Dependeny Pairs for Equational Rewriting
JurgenGiesl 1
andDeepakKapur 2
1
LuFGInformatikII,RWTHAahen,Ahornstr.55,52074Aahen,Germany,
gieslinformatik.rwth- aa h en .d e
2
ComputerSieneDept.,UniversityofNew Mexio,Albuquerque,NM87131,USA
kapurs.unm.edu
Abstrat. The dependenypairtehnique ofArtsand Giesl [1{3℄for
terminationproofsoftermrewritesystems(TRSs)isextendedtorewrit-
ingmoduloequations.Uptonow,suhanextensionwasonlyknownin
the speial ase of AC-rewriting [15,17℄. Inontrastto that, the pro-
posedtehniqueworksforarbitrarynon-ollapsingequations(satisfying
aertainlinearityondition).Withtheproposedapproah,itisnowpos-
sibletoperformautomatedterminationproofsformanysystemswhere
this was not possible before.Inotherwords, the powerof dependeny
pairsannowalsobeusedforrewritingmoduloequations.
1 Introdution
Termination of term rewriting (e.g., [1{3,9,22℄) and termination of rewriting
modulo assoiativityand ommutativity equations(e.g., [8,13,14,20,21℄)have
beenextensivelystudied. For equationsother than AC-axioms, however, there
areonlyafewtehniques availabletoprovetermination (e.g.,[6,10,16,18℄).
Thispaperpresents anextension ofthe dependeny pairapproah [1{3℄ to
rewriting modulo equations. In the speial ase of AC-axioms, our tehnique
orresponds to the methods of [15,17℄, but in ontrast to these methods, our
tehniqueanalsobeusediftheequationsarenotAC-axioms.Thisallowsmuh
more automatedtermination proofs for equational rewritesystems than those
possiblewith diretlyapplying simpliationorderingsforequationalrewriting
(likeequationalpolynomialorderingsorAC-versionsofpathorderings).
We rst review dependeny pairs for ordinary term rewriting in Set. 2.
In Set. 3, we show why a straightforward extension of dependeny pairs to
rewritingmodulo equationsisnotpossible.Therefore,wefollowanideasimilar
to the one of [17℄ for AC-axioms: Weonsider a restrited form of equational
rewriting,whihismoresuitableforterminationproofswithdependenypairs.
InSet.4,weshowhowto ensurethat termination of thisrestrited equa-
tionalrewriterelationisequivalenttoterminationoffullrewritingmoduloequa-
tions.UnderertainonditionsontheequationsE,weshowhowtoomputean
?
Proeedingsof the12thInternationalConfereneonRewritingTehniquesand Ap-
pliations, RTA-2001, Utreht, The Netherlands,Leture Notes in Computer Si-
ene, Springer-Verlag. Supported by the Deutshe Forshungsgemeinshaft Grant
GI274/4-1andtheNationalSieneFoundationGrantsnos.CCR-9996150,CDA-
9503064,CCR-9712396.
E
rewriterelationofExt
E
(R)moduloEisterminatingiRisterminatingmodulo
E. Thisisprovedfor(almost)arbitraryE-rewriting,thusgeneralizinga related
resultforAC-rewriting.Thisgeneralresultmaybeofindependentinterest,and
may also beuseful in investigating other properties of E-rewriting. Finally, in
Set.5,weextendthedependenypairapproahtorewritingmoduloequations.
2 Dependeny Pairs for Ordinary Rewriting
Thedependenypairapproah allowstheuseof standardmethods likesimpli-
ationorderings[9,22℄forautomatedterminationproofs wheretheywerenot
appliablebefore.Inthissetionwebrieysummarizethebasioneptsofthis
approah. All results in this setion aredue to Arts andGiesl and werefer to
[1{3℄forfurtherdetails,renements,andexplanations.
In ontrast to the standard tehniques for termination proofs, whih om-
pareleftandright-hand sidesofrules,inthisapproahone onentratesonthe
subtermsintheright-handsidesthathaveadened 1
rootsymbol,beausethese
aretheonlytermsresponsibleforstartingnewredutions.
Morepreisely,foreveryrulef(s
1
;:::;s
n
)!C[g(t
1
;:::;t
m
)℄(wherefandg
aredenedsymbols),weomparetheargumenttupless
1
;:::;s
n andt
1
;:::;t
m .
To avoid the handling of tuples, for every dened symbol f, we introdue a
freshtuple symbolF.Toeasereadability,weassumethattheoriginalsignature
onsists of lower ase funtion symbols only, whereas the tuple symbols are
denoted by the orresponding upper ase symbols. Now instead of the tuples
s
1
;:::;s
n andt
1
;:::;t
m
weomparetheterms F(s
1
;:::;s
n
)andG(t
1
;:::;t
m ).
Denition1 (Dependeny Pair [1{3℄). If f(s
1
;:::;s
n
)!C[g(t
1
;:::;t
m )℄
isaruleofaTRSRandgisadenedsymbol,thenhF(s
1
;:::;s
n );G(t
1
;:::;t
m )i
isadependenypairofR.
Example 2. As anexample, onsider the TRS fa+b! a+(b+)g, f. [17℄.
Terminationofthissystemannotbeshownbysimpliationorderings,sinethe
left-handsideoftheruleisembeddedintheright-handside.Inthissystem,the
denedsymbolis+andthus,weobtainthedependenypairshP(a;b);P(a;b+)i
andhP(a;b);P(b;)i(wherePisthetuplesymbolfortheplus-funtion\+").
Artsand Giesldevelopedthefollowingnewterminationriterion.As usual,
a quasi-ordering % is a reexive and transitive relation, and we say that an
ordering>isompatiblewith %ifwehave>Æ%>or%Æ>>.
Theorem 3 (Termination with Dependeny Pairs [1{3℄). A TRS R is
terminating i there exists a weakly monotoni quasi-ordering % and a well-
founded ordering > ompatible with %, where both % and > are losed under
substitution,suhthat
1
Rootsymbolsofleft-handsidesaredened andallotherfuntionsareonstrutors.
(2) l%r forallrules l!r of R.
Consider the TRS from Ex. 2 again. In order to prove its termination a-
ording toThm.3,wehavetonda suitablequasi-ordering%andordering>
suh thatP(a;b)>P(a;b+),P(a;b)>P(b;),anda+b%a+(b+).
Most standard orderings amenable to automation are strongly monotoni
(f. e.g. [9,22℄), whereas here we only need weak monotoniity. Hene, before
synthesizingasuitableordering,someoftheargumentsoffuntionsymbolsmay
be eliminated, f. [3℄. For example, in our inequalities, one may eliminate the
rstargumentof+.Theneveryterms+tintheinequalitiesisreplaedby+ 0
(t)
(where + 0
is a newunary funtionsymbol).Byomparing thetermsresulting
from this replaement instead of the original terms,wean takeadvantage of
the fat that + doesnot have to bestrongly monotoni in its rst argument.
Note that there are only nitely many possibilities to eliminate arguments of
funtionsymbols.Thereforeallthesepossibilitiesanbehekedautomatially.
Inthisway,weobtaintheinequalitiesP(a;b)>P(a;+ 0
()),P(a;b)>P(b;),
and + 0
(b) % + 0
(+
0
()). These inequalities are satised by the reursive path
ordering (rpo) [9℄ with the preedene a A b A A + 0
(i.e., we hoose % to
be %
rpo
and > to be
rpo
). So termination of this TRS an now be proved
automatially.Forimplementationsofthedependenypairapproahsee[4,7℄.
3 Rewriting Modulo Equations
For a set E of equations between terms, we write s !
E
t if there exist an
equationlrin E,a substitution, andaontext C suh thats=C[l℄and
t = C[r℄. The symmetrilosure of !
E
is denoted by `a
E
and thetransitive
reexivelosureof`a
E
isdenoted by
E
.Inthefollowing,werestritourselves
toequationsE where
E
isdeidable.
Denition4 (Rewriting ModuloEquations).LetRbeaTRSandletE be
aset of equations. Aterm s rewritestoa term t modulo E,denoteds!
R=E t,
ithereexist termss 0
andt 0
suhthats
E s
0
!
R t
0
E
t.TheTRSRisalled
terminatingmodulo E i theredoes notexist aninnite !
R=E
redution.
Example 5. AninterestingspeialaseareequationsE whih statethatertain
funtion symbols are assoiative and ommutative (AC).As an example,on-
sidertheTRSR=fa+b!a+(b+)gagainandletEonsistoftheassoiativity
andommutativityaxioms for+,i.e.,E=fx
1 +x
2 x
2 +x
1
;x
1 +(x
2 +x
3 )
(x
1 +x
2 )+x
3
g,f.[17℄.Risnotterminatingmodulo E,sinewehave
a+b!
R
a+(b+)
E
(a+b)+!
R
(a+(b+))+
E
((a+b)+)+!
R :::
Thereare,however,manyothersetsofequationsE apartfromassoiativity
andommutativity,whiharealsoimportantinpratie,f.[11℄.Hene,ouraim
istoextenddependenypairstorewritingmodulo(almost)arbitraryequations.
that whenever a termstarts aninnite redution,then one an also onstrut
an inniteredution where only terminating or minimal non-terminating sub-
terms are redued(i.e., one only appliesrules to redexeswithout proper non-
terminating subterms). The ontexts of minimal non-terminating redexes an
beompletelydisregarded.Ifa ruleisappliedattheroot positionofa minimal
non-terminating subterm s (i.e., s !
R
t where denotes the root position),
thensandeahminimalnon-terminatingsubtermt 0
oftorrespondtoadepen-
denypair. Hene,Thm.3 (1)implies s>t 0
.If a ruleisapplied at a non-root
position ofa minimal non-terminating subterm s(i.e., s!
>
R
t), then wehave
s %t by Thm. 3 (2). However, due to theminimality ofs, after nitely many
suh non-root rewrite steps,a rulemust beapplied at theroot positionof the
minimal non-terminating term. Thus, every innite redution of minimal non-
terminating subterms orresponds to an innite >-sequene. This ontradits
thewell-foundedness of>.
Soforordinary rewriting,anyinniteredution from aminimal non-termi-
natingsubterminvolvesanR-redutionattherootposition.Butasobservedin
[15℄, when extendingthedependenypairapproah to rewriting modulo equa-
tions, this is no longer true. For an illustration, onsider Ex. 5 again, where
a+(b+) is a minimal non-terminating term. However, in its innite R=E-
redutionnoR-stepiseverappliableattherootposition.(Insteadoneapplies
anE-stepattherootpositionandfurtherR-andE-stepsbelowtheroot.)
Intherestofthepaper,fromarewritesystemR,wegenerateanewrewrite
system R 0
with the following three properties: (i)the termination of a weaker
formofrewritingbyR 0
moduloE isequivalenttotheterminationofRmodulo
E,(ii)everyinniteredutionofaminimalnon-terminatingterminthisweaker
formofrewritingbyR 0
moduloEinvolvesaredutionstepattherootlevel,and
(iii) every suh minimal non-terminating termhas an inniteredution where
thevariablesoftheR 0
-rulesareinstantiatedwithterminatingtermsonly.
4 E-Extended Rewriting
Weshowedwhythedependenypairapproahannotbeextendedtorewriting
moduloequationsdiretly.Asasolutionforthisproblem,weproposetoonsider
arestritedformofrewritingmoduloequations,i.e.,theso-alledE-extendedR-
rewrite relation !
EnR
. (This approah was already takenin [17℄ for rewriting
moduloAC.)Therelation!
EnR
wasoriginallyintroduedin[19℄inordertoir-
umventtheproblemswithinniteorimpratiallylargeE-equivalenelasses.
2
Denition6 (E-extended R-rewriting [19℄). LetR beaTRS andletE be
aset ofequations.The E-extendedR-rewrite relationisdenedass!
EnR t i
sj
E
l andt =s[r℄
forsome rule l!r inR, someposition of s,and
somesubstitution .Wealso write!
EnR
instead of !
EnR .
2
In[12℄,therelation!
EnR
isdenoted\!
R;E
".
R=E EnR
again. We have already seen that !
R=E
is not terminating,sine a+b!
R=E
(a+b)+!
R=E
((a+b)+)+!
R=E
:::But!
EnR
isterminating,beause
a+b!
EnR
a+(b+),whih isanormalformw.r.t.!
EnR .
Theaboveexamplealsodemonstrates thatingeneral,terminationof!
EnR
is notsuÆient fortermination of!
R=E
. Inthissetion wewill showhow ter-
minationof!
R=E
anneverthelessbeensuredbyonlyregardinganE-extended
rewriterelationinduedbya largerR 0
R.
ForthespeialaseofAC-rewriting,thisproblemanbesolvedbyextending
Rasfollows:LetG bethesetofallAC-symbolsand
Ext
AC(G)
=R[ff(l;y)!f(r;y) j l!r2R;root(l)=f 2Gg;
where y is a newvariablenotourring in therespetiverule l!r.A similar
extension has also beenused in previous work onextending dependenypairs
toAC-rewriting[17℄.ThereasonisthatforAC-equationsE,theterminationof
!
R=E
isinfatequivalentto thetermination of!
EnExtAC(G)(R) .
For Ex. 5,weobtain Ext
AC(G)
(R) = fa+b! a+(b+);(a+b)+y !
(a+(b+))+yg.Thus,inordertoproveterminationof!
R=E
,itisnowsuÆient
toverifyterminationof!
EnExt
AC(G) (R)
.
Theaboveextensionof[19℄onlyworksforAC-axiomsE.A laterpaper[12℄
treatsarbitraryequations,butitdoesnotontainanydenition forextensions
Ext
E
(R),andterminationof!
R=E
isalwaysaprerequisitein [12℄.Thereason
is that [12℄ and also subsequent work on symmetrization and oherene were
devoted to the development of ompletion algorithms (i.e., here the goal was
to generateaonvergentrewritesystemand notto investigatethetermination
behaviorofpossiblynon-terminatingTRSs).Thus,thesepapersdidnotompare
theterminationbehavioroffullrewritingmoduloequationswiththetermination
ofrestritedversionsofrewritingmoduloequations.Infat,[12℄ fousesonthe
notionofoherene,whihisnotsuitableforourpurposesineohereneofEnR
moduloE doesnotimplythatterminationof!
R=E
isequivalenttotermination
of!
EnR .
3
To extend dependeny pairs to rewriting modulo non-AC-equations E, we
haveto omputeextensions Ext
E
(R) suh that termination of!
R=E
is equiv-
alentto terminationof !
EnExt
E (R)
.Theonlyrestritionwewill imposeonthe
equations in E is that theymust have idential unique variables. This require-
mentissatisedbymostpratialexampleswhereR=Eisterminating.Asusual,
a termt isalledlinear ifnovariableoursmore thanoneint.
Denition7 (EquationswithIdentialUniqueVariables[19℄).Anequa-
tion uv issaid to have idential uniquevariablesif u andv are both linear
andthe variablesinuarethesame asthe variablesinv.
3
In[12℄,EnRisoherentmoduloEiforalltermss;t;u,wehavethats
E t!
+
EnR u
implies s! +
EnR v
E w
EnR
uforsome v;w. ConsiderR=fa+b!a+(b+
); x+y!dgwithE beingtheAC-axiomsfor+.Theabovesystemisoherent,
sine s E t ! +
EnR
u implies s ! +
R d
R
u.However, !
EnR
is terminatingbut
!
R=E
isnotterminating.
E
usual,ÆisanE-unier ofsandtisÆ
E
tÆ andasetuni
E
(s;t)ofE-uniersis
omplete iforeveryE-unierÆ thereexistsa2uni
E
(s;t)anda substitution
suh that Æ
E
, f. [5℄. (\" is the omposition of and where is
appliedrstand\Æ
E
"meansthatforallvariablesxwehavexÆ
E x.)
To onstrutExt
E
(R),weonsideralloverlapsbetweenequationsuv or
vufromE andrulesl!rfrom R.Morepreisely,wehekwhethera non-
variable subterm vj
of v E-unies with l (where we alwaysassume that rules
in Rare variabledisjoint from equationsin E). Inthisaseone adds therules
(v[l℄
) ! (v[r℄
) for all 2 uni
E (vj
;l).
4
In Ex. 5,the subterm x
1 +x
2 of
theright-handsideofx
1 +(x
2 +x
3 )(x
1 +x
2 )+x
3
unieswiththeleft-hand
sideoftheonlyrulea+b!a+(b+).Thus, intheextensionofR,weobtain
therule(a+b)+y!(a+(b+))+y.
Ext
E
(R) is built via a kind of xpoint onstrution, i.e., we also have to
onsider overlaps between equations of E and the newly onstruted rules of
Ext
E
(R).Forexample,thesubtermx
1 +x
2
alsounies withtheleft-hand side
ofthenewrule(a+b)+y !(a+(b+))+y.Thus,onewouldnowonstrut
a newrule((a+b)+y)+z!((a+(b+))+y)+z.
Obviously,inthiswayoneobtainsaninnitenumberofrulesbysubsequently
overlapping equations with the newly onstruted rules. However, in order to
useExt
E
(R) forautomatedterminationproofs, ouraimisto restritourselves
to nitely many rules. It turns out that we do not have to inlude new rules
(v[l℄
)!(v[r℄
) in Ext
E
(R) ifu !
0
EnExtE(R) q
E (v[r℄
) alreadyholds
forsomeposition 0
ofuandsometermq(usingjusttheoldrulesofExt
E (R)).
Whenonstrutingtherule((a+b)+y)+z!((a+(b+))+y)+zabove,
theequationuv usedwas x
1 +(x
2 +x
3 )(x
1 +x
2 )+x
3
andtheunier
replaedx
1
by(a+b)andx
2
byy.Hene,hereuistheterm(a+b)+(y+x
3 ).
Butthistermredueswith! 1
EnExtE(R)
to(a+(b+))+(y+x
3
)whihisindeed
E
-equivalentto(v[r℄
),i.e.,to((a+(b+))+y)+x
3
.Thus,wedonothave
toinlude therule((a+b)+y)+z!((a+(b+))+y)+zin Ext
E (R).
Thefollowingdenition showshowsuitableextensionsan beomputedfor
arbitrary equations with idential unique variables. It will turn out that with
these extensions one an indeedsimulate !
R=E by !
EnExtE(R)
, i.e., s!
R=E t
implies s !
EnExtE(R) t
0
for some t 0
E
t. This onstitutes a ruial ontribu-
tion of the paper, sine it is the main requirement needed in order to extend
dependenypairstorewritingmoduloequations.
Denition8 (Extending R for Arbitrary Equations). Let R be a TRS
andletE beasetof equations.LetR 0
beasetontainingonlyrulesof theform
4
Obviously,uni
E (vj
;l)alwaysexists,butit anbe inniteingeneral.Sowhen au-
tomatingour approahforequational terminationproofs,we have to restritour-
selvestoequations E where uni
E (vj
;l)anbe hosento beniteforallsubterms
vj
ofequationsandleft-handsidesofrulesl.ThisinludesallsetsE ofnitaryuni-
ationtype,butourrestritionisweaker, sineweonlyneednitenessforertain
termsvj
andl.
C[l℄!C[r℄ (where C is aontext, isasubstitution, and l!r 2R). R
isan extensionofRfortheequationsE i
(a) RR 0
and
(b) for all l ! r 2 R 0
, u v 2 E and v u 2 E, all positions of v
and 2 uni
E (vj
;l), there is a position 0
in u and a q
E (v[r℄
) with
u!
0
EnR 0
q.
Inthefollowing,letExt
E
(R) alwaysdenote anarbitraryextensionofRforE.
InordertosatisfyCondition(b)ofDef.8,itisalwayssuÆienttoaddtherule
(v[l℄
)!(v[r℄
) to R 0
. Thereasonis thatthen wehaveu!
EnR 0
(v[r℄
).
But ifu !
0
EnR 0
q
E (v[r℄
) already holds with the otherrules of R 0
, then
therule(v[l℄
)!(v[r℄
) doesnothaveto beaddedtoR 0
.
Condition(b) ofDef. 8 also makessure that as long as the equationshave
idential unique variables, wedo nothave to onsider overlaps at variable po-
sitions.
5
The reason is that if vj
is a variable x 2 V, then we have u =
u[x℄
0
E u[l℄
0
!
R u[r℄
0
E v[r℄
=(v[r℄
),where 0
isthepositionof
xinu.Hene,suhrules(v[l℄
)!(v[r℄
) donothavetobeinludedinR 0
.
Overlapsat root positions donothaveto beonsideredeither.To see this,
assumethatisthetoppositionofv,i.e.,thatv
E
l.Inthisasewehave
u
E
v
E l!
R
randthus,u!
EnR
r=(v[r℄
).Soagain,suhrules
(v[l℄
)!(v[r℄
) donothavetobeinludedin R 0
.
Thefollowingproedureisusedtoomputeextensions.Here,weassumeboth
RandEtobenite,wheretheequationsEmusthaveidentialuniquevariables.
1.R 0
:=R
2.For alll!r 2R 0
,
alluv orvufrom E,
andallpositionsofv where6=and vj
62V do:
2.1.Let:=uni
E (vj
;l).
2.2.For all2do:
2.2.1.LetT :=fq j u!
0
EnR 0
qfora position 0
ofug:
2.2.2.Ifthereexistsa q2T with(v[r℄
)
E
q,then:=nfg.
2.3.R 0
:=R 0
[f(v[l℄
)!(v[r℄
) j 2g.
Thisalgorithmhasthefollowingproperties:
(a) Ifin Step 2.1, uni
E (vj
;l) isnite and omputable, thenevery stepin the
algorithmisomputable.
(b) Ifthealgorithmterminates, thenthenal valueofR 0
isan extensionofR
fortheequationsE.
5
Notethatonsideringoverlapsatvariablepositionsaswellwouldstillnotallowus
totreatequationswithnon-linearterms.AsanexampleregardE =ff(x)g(x;x)g
andR=fg(a;b)!f(a);a!bg.Here,!
EnExt
E (R)
iswellfoundedalthough Ris
notterminatingmoduloE.
E
(a+(b+))+yg.Ingeneral,ifE onlyonsistsofAC-axiomsforsomefuntion
symbolsG,thenDef.8\oinides"withthewell-knownextensionforAC-axioms,
i.e., R 0
= R[ff(l;y) ! f(r;y)jl ! r 2 R;root(l) = f 2 Gg satises the
onditions(a)and(b)ofDef.8.SoinaseofAC-equations,ourapproahindeed
orresponds tothe approahesof [15,17℄. However,Def.8 an also beused for
otherformsofequations.
Example 9. As anexample,onsiderthefollowingsystemfrom[18℄.
R=f x 0!x; E =f(uv)w (uw)vg
s(x) s(y)!x y;
0s(y)!0;
s(x)s(y)!s((x y)s(y))g
Byoverlappingthesubterm uw in theright-handside of theequation with
theleft-handsidesofthelasttwo rulesweobtain
Ext
E
(R)=R[ f (0s(y))z!0z;
(s(x)s(y))z!s((x y)s(y))zg:
Note that these are indeed all therules of Ext
E
(R). Overlapping thesub-
term uv of theequation's left-hand sidewith the third rule would resultin
(0s(y))z 0
! 0z 0
. But this new rule does not have to be inluded in
Ext
E
(R), sinethe orresponding other term of the equation, (0z 0
)s(y),
would !
EnExtE(R)
-redue with therule(0s(y))z !0z to 0z 0
.Over-
lappinguv withtheleft-handsideofthefourthruleisalsosuperuous.
Similarly, overlaps with the new rules (0s(y))z ! 0z or (s(x)
s(y))z ! s((x y)s(y))z also do not give rise to additional rules in
Ext
E
(R).To seethis, overlap thesubterm uwin the right-hand sideof the
equation with theleft-handside of (0s(y))z !0z.This gives therule
((0s(y))z)z 0
!(0z)z 0
.However,theorrespondingother termof
theequationis((0s(y))z 0
)z. Thisreduesatposition1 (orposition11)
to(0z 0
)z,whihisE-equivalentto(0z)z 0
.Overlapswiththeothernew
rule(s(x)s(y))z!s((x y)s(y))zarenotneededeither.
Nevertheless,theabovealgorithmforomputingextensionsdoesnotalways
terminate.Forexample,forR=fa(x)!(x)g,E =fa(b(a(x)))b(a(b(x)))g,
itan beshownthat allextensionsExt
E
(R)areinnite.
WeprovebelowthatExt
E
(R)(aordingtoDef.8)hasthedesiredproperty
neededto reduerewriting moduloequations toE-extendedrewriting. Thefol-
lowingimportantlemmastatesthatwheneversrewritestotwith!
R=E
modulo
E,then salsorewriteswith!
EnExt
E (R)
toatermwhih isE-equivalenttot.
6
6
Our extension Ext
E
has some similarities to the onstrutionof ontexts in [23℄.
However,inontrastto[23℄wealsoonsidertherulesofR 0
inCondition(b)ofDef.
8inorderto reduethenumberofrulesinExtE.Moreover, in[23℄equations may
alsobenon-linear(andthus,Lemma10doesnotholdthere).
R=E EnExt
E (R)
andletE beasetof equationswithidential uniquevariables.Ifs!
R=E t,then
thereexistsaterm t 0
E
tsuhthat s!
EnExt
E (R)
t 0
.
Proof. Let s!
R=E
t, i.e., there exist terms s
0
;:::;s
n
;p with n 0 suh that
s =s
n
`a
E s
n 1
`a
E ::: `a
E s
0
!
R p
E
t.For thelemma, it suÆes to show
that thereisat 0
E
psuhthats!
EnExtE(R) t
0
,sinet 0
E
pimpliest 0
E t.
We perform indution on n. If n = 0, we have s = s
n
= s
0
!
R
p. This
impliess!
EnExtE(R)
psineRExt
E
(R).Sowitht 0
=pthelaimisproved.
If n > 0,the indution hypothesis implies s =s
n
`a
E s
n 1
!
EnExt
E (R)
t 0
suh that t 0
E
p. So there exists an equation u v or v u from E and a
rulel!rfrom Ext
E
(R) suh that sj
=uÆ, s
n 1
=s[vÆ℄
, s
n 1 j
E
lÆ, and
t 0
=s
n 1 [rÆ℄
forpositions and anda substitution Æ.Wean usethesame
substitutionÆforinstantiatingtheequationuv(orvu)andtherulel!r,
sineequationsandrulesareassumedvariabledisjoint.Wenowperformaase
analysisdependingontherelationshipofthepositions and.
Case1:=forsome. Inthisase,wehavesj
=sj
[uÆ℄
`a
E sj
[vÆ℄
=
s
n 1 j
E
lÆ.This impliess!
EnExtE(R) s[rÆ℄
=s
n 1 [rÆ℄
=t 0
,as desired.
Case2:?. Nowwehavesj
=s
n 1 j
E
lÆandthus,s!
EnExt
E (R)
s[rÆ℄
=
s[rÆ℄
[uÆ℄
`a
E s[rÆ℄
[vÆ℄
=s[vÆ℄
[rÆ℄
=s
n 1 [rÆ℄
=t 0
.
Case3:=forsome. Thus,(vÆ)j
E
lÆ.Wedistinguishtwosub-ases.
Case3.1:uÆ!
EnExtE(R) q
E (v[r℄
)Æforsometermq. Thisimpliess=s[uÆ℄
!
EnExtE(R) s[q℄
E s[v[r℄
Æ℄
=(s[vÆ℄
)[rÆ℄
=s
n 1 [rÆ℄
=t 0
.
Case3.2:Otherwise. First assumethat =
1
2
where vj
1
is a variablex.
Hene,(vÆ)j
=Æ(x)j
2 . Let Æ
0
(y)=Æ(y) fory 6=x andletÆ 0
(x)=Æ(x)[rÆ℄
2 .
Sine u v (or v u) is an equation with idential unique variables, x also
ours inuat someposition 0
.This impliesuÆj
0
2
=Æ(x)j
2
E lÆ!
Ext
E (R)
rÆ. Hene, we obtain uÆ !
0
2
EnExtE(R) uÆ[rÆ℄
0
2
= uÆ 0
E vÆ
0
= (v[r℄
)Æ in
ontraditiontotheonditionofCase3.2.
Hene,isapositionofvandvj
isnotavariable.Thus,(vÆ)j
=vj
Æ
E lÆ.
Sinerulesandequationsareassumedvariabledisjoint,thesubtermvj
E-unies
withl.Thus,thereexists a2uni
E (vj
;l)suh thatÆ
E .
DuetotheCondition(b)ofDef.8,thereisatermq 0
suhthatu!
0
EnExt
E (R)
q 0
E (v[r℄
).Sine 0
isapositioninu,wehaveuj
0
E Æ!
Ext
E (R)
q 00
,where
q 0
=u[q 00
℄
0. Thisalso impliesuj
0Æ
E uj
0
E Æ!
ExtE(R) q
00
,and thus
uÆ!
0
EnExt
E (R)
uÆ[q 00
℄
0
E u[q
00
℄
0
=q 0
E (v[r℄
)
E (v[r℄
)Æ.Thisisa
ontraditiontotheonditionofCase3.2. ut
ThefollowingtheoremshowsthatExt
E
indeed hasthedesiredproperty.
Theorem 11 (TerminationofR=E byE-ExtendedRewriting). LetRbe
a TRS,let E be aset of equations with idential unique variables, and lett be
a term. Then t does not start an innite !
R=E
-redution i t does not start
EnExt
E (R)
(i.e., !
R=E
iswellfounded)i!
EnExt
E (R)
iswellfounded.
Proof. The \only if" diretion is straightforward beause !
Ext
E (R)
=!
R and
therefore,!
EnExt
E (R)
!
Ext
E (R)=E
=!
R=E .
Forthe\if"diretion, assumethattstartsaninnite!
R=E
-redution
t=t
0
!
R=E t
1
!
R=E t
2
!
R=E :::
Foreveryi2IN,letf
i+1
beafuntionfromtermsto termssuhthatforevery
t 0
i
E t
i ,f
i+1 (t
0
i
)isatermE-equivalenttot
i+1
suhthatt 0
i
!
EnExtE(R) f
i+1 (t
0
i ).
These funtions f
i+1
mustexist due to Lemma 10,sinet 0
i
E t
i and t
i
!
R=E
t
i+1
impliest 0
i
!
R=E t
i+1
.Hene,tstartsaninnite!
EnExt
E (R)
-redution:
t!
EnExt
E (R)
f
1 (t)!
EnExt
E (R)
f
2 (f
1 (t))!
EnExt
E (R)
f
3 (f
2 (f
1 (t)))!
EnExt
E (R)
::: ut
5 Dependeny Pairs for Rewriting Modulo Equations
In this setion we nally extend the dependeny pair approah to rewriting
modulo equations: To show that R modulo E terminates, one rst onstruts
the extension Ext
E
(R) of R. Subsequently, dependeny pairs an be used to
provewell-foundedness of!
EnExt
E (R)
(whihis equivalenttotermination ofR
moduloE).Theideafortheextensionofthedependenypairapproahissimply
tomodifyThm.3 asfollows.
1. Theequationsshouldbesatisedbytheequivaleneorrespondingtothe
quasi-ordering%,i.e.,wedemand uv forallequationsuv inE.
2. Asimilarrequirementisneededforequationsuv whentherootsymbols
of u and v are replaed by the orresponding tuple symbols. We denote
tuplesoftermss
1
;:::;s
n
bysandforanytermt=f(s)withadenedroot
symbolf,lett
℄
bethetermF(s).Hene,wealsohaveto demandu
℄
v
℄
.
3. Thenotionof\denedsymbols"mustbehangedaordingly.Asbefore,all
root symbols of left-handsides of rules are regardedas being dened,but
ifthere is an equation f(u)=g(v) in E and f is dened,then g must be
onsidereddened as well, as otherwise we would notbeable to trae the
redexinaredution byonlyregardingsubtermswithdened rootsymbols.
Denition12 (DenedSymbolsforRewritingModuloEquations).Let
Rbe aTRSandletE beasetof equations.Thenthe set of dened symbolsD
of R=E isthe smallestsetsuhthat D=froot(l) j l!r 2Rg[froot(v)ju
v2E orvu2E; root(u)2Dg.
Theonstraintsofthedependenypairapproahas skethed abovearenot
yetsuÆientforterminationof!
EnR
as thefollowingexampleillustrates.
Example 13. Consider R=ff(x)!xgand E=ff(a)ag.Thereisnodepen-
deny pair in this exampleand thus, theonly onstraints would be f(x) %x,
f(a) a, and F(a) A. Obviously, these onstraints are satisable (by using
an equivalene relation where all terms are equal). However, !
EnR is not
terminating sinewehavea`a
E f(a)!
R a`a
E f(a)!
R a`a
E :::
3) relieson thefatthatan inniteredutionfrom a minimalnon-terminating
termanbeahievedbyapplyingonlynormalizedinstantiationsofR-rules.But
for E-extended rewriting (or full rewriting modulo equations), this is nottrue
anymore.Forinstane,theminimalnon-terminatingsubtermainEx.13isrst
modied by applying an E-equation (resulting in f(a)) and then an R-rule is
applied whosevariableis instantiated withthenon-terminating terma.Hene,
the problemis that thenew minimalnon-terminating subterm a whih results
from appliation ofthe R-ruledoesnotorrespond to theright-hand side ofa
dependenypair, beausethis minimal non-terminating subterm is ompletely
insidetheinstantiationofavariable oftheR-rule.Withordinaryrewriting,this
situation anneverour.
InEx.13, theprobleman beavoided by addinga suitableinstane ofthe
rulef(x)!x(viz.f(a)!a)to R,sinethisinstane isusedin theinnitere-
dution.NowtherewouldbeadependenypairhF(a);Aiandwiththeadditional
onstraintF(a)>Atheresultinginequalitiesarenolongersatisable.
The following denition shows how to add the right instantiations of the
rulesin Rinorderto allowa soundappliation ofdependenypairs.As usual,
a substitution is alled a variable renaming i the range of only ontains
variablesandif(x)6=(y)forx6=y.
Denition14 (Adding Instantiations). Given a TRS R, a set E of equa-
tions, let R 0
be a set ontaining only rules of the forml !r (where isa
substitution andl!r 2R). R 0
isan instantiation ofRfortheequationsE i
(a) RR 0
,
(b) foralll!r 2R,alluv 2Eandvu 2E,andall2uni
E
(v;l),there
existsarulel 0
!r 0
2R 0
andavariable renaming suhthatl
E l
0
and
r
E r
0
.
Inthefollowing,letIns
E
(R)alwaysdenoteaninstantiationofRforE.
Unlike extensions Ext
E
(R), instantiations Ins
E
(R) are never innite if R
andE areniteandifuni
E
(v;l)isalwaysnite(i.e., theyarenotdenedviaa
xpointonstrution).Infat,onemightevendemandthatforalll!r2R,all
equations,andallfromtheorrespondingompletesetofE-uniers,Ins
E (R)
should ontainl ! r. The ondition that it is enough ifsome E-equivalent
variable-renamedruleisalreadyontainedinIns
E
(R)isonlyaddedforeÆieny
onsiderationsinordertoreduethenumberofrulesinIns
E
(R).Evenwithout
thisondition,Ins
E
(R)wouldstillbeniteandallthefollowingtheoremswould
holdas well.
However, theabove instantiation tehniqueonly servesits purpose if there
arenoollapsingequations(i.e.,noequationsuv orvuwithv2V).
Example 15. ConsiderR=ff(x)!xgandE =ff(x)xg.NotethatIns
E (R)
=R.Although!
EnR
islearlynotterminating,thedependenypairapproah
would falselyproveterminationof!
EnR
,sinethere isnodependenypair.
Nowwean presentthemain resultofthepaper.
Pairs).LetRbeaTRSandletE beasetofnon-ollapsingequationswithiden-
tialuniquevariables.RisterminatingmoduloE (i.e.,!
R=E
iswellfounded)if
thereexistsaweaklymonotoniquasi-ordering%andawell-foundedordering>
ompatiblewith %whereboth%and>arelosedunder substitution,suhthat
(1) s>tforall dependeny pairshs;tiof Ins
E (Ext
E (R)),
(2) l%r forallrules l!r of R,
(3) uv forallequations uv ofE,and
(4) u
℄
v
℄
forall equationsuv ofE whereroot(u) androot(v) aredened.
Proof. Suppose that there is a termt with an innite !
R=E
-redution. Thm.
11 implies that t also has an innite !
EnExtE(R)
-redution. By a minimality
argument, t = C[t 0
℄, where t 0
is an minimal non-terminating term (i.e., t 0
is
non-terminating, but allits subterms only havenite !
EnExtE(R)
-redutions).
We will show that there exists a term t
1
with t ! +
EnExtE(R) t
1 , t
1
ontains a
minimalnon-terminatingsubtermt 0
1 ,andt
0
℄
%Æ>t 0
1
℄
.Byrepeatedappliation
ofthis onstrutionweobtainaninnitesequenet! +
EnExtE(R) t
1
! +
EnExtE(R)
t
2
! +
EnExtE(R)
:::suh that t 0
℄
%Æ>t 0
1
℄
%Æ>t 0
2
℄
%Æ>:::. This,however,is
a ontraditiontothewell-foundedness of>.
Lett 0
havetheformf(u).Intheinnite!
EnExt
E (R)
-redutionoff(u),rst
some!
EnExt
E (R)
-stepsmaybeappliedtouwhihyieldsnewtermsv.Notethat
duetothedenitionofE-extendedrewriting,intheseredutions,noE-stepsan
beapplied outside of u. Due to the termination of u, after a nite number of
thosesteps,an!
EnExt
E (R)
-stepmustbeappliedontherootpositionoff(v).
Thus, thereexists arulel!r2Ext
E
(R)suh thatf(v)
E
landhene,
the redution yieldsr . Nowthe innite!
EnExtE(R)
-redution ontinues with
r ,i.e.,thetermrstartsaninnite!
EnExtE(R)
-redution,too.Souptonow
theredutionhasthefollowingform(where!
ExtE(R)
equals!
R ):
t=C[f(u)℄!
EnExt
E (R)
C[f(v)℄
E
C[l ℄!
ExtE(R) C[r ℄:
WeperformaaseanalysisdependingonthepositionsofE-stepsinf(v)
E l .
FirstonsidertheasewhereallE-steps inf(v)
E
ltakeplaebelowthe
root.Thenwehavel=f(w)andv
E
w .Lett
1
:=C[r ℄.Notethatvdonot
startinnite!
EnExtE(R)
-redutions andbyThm.11,theydonotstartinnite
!
R=E
-redutionseither.Butthenwalsoannotstartinnite!
R=E
-redutions
andthereforetheyalsodonotstartinnite!
EnExt
E (R)
-redutions.Thisimplies
thatforallvariablesxourringinf(w)theterms (x)areterminating.Thus,
sine r startsan innite redution,there ours a non-variable subterm s in
r, suh that t 0
1
:= s is a minimal non-terminating term. Sine hl
℄
;s
℄
i is a
dependenypair,weobtaint 0
℄
=F(u)%F(v)l
℄
>s
℄
=t 0
1
℄
.Here,F(u)%
F(v)holdssineu!
EnExt
E (R)
vandsinel%rforeveryrulel!r2Ext
E (R).
NowweonsidertheasewherethereareE-steps inf(v)
E
lattheroot
position. Thus wehavef(v)
E
f(q)`a
E p
E
l ,where f(q)`a
E
pistherst
in E suhthat f(q)isaninstantiationof v.
Notethatsinev
E
q,thetermsqonlyhavenite!
EnExt
E (R)
-redutions
(theargumentationissimilarasintherstase).LetÆbethesubstitutionwhih
operates likeon thevariablesof land whih yields vÆ =f(q).Thus, Æ isan
E-unieroflandv.SinelisE-uniablewithv,therealsoexistsaorresponding
omplete E-unier from uni
E
(l;v). Thus, there is also a substitution suh
that Æ
E
. As l is a left-hand side of a rule from Ext
E
(R),there is a rule
l 0
!r 0
in Ins
E (Ext
E
(R)) and a variablerenaming suh that l
E l
0
and
r
E r
0
.
Hene,v
E
vÆ=f(q),l 0
E
l
E
lÆ=l ,andr 0
E
r
E rÆ=
r . Soinsteadwenow onsider thefollowing redution (where!
InsE(ExtE(R))
equals!
R ):
t=C[f(u)℄!
EnExtE(R)
C[f(v)℄
E C[l
0
℄!
Ins
E (Ext
E (R))
C[r 0
℄=t
1 :
Sine all proper subterms of vÆ only have nite !
R=E
-redutions, for all
variablesxofl 0
,thetermxonlyhasnite !
R=E
-redutions andhene,also
onlynite!
EnExt
E (R)
-redutions.Toseethis,notethatsineallequationshave
idential uniquevariables,v
E
l
E l
0
impliesthat allvariablesofl 0
also
our in v. Thus, ifx is a variable from l 0
, then there exists a variable y in
v suh that xours in y. SineE doesnotontainollapsing equations,y is
a proper subterm of v and thus, yÆ is a proper subterm of vÆ. As all proper
subterms ofvÆonly havenite !
R=E
-redutions, this impliesthat yÆ onlyhas
nite!
R=E
-redutions, too.Butthen,sineyÆ
E
y,thetermyonlyhas
nite !
R=E
-redutions, too.Then this also holds for allsubterms of y, i.e.,
all!
R=E
-redutions ofxarealsonite.
Soforallvariablesxofl 0
,xonlyhasnite!
EnExtE(R)
-redutions.(Note
that this only holdsbeause is justa variablerenaming.) Siner startsan
innite!
EnExt
E (R)
-redution,r 0
E
rmuststartaninnite!
R=E
-redution
(and hene,aninnite!
EnExt
E (R)
-redution) as well. Asfor allvariables xof
r 0
, xis !
EnExt
E (R)
-terminating,there must be a non-variable subterm s of
r 0
, suh that t 0
1
:= s is a minimal non-terminating term. As hl 0
℄
;s
℄
i is a
dependeny pair, we obtain t 0
℄
=F(u) % F(v)
E l
0
℄
> s
℄
=t 0
1
℄
. Here,
F(v)
E l
0
℄
isa onsequeneofCondition(4). ut
Now termination of the division-system (Ex. 9) an be proved by depen-
deny pairs.Here wehave Ins
E (Ext
E
(R)) =Ext
E
(R) and thus, theresulting
onstraintsare
M(s(x);s(y))>M(x;y) Q(0s(y);z)>Q(0;z)
Q(s(x);s(y))>M(x;y) Q(s(x)s(y);z)>M(x;y)
Q(s(x);s(y))>Q(x y;s(y)) Q(s(x)s(y);z)>Q(x y;s(y))
Q(s(x)s(y);z)>Q(s((x y)s(y));z)
as well as l % r for all rules l ! r, (uv)w (uw)v, and Q(u
v;w) Q(u w;v). (Here, M and Q are the tuple symbols for the minus-
symbol\ "and thequot-symbol\".) Asexplainedin Set.2 onemayagain
In this example we will eliminate the seond arguments of , , M, and Q
(i.e.,everyterms tisreplaedby 0
(s),et.).Thentheresultinginequalities
are satised by the rpo with the preedene 0
A s A 0
, Q 0
A M 0
. Thus,
with the method of the present paper,one an now verify termination of this
exampleautomatially forthe rsttime. This examplealso demonstrates that
by using dependeny pairs,termination of equational rewritingan sometimes
even beshown byordinary base orderings (e.g., theordinary rpo whih onits
ownannotbeusedforrewritingmoduloequations).
6 Conlusion
Wehaveextendedthedependenypairapproahtoequationalrewriting.Inthe
speialaseofAC-axioms,ourmethodissimilartotheonespreviouslypresented
in [15,17℄. Infat,as longas theequationsonlyonsistofAC-axioms, onean
showthatusingtheinstanesIns
E
inThm.16isnotneessary.
7
(Hene,suha
oneptannotbefoundin[17℄).However,eventhentheonlyadditionalinequal-
itiesresultingfrom Ins
E
areinstantiationsofotherinequalitiesalreadypresent
andinequalitieswhiharespeialasesofanAC-deletionproperty(whihissat-
ised byallknownAC-orderingsand similarto theone requiredin [15℄). This
indiates that in pratialexamples with AC-axioms, ourtehniqueis at least
aspowerfulastheonesof[15,17℄(atually,weonjeturethatforAC-examples,
thesethreetehniquesarevirtuallyequallypowerful).Butomparedtotheap-
proahesof[15,17℄,ourtehniquehasamoreeleganttreatmentoftuplesymbols.
(For example,iftheTRSontainsarulef(t
1
;t
2
)!g(f(s
1
;s
2 );s
3
)werefandg
aredened AC-symbols,thenwedonothavetoextendtheTRSbyruleswith
tuple symbols like f(t
1
;t
2
)!G(f(s
1
;s
2 );s
2
) in [17℄. Moreover,we donotneed
dependenypairs where tuple symbols our outsidetheroot position suh as
hF(F(t
1
;t
2
);y);:::iin[17℄and[15℄andhF(t
1
;t
2 );G(F(s
1
;s
2 );s
3
)iin[15℄.Finally,
we alsodo notneed the\AC-markedondition" F(f(x;y);z)F(F(x;y);z)of
[15℄.) But most signiantly, unlike [15,17℄ our tehnique works for arbitrary
non-ollapsing equations E with idential uniquevariableswhere E-uniation
is nitary (for subterms of equations and left-hand sides of rules). Obviously,
animplementationofourtehniquealsorequiresE-uniationalgorithms[5℄for
theonretesetsofequationsE under onsideration.
In[1{3℄,ArtsandGieslpresentedthedependeny graph renementwhihis
basedon theobservation that itis possibleto treatsubsets ofthedependeny
pairsseparately.Thisrenementarriesovertotheequationalaseinastraight-
forwardway(byusingE-uniationtoomputeanestimationofthisgraph).For
details on this renement and for further examples to demonstratethe power
andtheusefulness ofourtehnique,thereaderisreferredto [11℄.
Aknowledgments.WethankA.Middeldorp,T.Arts,andtherefereesforomments.
7
ThenintheproofofThm.16,insteadofaminimalnon-terminatingtermt 0
onere-
gardsatermt 0
whihisnon-terminatingandminimaluptosomeextraf-ourrenes
onthetop(wheref isanAC-symbol).
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OrderingsFail,inPro.TAPSOFT'97,LNCS1214,261-272,1997.
2. T.ArtsandJ.Giesl,ModularityofTerminationUsingDependenyPairs,inPro.
RTA'98,LNCS1379,226-240,1998.
3. T. Arts and J. Giesl, Termination of Term Rewriting Using Dependeny Pairs,
TheoretialComputer Siene,236:133-178,2000.
4. T. Arts, SystemDesription: The Dependeny Pair Method,in Pro.RTA '00,
LNCS1833,261-264,2000.
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