligene,147(1-2):85117,2003.

(Chen

etal., 2009)

Yibin Chen, Sean Safarpour,

AndreasG.Veneris,andJoãoP.MarquesSilva.

Spatial and temporal design debug using par-

tialMaxSAT.InACMGreatLakesSymposium

onVLSI,pages345350,2009.

(

deGivryetal., 2003 )

Simon de Givry, Javier

Larrosa,Pedro Meseguer,andThomasShiex.

Solving Max-SAT as weighted CSP. In CP,

pages363376,2003.

(de

KleerandWilliams, 1987)

Johan de Kleer

and Brian Williams. Diagnosing multiple

Table2:OptimalityofUBCSATalgorithmsandDiorama/Safariandthenumberofsteps(inparentheses)inwhihthisoptimalityhasbeen ahieved SetName RGSAT Shöening CDRW URW SAMD IRoTS RoTS G2WSAT Novelty

+

G2WSAT A daptive Novelty

+

Novelty

+

Novelty WalkSAT/TABU

WalkSAT HWSAT HSAT GSAT/TABU GWSAT GSAT Diorama/

Safari

MAX3SAT/40

3 0 3 9 3 9 4 8 3 0 3 0 3 0 3 0 3 0 3 0 3 1 3 1 3 4 3 2 3 1 3 3 3 0 3 0 3 3 5 8 (8 0 1 ) (4 5 8 6 ) (4 4 2 5 ) (4 5 1 5 ) (1 8 3 ) (1 0 9 ) (1 0 8 ) (2 4 7 ) (2 4 0 ) (1 84 6 ) (2 8 5 2 ) (3 1 4 1 ) (4 2 7 2 ) (3 6 9 9 ) (1 5 0 5 ) (2 9 ) (2 3 1 ) (7 4 7 ) (3 2 ) ( − )

MAX3SAT/50

2 6 3 7 3 7 4 8 2 6 2 6 2 6 2 6 2 6 2 6 2 7 2 7 3 1 2 9 2 7 2 9 2 6 2 6 2 9 5 4 (2 1 9 2 ) (4 4 8 6 ) (4 6 5 9 ) (4 5 4 6 ) (4 0 6 ) (2 4 1 ) (2 5 5 ) (6 5 4 ) (6 9 5 ) (2 3 8 8 ) (3 3 7 4 ) (3 4 8 4 ) (3 9 9 9 ) (3 8 9 2 ) (1 5 2 2 ) (4 7 ) (3 8 4 ) (1 5 1 3 ) (7 7 ) ( − )

MAX3SAT/60

2 2 3 4 3 4 4 8 2 1 2 1 2 1 2 1 2 1 2 2 2 3 2 3 2 6 2 5 2 3 2 4 2 1 2 1 2 4 4 8 (2 9 4 7 ) (4 6 6 7 ) (4 6 2 2 ) (4 7 5 5 ) (3 2 8 ) (2 8 9 ) (2 7 1 ) (7 5 5 ) (8 6 7 ) (2 1 6 1 ) (3 0 9 4 ) (3 2 0 9 ) (3 8 6 8 ) (3 6 3 3 ) (1 6 9 6 ) (8 2 ) (3 1 0 ) (1 4 8 8 ) (1 0 8 ) ( − )

MAX3SAT/70

1 9 3 2 3 1 4 8 1 8 1 8 1 8 1 8 1 8 1 9 2 0 2 0 2 3 2 2 2 0 2 1 1 8 1 8 2 1 4 3 (3 7 3 5 ) (4 4 8 0 ) (4 5 9 1 ) (4 9 7 6 ) (8 4 7 ) (6 3 1 ) (6 2 2 ) (1 5 2 7 ) (1 6 9 7 ) (2 5 1 6 ) (3 2 0 5 ) (3 0 9 3 ) (3 6 4 8 ) (3 4 6 4 ) (1 6 0 4 ) (1 4 7 ) (7 7 0 ) (2 3 1 6) (2 7 9 ) ( − )

MAX3SAT/40

3 1 3 9 3 9 4 8 3 1 3 1 3 1 3 1 3 1 3 1 3 2 3 2 3 4 3 3 3 2 3 4 3 1 3 1 3 4 5 8 (8 1 2 ) (4 4 1 1 ) (4 3 9 8 ) (4 5 4 0 ) (2 0 9 ) (1 3 0 ) (1 2 6 ) (2 7 7 ) (2 9 6 ) (2 00 7 ) (2 8 4 1 ) (2 8 1 9 ) (3 6 5 4 ) (3 3 7 1 ) (1 3 8 6 ) (4 0 ) (2 2 0 ) (8 2 9 ) (4 1 ) ( − )

MAX3SAT/60

2 3 3 4 3 4 4 9 2 2 2 2 2 2 2 2 2 2 2 3 2 4 2 4 2 7 2 6 2 4 2 5 2 2 2 2 2 5 4 9 (2 7 6 8 ) (3 9 6 2 ) (3 9 5 7 ) (4 3 2 4 ) (3 3 7 ) (2 3 4 ) (2 4 7 ) (7 3 5 ) (8 0 3 ) (2 1 7 6 ) (3 0 2 2 ) (3 1 7 8 ) (3 3 6 7 ) (3 2 4 5 ) (1 2 5 9 ) (1 2 8 ) (3 6 3 ) (1 4 4 2 ) (1 4 2 ) ( − )

MAX3SAT/80

1 8 3 1 3 1 4 9 1 7 1 7 1 7 1 7 1 7 1 8 1 9 1 9 2 1 2 1 1 8 2 0 1 7 1 7 2 0 4 0 (3 3 7 6 ) (3 7 1 1 ) (3 7 2 2 ) (4 3 8 0 ) (5 3 4 ) (5 3 4 ) (4 5 8 ) (1 7 2 7 ) (1 6 8 9 ) (1 9 7 2 ) (2 5 8 1 ) (2 7 6 9 ) (3 3 5 6 ) (3 0 4 9 ) (1 4 2 7 ) (1 4 8 ) (5 9 3 ) (2 1 0 9) (2 3 5 ) ( − )

MAX2SAT/60

9 3 1 1 0 1 1 0 1 3 2 9 3 9 3 9 3 9 3 9 3 9 7 1 0 2 1 0 2 1 0 4 1 0 2 9 3 9 3 9 3 9 3 9 3 1 4 3 (7 9 4 ) (4 7 5 4 ) (4 6 7 3 ) (4 7 6 5 ) (8 8 ) (6 3 ) (6 4 ) (2 2 0 4 ) (2 1 3 6 ) (3 7 74 ) (4 3 4 8 ) (4 4 4 5 ) (4 4 4 2 ) (4 3 4 6 ) (4 4 4 ) (5 8 ) (9 5 ) (3 8 7 ) (8 4 ) ( − )

MAX2SAT/100

8 1 1 0 5 1 0 5 1 3 8 8 0 8 0 8 0 8 2 8 3 8 7 9 6 9 5 9 5 9 5 8 1 8 1 8 0 8 0 8 1 1 3 1 (3 3 2 0 ) (4 6 1 1 ) (4 6 5 9 ) (4 7 1 0 ) (2 4 6 ) (2 3 8 ) (2 2 1 ) (2 9 3 0 ) (3 0 8 0 ) (3 6 5 1 ) (4 1 7 8 ) (4 2 3 7 ) (4 4 4 4 ) (4 1 8 4 ) (5 7 4 ) (2 3 3 ) (2 4 8 ) (1 2 2 7 ) (3 5 7 ) ( − )

MAX2SAT/140

6 8 9 3 9 3 1 3 9 6 6 6 6 6 6 7 0 7 0 7 3 8 3 8 3 8 0 8 1 6 6 6 6 6 6 6 6 6 7 1 1 3 (4 0 8 0 ) (4 4 0 4 ) (4 3 8 4 ) (4 7 5 2 ) (5 5 4 ) (5 2 2 ) (6 8 9 ) (3 0 5 6 ) (2 9 1 5 ) (3 5 3 3 ) (3 9 7 5 ) (3 9 6 6 ) (3 9 6 0 ) (4 0 0 1 ) (7 6 4 ) (5 7 4 ) (5 4 2 ) (1 8 2 4 ) (6 7 2 ) ( − )

SPINGLASS

6 4 8 6 8 7 1 1 3 5 8 5 8 5 8 6 3 6 3 6 6 7 5 7 4 7 0 6 9 6 0 6 0 5 8 5 9 6 0 1 0 5 (3 6 2 8 ) (4 2 4 9 ) (4 4 8 4 ) (4 4 9 2 ) (6 9 2 ) (4 3 1 ) (7 9 2 ) (3 4 8 8 ) (3 7 7 1 ) (3 3 6 8 ) (4 4 1 8 ) (4 3 4 5 ) (4 1 0 0 ) (4 3 6 9 ) (9 1 4 ) (5 8 7 ) (9 6 4 ) (2 7 3 9 ) (7 1 2 ) ( − )

RAMSEY

1 1 1 6 1 6 2 1 1 0 9 9 1 0 1 0 1 0 1 1 1 1 1 4 1 4 1 0 1 1 1 0 9 1 1 6 7 3 (1 2 4 3 ) (1 7 2 1 ) (1 6 2 5 ) (2 5 9 6 ) (1 8 9 ) (3 6 8 ) (2 6 1 ) (8 1 8 ) (7 2 4 ) (6 7 2 ) (1 0 6 3 ) (1 1 3 2 ) (1 1 1 5 ) (1 2 1 4 ) (6 5 7 ) (4 0 ) (1 9 8 ) (6 8 7 ) (4 5 ) ( − )

DIMACS

2 3 3 2 4 8 2 4 8 2 5 2 2 3 3 2 3 3 2 3 3 2 3 3 2 3 3 2 3 8 2 4 2 2 4 2 2 4 3 2 4 2 2 3 4 2 3 5 2 3 3 2 3 3 2 3 5 2 6 7 (6 0 1 ) (3 8 7 8 ) (3 9 7 5 ) (4 1 5 3 ) (8 9 ) (5 9 ) (6 7 ) (3 9 1 ) (3 6 7 ) (3 3 6 5 ) (3 8 8 5 ) (3 6 8 0 ) (3 9 7 4 ) (3 7 8 5 ) (8 6 3 ) (2 2 ) (1 1 1 ) (5 1 0 ) (2 5 ) ( − )

RANDOM

1 8 2 2 1 2 2 1 2 2 2 3 1 8 2 1 8 2 1 8 2 1 8 2 1 8 2 1 9 3 2 0 1 2 0 1 2 0 2 2 0 2 1 8 5 1 8 8 1 8 2 1 8 2 1 8 7 2 3 9 (2 4 5 9 ) (4 7 1 0 ) (4 8 9 6 ) (4 5 3 4 ) (3 0 3 ) (2 0 5 ) (2 0 4 ) (2 6 3 8 ) (2 6 9 8 ) (4 2 9 9 ) (4 5 4 4 ) (4 8 1 8 ) (4 7 1 8 ) (4 8 6 5 ) (1 2 9 8 ) (3 3 ) (3 4 4 ) (1 9 3 1 ) (3 7 ) ( − )

SPINGLASS

1 1 3 1 5 6 1 5 5 1 9 9 9 8 9 6 9 7 1 1 2 1 1 2 1 1 6 1 3 6 1 3 6 1 1 8 1 2 6 9 9 9 9 9 8 9 9 1 0 0 1 7 2 (4 5 5 8 ) (4 0 0 8 ) (4 3 7 0 ) (4 8 8 7 ) (8 9 9 ) (1 9 8 1 ) (2 2 8 9 ) (3 2 4 6 ) (3 7 24 ) (3 6 7 3 ) (3 9 3 1 ) (3 6 1 5 ) (3 5 5 6 ) (4 5 4 3 ) (1 8 2 6 ) (1 1 5 3 ) (1 0 5 2 ) (3 5 5 3 ) (1 2 0 2 ) ( − )

Table3: Optimality of theUBC SLS-basedMax-SATalgorithms and Safari/Diorama on small in-

dustrialMax-SAT2009instanes

3 5315 6288 7552 mot_omb1 mot_omb2 mot_omb3 s15850

RGSAT

1 215 526 440 647 531 1362 1 758 3 483

Shöening

1 556 146 410 262 1 805 2 534 6 055

CDRW

1 532 145 423 255 2 832 2 533 6 012

URW

4 936 1 098 1 939 1 520 1 432 3 654 6 838 12 093

SAMD

342 500 132 695 9 86 598 2 175

IRoTS

261 78 86 86 24 83 436 1 972

RoTS

347 129 130 122 5 108 565 2 170

G2WSATNovelty

+ 237 13 99 54 1 4 544 2 124

G2WSAT

206 16 101 60 1 3 484 2 084

AdaptiveNovelty

+ 238 38 106 64 1 184 522 2 504

Novelty

+ 339 13 111 66 1 49 800 3 367

Novelty

314 16 121 63 1 46 769 3 335

WalkSAT/TABU

368 24 129 84 1 676 810 2 819

WalkSAT

555 21 177 94 1 27 1 088 3 281

HWSAT

351 417 123 547 4 66 579 2 187

HSAT

354 498 130 798 18 116 576 2 145

GSAT/TABU

354 98 126 98 7 147 589 2 144

GWSAT

446 371 117 474 1 229 839 2 629

GSAT

372 392 130 347 18 149 596 2 191

Diorama/Safari

1 2 3 1 2 2 2 1

faults. Artiial Intelligene, 32(1):97130,

1987.

(de

Kleeretal., 1992)

Johan de Kleer, Alan

Makworth,and Raymond Reiter. Charater-

izingdiagnoses andsystems. Artiial Intelli-

gene,56(2-3):197222,1992.

(F

eldman etal., 2010)

Alexander Feldman, Gre-

gory Provan, and Arjan van Gemund. Ap-

proximate model-baseddiagnosis usinggreedy

stohastisearh. JAIR,2010.

(F

u andMalik, 2006)

Zhaohui Fu and Sharad

Malik.Onsolvingthepartialmax-satproblem.

InSAT,pages252265,2006.

(Gent

andWalsh, 1993)

Ian P. Gent and Toby

Walsh. Towards an understanding of hill-

limbing proedures for SAT. In Pro.

AAAI'93,pages2833,1993.

(Giunhiglia

andMaratea, 2006)

Enrio

Giunhiglia and Maro Maratea. Solving

optimization problems with DLL. In Pro.

ECAI'06, pages377381,2006.

(

Gomesetal.,2007 )

Carla P. Gomes, Henry

Kautz, Ashish Sabharwal, and Bart Selman.

Satisability solvers. Handbook of Knowledge

Representation,2007.

(Hoos

andStützle, 2004)

Holger Hoos and

Thomas Stützle. Stohasti Loal Searh:

Foundations and Appliations. Morgan

KaufmannPublishersIn.,2004.

(Hutter

et al., 2002)

Frank Hutter, Dave A. D.

Tompkins, and Holger H. Hoos. Saling and

probabilisti smoothing: Eient dynami lo-

alsearhforSAT. InPro. CP'02,pages233

248,2002.

(Iyer

etal., 2003)

Madhu K. Iyer, Ganapathy

Parthasarathy, and Kwang Ting Ting Cheng.

SATORI-afastsequentialSATengineforir-

uits.InPro.ICCAD'03,pages320325,2003.

(Kutsuna

et al., 2009)

Takuro Kutsuna, Shuihi

Sato,andNaoyaChujo.Diagnosingautomotive

ontrolsystemsusingabstratmodel-baseddi-

agnosis. InPro.DX'09, pages99105,2009.

(Liton

andSakallah, 2005)

Mark H. Liton

and Karem A. Sakallah. On nding all min-

imally unsatisable subformulas. In Pro.

SAT'05,pages173186,2005.

(Papadimitriou, 1991)

Christos Papadimitriou.

On seletingasatisfyingtruth assignment. In

Pro. FOCS'91,pages163169,1991.

(Sang

et al., 2007)

Tian Sang, Paul Beame, and

HenryA.Kautz.AdynamiapproahforMPE

and weighted MAX-SAT. In Pro. IJCAI'07,

pages173179,2007.

(T

ompkinsandHoos, 2005)

Dave A. D. Tomp-

kinsandHolgerH.Hoos. UBCSAT:Animple-

mentation and experimentation environment

forSLSalgorithmsforSATandMAX-SAT. In

Pro. SAT'04,pages306320,2005.

(Williams

andRagno, 2007)

Brian Williams and

RobertRagno.Conit-diretedA*anditsrole

in model-basedembeddedsystems. Journalof

Disrete Applied Mathematis, 155(12):1562

1595,2007.