Value Withdrawal Explanations: a Theoretical Tool for Programming Environments

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Programming Environments

Willy Lesaint

To cite this version:

Willy Lesaint. Value Withdrawal Explanations: a Theoretical Tool for Programming Environments.

12th Workshop on Logic Programming Environments, 2002, Denmark. pp.17-33. �hal-00459195�

WillyLesaint

Laboratoired'InformatiqueFondamentaled'Orleans

rueLeonarddeVini{BP6759 {F-45067OrleansCedex2{Frane

Willy.Lesaintlifo.univ-orlea ns.f r

Abstrat. Constraintlogiprogrammingombinesdelarativityandef-

ienythanks to onstraint solvers implementedfor spei domains.

ValuewithdrawalexplanationshavebeeneÆientlyusedinseveralon-

straintsprogrammingenvironmentsbuttheredoesnotexistanyformal-

izationofthem.Thispaperisanattempttollthis lak.Furthermore,

wehopethatthistheoretialtoolouldhelptovalidatesomeprogram-

mingenvironments.Avaluewithdrawalexplanationisatreedesribing

thewithdrawalofavalueduringadomainredutionbyloalonsisteny

notionsandlabeling.Domainredutionisformalizedbyasearhtreeus-

ingtwokindsofoperators:forloalonsistenynotionsandforlabeling.

Theseoperators are dened by setsof rules.Proof trees are builtwith

respettotheserules.Foreahremovedvalue,thereexistssuhaproof

treewhihisthewithdrawalexplanationofthisvalue.

1 Introdution

Constraintlogiprogrammingisoneofthemostimportantomputingparadigms

of the last years. It ombines delarativity and eÆieny thanks to onstraint

solvers implemented for spei domains. The needs in programming environ-

ments is growing. But logi programming environments are not always suÆ-

ient to deal with the onstraint side of onstraint logi programming. Value

withdrawal explanations have been eÆiently used in several onstraints pro-

gramming environments but there does not exist any formalization of them.

This paperisanattemptto llthislak.This workis supportedbythefrenh

RNTL 1

projetOADymPPaC 2

whoseaimistoprovideonstraintprogramming

environments.

Avaluewithdrawalexplanationisatreedesribingthewithdrawalofavalue

duringadomainredution.Thisdesriptionisdoneintheframeworkofdomain

redutionofnite domainsbynotionsofloalonsistenyandlabeling. Arst

work[7℄dealtwithexplanationsin theframeworkofdomainredutionbyloal

onsisteny notions only. A value withdrawal explanation ontains the whole

1

ReseauNationaldesTehnologiesLogiielles

2

Outils pour l'Analyse Dynamique et la mise auPoint de Programmes ave Con-

informationaboutaremovalandmaythereforebeausefultoolforprogramming

environments.Indeeditallowsperforming:

{ failureanalysis:afailureexplanationbeingasetofvaluewithdrawalexpla-

nations;

{ onstraintretration:explanationsprovidethevalueswhihhavebeenwith-

drawndiretlyorindiretlybytheonstraintandthenallowtoeasilyrepair

thedomains;

{ debugging:anexplanationbeingakindofdelarativetraeofavaluewith-

drawal,itanbeusedtondanerrorfromasymptom.

Therstandseond itemhavebeenimplementedinthePaLMsystem[8℄. PaLM

is basedon the onstraint solver hoo [9℄ where labeling is replaed by the

use ofexplanations. Note that theonstraintretration algorithmof PaLMhas

beenproved orret thanksto ourdenition of explanations, and moregener-

allya largefamily ofonstraintretration algorithms arealso inluded in this

framework.

Themainmotivationofthisworkisnotonlytoprovideaommonmodelfor

the partners of theOADymPPaC projet but also to use explanationsfor the

debuggingofonstraintsprograms.Nevertheless,theaimofthispaperisnotto

desribetheappliationsofvaluewithdrawalexplanationsbuttoformallydene

thisnotionofexplanation.

ThedenitionofaConstraintSatisfationProblemisgiveninthepreliminary

setion.Inthirdandfourthsetionsatheoretialframeworkfortheomputation

ofsolutionsisdesribedinsetions3and4.Aomputationisviewedasasearh

tree where eah branh is an iteration of operators. Finally, explanations are

presentedinthelastsetionthankstothedenitionofrulesassoiatedtothese

operators.

2 Preliminaries

Following[10℄,aConstraintSatisfation Problem (CSP)is made oftwoparts:

a syntati part and a semanti part. The syntati part is a nite set V of

variables, a nite set C of onstraints and afuntion var: C ! P(V), whih

assoiatesasetofrelatedvariablestoeahonstraint.Indeed,aonstraintmay

involveonlyasubsetof V.

Forthesemantipart,weneedtointroduesomepreliminaryonepts.We

onsidervariousfamiliesf =(f

i )

i2I

.Suhafamilyisreferredtobythefuntion

i7!f

i

orbytheset f(i;f

i

)ji2Ig.

Eahvariableis assoiatedwith aset ofpossiblevalues.Therefore,weon-

siderafamily(D

x )

x2V

whereeahD

x

isanite nonempty set.

Wedene the domain by D = S

x2V

(fxgD

x

). This domain allowssim-

pleand uniformdenitionsof (loalonsisteny)operatorsonapower-set.For

domain redution,weonsider subsetsdof D. Suh asubset isalled an envi-

ronment.Wedenote by dj

W

therestritionof asetdD to asetof variables

W V, that is, dj = f(x;e)2 dj x 2 Wg.Any d D is atually afamily

(d

x )

x2V

with d

x D

x

:forx2V, wedened

x

=fe2D

x

j(x;e)2dgandall

ittheenvironmentof x.

Constraintsare dened bytheirset of allowed tuples.A tuple t onW V

is a partiular environment suh that eah variable of W appears only one:

t Dj

W

and 8x2 W;9e2 D

x

;tj

fxg

=f(x;e)g. Foreah 2 C, T

is aset of

tuplesonvar(), alledthesolutionsof.

Wean nowformallydeneaCSP.

Denition1. A ConstraintSatisfationProblem (CSP)isdenedby:

{ anite setV ofvariables;

{ anite setC of onstraints;

{ afuntion var:C!P(V);

{ thefamily (D

x )

x2V

(thedomains);

{ afamily (T

)

2C

(theonstraintssemantis).

Notethatatuplet2T

isequivalenttothefamily(e

x )

x2var()

andthattis

identiedwithf(x;e

x

)jx2var()g.

A user is interested in partiular tuples (onV) whih assoiatea value to

eahvariable,suh thatalltheonstraintsaresatised.

Denition2. A tuplet onV isa solutionofthe CSP if82C ;tj

var() 2T

.

Example 1. Confereneproblem

Mike, Peter and Alan want to give a talk about their work to eah other

during threehalf-days.PeterknowsAlan'sworkandvie versa.There arefour

talks(andsofourvariables):MiketoPeter(MP),PetertoMike(PM),Miketo

Alan(MA)andAlantoMike(AM). Notethat MikeannotlistentoAlanand

Petersimultaneously(AM6=PM).Mikewantsto knowtheworksofPeterand

Alanbeforetalking(MA>AM,MA>PM,MP>AM,MP>PM).

Thisanbewritten inGNU-Prolog[4℄(withalabelingonPM)by:

onf(AM,MP,PM,MA):-

fd_domain([MP,PM,MA,AM℄,1,3),

MA #> AM,

MA #> PM,

MP #> AM,

MP #> PM,

AM #\= PM,

fd_labeling(PM).

Thevalues1;2;3orrespondstotherst,seondandthirdhalf-days.Notethat

thelabelingonPM issuÆientto obtainthesolutions.Withoutthis labeling,

thesolverprovidesredueddomainsonly(nosolution).

Thisexamplewillbeontinuedthroughoutthepaper.

Theaimofasolveristoprovideone(ormore)solutions.Inordertoobtain

notionsandlabeling.Therstoneisorretwithrespettothesolutions,thatis

itonlyremovesvalueswhihannotbelongtoanysolution,whereastheseond

oneis usedto restritthesearhspae. Notethat to doalabelingamounts to

utting aproblemin severalsub-problems.

Inthenextsetion,wedonotonsiderthewholelabeling(thatisthepassage

from aproblem toaset ofsub-problems)but onlythepassagefrom aproblem

tooneofitssub-problems.Thewholelabelingwillbeonsiderinsetion4with

thewell-knownnotionofsearhtree.

3 Domain redution mehanism

Inpratie, operatorsare assoiatedwith theonstraintsandare applied with

respettoapropagationqueue.Thismethodisinterleavedwithsomerestrition

(dueto labeling).Inthis setion,thisomputationof areduedenvironmentis

formalized thanksto ahaoti iterationof operators. Theredution operators

anbeoftwotypes:operatorsassoiatedwithaonstraintandanotionofloal

onsisteny,andoperatorsassoiatedwith arestrition.Theresultingenviron-

mentisdesribedin termsoflosureensuringonuene.

Domainredutionwithrespettonotionsofonsistenyanbeexpressedin

termsof operators.Suh an operatoromputes aset of onsistent valuesfora

set ofvariables W

out

aordingtothe environmentsofanother setof variables

W

in .

Denition3. Aloalonsistenyoperatoroftype(W

in

;W

out

),withW

in

;W

out

V isamonotoni funtion f :P(D)!P(D) suhthat:8dD,

{ f(d)j

VnW

out

=Dj

VnW

out ,

{ f(d)=f(dj

Win )

Notethat therst itemensures that theoperator isonlyonerned bythe

variables W

out

. The seond one ensures that this result only depends on the

variableW

in .

TheseoperatorsareassoiatedwithonstraintsoftheCSP.Soeahoperator

mustnotremovesolutionsofitsassoiatedonstraint(andofourseoftheCSP).

Thesenotionsoforretionaredetailedin[6℄.

Example 2. In GNU-Prolog, two loal onsisteny operators are assoiated

with the onstraint MA #> PM: the operator whih redues the domain of MA

withrespettoPMand theonewhihredues thedomainofPMwith respet

to MA.

Fromnowon,wedenotebyLasetofloalonsistenyoperators(thesetof

loalonsistenyoperatorsassoiatedwiththeonstraintsoftheCSP).

Domain redution by notions of onsisteny alone is not always suÆient.

Theresultingenvironmentisanapproximationof thesolutions(thatis allthe

(forexample,bythehoie ofavalueforavariable).Ofourse,suharestri-

tion(formalizedbytheappliationof arestritionoperator) doesnothavethe

propertiesoforretnessofaloalonsistenyoperator:theappliationofsuh

anoperatormayremovesolutions.But,inthenextsetion,theseoperatorswill

beonsidered asaset (orresponding to thewhole labeling onavariable). In-

tuitively, if weonsider alabelingsearh tree, this setiondealswith onlyone

branh ofthistree.

Inthe same way loal onsisteny operators have beendened, restrition

operatorsarenowintrodued.

Denition4. A restrition operator on x 2 V is a onstant funtion f :

P(D)!P(D) suhthat:8dD;f(d)j

Vnfxg

=Dj

Vnfxg .

Example 3. The funtion f suhthat 8d 2D;f(d) =Dj

VnfPMg

[f(PM;1)gis

arestritionoperator.

FromnowonwedenotebyRasetofrestritionoperators.

Thesetwokindofoperatorsaresuessivelyappliedtotheenvironment.The

environmentis replaedbyitsintersetion withtheresultoftheappliation of

theoperator.WedenotebyF theset ofoperatorsL[R .

Denition5. The redutionoperatorassoiatedwiththeoperator f 2F isthe

monotoni andontratingfuntion d7!d\f(d).

A ommon x-point of theredution operators assoiated with F starting

from an environmentd is an environmentd 0

dsuh that 8f 2 F;d 0

= d 0

\

f(d 0

),thatis8f 2F;d 0

f(d 0

). Thegreatestommonx-pointisthisgreatest

environmentd.Tobemorepreise:

Denition6. The downwardlosureof dby F ismaxfd 0

D jd 0

d^8f 2

F;d 0

f(d 0

)g andisdenotedbyCL#(d;F).

NotethatCL#(d;;)=dandCL#(d;F)CL#(d;F 0

)ifF 0

F.

Inpratie, the order of appliation of these operators is determined by a

propagation queue. It is implemented to ensuresto never forget any operator

and to alwaysreah thelosure CL #(d;F). From atheoretialpoint ofview,

thislosure analsobeomputedbyhaoti iterations introduedforthis aim

in [5℄.Thefollowingdenition istakenfrom Apt[2℄.

Denition7. A run is an innite sequene of operators of F, that is, a run

assoiates with eah i 2 IN (i 1) an element of F denoted by f i

. A run is

fair if eah f 2F appears init innitely often, that is, 8f 2F;fijf =f i

gis

innite.

The iteration of the set of operators F from the environment d D with

respettoaninnitesequeneofoperatorsofF:f 1

;f 2

;:::istheinnitesequene

d 0

;d 1

;d 2

;:::indutively denedby:

0

2. foreahi2IN,d i+1

=d i

\f i+1

(d i

).

Its limitis\

i2IN d

i

.

Ahaotiiteration isaniteration with respet toasequene ofoperators of

F (with respettoF,inshort)where eahf 2F appearsinnitely often.

Note that an iteration may start from a domain d whih an be dierent

from D. Thisis moregeneralandonvenientforalotof appliations(dynami

aspetsofonstraintprogramming,forexample).

Thenextwell-knownresultofonuene[3,5℄ensuresthatanyhaotiiter-

ation reahes thelosure. Notethat, sineis awell-foundedordering(i.e. D

is anite set), everyiterationfrom d D is stationary, that is, 9i 2 IN ;8j

i;d j

=d i

.

Lemma1. Thelimitd F

ofeveryhaoti iteration ofasetof operatorsF from

dD isthe downwardlosureof dby F.

Proof. Let d 0

;d 1

;d 2

;::: be a haoti iteration of F from d with respet to

f 1

;f 2

;:::

[CL#(d;F)d F

℄Foreahi,CL#(d;F)d i

,byindution:CL#(d;F)

d 0

=d.Assume CL#(d;F)d i

,CL#(d;F)f i+1

(CL#(d;F))f i+1

(d i

)

bymonotoniity.Thus,CL#(d;F)d i

\f i+1

(d i

)=d i+1

.

[d F

CL #(d;F)℄ There exists k 2 IN suh that d F

= d k

beause isa

well-foundedordering.Theiterationishaoti, hened k

isaommonx-point

of theset of operators assoiated withF, thusd k

CL #(d;F)(the greatest

ommonx-point).

Inorder toobtain alosure,it isnot neessaryto haveahaotiiteration.

Indeed,sinerestritionoperatorsareonstantfuntions,theyanbeapplyonly

one.

Lemma2. d L[R

=CL#(CL#(d;R );L)

Proof. d L[R

=CL#(d;L[R )bylemma1andCL#(d;L[R )=CL#(CL#

(d;R );L)beauseoperatorsofRareonstantfuntions.

Assaidabove,wehaveonsidered in thissetionaomputation inasingle

branh of a labeling searh tree. This formalization is extended in the next

setioninorder totakethewholesearhtreeintoaount.

4 Searh tree

A labelingonavariable anbe viewedasthepassagefromaproblem to aset

ofproblems.Theprevioussetionhastreatedthepassagefromthisproblem to

oneofitssub-problemsthankstoarestritionoperator.Inordertoonsiderthe

whole set ofpossiblevaluesfor thelabelingonavariable, restritionoperators

on asame variable must be grouped together. The unionof the environments

of the variable (the variable onerned by the labeling) of eah sub-problem

obtainedbytheappliationofeahoftheseoperatorsmustbeapartitionofthe

Denition8. A setfd

i

j1ingisa partitionofdonx if:

{ 8i;1in; dj

Vnfxg d

i j

Vnfxg ,

{ dj

fxg [

1in d

i j

fxg ,

{ 8i;j;1in;1jn;i6=j;d

i j

fxg

\d

j j

fxg

=;.

Inpratie,environmentredutionsbyloalonsistenyoperatorsandlabel-

ingareinterleavedtobethemosteÆient.

A labeling onx 2 V an bea omplete enumeration (eah environment of

the partition isredued to asingleton)ora splitting.Note that the partitions

alwaysverify:8i;1in;d

i j

fxg 6=;.

Example 4. fDj

VnfPMg

[f(PM;1)g;Dj

VnfPMg

[f(PM;2)g;Dj

VnfPMg

[f(PM;3)g

isapartitionofD.

Next lemma ensures that no solution is lost during a labeling step (eah

solutionwillremaininexatlyonebranh ofthesearhtreedenedlater).

Lemma3. If tdisasolutionof the CSPandfd

i

j1ingisapartition

of dthent[

1in

CL#(d

i

;L).

Proof. straightforward.

Eahnodeofasearhtreeanbeharaterizedbyaquadrupleontainingthe

environmentd(whihhavebeenomputeduptonow),thedepthpinthetree,

the operator f (loal onsisteny operator or restrition operator) onneting

itwith itsfatherandtherestritedenvironmente.Therestritedenvironment

isobtainedfromtheinitial environmentwhenonlytherestritedoperatorsare

applied.

Denition9. A searh node is a quadruple (d;e;f;p) with d;e 2 P(D) , f 2

F[f?gandp2IN.

Thedepth andthe restritedenvironmentallowto loalizethenodein the

searhtree.

Thereexiststwokindsoftransition inasearhtree:thoseaused byaloal

onsisteny operatorwhih ensurethe passageto oneonly sonand thetransi-

tionsaused by alabelingwhih ensurethepassage to somesons(as many as

environmentsinthepartition).

Denition10. A searh tree is a tree for whih eah node is a searh step

indutivelydenedby:

{ (D;D;?;0) isthe rootof the tree,

{ if(d;e;op;p) isanon leavenode thenithas:

oneson:(d\f(d);e;f;p+1)with f 2L;

nsons:(d\f

i

(d);e\f

i (d);f

i

;p+1)with ff

i

(d)j1ingapartition

of dandf

i 2R .

Denition11. A searh tree is said omplete if eah leaf (d;e;f;p) is suh

that:d=CL#(e;L).

This setion hasformally desribedthe omputation of solversin terms of