Automatic extraction of functional dependencies

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Automatic extraction of functional dependencies

Eric Gregoire, Richard Ostrowski, Bertrand Mazure, Lahkdar Sais

To cite this version:

Eric Gregoire, Richard Ostrowski, Bertrand Mazure, Lahkdar Sais. Automatic extraction of functional dependencies. Theory and Applications of Satisfiability Testing: 7th International Conference (SAT 2004), 2005, Vancouver, Canada. pp.122-132. �hal-00397450�

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ÉriGrégoire,RihardOstrowski,BertrandMazure,andLakhdarSaïs

CRILCNRSUniversitéd'Artois

rueJeanSouvrazSP-18

F-62307LensCedexFrane

{gregoire,ostrowski,mazure,sa is} ril. univ- arto is.fr

Abstrat. Inthispaper,anewpolynomialtimetehniqueforextratingfuntionalde-

pendeniesinBooleanformulasisproposed.Itmakesanoriginaluseofthewell-known

Booleanonstraintpropagationtehnique(BCP)inanewpreproessingapproahthat

extratsmorehiddenBooleanfuntionsanddependentvariablesthanpreviouslypub-

lishedapproahesonmanylassesofinstanes.

Keywords:SAT,Booleanfuntion,propositionalreasoningandsearh.

1 Introdution

Reent impressiveprogress in the pratial resolution of hard and large SAT instanes al-

lows real-world problems that are enoded in propositional lausal normal form (CNF) to

be addressed(see e.g. [11,7,18℄). Whilethere remainsa strongompetition aboutbuilding

more eient provers dediated to hard random k-SAT instanes [6℄, there is also a real

surgeof interestin implementingpowerfulsystemsthat solvediultlargereal-worldSAT

problems.Manybenhmarkshavebeenproposedandregularompetitions(e.g.[4,1,14,15℄)

areorganizedaroundthese speiSATinstanes, whih are expeted toenodestrutural

knowledge,atleasttosomeextent.

Clearly,enodingknowledgeundertheformofaonjuntionofpropositionallausesan

attensomestruturalknowledgethatwouldbemoreapparentinmoreexpressiveproposi-

tionallogirepresentationformalisms,andthatouldproveusefulintheresolutionstep[13,

8℄.

Inthis paper,anew pre-proessingstepis proposed in theresolution ofSATinstanes,

that extratsand exploits somestruturalknowledgethat is hiddenin the CNF.The teh-

niquemakesanoriginaluseofthewell-knownBooleanonstraintpropagation(BCP)proess.

WhereasBCPistraditionallyusedtoprodueimpliedand/orequivalentliterals,inthis pa-

peritisshownhowitanbeextended sothatitdeliversanhybridformulamadeoflauses

together with aset of equations of the form y = f(x

1

;:::;x

n

) where f is astandard on-

netiveoperatoramongf_,^g andwhere y andx

i

areBoolean variablesoftheinitialSAT

instane. These Boolean funtions allowus to detet asubset of dependent variables, that

anbeexploited bySATsolvers.

This paper extends in a signiant way the preliminary results that were published in

[12℄in that it desribesatehniquethat allowsmoredependent variablesand hidden fun-

tional dependenies to be deteted in several lassesof instanes. We shallsee that theset

offuntionaldependeniesanunderlie yles.Unfortunately, highlightingatualdependent

variablestakingpartintheseylesanbetime-onsumingsineitoinidestotheproblem

ofndingaminimalyleutsetofvariablesinagraph,whihisawell-knownNP-hardprob-

lem.Aordingly,eientheuristisareexploredtouttheseylesanddelivertheso-alled

dependentvariables.

Thepaperisorganizedasfollows.After somepreliminary denitions,Booleangatesand

theirpropertiesarepresented.Itisthenshownhowmorefuntionaldependeniesthan[12℄an

bededuedfromtheCNF,usingBooleanonstraintpropagation.Then,atehniqueallowing

ustodeliverasetofdependentvariablesispresented,allowingthesearhspaetoberedued

in anexponentialway.Experimental resultsshowingthe interestof the proposed approah

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2 Tehnial preliminaries

LetBbeaBoolean(i.e.propositional)languageofformulasbuiltin thestandardway,using

usualonnetives(_,^,:,),,)andasetofpropositionalvariables.

A CNFformula is aset (interpretedasaonjuntion) oflauses, where alauseis a

set (interpretedas adisjuntion) of literals.A literal is apositiveor negatedpropositional

variable. We note V() (resp. L()) the set of variables (resp. literals) ourring in . A

unitlauseis alauseformed withoneuniqueliteral. Aunitliteral istheuniqueliteralofa

unitlause.

Inaddition to theseusual set-basednotations, wedene thenegationof aset ofliterals

(:fl

1

;:::;l

n

g)asthesetof theorrespondingoppositeliterals(f:l

1

;:::;:l

n g).

An interpretation ofaBooleanformulais anassignmentoftruth valuesftrue;falsegto

itsvariables.Amodelofaformulaisaninterpretationthatsatisestheformula.Aordingly,

SATonsistsinndingamodelofaCNFformulawhensuhamodeldoesexistorinproving

thatsuhamodel doesnotexist.

Let

1

be alause ontaininga literal a and

2

a lause ontaining the opposite literal

:a,oneresolvent of

1 and

2

isthe disjuntionof allliteralsof

1 and

2

lessaand :a.A

resolventisalled tautologial whenitontainsoppositeliterals.

Let us reall here that any Boolean formula an be translated thanks to alinear time

algorithmintoCNF,equivalentwithrespettoSAT(butthatanuseadditionalpropositional

variables). Most satisability heking algorithms operate on lauses, where the strutural

knowledge of the initial formulas is thus attened. In the following,CNF formulas will be

representedasBooleangates.

3 Boolean gates

A (Boolean) gate is an expression of the form y = f(x

1

;:::;x

n

), where f is a standard

onnetiveamongf_,^,,gandwherey andx

i

arepropositionalliterals,thatisdenedas

follows:

y=^(x

1

;:::;x

n

)representsthesetoflausesfy_:x

1

_:::_:x

n

;:y_x

1

;:::;:y_x

n g,

translatingtherequirementthatthetruthvalueofyisdeterminedbytheonjuntionof

thetruthvaluesofx

i

s.t.i2[1::n℄;

y=_(x

1

;:::;x

n

)representsthesetoflausesf:y_x

1

_:::_x

n

;y_:x

1

;:::;y_:x

n g;

y =, (x

1

;:::;x

n

) represents the following equivalene hain (also alled bionditional

formula) y , x

1

,::: , x

n

, whih is equivalent to the set of lausesfy_x

1 _:::_

x

n

;y_:x

1

_:::_:x

n

;:y_x

1 _:x

2

_:::_:x

n

;:::;:y_:x

1

_:::_:x

n 1 _x

n g.

Inthefollowing,weonsidergates oftheformy=f(x

1

;:::;x

n

)wherey isavariableor

theBooleanonstanttrue,only.

Indeed,anylauseanberepresentedasagateoftheformtrue=_(x

1

;:::;x

n

).Moreover,

agate:y =^(x

1

;:::;x

n

) (resp.:y = _(x

1

;:::;x

n

)) is equivalentto y =_(:x

1

;:::;:x

n )

(resp. y = ^(:x

1

;:::;:x

n

) ). Aording to the well-known property of equivalene hain

assertingthateveryequivalenehainwithanodd(resp.even)numberofnegativeliteralsis

equivalentto the hain formed with thesameliterals, but allin positive(resp. exeptone)

form,everygateof theformy =,(x

1

;:::;x

n

)analwaysberewrittenintoagatewherey

isapositiveliteral.Forexample,:y=,(:x

1

;x

2

;x

3

)isequivalenttoy=,(x

1

;x

2

;x

3 )and

:y=,(:x

1

;x

2

;:x

3

)isequivalenttoe.g.y=,(x

1

;x

2

;:x

3 ).

Apropositionalvariabley(resp.x

1

;:::;x

n

)isanoutputvariable(resp.areinputvariables)

ofagateof theform y=f(x 0

1

;:::;x 0

n

),wherex 0

i 2fx

i

;:x

i g.

A propositionalvariable z is an output (dependent) variable of a set of gates i z is an

output variable of at least one gatein theset. An input (independent) variable of a set of

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AgateissatisedunderagivenBooleaninterpretationitheleftandrighthandsidesof

thegatearesimultaneouslytrueorfalseunderthisinterpretation.Aninterpretationsatises

asetofgatesieahgateissatisedunderthisinterpretation.Suhaninterpretationisalled

amodel ofthis setofgates.

4 From CNF to gates

Pratially,wewanttondarepresentationofaCNFusinggatesthathighlightsamaximal

numberofdependentvariables,in orderto dereasetheatualomputationalomplexityof

heking the satisability of . Atually, weshall desribea tehniquethat extrats gates

thatanbededuedfrom,andthatthusoverasubsetoflausesof.Remaininglauses

ofwillberepresentedasor-gatesoftheformtrue=_(x

1

;:::;x

n

),inordertogetauniform

representation.

Moreformally,assumethatasetGofgateswhoseorrespondinglausesCl(G)arelogial

onsequenesofaCNF,theset

unovered(G)

ofunoveredlausesofw.r.t.Gistheset

oflausesofnCl(G).

Aordingly,

unovered(G)

[Cl(G).

Nottrivially,weshallseethattheadditionallausesCl(G)nanplayanimportantrole

infurtherstepsofdedutionorsatisabilityheking.

Knowing output variables anplay an important role in solvingthe onsisteny status

of a CNF formula. Indeed, the truth-value of an y output variable of a gate depends on

the truth value of the orresponding x

i

input variables. The truth value of suh output

variablesanbeobtainedbypropagation,andtheyanbeomittedbyseletionheuristisof

DPLL-likealgorithms[3℄.Inthegeneralase,knowingn 0

outputvariablesofagate-oriented

representationofaCNFformulausingnvariablesallowsthesizeofthesetofinterpretations

tobeinvestigatedtodereasefrom2 n

to2 n n

0

.Obviously,theredutioninthesearhspae

inreaseswiththenumberofdeteteddependentvariables.

Unfortunately,toobtainsuharedutioninthesearhspae,onemightneedtoaddress

thefollowingproblems:

Extrating gates from a CNF formula anbe a time-onsuming proess in the general

ase,unlesssomedepth-limitedsearhresouresorheuristiriteriaareprovided.Indeed,

showingthaty=f(x

1

;:::;x

i

)(wherey;x

1

;:::;x

i

belongto)followsfromagivenCNF

,isoNP-omplete.

when the set of deteted gates ontains reursivedenitions (like y = f(x;t) and x =

g(y;z)), assigning truth values to the set of independent variables is not suient to

determinethetruthvaluesofallthedependentones.Handlingsuhreursivedenitions

oinidestothewell-knownNP-hardproblemofndingaminimalyleutsetinagraph.

In this paper, these two omputationally-heavy problems are addressed. The rst one

byrestriting dedution to Boolean onstraintpropagation,only. The seond one byusing

graph-orientedheuristis.

Letusrstreallsomeneessarydenitions aboutBooleanonstraintpropagation.

5 Boolean onstraint propagation (BCP)

Booleanonstraintpropagationorunitresolution,isoneofthemostusedandusefullookahead

algorithmforSAT.

Let beaCNFformula,BCP()istheCNFformulaobtainedbypropagatingallunit

literalsof.Propagatingaunit literallof onsistsinsuppressing alllauses of suh

thatl2 andreplaing alllauses 0

of suhthat :l2 0

by 0

nf:lg.TheCNFobtained

insuhawayis equivalentto withrespetto satisability.

Theset of propagated unitliterals of using BCP isnotedUP(). Obviously,wehave

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ItisalsoompleteforHornformulas.InadditiontoitsuseinDPLLproedures,BCPisused

in manySAT solversasaproessingstep todedue further interestinginformation suh as

implied[5℄ and equivalentliterals [2℄[9℄.Loal proessingbased-BCP is alsoused to deliver

promisingbranhingvariables(heuristiUP [10℄).

Inthesequel,itisshownthatBCPanbefurtherextended,allowingmoregeneralfun-

tionaldependeniestobeextrated.

6 BCP and funtional dependenies

Atually,BCPanbeusedtodetethiddenfuntionaldependenies.Themainresultofthe

paperisthepratialexploitationofthefollowingoriginalproperty:gatesanbeomputed

usingBCP only, while heking whether agateis alogialonsequeneof aCNFis oNP-

ompleteinthegeneralase.

Property1. Let beaCNFformula,l2L(),and2 s.t.l2.Ifnflg:UP(^l)

thenl=^(:fnflgg).

Proof. Let = fl;:l

1

;:l

2

;:::;:l

m

g 2 s.t. nflg = f:l

1

;:l

2

;:::;:l

m

g :UP(^l).

The Boolean funtion l = ^(:fnflgg) an be written as l = ^(l

1

;l

2

;:::;l

m

). To prove

that l = ^(l

1

;l

2

;:::;l

m

), we need to showthat every model of , is also a model of

l=^(l

1

;l

2

;:::;l

m

).LetI beamodelof,then

1. liseithertrueinI :I isalsoamodelof^l.Asf:l

1

;:l

2

;:::;:l

m

g:UP(^l),we

havefl

1

;l

2

;:::;l

m

gUP(^l), then fl

1

;l

2

;:::;l

m

g aretrue in I. Consequently, I is

alsoamodelofl=^(l

1

;l

2

;:::;l

m gg);

2. orlisfalseinI:as=fl;:l

1

;:l

2

;:::;:l

m

g2thenIsatises=f:l

1

;:l

2

;:::;:l

m g2

. So, at least one the literals l

i

;i 2 f1;:::;mg is true in I. Consequently, I is also a

modelofl=^(l

1

;l

2

;:::;l

m gg)

Clearly,dependingonthesignoftheliterall,and-gatesoror-gatesanbedeteted.Forex-

ample,theand-gate:l=^(l

1

;l

2

;:::;l

n

)isequivalenttotheor-gatel=_(:l

1

;:l

2

;:::;:l

n ).

Letusalsonote that thispropertyoversbinaryequivalene sinea=^(b)is equivalentto

a,b.

Atually, this property allows gates to be deteted, whih were not in the sope the

tehniquedesribedin[12℄.Letusillustratethisbymeansofanexample.

Example1. Let

1

fy_:x

1 _:x

2 _:x

3

;:y_x

1

;:y_x

2

;:y_x

3 g.

Aording to [12℄,

1

an be representedby agraphwhere eah vertex representsalause

andwhere eah edgeorrespondsto theexisteneof tautologialresolventbetweenthetwo

orrespondinglauses. Eah onneted omponent might be agate.As we ansee the rst

four lausesbelong to asame onnetedomponent. Thisis a neessaryonditionfor suh

a subset of lauses to represent a gate. Suh a restrited subset of lauses (namely, those

appearinginthesameonnetedomponent)isthenhekedsyntatiallytodetermineifit

representsanand/orgate.Suhapropertyanbehekedinpolynomialtime. Intheabove

example,wethushavey=^(x

1

;x

2

;x

3 ).

Now,letusonsider,thefollowingexample,

Example2.

2

fy_:x

1 _:x

2 _:x

3

;:y_x

1

;:x

1 _x

4

;:x

4 _x

2

;:x

2 _x

5

;:x

4 _:x

5 _x

3 g.

Clearly,thegraphialrepresentationofthislaterexampleisdierentandtheaboveteh-

niquedoesnothelpus indisoveringthey=^(x

1

;x

2

;x

3

)gate.Indeed,theaboveneessary

butnotsuientonditionisnotsatised.

Now,aordingto Property1,boththeand-gatesbehindExample1and Example2an

bedeteted.Indeed,UP( ^y)=fx ;x ;x g(resp.UP( ^y)=fx ;x ;x ;x ;x g)and

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92

1

,(resp.

0

2

),=(y_:x

1 _:x

2 _:x

3

)(resp.

0

=(y_:x

1 _:x

2 _:x

3 ))suh

thatnfyg:UP(

1

^y)(resp.

0

nfyg:UP(

2

^y)).

Aordingly,apreproessingtehniquetodisovergatesonsistsinhekingtheProperty

1foranyliteralourringin .A furthersteponsistsin ndingdependentvariablesofthe

originalformulas, as theyan be reognisedin the disoveredgates.A gatelearlyexhibits

onedependentliteral with respetto theinputs whih are onsideredindependent, asfara

singlegate is onsidered.Now, when several gates share literals, suh a haraterisationof

dependent variables doesnotapply anymore.Indeed, forms of yleanour asshown in

thefollowingexample.

Example3.

3

fx=^(y;z);y=_(x;:t)g.

Clearly,

3

ontainayle.Indeed,xdependsonthevariablesyandz,whereasydepends

onthe variables x and t. Whena singlegate is onsidered,assigning truth valuesto input

variables determines the truth value of the output, dependent, variable. As in Example 3,

assigningtruthvaluesto inputvariablesthat arenotoutputvariablesfor othergatesis not

enoughtodeterminethetruthvalueofallinvolvedvariables.Intheexample,assigningtruth

valuesto z and t is notsuient to determinethe truth valueof x and y.However, in the

example,when we assign atruth valueto an additionalvariable (x, whih isalled ayle

utsetvariable) intheyle,thetruth valueof y isdetermined.Aordingly,weneedto ut

suh aform ofyle in orderto determinateasuient subsetofvariables that determines

thevaluesof all variables. Suh aset is alled astrong bakdoorin [17℄.InExample 3, the

strongbakdoororrespondsto thesetof fxg[fz;tg.In thisontext, astrong bakdooris

theunionofthesetofindependentvariablesandofthevariablesoftheyleutset.Finding

theminimalsetofvariablesthatutsalltheylesinthesetofgatesisanNP-hardproblem.

Thisissueisinvestigatedin thenextsetion.

7 Searhing for dependent variables

Inthefollowing,agraphrepresentationoftheinterationofgatesisonsidered.Moreformally,

Aset ofgatesanberepresentedbyabipartitegraphG=(O[I;E)asfollows:

foreahgateweassoiatetwoverties, therstone o2O representsthe outputof the

gate,andtheseondonei2I representsthesetofitsinputvariables.Sothenumberof

vertexislessthan2#gates,where#gates isthenumberofgates;

Foreah gate, anedge (o;i)betweenthe two vertieso and irepresenting theleft and

therighthandsidesofagateisreated.Additionaledgesarereatedbetweeno2Oand

i2I ifoneoftheliteralsoftheoutputvariableassoiatedtothevertexobelongstothe

setofinputliteralsassoiatedto thevertexi.

Finding a smallest subset V 0

of O s.t. the subgraph G 0

= (V 0

[O;E 0

) is ayli is a

well-knownNP-hardproblem.

Atually, any subset V 0

that makes thegraphayli is therepresentation of theset of

variables,whihtogetherwithalltheindependentones,allowsallvariablestobedetermined.

WhenV 0

is of size ,and theset of dependent variables is of size d, then thesearhspae

isredued from 2 n

to 2 n (d )

, where nis the number ofvariable ourring in theoriginal

CNFformula.

Wethusneedtondatrade-obetweenthesizeofV 0

,whihinuenestheomputational

ostto ndit,andtheexpetedtimegaininthesubsequentSAThekingstep.

In thefollowing,twoheuristis are investigated in order to nda yle-utset V 0

. The

rst-oneisalledMaxdegree.ItonsistsinbuildingV 0

inrementallybyseletingvertieswith

thehighestdegreerst,untiltheremainingsubgraphbeomesayli.

The seond one is alled MaxdegreeCyle. It onsists in building V 0

inrementally by

seletingrstavertexwiththehighestdegreeamongthevertiesthatbelongtoayle.This