Well-Terminating, Input-Driven Logi Programs
Jan{GeorgSmaus
UniversityofKentatCanterbury,Canterbury,CT27NF,UnitedKingdom,
telephonexx44/1227/827553, fax762811,j.g.smausuk .a .uk
Abstra t. Weidentifya lassofpredi atesforwhi hterminationdoes
notdependonleft-to-rightexe ution.Allthatisrequiredisthat
deriva-tionsare input-driven, that is, inea h derivationstep, the input
argu-ments of the sele ted atom do not be ome instantiated. The method
ofshowingthat apredi ateis inthat lassisbasedonlevelmappings,
loselyfollowingthetraditionalapproa hforLDderivations.Many
pred-i atesterminateundersu hweakassumptions.Knowingthesepredi ates
anbe averyusefulpart of amore omprehensive methodofshowing
termination,whi hwouldhavetomakemorespe i assumptionsabout
thesele tionrule.
Keywords:Logi programming,termination,modes,sele tion rules.
1 Introdu tion
Terminationoflogi programshasbeenwidelystudiedforLDderivations, that
is derivationswhere theleftmost atomin aqueryis alwayssele ted[1,3,8{11,
13℄.Alloftheseworksarebasedonthefollowingidea:atthetimewhenanatom
ainaqueryissele ted,itispossibletopindown thesize 1
ofa.Thissize annot
hangeviafurtherinstantiation.Itisthenshownthatfortheatomsintrodu ed
inthisderivationstep,itisagainpossibletopindowntheirsizewheneventually
theyaresele ted,andthattheseatomsaresmallerthana.
Thisideahasalsobeenappliedtoarbitraryderivations[6℄.Sin eno
restri -tionisimposedonwhenanatom anbesele ted,itisrequiredthatinea hquery
inaderivation,thesizeofea hatomisalwaysbounded.Programsthatful llthis
requirementare alled strongly terminating. The lass of stronglyterminating
programsisverylimited.
Formostprograms,itisne essaryforterminationtorequirea ertaindegree
ofinstantiationofanatombeforeit anbesele ted.This anbea hievedusing
delay de larations [2,16{21℄.The problem is that, depending on what kind of
delayde larationsandsele tionruleareused, itisnotpossibletopindownthe
size ofthesele tedatom, sin ethissize maydepend ontheresolutionof other
atoms in the query that are not yet resolved. Nevertheless, [17,18℄ and to a
limitedextent[16℄arebasedontheideades ribedabove,whereas[19{21℄avoid
anyexpli itmentionof\size"andinsteadtrytoredu etheproblemtoshowing
termination forLDderivations.
1
Thete hni almeaningof\pinningdownthesize" di ersamongdi erentmethods.
CORE
Metadata, citation and similar papers at core.ac.uk
thesele tionruleontheonehand andmakingveryspe i assumptionsonthe
other.Webelievethatareasonableminimalrequirementfortermination anbe
formulatedintermsofmodes:
Inea hderivationstep,theinputargumentsofthesele tedatom annot
be omeinstantiated.
Inotherwords,anatominaquery anonlybesele tedwhenitissuÆ iently
in-stantiatedsothatthemostgeneraluni erwiththe lauseheaddoesnotbindthe
inputargumentsoftheatom. We allderivations whi hmeetthis requirement
input-driven.
Thispaperisaboutidentifyingpredi atesforwhi hallinput-driven
deriva-tionsare nite.Otherworksinthisareahaveusuallymadespe i assumptions
about thesele tion ruleand thedelayde larations,for examplelo al sele tion
rules[17℄,delayde larationsthattestargumentsforgroundnessorrigidness[16,
18℄,orthedefaultleft-to-rightsele tionruleofmostPrologimplementations[19{
21℄.In ontrast,weshowhowpreviousresultsaboutLDderivations anbe
gen-eralised,theonlyassumptionaboutthesele tionrulebeingthatderivationsare
input-driven.We loselyfollow[13℄.
Weexploit that under ertain onditions, itis enough to relyon arelative
de reaseinthesizeofthesele tedatom,eventhoughthissize annotbepinned
down.
Example 1.1. Considertheusualappendprogramandthefollowinginput-driven
derivation,where thesele tedatomisunderlined:
append([1℄;[℄;As);append(As;[℄;Bs)!
append([℄;[℄;As 0 );append([1jAs 0 ℄;[℄;Bs)! append([℄;[℄;As 0 ) ;append(As 0 ;[℄;Bs 0 )! append([℄;[℄;Bs 0 )!2: When append([1jAs 0
℄;[℄;Bs) is sele ted, it is not possible to pin down its size
in any meaningful way. In fa t, nothing an be said about the length of the
derivation asso iated with append([1jAs 0
℄;[℄;Bs) without knowing about other
atoms whi h might instantiate As 0
. However, the derivation ould be in nite
onlyifthederivationasso iatedwithappend([℄;[℄;As 0
)wasin nite.Ourmethod
isbasedonsu hadependen ybetweentheatomsofaquery.
Notsurprisingly,the lassofprogramsforwhi hallinput-drivenderivationsare
nite isquite limited,although itis obviouslylargerthan the lass ofstrongly
terminatingprograms.Realisti ally,a omprehensivemethodforproving
termi-nationwouldhavetomakestrongerassumptions.However,withintheframework
ofsu hamethod,it anbeusefultoknowforwhi hpredi atestermination an
alreadybeensuredonlyassuminginput-drivenderivations.Thisisdemonstrated
in[21℄,butapartfromthat,webelievethatithasnotbeenre ognisedpreviously.
Example 1.2. Consider the following program whi h permutes a list. Assume
permute(Y, [U | X℄) :-delete(Y, U, Z), permute(Z, X). delete([X|Z℄, X, Z). delete([U|Y℄, X, [U|Z℄) :-delete(Y, X, Z).
Thenwehavethefollowingin niteinput-drivenderivation:
permute([1℄;W)! delete([1℄;U 0 ;Z 0 );permute(Z 0 ;X 0 )! delete([℄;U 0 ;Z 00 );permute([1jZ 00 ℄;X 0 )! delete([℄;U 0 ;Z 00 );delete([1jZ 00 ℄;U 00 ;Z 000 );permute(Z 000 ;X 00 )! delete([℄;U 0 ;Z 00 );delete(Z 00 ;U 00 ;Z 0000 );permute([1jZ 0000 ℄;X 00 )!:::
Therestofthispaperisorganisedasfollows.Thenextse tion xesthenotation.
Se tion 3introdu eswellandni elymodedprogramsand Se tion4showsthat
for these, it is suÆ ient to prove termination for one-atom queries. Se tion 5
then deals with how one-atom queries an be proven to terminate. Se tion 6
dis ussestheresultsandtherelatedwork.
2 Preliminaries
Ournotation follows [1,13℄. Forthe examplesweuse Prologsyntax. Were all
someimportantnotions.Thesetof variablesin asynta ti obje toisdenoted
asvars (o).Thedomainofasubstitutionisdenotedasdom().Therestri tion
ofasubstitutiontothevariableso urringinasynta ti alobje toisdenoted
asjo.A synta ti obje tislinearifeveryvariableo ursinitat moston e.
Forapredi atep=n,amodeisanatomp(m
1 ;:::;m n ),where m i 2fI;Og
for i 2f1;:::;ng. Positions with I are alled input positions,and positions
with O are alled output positions of p. We assume that a xed mode is
asso iatedwithea hpredi atein aprogram.Tosimplifythenotation,anatom
writtenasp(s;t)means:sistheve torofterms llingtheinputpositions,andt
istheve torofterms llingtheoutputpositions.Anatomp(s;t)isinput-linear
ifsislinear.
Aqueryisa nitesequen eofatoms.AderivationstepforaprogramP is
apairhQ;i;hR ;i, whereQ=Q
1 ;p(s;t);Q 2 andR=Q 1 ;B;Q 2 arequeries;
is asubstitution; p(v ;u) B a renamedvariantof a lausein P and the
mostgeneraluni erofp(s;t) andp(v ;u). We allp(s;t) thesele ted atom
andR theresolventofQandh B.Aderivationstepisinput-drivenif
dom()\vars(s)=;.
Aderivation foraprogram P is asequen e hQ
0 ; 0 i;hQ 1 ; 1 i;:::, where
ea hsu essivepairhQ
i ; i i;hQ i+1 ; i+1
iin isaderivationstep.Alternatively,
wealsosaythat isaderivationofR[fQ
0
0
0 0 1 1
alwaystheleftmostatominaquery.Aninput-drivenderivationisaderivation
onsistingofinput-drivenderivationsteps.
If Q;a;R ;(Q;B;R ) is a step in a derivation, then ea h atom in B is a
dire t des endantofa,and bisadire tdes endantof bforallb2Q;R .
Wesaybis ades endant ofa if(b;a)isin there exive,transitive losure of
therelationisadire tdes endant.Thedes endantsofasetofatomsarede ned
intheobviousway.If,foraderivation:::Q;:::;Q 0
;Q 00
:::,thesele tedatomin
Q 0
;Q 00
isades endantofanatomain Q,thenQ 0
;Q 00
isana-step.
3 Modes
In this se tion weintrodu e thenotions ofwell moded and ni ely moded
pro-grams.Well-modednesshasbeenusedbeforetoshowterminationofLD
deriva-tions[13℄.Inthe ontextofarbitraryinput-drivenderivations,itisalso ru ial
torequirethatprogramsareni elymoded.
Well-modednesshasbeenintrodu edin [12℄andwidelyusedforveri ation
sin e.InMer uryit isevenmandatorythatprogramsarewellmoded,whi h is
oneof thereasonsforitsremarkableperforman e[22℄.
De nition3.1 (well moded).A queryQ=p
1 (s 1 ;t 1 );:::;p n (s n ;t n )is well
modedifforalli2f1;:::;ng
vars(s i ) [ j<i vars (t j ) (1) The lausep(t 0 ;s n+1
) Qiswellmodedif(1)holdsforalli2f1;:::;n+1g.
Aprogramiswellmodedifallofits lausesarewellmoded.
Another ommon on eptforveri ationisthefollowing.
De nition3.2 (ni ely moded). A query Q = p
1 (s 1 ;t 1 );:::;p n (s n ;t n ) is ni ely modedift 1 ;:::;t n
isalinearve toroftermsandforalli2f1;:::;ng
vars (s i )\ [ ji vars (t j )=;: (2) The lauseC=p(t 0 ;s n+1
) Qisni elymodedifQisni elymodedand
vars (t 0 )\ n [ j=1 vars (t j )=;: (3)
AprogramP isni elymodedifallofits lausesareni elymoded.
Note that other authors have denoted the lause head of C as p(s
0 ;t n+1 ) or p(s 0 ;t 0
),whi hallowsforamore ompa tde nition[13℄ 2
,furthermore
suggest-ingthat thereisananalogybetween(2)and(3)above[2℄.Wehave hosennot
2
We refer to the de nitionof simply moded here, but this is very similar to ni ely
ondly,theanalogyismisleading.Itwouldbemoreappropriatetoseeananalogy
between(3)andtherequirementthatt
1 ;:::;t
n
isalinearve toroftermsthan
between(3)and(2).
Example 3.1. TheprograminEx.1.2iswellandni elymoded.Itisneitherwell
moded nor ni elymoded in reverse mode, howeverit an easily be made well
andni elymoded byinter hangingthetwobodyatomsinthese ond lause.
Theexampleshowsthatmultiplemodesofapredi ate anbeobtainedbyhaving
multiple(renamed)versionsofapredi ate.Thisiswhyitisoftenassumedthat
ea h predi ate has a xed mode [2,13,19,22℄. However, this usually implies
a tual odedupli ationand isthereforeareallossofgenerality.
In this paper, assuming a xed mode for ea h predi ate is not a loss of
generality, but merely anotational onvenien e. We onsider arbitrary
input-drivenderivations.Thetextualpositionofanatomwithin aqueryisirrelevant
foritssele tion.Anyresultthatholdsforawellmodedprogramalsoholdsfora
programwheretheatomsinea h lausebodyarepermutedinanarbitraryway.
Inthissense,we anassumethattheprogramofEx.3.1iswellmodedandni ely
modedin bothmodes.Wheneverone onsidersspe i sele tionruleswherethe
textualpositionisrelevant,onehastotreatmultiplemodesexpli itly[21℄.
The following lemma states a persisten e property of well-modedness [2,
Lemma16℄. IthasbeenshownpreviouslyforLDderivations[5℄.
Lemma3.1. EveryresolventofawellmodedqueryQandawellmoded lause
C,wherevars (C)\vars (Q)=;,iswellmoded.
Forni ely-modedness,thereisasimilarpersisten eproperty.Ithasbeenshown
previouslyforLDresolvents[5℄andarbitraryresolvents[2℄.Howeverinthe
lat-ter ase, it was requiredthat the lause head is input-linear. Forinput-driven
derivations, this is notne essary. Itis assumed that thesele ted atom is
suÆ- ientlyinstantiated, so that a multiple o urren e of the samevariable in the
inputargumentsofthe lause head annot auseanybindingsto thequery.
Lemma3.2. Everyresolventof a ni elymoded queryQ and a ni elymoded
lauseC,wherethederivationstepisinput-drivenandvars (C)\vars(Q)=;,
isni elymoded.
Proof. Let C = h B. We want to use [2, Lemma 11℄. Therefore we must
showthat we an assumewithout lossof generality that his input-linear.Let
C 0
= h 0
Eq;B be the lause obtained from C by repeatedly applying the
followingtransformation:Iftheinputargumentsofthe lausehead ontaintwo
o urren esof avariablex,repla e oneo urren e withafresh variabley and
addtheequationx=yatthebeginningofthe lause body.
Thenh 0
isinput-linear,andC 0
isni elymoded,wherethepredi ate=inEq
isusedin mode(I;I).Furthermore,ifQ=Q
1 ;a;Q 2 and hQ;;i;hQ 1 ;B;Q 2 ; i
tion hQ;;i;hQ 1 ;Eq;B;Q 2 ; 0 i;:::;hQ 1 ;B;Q 2 ; i usingC 0
andthe lause\z=z:"(whi his on eptuallythede nition of=).
By [2, Lemma 11℄, (Q 1 ;Eq;B;Q 2 ) 0
is ni ely moded. Furthermore, sin e
hQ 1 ;Eq;B;Q 2 ; 0 i;:::;hQ 1 ;B;Q 2
; i is input-driven, it follows that Eq 0
is a
sequen eofatomsoftheforms=s.Therefore 0 =and(Q 1 ;B;Q 2 )isni ely moded. ut
Forani elymodedprogramandquery,itisguaranteedthateveryinput-driven
derivationsteponlyinstantiatesotheratomsinthequerythato urtotheright
ofthesele tedatom.
Lemma3.3. Let P be a ni ely moded program, Q = Q
1
;p(s;t);Q
2
a ni ely
moded query, and hQ;;i;hQ
1 ;B;Q
2
; i an input-driven derivation step. Then
dom()\vars(Q
1 )=;.
Proof. Sin e the derivation step is input-driven, it follows that dom()jQ
vars (t).Thussin eQisni elymoded,dom ()\vars (Q
1
)=;. ut
4 Controlled Coroutining
Inthisse tionwede newell-terminatingpredi ates,thatispredi atesforwhi h
all one-atomquerieshave nite derivations. Asin [13℄,wethen show that
ter-minationforone-atomqueriesimpliesterminationforarbitraryqueries.
ForLDderivations, this isalmost obvious andonlyrequires that programs
and queriesare well moded [13,Lemma 4.2℄.Given aderivation for aquery
a
1 ;:::;a
n
,thesub-derivationsforea ha
i
donotinterleave,andtherefore an
beregardedasaderivationfora
1
followedbyaderivationfora
2
andsoforth.The
followingexampleillustrates thatin the ontext ofinterleavingsub-derivations
( oroutining),thisismu hlessobvious.
Example 4.1. Considertheusualappend program
append([℄,Y,Y).
append([X|Xs℄,Ys,[X|Zs℄)
:-append(Xs,Ys,Zs).
in modeappend(I;I;O)andthequery
append([℄;[℄;As);append([1jAs℄;[℄;Bs);append(Bs;[℄;As):
This query is well moded but not ni ely moded. Then we have the following
in niteinput-drivenderivation:
append([℄;[℄;As);append([1jAs℄;[℄;Bs);append(Bs;[℄;As)!
append([℄;[℄;As);append(As;[℄;Bs 0 );append([1jBs 0 ℄;[℄;As)! append([℄;[℄;[1jAs 0 ℄);append([1jAs 0 ℄;[℄;Bs 0 );append(Bs 0 ;[℄;As 0 )!:::
This well-known termination problem of programs with oroutining has been
De nition4.1 (well-terminating predi ate/atom). Let P be a well and
ni elymodedprogram.Apredi atepinP iswell-terminatingifforea hwell
andni elymodedqueryp(s;t),allinput-drivenderivationsofP[fp(s;t)gare
nite.An atomiswell-terminatingifitspredi ateiswell-terminating.
Thefollowinglemma saysthat awell-terminating atom annotpro eed
inde -nitelyunlessitisrepeatedlyfed bysomeotheratom.
Lemma4.1. Let P be a well and ni ely moded program and F;b;H a well
and ni ely moded query where b is a well-terminating atom. An input-driven
derivationofP[fF;b;Hg anhavein nitelymanyb-stepsonlyifithasin nitely
manya-steps,forsomea2F.
Proof. Inthis proof,we allana-stepforsomea2F anF-step, andlikewise
forH.ByLemma3.3,anyH-stepdoesnotinstantiateanydes endantofb.Thus
theH-steps an bedisregarded,andwithoutlossofgenerality,weassumeH is
empty.Let =hF;b; ;i;:::;hQ 0 ; 0 i;hQ 1 ; 1 i:::
beaninput-drivenderivationsu hthathQ
0 ; 0 i;hQ 1 ; 1 i::: ontainsnoF-steps
(thatis, ontainsonly nitely manyF-steps).Sin ebyLemma3.3, nob-step
aninstantiateanydes endantofF,thereexistsaninput-drivenderivation
2 =hF;b; ;i;:::;hR ;i;:::;hQ 0 ; 0 i;hQ 1 ; 1 i:::
su hthathF;b; ;i;:::;hR ;i ontainsonlyF-stepsandhR ;i;:::;hQ
0 ;
0 i
on-tainsonly b-steps (thatis, the F-stepsare moved forwardusing theSwit hing
Lemma[15℄).Sin eR=R 0
;b forsomeR 0
,thereexistsaninput-drivenderivation
3 =hb;i;:::;hI 0 ; 0 i;hI 1 ; 1 i:::
obtainedfromhR ;i;:::;hQ
0 ; 0 i;hQ 1 ; 1
i:::byremovingthepre xR 0 inea h query. Lett 1 ;:::;t m
betheve torofoutputargumentsofR 0 andasubstitution su hthat(t 1 ;:::;t m
) isground.ThenbyDef.3.1,bisawellmodedquery.
ByLemma 3.3, nob-step instantiates t
1 ;:::;t m .Therefore from 3 we an
onstru taninput-drivenderivation
4 =hb;i;:::;hI 0 ; 0 i;hI 1 ; 1 i:::
Sin ebisawellandni elymodedqueryandbiswell-terminating,
4 is nite. Therefore 3 , 2
, and nally are nite. ut
The following lemma is a onsequen eand states that well-terminating atoms
ontheirown annotprodu eanin nitederivation.
Lemma4.2. LetPbeawellandni elymodedprogramandQawellandni ely
modedquery.Aninput-drivenderivationofP[fQg anbein niteonlyifthere
derivation forQ.Weshowthat anhavein nitelymanyb-stepsonlyif has
in nitelymanystepswhereanatomisresolvedthatisnotwell-terminating.
Theproofisbyindu tiononthelengthofF.IfF isempty,theresultfollows
from Lemma4.1.Nowsuppose F ontainsatleast oneatom.ByLemma4.1,
anhave in nitelymany b-stepsonly if forsome a2F, has in nitelymany
a-steps.Ifaisnotwell-terminating,theresultfollowsimmediately.Ifa is
well-terminating,letF =F 1 ;a;F 2 .Sin eF 1
ontainsfeweratomsthanF,theresult
followsfrom theindu tivehypothesis. ut
Lemma4.2providesuswiththeformaljusti ationforrestri tingourattention
to one-atomqueries.However,itrequires thatprogramsand queriesareni ely
moded. Thisisnotne essaryforLDderivations[13℄.
We now de ne well-terminating programs. The de nition di ers from the
orrespondingonein[13℄in thatthey onsideronlyLDderivations.
De nition4.2 (well-terminating program). Let P be a well and ni ely
moded programandQawellandni elymodedquery. P is well-terminating
ifallinput-drivenderivationsofP[fQgare nite.
Thefollowingisanobvious orollaryofLemma4.2.
Corollary 4.3. AprogramP iswell{terminatingifandonlyifallitspredi ates
arewell-terminating.
5 Showing Weak Termination
All of thementionedapproa hesto termination [1,3,8{11,13℄moreorless
ex-pli itlyrelyonmeasuringthesizeoftheinputin aquery.Weagree withEtalle
etal.[13℄that itisreasonabletomakethisdependen yexpli it.Thisgivesrise
tothe on eptofmodedlevelmapping[13℄,whi hisaninstan eoflevelmapping
rst introdu ed in [6,7℄. B
P
denotes the set of groundatoms using predi ates
o urringin P.
De nition5.1 (modedlevelmapping).LetPbeaprogram.j:jisamoded
level mappingif
1. itis alevelmapping,thatisafun tion j:j:B
P !IN,
2. foranytandu,jp(s;t)j=jp(s;u)j.
Fora2B
P
,jaj isthelevelofa.
Thusthelevelofanatomonlydependsonthetermsin theinputpositions.
Thefollowing on ept,adoptedfrom[1℄,isusefulforprovingterminationfor
awholeprogramin rementally,byprovingitforonepredi ateatatime.
De nition5.2 (dependson).Letp;qbepredi atesin aprogramP.Wesay
that prefersto qifthereisa lauseinP withpinitsheadandqinitsbody,
andpdependsonq(writtenpwq)if(p;q)isinthere exive,transitive losure
q(u;v )standsforpwq,andlikewisefor=and.Furthermore,wedenotethe
equivalen e lassofapredi atepwithrespe ttoas[p℄
.
Thefollowingde nitionprovidesuswitha riteriontoprovethatapredi ate
iswell-terminating.
De nition5.3 (well-a eptable). LetP beaprogramandj:jamodedlevel
mapping. A lause C = h B is well-a eptable (with respe t to j:j) if
foreverysubstitutionsu hthatC isground,andforeveryain B su h that
ah,wehavejhj>jaj.
A set of lauses is well-a eptable with respe t to j:j if ea h lause is
well-a eptablewithrespe ttoj:j.
Let us ompare this on ept to some similar on epts in the literature:
re ur-rent[6℄,well-a eptable [13℄anda eptable [4,11℄programs.
Like[11,13℄ andunlike[4,6℄,werequirejhj>jajonly foratomsa where
ah.Thisis onsistentwiththeideathatterminationshouldbeproven
in re-mentally:toshowterminationforapredi atep,itisassumedthatallpredi ates
q with p=qhavealreadybeenshown to terminate. Therefore we anrestri t
ourattentiontothepredi atesqwhereqp.
Like[6℄andunlike[4,11,13℄,ourde nitiondoesnotinvolvemodelsor
om-putedanswersubstitutions.Traditionally,thede nition ofa eptableprograms
is based on a model M of the program, and for a lause h a
1 ;:::;a n , jhj > ja i j is only required if M j= (a 1 ;:::;a i 1
). The reason is that for
LD derivations, a
1 ;:::;a
i 1
must be ompletely resolved before a
i
is sele ted.
Bythe orre tnessofLDresolution[15℄andwell-modedness[5℄,thea umulated
answer substitution , just before a
i
is sele ted, is su h that (a
1 ;:::;a i 1 ) is groundandMj=(a 1 ;:::;a i 1 ).
Su h onsiderations ountforlittlewhenderivations aremerelyrequiredto
beinput-driven.ThisisillustratedinEx.1.2.Inthethirdlineofthederivation,
permute([1jZ 00
℄;X 0
)issele ted,although thereisnoinstan eofdelete([℄;U 0
;Z 00
)
in the model of theprogram. This problem has been des ribedby saying that
delete makes a spe ulative output binding [19℄. Programs that do not make
spe ulativeoutputbindingsare onsideredin[20℄.
Theorem 5.1. Let P be a well and ni elymoded program and pbe a
pred-i ate in P. Suppose all predi ates q with p = q are well-terminating,and all
lausesde ningpredi atesq2[p℄
arewell-a eptable.Thenp,andhen eevery
predi atein[p℄
,iswell-terminating.
Proof. Supposethesetof lausesde ningthepredi atesq2[p℄
is
well-a ept-ablewithrespe ttothemodedlevelmappingj:j.Foranatomausingapredi ate
in[p℄
,wede nejjajj=sup(fjajjaisgroundg),ifthesetfjajjaisgroundg
isbounded.Otherwisejjajjis unde ned.Observethat
ea h atom whose predi ateis in [p℄
. The multisetis formalised asafun tion
Size,whi htakesasargumentsaqueryandanaturalnumber.
Size(Q)(n)=#fq(u;v )jq(u;v )is anatomin Q;qpandjjq(u;v )jj=ng
Note that ifaquery ontainsseveral identi alatoms, ea h o urren emust be
ounted.Wede neSize(Q)<Size(R )ifandonlyifthereisanumberlsu hthat
Size(Q)(l)<Size(R )(l)andSize(Q)(l 0 )=Size(R )(l 0 )foralll 0 >l.Intuitively,
ade rease with respe t to < is obtainedwhen anatom in a queryis repla ed
with a nite numberof smaller atoms. It is easy to see (Konig's Lemma [14℄)
that alldes ending hainswithrespe t to<are nite.
Let Q
0
= p(s;t) bea welland ni ely moded query. Then s is ground and
thus jjQ 0 jj is de ned. Let = Q 0 ;Q 1 ;Q 2 ::: be an input-driven derivation of P[fQ 0 g.
Sin eallpredi atesqwithp=qarewell-terminating,itfollowsbyLemma4.2
that there annot be an in nite suÆx of without any steps where an atom
q(u;v )su hthatqpisresolved.Weshowthatforalli0,ifthesele tedatom
in Q
i ;Q
i+1
is q(u;v ) and q p, then Size(Q
i+1 ) < Size(Q i ), and otherwise Size(Q i+1 )Size(Q i
). This impliesthat is nite, and, as the hoi e of the
initialqueryQ
0
=p(s;t) wasarbitrary,piswell-terminating.
Consideri0andletC=q(v
0 ;u m+1 ) q 1 (u 1 ;v 1 );:::;q m (u m ;v m )bethe
lause,q(u;v )thesele tedatomandthemostgeneraluni erusedinQ
i ;Q i+1 . Ifp=q,then p=q j
forallj 2f1;:::;mgandhen eby() itfollowsthat
Size(Q
i+1
)Size(Q
i ).
Now onsider qp.Sin e C is awell-a eptable lause, jjq(v
0 ;u m+1 )jj> jjq j (u j ;v j
)jjforallj withq
j
p.Thistogetherwith()impliesSize(Q
i+1 )<
Size(Q
i
). ut
Example 5.1. Wenow give afew examples of well-terminating predi ates. We
denote the term size of aterm t,that is the numberof fun tion and onstant
symbolsthat o urint,asTSize(t).
The lauses de ning append(I;I;O) (Ex. 4.1) are well-a eptable, where
jappend(s 1 ;s 2 ;t)j = TSize(s 1
). Thus append(I;I;O) is well-terminating.The
sameholdsforappend(O;O;I),de ningjappend(t
1 ;t
2
;s)j=TSize(s).
The lauses de ning delete(I;O;O) (Ex. 1.2) are well-a eptable, where
jdelete(s;t
1 ;t
2
)j = TSize(s). Thus delete(I;O;O) is well-terminating. The
sameholdsfordelete(O;I;I),de ningjdelete(t;s
1 ;s 2 )j=TSize(s 2 ).
Inasimilarway,we anshowthat permute(O;I)iswell-terminating.
How-ever,permute(I;O)isnotwell-terminating.
Figure 1shows afragmentfrom aprogram for then-queens problem. The
mode is fnqueens(I;O); sequen e(I;O); safe(I); permute(O;I); <(I;I) ;
is(O;I); safeaux(I;I;I); nodiag(I;I;I); =\=(I;I)g. Again using as level
mappingthetermsizeofoneofthearguments,one anseethatthe lauses
de n-ingfnodiag;safeaux;safegarewell-a eptableandthusthesepredi atesare
well-terminating.Thisinformationisusefulsin ethisprogramreliesonnon-LD
sequen e(N,Seq), safe(Sol), permute(Sol,Seq). safe([℄). safe([N|Ns℄) :-safe_aux(Ns,1,N), safe(Ns). safe_aux([M|Ms℄,Dist,N) :-no_diag(N,M,Dist), Dist2 is Dist+1, safe_aux(Ms,Dist2,N). no_diag(N,M,Dist) :-Dist =\= N-M, Dist =\= M-N.
Fig.1.Aprogramforn-queens
Asamore omplexexample, onsiderthefollowingprogram,whosemodeis
fplusone(I);minustwo(I);minusone(I);g. This example uses the su essor
notationfornaturalnumbers.
plus_one(X) :- minus_two(s(X)). minus_two(s(X)) :- minus_one(X). minus_two(0). minus_one(s(X)) :- plus_one(X). minus_one(0). Wede ne
jplusone(s)j=3TSize(s)+4
jminustwo(s)j=3TSize(s)
jminusone(s)j=3TSize(s)+2
Thentheprogramiswell-a eptableandthuswell-terminating.
We see that wheneverin some argument position of a lause head, there is a
ompoundtermofsomere ursivedatastru ture,su has[XjXs℄,andallre ursive
allsinthebodyofthe lausehaveastri tsubtermofthatterm,su hasXs,in
the sameposition | then the lause is well-a eptable using aslevelmapping
thetermsize of thatargumentposition.Sin e thissituation o urs veryoften,
it an be expe ted that an average program ontains many well-terminating
predi ates. However,it is unlikely that in anyreal program,allpredi ates are
well-terminating.
Thelastexampleshowsthatmore omplexs enariosthantheonedes ribed
abovearepossible,butwedoubtthattheywouldofteno urinpra ti e.
There-forelevelmappingssu hastheoneusedin theexamplewillrarelybeneeded.
ConsideragainDef.5.3.Givena lause h a
1 ;:::;a n andanatoma i h, werequirejhj>ja i
j forallgroundingsubstitutions , ratherthanonlyfor
su hthat(a
1 ;:::;a
i 1
)isin a ertainmodeloftheprogram.Thisisof ourse
aseriousrestri tion.InEx.1.2,assumingmodepermute(I;O),there anbeno
moded level mapping su h that jpermute(Y;[UjX℄)j > jpermute(Z;X)j for all
. It mightbe possible to relax Def. 5.3 to allowmore programs,but thefa t
Wehave identi edthe lassof programsfor whi h all input-driven derivations
are nite.Aninput-drivenderivationisaderivationwhereinea hstep,theinput
arguments of thesele ted atom are not instantiated. Predi ates anbe shown
tobeinthat lassusing thenotionsoflevel mappinganda eptable lauseina
verysimilarwayto methodsforLDderivations[8,11,13℄.
Thispaper loselyfollows[13℄.Thereastatementisshownwhi hisessentially
the onverse of Thm. 5.1. It says that if apredi ate is well-terminating, then
there is a level mappingsu h that the lausesde ning the predi ate are
well-a eptable. Itwouldbe interestingto showasimilarresultforarbitrary
input-drivenderivations, but webelievethat it mustbediÆ ult,sin eour de nition
ofa eptabilityismu hmorerestri tive.
We have laimed that most other approa hes to termination rely on the
idea that the size of an atom an be pinned down when the atom is sele ted.
Te hni ally,this usually meansthat theatom is boundedwith respe t to some
levelmapping[4,6,13,18℄.This isdi erentin[9,11℄,wheretermination anbe
shownforthequery,say,append([X℄;[℄;Zs)usingaslevelmappingthetermsize
ofthe rstargument,eventhoughthetermsizeof[X℄isnotbounded. However,
themethodonlyworksforLDderivationsandreliesonthefa t thatanyfuture
instantiationofX annota e tthederivationforappend([X℄;[℄;Zs).Thereforeit
ise e tivelypossibletopindownthesize ofappend([X℄;[℄;Zs).
In ontrast,weshowthatunder ertain onditions,itisenoughtorelyona
relativede reaseinthesizeofthesele tedatom,eventhoughthissize annotbe
pinneddown.More on retely,weusethat anatom in aquery annot pro eed
inde nitely unless itis repeatedly fed by someother atom o urring earlierin
thequery.This impliesthat everyderivationforthequeryterminates.
Bezem[6℄hasidenti edthe lassofstronglyterminatingprograms,whi hare
programsthatterminate underany sele tionrule. Whileitisshownthatevery
totalre ursivefun tion anbe omputedbyastronglyterminatingprogram,this
does not hange thefa t that few existing programs are stronglyterminating.
Transformationsareproposedforthreeexampleprogramstomakethemstrongly
terminating,butthetransformationsare ompli atedandad-ho .
Thispaperismoreabstra tthantheliteratureonprogramswithdelay
de -larations [2,16{21℄. We are not on erned with the details of parti ular delay
onstru ts.Instead, weonlyassumewhat we seeasthe basi purposeof delay
de larations:ensuringthatderivationsareinput-driven.Notethatdependingon
what kindof onstru tsare used, ensuringthat derivations are input-driven is
a tuallyquitesubtle[21℄.Nevertheless,delayde larationsare learlyapowerful
instrumentforthispurpose.
Onthewhole,thereseemstobeastrongrelu tan etogiveuptheideathat
thesizeofanatommustbepinneddownwhentheatomissele ted.Thisistrue
even for[6℄, where noassumptions at allare made aboutthesele tion rule. It
isalsotruefor[17℄,wherealo al sele tion ruleisassumed,that isaruleunder
whi honlymostre entlyintrodu edatoms anberesolvedinea hstep.In[18℄,a
there is mainly to generate delay de larations automati ally rather thanprove
termination. 3
Howeverinsome ases,thedelayde larationsthat aregenerated
requireanargumentofanatomtobearigidlistbeforethatatom anbesele ted,
whi hissimilarto[17,18℄.Su husesofdelayde larationsgofarbeyondensuring
that derivationsareinput-driven.
Wedo not laimto present a omprehensivemethod for showing
termina-tion.Inanaverageprogram,somepredi atesarewell-terminatingbutsomeare
not.Ingeneral,onehastomakestrongerassumptionsaboutthesele tionrule.
Nevertheless,it isuseful toknowwhi hpredi ates arewell-terminating,
essen-tially be ause it means that one has to make the stronger assumptions only
forthepredi atesthatarenotwell-terminating.Forexample,requiringground
orrigid arguments[16,18℄ ouldbelimited to atomswhose predi ates arenot
well-terminating.
In[21℄,well-terminatingpredi atesare onsideredinamore on retesetting
than here and are alled robust predi ates. The default left-to-right sele tion
ruleofmostPrologimplementationsisassumed.Itisexploitedthatthetextual
position of atomsusing robustpredi ates in lause bodies is irrelevantfor
ter-mination.Theotheratomsmustbepla edsu hthattheatomsprodu ingtheir
inputo urearlier.
A knowledgements
TheauthorwouldliketothankFloren eBenoyforproofreadingthispaper.This
workwasfunded byEPSRCGrantNo.GR/K79635.
Referen es
1. K.R.Apt. From Logi ProgrammingtoProlog. Prenti eHall,1997.
2. K.R.AptandI.Luitjes. Veri ationoflogi programswithdelayde larations.In
Pro eedingsofAMAST'95,LNCS,Berlin,1995.Springer-Verlag.InvitedLe ture.
3. K. R. Apt and D. Pedres hi. Studies in Pure Prolog: Termination. In J. W.
Lloyd,editor,Pro eedingsoftheSymposiuminComputationalLogi ,LNCS,pages
150{176.Springer-Verlag,1990.
4. K.R.AptandD.Pedres hi. Modularterminationproofsforlogi andpureProlog
programs. InG.Levi,editor,Advan esinLogi ProgrammingTheory,pages183{
229.OxfordUnversityPress,1994.
5. K. R. Apt and A. Pellegrini. On the o ur- he k free Prolog programs. ACM
Toplas,16(3):687{726,1994.
6. M. Bezem.Strongterminationoflogi programs. JournalofLogi Programming,
15(1&2):79{97,1993.
7. L. Cavedon. Continuity, onsisten y and ompleteness properties for logi
pro-grams. InG.Leviand M. Martelli, editors, Pro eedings of the 6thInternational
Conferen e onLogi Programming,pages571{584.MITPress, 1989.
3
For thereaderfamiliar with[16℄, it is notsaidhowit isshownthat programsare
story. Journalof Logi Programming,19/20:199{260, 1994.
9. D.DeS hreye,K.Vers haetse,andM.Bruynooghe.Aframeworkforanalysingthe
terminationofde nitelogi programswithrespe tto allpatterns.InPro eedings
of FGCS,pages481{488.ICOTTokyo,1992.
10. S. De orte and D. De S hreye. Automati inferen e of norms: a missing link
in automati termination analysis. In D. Miller, editor, Pro eedings of the 10th
InternationalLogi Programming Symposium,pages420{436.MITPress,1993.
11. S.De orteandD.DeS hreye. Terminationanalysis:Somepra ti alpropertiesof
thenormandlevelmappingspa e.InJ.Ja ar,editor,Pro eedingsofthe15thJoint
International Conferen e andSymposiumonLogi Programming,pages 235{249.
MITPress,1998.
12. P. Dembinski and J. Ma luszynski. AND-parallelism with intelligent
ba ktra k-ing for annotatedlogi programs. InPro eedings of the 2nd International Logi
ProgrammingSymposium,pages29{38.MITPress,1985.
13. S.Etalle, A. Bossi,and N.Co o. Well-terminating programs. Journal of Logi
Programming,1998. A eptedforpubli ation.
14. M. Fitting. First-order Logi and Automated TheoremProving. Springer-Verlag,
1996.
15. J.W.Lloyd. Foundations ofLogi Programming. Springer-Verlag,1987.
16. S.Luttringhaus-Kappel. Controlgenerationforlogi programs. InD.S.Warren,
editor, Pro eedings of the 10th International Conferen e onLogi Programming,
pages478{495.MITPress,1993.
17. E. Mar hioriand F.Teusink. Provingterminationof logi programs withdelay
de larations. InJ. W.Lloyd,editor, Pro eedings of the12th International Logi
ProgrammingSymposium,pages447{461.MITPress,1995.
18. J. C.Martin and A. M. King. GeneratingeÆ ient, terminating logi programs.
InM.BidoitandM.Dau het,editors,Pro eedingsofTAPSOFT'97,LNCS,pages
273{284.Springer-Verlag,1997.
19. L. Naish. Coroutiningand the onstru tionofterminatinglogi programs.
Te h-ni alReport92/5,UniversityofMelbourne,1992.
20. J.-G. Smaus, P. M. Hill, and A. M. King. Preventing instantiation errors and
loops for logi programs withseveral modesusing blo kde larations. In Pierre
Flener,editor,Pre-pro eedingsofthe8thInternationalWorkshoponLogi Program
Synthesis andTransformation, numberUMCS-98-6-1, pages72{29. University of
Man hester,1998. Extendedabstra t.
21. J.-G. Smaus, P. M. Hill, and A. M. King. Termination of logi programs with
blo kde larations runninginseveralmodes. InC.Palamadessi,editor,
Pro eed-ingsofthe10thSymposiumonProgrammingLanguageImplementationsandLogi
Programming,LNCS.Springer-Verlag,1998.
22. Z.Somogyi,F.Henderson,andT.Conway. Theexe utionalgorithmofMer ury,
aneÆ ientpurelyde larativelogi programminglanguage. JournalofLogi
