A Practical View on Renaming

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by  Marija Kulaš

A Practical View on Renaming
MarijaKulasˇ
FernUniversita¨tinHagen,WissensbasierteSysteme,58084Hagen,Germany
[email protected]
Werevisitvariablerenamingfromapractitioner’spointofview,presentingconceptswefounduse-
fulindealingwithoperationalsemanticsofpureProlog.Aconceptofrelaxedcorerepresentationis
introduced,uponwhichaconceptofprenamingisbuilt. Prenamingformalizestheintuitivepractice
ofrenamingtermsbyjustconsideringthenecessarybindings,wherenowsomepassive”bindings”
x/xmaybenecessaryaswell. Asanapplication,aconstructiveversionofvariantlemmaforimple-
mentedHornclauselogic hasbeenobtained. There,prenamingsmadeitpossibleto incrementally
handlenew(local)variables.
1 Introduction
Forlogic program analysis orformal semantics, the issue of renaming variables and generally handling
substitutions is inevitable. Yet the image of substitutions in logic programming research is a somewhat
taintedone,atleastsinceithasbeenpointedoutbyH.-P.Ko[16,p.148]thattheoriginalclaimofstrong
completeness of SLD-resolution needs to be amended, due to a counter-example using the fact that
x isnotmoregeneralthan x . Theexamplemaylookcounter-intuitive, butitcomplieswith
(cid:16)f(y,z)(cid:17) (cid:16)f(a,a)(cid:17)
the definition of substitution generality. Also, by composing substitutions, properties like equivalence,
idempotencyorrestrictionarenotpreserved. Lastly,duetogroupstructureofrenamings, permutingany
number ofvariables amounts to”doing nothing”, asin xy ∼ e ,andsuch equivalences arealsofeltto
(cid:16)yx(cid:17)
be counter-intuitive. Hence the prevalent sentiments that substitutions are ”a quite hard matter to deal
with”([13])or”verytricky”([16]). Asaremedy,inthecontextofaggregatingmostgeneralunifiersina
logic programming computation some helpful new concepts and operators were proposed, like parallel
composition insteadoftraditional composition([13])andresultant insteadofanswersubstitution ([12]).
Still, for almost anyone embarking on a journey of logic program analysis or formal semantics, sooner
orlatertheneedforrenamingvariables andgenerally handlingsubstitutions inanewcontextarises.
In case of this author, the need arose while trying to prove adequacy of an operational semantics
for pure Prolog, S1:PP [9], and the context was one of extensibility: Given is a pair of queries that are
alphabeticvariantsofeachother. AstheirrespectiveS1:PPderivationsproceedtodevelop,newvariables
maycropup, duetostandardization-apart (herecalled local variables, Subsection 6.1),butthestatus of
being variant should hold. This setup is known from the classical variant lemma ([11]). Additionally,
thecorresponding variables needtobecollected, obtaining ateachstepthetemporary variance between
the derivations. As an example, assume the first query is p(z,u,x) and the second p(y,z,x). There is
only one relevant renaming, r = zuy . Now assume in the next step the first derivation acquires the
(cid:16)yzu(cid:17)
variabley,andthesecondw. Therelevantrenamingthistimewouldber ′= zu y w . Clearly,r ′ isnot
(cid:16)yzw u(cid:17)
anextension ofr ,whichmakesitseemunsafetoproceed: aresomeproperties ofthepreviousstepnow
indanger? Sothequestion is,howto”safely” extendavariable mapping. Forthispurpose, inSection5
we introduce a slight generalization of renaming, called prenaming. It is a mathematical underpinning
S.SchwarzandJ.Voigtla¨nder(Eds.):29thand30thWorkshops
on(Constraint)LogicProgrammingand24thInternational (cid:13)c M.Kulasˇ
WorkshoponFunctionaland(Constraint)LogicProgramming Thisworkislicensedunderthe
(WLP’15/’16/WFLP’16). CreativeCommonsAttributionLicense.
EPTCS234,2017,pp.27–41,doi:10.4204/EPTCS.234.3
28 APracticalViewonRenaming
oftheintuitive practice ofrenaming termsbyjustconsidering thenecessary bindings, andnotworrying
whethertheresultisapermutation. Intheaboveexample,renaming p(z,u,x)to p(y,z,x)meansmapping
z7→ y,u7→ z and x7→x. Intuitively, only z7→ y,u7→ z are considered necessary bindings, giving the
”renaming” zu . Forprenaming,x7→xisnecessaryaswell. Itisbasedonrelaxedcorerepresentation,
(cid:16)yz(cid:17)
whichisnothingelsethanallowingsomex7→xpairsalongside ”real”bindings, asplaceholders.
Prenamings relate to and are inspired by previous work as follows. In [14], the concept of transla-
tion is defined, upon which alphabetic variance and standardization apart are built; this is the same as
prenaming but for relaxed core (page 31). A safe prenaming is more general than renaming for a term
from [11], and it maximizes W in the notion of W-renaming from [6] (page 33). Also, it generalizes
substitution renamingfrom[1](subsubsection 5.3.2).
In Section6, prenamings are used to express and prove a propagation claim for implemented Horn
clause logic, by means of local variable extension (Lemma6.3). As a corollary, a variant lemma is
obtained (Theorem6.5). Underway, wetouch onthediscrepancy betweentherather abundant theory of
logicprogramming andascarcity ofmathematical claimsforimplemented logicprogramming systems.
While there are some formal proofs of properties like nominal unification [17], for logic programming
systems or their compilation such are still few and far between, a notable exception being [15]. New
concepts likeprenaming maybeofhelp.
2 Substitution
First we need a bit of notation. Assume two disjoint sets: a countably infinite setVof variables and a
set Funof shapes. IfW ⊆V, any mapping F with F(W)⊆Vshall be called variable-pure onW. A
mapping variable-pure on thewhole set ofvariablesVshall be simply called variable-pure. IfV\W is
finite,W issaidtobeco-finite. AmappingF isinjectiveonW,ifwheneverF(x)=F(y)forx,y∈W also
holds x=y. Each f/n∈Funconsists of a functor f and an associated number of arguments n, called
arity. Functorsofarity0arecalledconstants. StartingfromVandFun,dataobjects orterms1 arebuilt:
Anyvariable x∈Vis aterm. Ift ,...,t are terms and f/n∈Fun, then f(t ,...,t )is aterm with shape
1 n 1 n
f/n and constructor f. In case of f/0, the term shall be written without parentheses. If a term soccurs
withinatermt,wewrites∈t. Theorderedpairoftermshandt iswrittenas[h|t],wherehiscalledthe
head andt thetailofthepair. Aspecial caseisanon-empty list,distinguished byitstailbeing aspecial
term [] called the empty list, or a non-empty list itself. A list of n elements is the term [t |[t |[...[t |[]]]]],
1 2 n
conveniently written as [t ,...,t ]. Let Vars(t) be the set of variables in the term t. If the terms s and t
1 n
shareavariable, thatshallbewrittens⊲⊳t. Otherwise,wesays,t arevariable-disjoint, writtenass⊲6 ⊳t.
A recurrent theme in this paper shall be ”relevance”, meaning ”no extraneous variables” relative to
some term or terms. It was used in [2, p.3˙8] in unary sense, i.e. no extraneous variables relative to
(one) term. This usage shall be reflected in the text as follows: A renaming r embedding a prenaming
a is a relevant embedding, if Vars(r )⊆Vars(a ) (Figure1). Additionally, relevance in a binary sense,
concerning two terms, shall also be needed: A mapping F is relevant fort tot , if Dom(F)⊆Vars(t )
1 2 1
andRange(F)⊆Vars(t )(Figure2,Lemma6.3).
2
Definition 2.1 (substitution). A substitution q is a function mapping variables to terms, which is iden-
tity almost everywhere. In other words, it is a function q with domain Dom(q )=Vsuch that the set
Core(q )··={x∈V|q (x)6=x}isfinite.2
1InProlog,everythingisaterm,andsoshalltermbeherethetopmostsyntacticconcept.
2[7]speaksoffinitesupport.
M.Kulasˇ 29
ThesetCore(q )shallbecalledtheactivedomain3 orcoreofq ,anditselementsactivevariables4 of
q . ThesetRan(q )··=q (Core(q ))istheactiverangeofq . Avariablexsuchthatq (x)=xshallbecalled
apassive variable, orafixpoint, forq . Also, wesaythat q isactive onthevariables from Core(q ), and
passiveonalltheothervariables. IfCore(q )={x ,...,x },wherex ,...,x arepairwisedistinctvariables,
1 k 1 k
andq mapseachx tot,thenq shallhavethecorerepresentation {x /t ,...,x /t },ortheperhapsmore
i i 1 1 k k
visual x1 ...xk . Eachpairx,t iscalled thebinding forx inq , denoted byx/t ∈q . Oftenweidentify
(cid:16)t1 ...tk(cid:17) i i i i i
a substitution with its core representation, and thus regard itas a syntactical object, a term representing
afiniteset. Sothesetofvariablesofasubstitution isdefinedasVars(q )··=Core(q )∪Vars(Ran(q )).
Thenotionsofrestrictionandextensionofamappingshallalsobetransportedtocorerepresentation:
ifq ⊆s ,wesayq isarestrictionofs ,ands isanextensionofq . Therestrictionq ↾ ofasubstitutionq
W
onasetofvariablesW ⊆Visdefinedasfollows: ifx∈W thenq ↾W(x)··=q (x),otherwiseq ↾W(x)··=x.
Therestriction ofq uponthevariables oft isabbreviated asq ↾t ··=q ↾Vars(t).
The composition q ·s of substitutions q and s is defined by (q ·s )(x)··=q (s (x)). Composition
maybeiterated,writtenass n··=s ·s n−1forn≥1,ands 0··=e . Heree ··=()istheidentityfunctionon
V. Incase avariable-pure substitution r isbijective, itsinverse shall bedenoted asr −1. Asubstitution
q satisfying theequalityq ·q =q iscalledidempotent.
Definitionofsubstitution isenhancedfromvariablestoarbitrarytermsinastructure-preserving way
byq (f(t1,...,tn))··= f(q (t1),...,q (tn)). Ift isaterm,thenq (t)isaninstance oft viaq .
Example2.2. xwu v · u v xyzw = 6u6vxy z w6x6w6u6v = xy z w .
(cid:0)u v xw(cid:1) (cid:16)xwyuv z(cid:17) (cid:16)6u6vyxw z6u6v6x6w(cid:17) (cid:16)yxw z(cid:17)
3 Renaming
Definition3.1(renaming). Arenamingofvariables isabijectivevariable-pure substitution.
In [6], it is synonymously called ”permutation”. We shall reserve the word for the general case where
movement of infinitely many variables is possible. Here we synonymously speak of finite permutation
duetothefactthat,beingasubstitution, anyrenaminghasafinitecore,andLegacy3.3holds.
Duetostructurepreserving, ifs∈t thens (s)∈s (t). Forbijectivesubstitutions (i.e.renamings), the
converse property holdsaswell,giving
Lemma 3.2 (renaming stability of ”=”, ”∈”, ”⊲6 ⊳”). Let r be a renaming and s,t be terms. Then s=t
iff r (s)=r (t),andalsos∈t iff r (s)∈r (t). Asaconsequence, s⊲6 ⊳t iff r (s)⊲6 ⊳r (t).
Legacy3.3([10]). Asubstitution r isarenamingiffr (Core(r ))=Core(r ).
Legacy3.4([6]). Everyinjectivevariable-pure substitution isarenaming.
So composition of renamings is a renaming. The next property is about cycle decomposition of a
finitepermutation.
Lemma 3.5 (cycles). Let s be a variable-pure substitution. It is injective iff for every x∈Vthere is
n∈Nsuchthats n(x)=x.
Proof. Assumes injective,andchoosex ∈V. Ifs (x )=x ,wearedone. Otherwise,s i(x )6=s i−1(x )
0 0 0 0 0
forall i≥1, due toinjectivity. Hence, s i−1(x )∈Core(s )foreveryi≥1. Because ofthefiniteness of
0
Core(s ), there is m>k≥1 such that s m(x )=s k(x ). Due to injectivity, s m−1(x )=s k−1(x ). By
0 0 0 0
3Traditionallycalledjustdomain. Thismaybeconfusing,sinceintheusualmathematicalsenseitisalwaysthewholeV
thatisthedomainofanysubstitution.
4Thenameactivevariableappearsin[8].
30 APracticalViewonRenaming
iteration we get n··=m−k. For the other direction, assume s (x)=s (y), and minimal m,n such that
s n(x)=x, s m(y)=y. Considerthecasem6=n,saym>n. Thens m−n(y)=s m−n(x)=s m−n(s n(x))=
s m−n(s n(y))=s m(y)=y,contradicting minimalityofm. Hencem=n,sox=s n(x)=s n(y)=y. ♦
4 Relaxed core representation
If there is a substitution s mapping a term s on a term t, then it is mapping each variable in s on a
subterm of t. It is possible that a variable stays the same, so if we want our mapping to explicitely
coverallvariablesins,asinthepromisedapplication (Section6),thennecessarilyx/xwouldhavetobe
tolerated asa”binding”.
To cater for such wishes, the core of the substitution s can be relaxed to contain some passive
variables, raising those above the rest, asitwere. Thissimple technique isuseful beyond thecontext of
renaming, soweassumearbitrarysubstitutions.
Definition 4.1 (relaxed core). If Core(s )⊆{x ,...,x }, where variables x ,...,x are pairwise distinct,
1 n 1 n
then{x ,...,x }shallbecalledarelaxedcoreand x1 ... xn shallbecalledarelaxedcorerepresen-
1 n (cid:16)s (x1)...s (xn)(cid:17)
tationfors . Ifwefixarelaxedcorefors ,itshallbedenotedC(s )··={x1,...,xn}. Theassociatedrange
s (C(s )) wedenote asR(s ). The setof variables ofs isas expected,V(s )··=C(s )∪Vars(R(s )). To
getbacktothetraditional representation, wedenoteby[s ]the(non-relaxed) corerepresentation ofs .
Forextending, substitutions aretreatedlikesetsofactivebindings, so(disjoint) unionmaybeused:
Definition 4.2 (sum of substitutions). If s = x1 ...xn and q = y1...ym are substitutions in relaxed
(cid:16)s1 ...sn(cid:17) (cid:16)t1 ...tm(cid:17)
representation suchthat{y1,...,ym}⊲6 ⊳{x1,...,xn},thens ⊎q ··=(cid:16)xs11......xsnnyt11 ......ytmm(cid:17)isthesumofs andq .
ForSubsection 6.2,backwardcompatibility ofanextensionshallbeneeded.
Lemma4.3(backwardcompatibility). Lets ,q besubstitutions andxbeavariable. Then(s ⊎q )(x)=
s (x) iff q (x)=x.
Proof. Ifx6∈C(q ),thenq (x)=x,and(s ⊎q )(x)=s (x). Ifx∈C(q ),then(s ⊎q )(x)=q (x)andalso
x6∈C(s ),hences (x)=x. Thecondition (s ⊎q )(x)=s (x)collapses toq (x)=x. ♦
Passivityofq onatermt isguaranteed ifs is”complete”fort,i.e.laysclaimtoallitsvariables:
Definition 4.4 (complete for term). Let s be given in relaxed core representation. We say that s is
completefort ifVars(t)⊆C(s ).
Insuchacasethereisnodangerthatanextensionofs mightmapt differently froms :
Corollary 4.5(backward compatibility). Ifs iscomplete fort,thenforanyq holds: s ⊎q iscomplete
fort and(s ⊎q )(t)=s (t).
5 Prenaming
In practice, one would like to change the variables in a term without bothering to check whether this
changeisapermutationofvariables,i.e.arenaminginthesenseofDefinition3.1. Forexample,theterm
p(z,u,x) can be changed to p(y,z,x) using mapping z7→y, u7→z, x7→x. Let us call such a mapping
prenaming5.
5Findinganappropriatenamecanbeastruggle.Shortlistedwerepre-renamingandproto-renaming.
M.Kulasˇ 31
Likeanysubstitution, aprenaming a shallalsoberepresented finitely, but inrelaxed core represen-
tation,inordertocapturepossiblex7→xpairings. ThesetC(a )isfixedbythetermstomap. Obviously,
injectivity isimportantforsuchamapping, since p(z,u,x)cannotbemappedon p(y,y,x)withoutlosing
avariable. Hence,
Definition 5.1 (prenaming). A prenaming a is a variable-pure substitution injective on a finite set of
variablesC(a )⊇Core(a ).
Clearly, anyrenamingisaprenaming. ForTheorem6.5,weneedtoextendagivenprenaming.
Lemma 5.2 (extension of prenaming). Let a = x1...xn and b = u1 ...um be prenamings such that
(cid:16)y1...yn(cid:17) (cid:16)v1 ...vm(cid:17)
{u ,...,u }⊲6 ⊳{x ,...,x } and {v ,...,v }⊲6 ⊳{y ,...,y }. Then a ⊎b = x1...xn u1...um is also a pre-
1 m 1 n 1 m 1 n (cid:16)y1...yn v1 ...vm(cid:17)
naming, withC(a ⊎b )=C(a )⊎C(b )andR(a ⊎b )=R(a )⊎R(b ).
Plotkin’s concept of translation [14, p.46] corresponds to prenaming without passive bindings.
There,theinversetranslationfort ··=(cid:16)yzuz(cid:17)wouldbetinv··=(cid:0)yzuz(cid:1). Clearly,tinv·t =(cid:0)yzuz(cid:1)·(cid:16)yzuz(cid:17)=(cid:0)yz(cid:1),
which is not identity substitution. Although (t ·t )↾ =e , so t is reversible and thus ”safe” to use
inv {z,u}
on{z,u},onemightinstinctivelybewaryofthepossibilitythathandlingseveraltranslations inthesame
computation could somehowproduce ”unsafety”. Presumably forthatreason, theconcept oftranslation
didnotcatchon,anditismeanwhilecustomarytodefinealphabeticvarianceusingrenamingratherthan
translation ([2]). Werevisit Plotkin’s concept, enriched with passive bindings and deemed fit for a new
name, prenaming, and show that its safe application on a term and safe (even backward-compatible)
extension areeasilyachievable, thusjustifying theintuitivepractice.
5.1 The question ofinverse
Soaprenaming ismorenatural inpractice, buta”full” renamingisbetter mathematically tractable, due
toitsbeinginvertibleonV. Thenextpropertyshowshowtoextendaprenaminga toobtainarenaming,
and arelevant one atthat, i.e. active only on the variables fromV(a ). The claim is essentially given in
[12], [2] and [1] with emphasis on the core6 of such an extension. Originally the claim appears in [6],
withemphasisontheextentofcoincidence7,whichisourconcernaswell. Werephrasetheclaimaround
thenotionofprenaming, andprovideaconstructive proofbasedonLemma3.5.
Theorem 5.3 (embedding). If a is a prenaming, there is a renaming a which coincides with a on
V\(R(a )\C(a ))suchthatVars(a )⊆V(a ). Additionally,ifa (x)6=xonC(a ),thena (x)6=xonV(a ).
a (x), ifx∈C(a )

a (x)··=z, ifx∈R(a )\C(a )anda m(z)=xformaximalm≤n

x, outside ofC(a )∪R(a )


Figure1: Closure, thenatural relevantembedding
6[2,p.23]: ”Everyfinite1-1mapping f fromAontoBcanbeextendedtoapermutationgofA∪B. Moreover,if f hasno
fixpoints,thenitcanbeextendedtoagwithnofixpoints.”
7[6, p.35]: ”LetWbeaco-finitesetof variables(...) andlets beaW-renaming. Thenthereisapermutationp which
coincideswiths onthesetW.”
32 APracticalViewonRenaming
Proof. If a is a prenaming, then C(a ) and R(a ) are sets of n distinct variables each. The wanted
renamingisconstructed inFigure1,withtheintentiontoclosethepossiblyopenchainx,a (x),a 2(x),...
Soletusseewhetherforeveryxthereisa jsuchthata j(x)=x. Ifx∈C(a ),westartasintheproofof
Lemma3.5, and consider the sequence x,a (x),a 2(x),... SinceC(a ) is finite, either we get two equals
(and proceed as there), or we get a k(x)6∈C(a ) and are stuck. For y··=a k(x) we know a (y)=z such
thata m(z)=ywithmaximalm,som≥k. Therefore, a m(a (y))=y=a k(x). Duetoinjectivity ofa on
C(a ),wegeta m−k(a (a k(x)))=x,andhencea m+1(x)=x.
The cases x ∈ R(a )\C(a ) or x 6∈ C(a )∪R(a ) are easy. By Lemma3.5, a is injective. By
Legacy3.4,a isarenaming. Thediscussion ofthecasea (x)6=xonC(a )isstraightforward. ♦
Definition 5.4 (closure of a prenaming). The renaming a constructed in Figure1 shall be called the
closureofa .
Remark 5.5 (relevant embedding is not unique). Let a = zuyw1 , and let us embed it in a relevant
(cid:16)yzxw2(cid:17)
renaming. The Figure1 gives a = zuyw1 xw2 . But r = zuyw1 x w2 is also a relevant renaming
(cid:16)yzxw2uw1(cid:17) (cid:16)yzxw2w1 u (cid:17)
which is embedding a . In the usual notation for cycle decomposition, r = {(x,w ,w ,u,z,y)} and
1 2
a ={(x,u,z,y),(w ,w )}.
1 2
If we reverse the prenaming, the closure algorithm shall be closing the same open chains but in the
opposite direction, hence
Lemma5.6(reverseprenaming). Leta ··=(cid:16)xy11......xynn(cid:17)anda inv ··=(cid:16)yx11 ......yxnn(cid:17). Thena inv =a −1.
Remark 5.7 (closure is not compositional). Takea ··= zuy and r ··= xy . Then a = zuyx , r ·
(cid:16)yzx(cid:17) (cid:16)yx(cid:17) (cid:16)yzxu(cid:17)
a = zux , r ·a = zux andr ·a = zuxy .
(cid:0)xzu(cid:1) (cid:16)xzy(cid:17) (cid:16)xzyu(cid:17)
Remark 5.8 (closure is not monotone). If a ⊇a ′, then not always a ⊇a ′. To see this, let a = zuy
(cid:16)yzx(cid:17)
anda ′= zu . Thena ′= zuy anda = zuyx .
(cid:16)yz(cid:17) (cid:16)yzu(cid:17) (cid:16)yzxu(cid:17)
5.2 Staying safe
Let us look more closely into Remark5.8: a (y) = xanda (x) = x, so y and x may not simultane-
ously occur in the candidate term. Otherwise, a variable shall be lost, which we call ”aliasing”, like
in y (p(x,f(y)))= p(x,f(x)).
(cid:0)x(cid:1)
Definition5.9(aliasing). Leta beaprenaming. Ifx6=ybuta (x)=a (y),thena isaliasingxandy.
SowhatRemark5.8meansis: ifwewanttousea onalarger setthanC(a ), thenthesetPit(a )··=
R(a )\C(a )shouldbeavoided, becausealiasing mayhappen. But,luckily, itscomplementissafe:
Lemma5.10(larger set). Aprenaming a isinjective ontheco-finite setV\Pit(a ). Thesetismaximal
containingC(a ).
Proof. Let x,y ∈V\Pit(a ). Is it possible that a (x)=a (y)? Possible cases: If x,y∈C(a ), then by
definition of prenaming a (x)6=a (y). If x,y6∈C(a ), then a (x)=x6=y=a (y). It remains to consider
the mixed case x∈C(a ),y6∈C(a ). We have a (x)∈R(a ) and a (y)=y. So is a (x)=y possible? If
yes,theny∈R(a ),butsincey6∈C(a ),thatwouldmeany∈Pit(a ). Contradiction.
The set cannot be made larger: if y∈ Pit(a ), then there is x ∈C(a ) with x 6= y and a (x) = y=
a (y). ♦
M.Kulasˇ 33
Definition5.11(injectivitydomain). SinceInDom(a )··=V\Pit(a )isthelargestco-finitesetcontaining
C(a )onwhicha isinjective, itshallbecalledtheinjectivity domainofa .
Theinjectivity domainofaprenaming isclearlytheonlysafeplaceforittobemappingtermsfrom.
Definition5.12(safetyofprenaming). Aprenaminga issafe8 foratermt ifVars(t)⊆InDom(a ).
Clearly, InDom(a )=C(a )∪(V\R(a )),soa issafeforitsrelaxedcore. Hence,
Corollary 5.13(completeandsafe). Ifaprenaming iscompleteforaterm,itissafeforthatterm.
Foraprenaminga withthequalityR(a )=C(a ),i.e.arenaming,itisnosurprisethatInDom(a )=V
andhencesafetyisguaranteed foranyterm.
A prenaming behaves like a renaming on its injectivity domain, since it coincides with its closure
there. Thisfollowsimmediately fromTheorem5.3:
Lemma5.14(injectivity domain). Letx∈InDom(a ). Thena (x)=a (x).
Corollary5.15(prenamingstability). Ageneralization ofLemma3.2holds: Lets,t betermsanda bea
prenamingsafefors,t. Thens=t iffa (s)=a (t)andalsos∈t iffa (s)∈a (t). Asaconsequence, s⊲6 ⊳t
iffa (s)⊲6 ⊳a (t).
Ourdefinition ofprenaming wasinspired bythefollowingmoregeneralnotionfrom[6].
Definition 5.16 (W-renaming, [6]). LetW ⊆V. Asubstitution s is aW-renaming ifs isvariable-pure
onW,ands isinjective onW.
With this notion, Lemma5.10 can be summarized as: InDom(a ) is a co-finite set of variables, and
thelargest setW ⊇C(a )suchthata isaW-renaming.
Whataboutsafety ofextension? Ifa issafefort,a ⊎b doesnothavetobe,evenifb (t)=t,asthe
following exampleshows: a ··= v , b ··= zuy , t ··= p(x)(herenoaliasinghappened, though). The
(cid:0)w(cid:1) (cid:16)yzx(cid:17)
nexttwoclaimsaddress safetyofextension.
Lemma5.17(monotonicity). Assumea ⊎b isdefined. Then
1. InDom(a )∪InDom(b )=V
2. InDom(a )∩InDom(b )⊆InDom(a ⊎b )
Proof. Since(V\A)∪(V\B)=V\(A∩B)),andPit(a )⊲6 ⊳Pit(b ),wegetInDom(a )∪InDom(b )=V.
Further, (V\A)∩(V\B)=V\(A∪B)and soPit(a ⊎b )=(R(a )⊎R(b ))\(C(a )⊎C(b ))⊆(R(a )\
C(a ))∪(R(b )\C(b ))=Pit(a )∪Pit(b ). ♦
In Remark5.8, Pit(a ′) = {y}, Pit( y ) = {x}, and Pit(a ) = {x}, hence InDom(a ′) = V\{y},
InDom( y )=V\{x}andInDom(a )=(cid:0)Vx(cid:1)\{x}.
By t(cid:0)hxe(cid:1)last claim, staying within InDom(a ) and InDom(b ) ensures staying within InDom(a ⊎b ).
By assuming a bit more about a than just safety, we may ignore the nature of extension b , and still
ensuresafetyandevenbackwardcompatibility ofa ⊎b . ThisshallbeusedinSection6.
Theorem5.18(safetyofextension). Assumea ⊎b isdefined.
1. Ifa issafefort and b issafefort,thena ⊎b issafefort.
2. Ifa iscompletefort,thena ⊎b iscomplete(hencesafe)fort,and(a ⊎b )(t)=a (t).
ThefirstpartfollowsfromLemma5.17andthesecondfromCorollary4.5andCorollary 5.13.
8Safeprenamingismoregeneralthanrenamingforatermin[11,p.22],sincewedonotrequireCore(a )⊆Vars(t).
34 APracticalViewonRenaming
5.3 Variantofterm andsubstitution
The traditional notion ofterm variance, which is term renaming, shall be generalized toprenaming. As
a special case, substitution variance is defined, inspired by substitution renaming from [1]. For this,
substitution shall once again be regarded as a special case of term. The term is of course the relaxed
core representation. This concept shall come in handy for proving properties of renamed derivations
(Subsection 6.2).
5.3.1 Termvariant
Definition5.19(termvariant). Ifa isaprenamingsafefort,thena (t)isavariantoft,writtena (t)∼=t.
Theparticular variance andthedirectionofitsapplication maybeexplicated bys=a t iffs=a (t).
If s ∼= t, then there is a unique a mapping s to t in a complete and relevant9 manner, i.e. map-
ping each variable pair and nothing else, as computed by Figure2. The algorithm makes do with only
one set for equations and bindings, thanks to different types. Termination can be seen from the tuple
(lfun (E),card (E)) decreasing in lexicographic order with each rule application, where lfun (E) is
= = =
thenumberoffunction symbolsinequations inE,andcard (E)isthenumberofequations inE.
=
Start from the set E ··={s=t} and transform according to the following rules. The transformation is
boundtostop. Ifthestopwasnotduetofailure,thenthefinalsetE istheprenamingofstot,Pren(s,t).
elimination E⊎{x=y} E, ifx/y∈E
failure: alias E⊎{x=y} failure, if(x/z∈E,z6=y)or(z/y∈E,z6=x)
binding E⊎{x=y} E∪{x/y}, if(x/ 6∈E)and( /y6∈E)
failure: instance E⊎{x=t} failure, ift 6∈V; E⊎{t =x} failure, ift 6∈V
decomposition E⊎{f(s ,...,s )= f(t ,...,t )} E∪{s =t ,...,s =t }
1 n 1 n 1 1 n n
failure: clash E⊎{f(s ,...,s )=g(t ,...,t )} failure, if f 6=gorm6=n
1 n 1 m
Figure2: Computingtheprenaming ofstot
Notation 5.20(epsoid). Theprenaming constructed inFigure2shall besimply called theprenaming of
stot,anddenoted Pren(s,t). Itiscompleteforsandrelevantforstot.
In case s =t, we obtain for Pren(s,t) essentially the identity substitution. However, regarded as
prenamings, Pren(t,t) and e are not the same. A prenaming a with relaxed core W mapping each
variable onitself(inother words,C(a )=W and[a ]=e )shallbecalled theW-epsoid anddenoted e .
W
Foratermt,weabbreviate et ··=eVars(t).
Regarding composition, an epsoid behaves just like e . Its use is for providing completeness, and
henceextensibility, bymeansofplaceholder bindingsx/x.
5.3.2 Specialcase: substitutionvariant
Even substitutions themselves can berenamed. Torename a substitution, one regards itas asyntactical
object, a set of bindings, and renames those bindings. If r is a renaming and s is a substitution, [1]
9”Relevant”inthebinarysense(page28).Incaseofprenaming,wenaturallyuseCasDomandRasRange.
M.Kulasˇ 35
defines substitution renaming by r (s )··={r (x)/r (s (x)) | x∈Core(s )}. It is easy tosee that r (s )is
a substitution in core representation. For this only two properties of r were needed: variable-purity on
Vars(s )andinjectivity onVars(s ). Theserequirements areclearlyfulfilledbyprenamings safeons as
well. Hence,
Definition 5.21 (substitution variant). Let s be a substitution and let a be a prenaming safe for s , i.e.
Vars(s )⊆InDom(a ). Thenavariantofs bya is
a (s )··={a (x)/a (s (x)) |x∈Core(s )} (1)
We may write q =a s if q =a (s ), as with any other terms. As can be expected, the concept of
variance by prenaming is well-defined, owing to safety. Otherwise, the result of prenaming would not
evenhavetobeasubstitution again,asinthecaseofa = y ands = xy .
(cid:0)x(cid:1) (cid:0)ab(cid:1)
Lemma 5.22 (well-defined). Substitution variant is well-defined, i.e. (1) is a core representation of a
substitution, anda doesnotintroduce aliasing.
Proof. Let Core(s ) = {x ,...,x }. Due to injectivity of a on Vars(s ), if a (x) = a (x ), then x =
1 n i j i
x , so i = j. To finish the proof that (1) a core representation, observe x ∈ Core(s ) iff x 6= s (x) iff
j
a (x)6=a (s (x)), due to injectivity again. Realiasing, by Corollary5.15, if a (s (x))⊲⊳a (s (x )), then
i j
s (x)⊲⊳s (x ),meaningthata doesnotintroduce aliasing. ♦
i j
FromDefinition5.21andLemma5.14follows
Lemma5.23. Lets beasubstitution, a ,b beprenamingsanda (s )and(a ·b )(s )bedefined. Then
1. (a ·b )(s )=a (b (s ))
2. a (s )=a (s )
Forthecaseof”full”renaming, thereisawaytodissolvethenewexpression:10
Legacy5.24([1]). Foranyrenamingr andsubstitution s
r (s )=r ·s ·r −1
Wouldsuchaclaimholdfortheweakened case,prenamings?
Theorem5.25(substitution variant). Lets beasubstitution anda beaprenamingsafefors . Then
1. a (s )·a =a ·s
2. a (s )=a ·s ·a −1
Proof. First part: According to Definition5.21, for every x∈Vholds (a (s )·a )(x)=a (s (x)). Since
any substitution is structure-preserving, the claim holds for any term t as well. Second part: From the
firstpartweknowa (s )·a =a ·s ,hencea (s )=a ·s ·a −1. ByLemma5.14,a (s )=a (s ). ♦
It is known that idempotence and equivalence of substitutions are not compatible with composition
[6]. Luckily, theconcept ofvariance, withconstantprenaming, doesnotsharethishandicap:
Theorem5.26(compositionality). Lets ,q besubstitutions anda betheirsafeprenaming. Then
a (s ·q )=a (s )·a (q )
Proof. Since Vars(s ·q ) ⊆ Vars(s )∪Vars(q ), clearly Vars(s ·q ) ⊆ InDom(a ). By Theorem5.25,
a (s )·a (q )=a ·s ·a −1·a ·q ·a −1=a ·s ·q ·a −1=a (s ·q ). ♦
10animmediateconsequencebeingr (s )6=r ·s
36 APracticalViewonRenaming
6 Application
ImplementinglogicprogrammingmeansthatthefreedomofHornclauselogic(HCL)mustberestrained:
• mostgeneralunifier(mgu)isprovidedbyafixedalgorithm,
• standardization-apart isprovided byafixedalgorithm.
Every implementation of HCL is parametrized by the two algorithms. Here we shall consider only the
unificationalgorithm,sobyHCL(U)animplementationofHCLusingunificationalgorithmUisdenoted.
Fromtheliterature (variant lemma)weknowthatsuch arestriction isnotcompromising soundness and
completenessofSLD-resolution. Yet,theremaybelotsof”lowlier”claimswhichmoreorlessimplicitly
rely on freedom of mgus and standardization-apart. For example, with both choices fixed we may not
anymorejustrename anSLD-derivation wholesale (theresolvents, themgus, theinputclauses), aswas
possibleinHornclauselogic,basedonLemma3.2. Thisisbecausethetwoalgorithmsdonothavetobe
renaming-compatible. Infact,thesecondonecannotbe,whichmakesclaimslikeLemma6.3necessary.
Let us cast a look at the first restriction. For any two unifiable terms s,t holds that the set of their
mgus, written as Mgus(s,t), is infinite. On the other hand, in practice any unification algorithm U
produces, for the given two unifiable terms, just one deterministic value as their mgu. We shall denote
thisparticular mguofsandt asU(s,t),thealgorithmic mguofsandt obtainedbyU.
Theabundancyofmgusisnotonlygood,italsostandsinthewayofproofs. Thesimplestunification
problem p(x) = p(y) has among others two equally attractive candidate mgus, (x) and (y). Assume
y x
ourunification algorithm decided upon(x). Assumefurther thatwerename theprotagonists andobtain
y
p(x)= p(z). Whatmgu shall be chosen this time? Toensure some dependability in this issue, weshall
requireofanyunificationalgorithm thefollowingsimplerequirement, postulated asanaxiom:
Axiom 6.1 (renaming compatibility). Let Ube a unification algorithm. For any renaming r and any
equation E,ithastoholdU(r (E))=r (U(E)).
Sinceclassical unification algorithms likeRobinson’s andMartelli-Montanari’s donotdepend upon
theactualnamesofvariables (asobservedin[1]),thisrequirement isinpractice alwayssatisfied.
Remark 6.2 (renaming compatibility of Mgus). For every r and E holds Mgus(r (E))=r (Mgus(E)).
This is due to Theorem5.25 and Lemma3.2. Assume s ∈Mgus(s,t), then r (s )(r (s))=r (s (s))=
r (s (t))=r (s )(r (t)). Further, ifq isaunifierofr (s),r (t),thenq ·r isaunifierofs,t,hencethereis
arenamingd withq ·r =d ·s ,givingq =d ·s ·r −1=d ·r −1·r ·s ·r −1=(d ·r −1)·r (s ),meaning
r (s )∈Mgus(r (E)). Fortheotherdirection, observeq =r ·r −1·d ·s ·r −1=r (r −1·d ·s ).
6.1 Handling localvariables inHCL(U)
With Ucomplying to Axiom6.1 and yielding relevant mgus, that is to say with practically any U,11
a propagation result for SLD-derivations can be proved, which leads to a constructive and incremental
versionofthevariantlemma.
Regarding SLD-derivations, for the most part we shall assume traditional concepts as given in [11]
and [2], but with some changes and additions listed below. An input clause K obtained from a pro-
i
gram clause K¯ byreplacing thevariables inorder ofappearance witht ,...,t maybedenoted asK =
1 n i
K¯[t1,...,tn]. AssumenowanSLD-derivationDforGoftheformG֒−⊲K1:s 1 G1֒−⊲K2:s 2 ...֒−⊲Kn:s n Gn.
11ClassicalunificationalgorithmsnotonlysatisfyAxiom6.1butalsoyieldidempotentmgus. Idempotentmgusarealways
relevant([2]).