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9.1 RuleApplication
107
RandomSelection
-
$
-
%
-
Variable
-
Variable
-
,
-
Variable
-
*
-
RuleModier
-
%
-
?
-
Rule
-
RuleIdent
-
(
-
Variable
,
-
)
-
FilterCalls
-
TheRuleExecutionclauseappliesasingleruleor test. . Incaseofarule,therstfound
patternmatchwillberewritten.Applicationwillfailincasenomatchwasfoundandsucceed
otherwise. Variablesandnamedgraphelementscanbepassedintotherule. . Thereturned
graphelementscanbeassignedtovariablesagain.Therulemodier? switchestheruletoa
test,i.e.theruleapplicationdoesnotperformtherewritepartoftherulebutonlytestsifa
matchexists.Therulemodier%isamulti-purpose ag. IntheGrShell(seeChapter20)
it dumps the matchedgraphelementsto stdout; ; indebug-mode (seeChapter 21) it acts
asabreakpoint (whichisitsmainuse infact);youare alsoabletousethis agforyour
ownpurposes,whenusingGrGen.NETviaitsAPIinterface(seeSection2.2.3). Thelter
calls (whichallowe.g. . toorderthematcheslistandrewriteonlythetopones,ortolter
symmetricmatches)areexplainedin15.
TheRuleExecutionclausecanbeappliedtoadenedsequence(cf.18.1),oranexternal
sequence(cf.25.5),too.Applicationwillsucceedorfaildependingontheresultofthebody
ofthe sequencedenitioncalled. . Incaseofsuccess, , the output t variables of thesequence
denitionarewrittentothe destinationvariables of the assignment. . Incase e offailure,no
assignmenttakesplace,sosequencecallsbehavethesameas rulecalls. . Thebreakpoint t %
canbeappliedtoasequencecall,butneitherthe? operatornorallbraces([]).
Thesquarebraces([])introduceaspecialkindofmultipleruleexecution: Everypattern
matchproducedbytherulewillberewritten;ifatleastonewasfound,ruleexecutionwill
succeed,otherwiseitwillfail. IfRule e returnsvalues,thevalueofeachruleapplicationwill
bereturned.Thishappensintheformofanarray,orbetteronearrayperreturnparameter,
withtypearray<T>forreturnparametertypeT.Thereturnparameterarraysarelledwith
asmanyvaluesasthereweresucceedingruleapplications.
Thecountedallbracketingcount[r]=>cassigns the count ofmatchesof rule rtothe
variable c, and applies s r r on n all the matches. . With h count[?r]=>c the e matches are e only
counted,norewritesarecarriedout.
Therandommatchselector$vsearches for allmatches andthenrandomly selects vof
themtoberewritten(butat mostasmuchas areavailable),with$[r1] beingequivalent
toanonymousTempVar=1 & & $anonymousTempVar[r1]. . Rule e applicationwill failin case no
matchwasfoundandsucceedotherwise. Youmaychange e the lower boundfor successby
giving a variable containing the value to apply before thecomma-separated upper bound
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108
RuleApplicationControlLanguage(Sequences)
variable. Incasealowerboundisgiventheupperboundmaybesettounlimitedwiththe
*. An%appendedtothe$denotesachoicepointallowingtheusertochoosethematchto
beappliedfromtheavailableonesinthedebugger(seeChapter 21).
EXAMPLE(55)
The sequence (u,v)=r(x,y) applies therule r with input fromthe variables x and y on
thehostgraphandassignsthereturnelementsfromtheruletothevariables uandv(the
parenthesis around the e out t variables are e always needed, even if there’s s only y one variable
assignedto). Thesequence$[t]determinesallmatchesoftheparameterlessruletonthe
hostgraph,thenoneofthematchesisrandomlychosenandexecuted.Rulerappliedwithall-
bracketingmaylooklike(u:array<Node>,v:array<int>)=[r(x,y)],notethereturnvalue
arrays,comparedtoasingleapplicationwith(u:Node,v:int)=r(x,y).
9.2 LogicalandSequentialConnectives
RewriteSequence
-
RewriteNegTerm
-
$
-
%
-
&
-
^
-
|
-
&&
-
||
-
;>
-
<;
-
RewriteNegTerm
-
(
-
RewriteSequence
-
)
-
RewriteNegTerm
-
!
-
RewriteTerm
-
=
-
>
-
&
-
>
-
|
-
>
-
Variable
-
Agraph rewrite sequenceconsists s ofseveral l rewriteterms s linked d by y operators. . Table
9.1givestheprioritiesandsemanticsoftheoperators,prioritiesinascendingorder. Forcing
executionorderagainsttheprioritiescanbeachievedbyparentheses.Themodier$changes
thesemanticsofthefollowingoperatortorandomlyexecutetheleftortherightoperandrst
(i.e. agstheoperatortoactcommutative);usuallyoperandsareexecuted/evaluatedfrom
left toright if not alteredbybracketing. . In n contrast t the sequences s s, t,u u ins s $<op> t
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9.3 SimpleVariableHandling
109
$<op> u u are executed/ evaluated d inarbitrary y order. . The e modier r % % appendedto the e $
overridestherandomselectionbyauserselection(cf. seeChapter21,choicepoints).
Theassign-tooperator=>optionallyavailableattheendoftheRewriteNegTermassigns
the(negatedincaseof !) ) resultofthe e RewriteTermexecutiontothegivenvariable;the
and-to&>operatorassignstheconjunctionandtheor-to|>operatorassignsthedisjunction
oftheoriginalvalueofthevariablewiththesequenceresulttothevariable.
Operator
Meaning
s1 <; s2
Then-Left,evaluatess1thens2andreturns(/projectsout)theresultof s1
s1 ;> s2
Then-Right,evaluatess1thens2andreturns(/projectsout)theresultofs2
s1 || s2
LazyOr,theresultisthelogicaldisjunction,evaluatess1,onlyif s1isfalse
s2getsevaluated
s1 && s2
LazyAnd,theresultisthelogicalconjunction,evaluatess1,onlyifs1istrue
s2getsevaluated
s1 | | s2
StrictOr,evaluatess1thens2,theresultisthelogicaldisjunction
s1 ^ ^ s2
StrictXor,evaluatess1thens2,theresultisthelogicalantivalence
s1 & & s2
StrictAnd,evaluatess1thens2,theresultisthelogicalconjunction
!s
Negation,evaluatessandreturnsitslogicalnegation
Table9.1: Semanticsandprioritiesofrewritesequenceoperators
9.3 SimpleVariableHandling
SimpleVariableHandling
-
SimpleAssignmentTarget
-
=
-
SimpleOrInteractiveExpression
-
VariableDeclaration
-
Variable
-
SimpleAssignmentTarget
-
Variable
-
VariableDeclaration
-
VariableDeclaration
-
Variable
-
:
-
Type
-
SimpleOrInteractiveExpression
-
Variable
-
Literal

-
%
-
BoolLit
-
$
-
%
-
(
-
Number
-
1.0
-
)
-
$
-
%
-
(
-
Type
-
)
-
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110
RuleApplicationControlLanguage(Sequences)
Variable
-
Word
-
::
-
Word
-
Thesimplevariablehandlinginthesequences allowstoassignavariableoraconstant
toavariable,tointeractivelyqueryforanelementofagiventypeoranumber andassign
ittoavariable,ortodeclarealocalvariable;theseconstructsalwaysresultintrue/success.
Inaddition,abooleanvariablemaybeusedasapredicate;usingsuchavariablepredicate
together with h the sequenceresult t assignment allows todirectly transmit executionresults
fromonepartofthesequencetoanotherone.Furtheron,abooleanconstantmaybeusedas
apredicate.Thesesequenceconstantsbeingoneofthebooleanliteralstrueorfalsecome
inhandyifasequenceistobeevaluatedbutitsresultmustbeapredenedvalue;furtheron
abreakpointmaybeattachedtothem.
Variablescanholdgraphelements,orvaluesofvalue/attributetypes,includingbooleans.
Thetypedexplicitdeclaration(whichmaybegivenatanassignment,renderingthatassign-
mentintoaninitialization)introducesasequencelocalvariable,thenamealonereferences
asequencelocalvariable. Aglobalvariableisaccessedwiththedoublecolonprex,itgets
implicitlydeclaredifnotexistingyet(youcan’tdeclareagraphglobalvariable). Theran-
domnumber assignment v=$(42) assigns anintegerrandomnumber inbetween n 0and41
(42excluded)tothevariablev. Therandomnumberassignmentv=$(1.0)assignsadouble
randomnumber inbetween0.0and1.0exclusivetothevariable v (hereyoucan’t change
theupperboundaswiththeintegerassignment). Appendinga%changesrandomselection
touserselection(deningachoicepoint). Theuserinputassignmentv=$%(string)queries
the user r for r a string value { this s only y works in n the GrShell. . The e user input t assignment
v=$%(Node)queriestheuserforanodefromthehostgraph{thisonlyworksintheGrShell
in debugmode. . The e non simple variable handlingis s givenin 17.1, evenfurther r variable
handlingconstructsaregivenin17.4.
EXAMPLE(56)
Thesequence (x)=s s || (x)=t ;> [r(x)] ] & & !u(::k,::l) ) isexecutedintwohalves,rst
(x)=s || (x)=t, , then n [r(x)] & & !u(::k,::l), , as s the then-right t operator r exercises s the
weakest binding g of f the used operators. . The e evaluation result of the rst t part t is s thrown
away afterit was computed,only the resultof the secondpartdenes theoutcomeof the
sequence. Fromtherstpart,rstsisexecuted,writingxincaseofsuccess. If smatches,
executionoftheleftpartiscompleteastheoutcomeofthelazyorisdeterminedtobetrue,
andtnotneededanymore.Ifitdoesnotmatch,tisexecuted,deningx(incaseofsuccess,
whichweassume).Thentherightpartisexecuted,rstapplyingronallmatchesfoundfor
thepreviouslywrittenxargument,thenuonthevalues storedintheglobalvariablesused
asarguments. Here,uisexecutedduetotheeager r operatorevenif rwasnotfound,thus
forcingtheresulttobefalse. Theresultoftheentiresequence e istruei  rwas s foundat
leastonce,anduwasnotfound(becauseofthenegation).Buttoreallygiveavalidsequence,
weneedtorstdeclarethelocalvariablexspecifyingitstype,incontrasttothedynamically
typed::kand::lglobalvariablesthatwecanuserightaway,sothecompletesequenceis
x:Node ;> > (x)=s || | (x)=t ;> [r(x)] ] & & !u(::k,::l).
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9.4 DecisionsandLoops
111
9.4 DecisionsandLoops
ExtendedControl
-
if
-
f
-
Condition
-
;
-
TrueCase
-
;
-
FalseCase
-
g
-
if
-
f
-
Condition
-
;
-
TrueCase
-
g
-
Theconditionalsequences,orconditionexecution(/decision)statement ifexecutesthe
conditionsequence,andifityieldedtrueexecutesthetruecasesequence,otherwisethefalse
casesequence.Thesequenceif{Condition;TrueCase}isequivalenttoif{Condition;TrueCase;true},
thusgivingalazyimplication.
RewriteTerm
-
RewriteFactor
-
*
-
+
-
[
-
Number
-
]
-
[
-
Number
-
:
-
Number
-
*
-
]
-
Arewritetermconsistsof arewritefactor whichcanbeexecutedmultipletimes. . The
star(*)executesasequencerepeatedlyaslongasitsexecutiondoesnotfail.Suchasequence
alwaysreturnstrue.Asequences+isequivalenttos && s*. . Thebrackets([m])executea
sequencerepeatedlyaslongasitsexecutiondoesnotfailbutmtimesatmost;themin-max-
brackets([n:m])additionallyfailiftheminimumamountnofiterationswasnotreached.
EXAMPLE(57)
Thesequenceif{ (x:Node)=s; ; (::cnt)=r(x,::cnt)* * ; ; count[q(x)]=>::cnt ;> true }
executesrst s,writingtheoutputvaluetothevariablexthatis declaredon-the- y. . The
resultofsisusedtodecidewhichparttoexecute,incaseoftrueit’s(::cnt)=r(x,::cnt)*,
incaseoffalseit’scount[q(x)]=>::cnt ;> > true.Theformerexecutestherulerasoften
as itmatches,incrementing avariable ::cnt usedas counter eachtime(assumingthat t r
containsareturn(incnt+1) for aninput parameterincnt). . Eachmatchis s soughtinthe
host graphat t thestate left behindby the executionof the rule inthe previous iteration
step. Thelattersequencepartexecutestheruleqonallmatchesinthecurrenthostgraph
atonce,assigningthenumberofmatchesfound(equallingthenumberofrewrites)to::cnt.
The result of the overallsequence is always true,for one becausethe star operator used
inthetrue-casealwayssucceeds,andfortheotherbecausethefalse-caseexplicitlyxesits
resulttotheconstanttrue.
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112
RuleApplicationControlLanguage(Sequences)
NOTE(34)
Consider all-bracketing introduced d in the e rst sectionfor rewriting g all matches s of f a rule
insteadofiterationiftheyareindependent. Attention: : Theallbracketingis s notequalto
Rule*. Insteadthisoperator r collects allthematches rstbeforestartingtorewrite. . Soif
onerewritedestroysothermatchesorcreatesnewmatchopportunitiesthesemanticsdier;
in particular r the semantics is unsafe, , i.e.one needs s toavoiddeleting or retyping agraph
elementthatisboundbyanothermatch(willbedeleted/retypedthere).Ontheotherhand
this versionismoreecientandallowsone togetalongwithout markingalready handled
situations (toprevent arule matchingagainandagainbecausethematchsituationisstill
thereaftertherewrite;normallyyouwouldneedsomematchpreventingdevicelikeanegative
orvisited agstohandlesuchasituation).
9.5 Quickreferencetable
Table9.2liststhebasicoperationsofthegraphrewritesequencesataglance.
s ;> t
Executesthent.Successif tsucceeded.
s <; t
Executesthent.Successif ssucceeded.
s | t
Executesthent.Successif sortsucceeded.
s || t
Thesameass | tbut t withlazyevaluation,i.e. . if f sissuccessful,
twillnotbeexecuted.
s & t
Executesthent.Successif sandtsucceeded.
s && t
Thesameass & & tbutwithlazyevaluation,i.e.ifsfails,twillnot
beexecuted.
s ^ t
Executesthent.Successif sortsucceeded,butnotboth.
iffr;s;tg
Execute r. . If f r succeeded, , execute s s and d returnthe result t of f s.
Otherwiseexecutetandreturntheresultof t.
iffr;sg
Sameasiffr;s;trueg
!s
Switchtheresultof sfromsuccessfultofailandviceversa.
$<op>
Userandominsteadofleft-associativeexecutionorderfor<op>.
s*
Executesrepeatedlyaslongasitsexecutiondoesnotfail.
s+
Sameass && s*.
s[n]
Execute s s repeatedly y as long as its execution n does s not failbut t n
timesatmost.
s[m:n]
Sameass[n]butfailsifexecutedlessthanmtimes.
s[m:*]
Sameass*butfailsifexecutedlessthanmtimes.
?Rule
SwitchesRule toatest.
%Rule
Thisisthemulti-purpose agwhenaccessedfromlibGr.Alsoused
forgraphdumpingandbreakpoints.
[Rule]
RewriteeverypatternmatchproducedbytheactionRule.
(v:array<T>)=[r] Searchesforallmatchesandrewritesthemall,forrreturningT.
count[a]=>v
Rewrite every patternmatchproducedbythe action a, andwrite
thecountofthematchesfoundtothevariablev.
true
Aconstantactingasasuccessfulmatch.
false
Aconstantactingasafailedmatch.
v
Abooleanvariableactingasapredicate.
Letr,s,tbesequences,u,v,wvariableidentiers,<op>2f|;^;&;||;&&g
Table9.2: Sequencesataglance
CHAPTER 10
ADVANCED MATCHING AND REWRITING
ThefollowingrewriterulementionedinGeiet al.[GBG
+
06]shows someoftheadvanced
constructs. In n this chapter we e explain n the constructs of f Example 58 that are e exceeding
thebasics alreadyintroducedinChapter5,andthenestednegativesalreadydescribedin
Chapter7. Furthermore,we’llvisitedsomeadditionaladvancedconstructs.
EXAMPLE(58)
1
#using"SomeModel.gm"
2
3
ruleSomeRule{
4
n1:NodeTypeA;
5
n2:NodeTypeA;
6
hom(n1, n2);
7
n1--> n2;
8
n3:NodeTypeB;
9
negative{
10
n3-e1:EdgeTypeA-> n1;
11
if{n3.a1==42*n2.a1;}
12
}
13
negative{
14
n4:Node\(NodeTypeB);
15
n3-e1:EdgeTypeB-> n4;
16
if{typeof(e1)>=EdgeTypeA;}
17
}
18
replace{
19
n5:NodeTypeC<n1>;
20
n3-e1:EdgeTypeB-> n5;
21
eval {n5.a3 =n3.a1*n1.a2;}
22
}
23
}
Theadvancedmodiersintroducedinthefollowingsectionallowtoannotatepatternsor
actionswithkeywordswhichrestrictwhatgraphpatternsareacceptedasmatches(someof
themindependentoftherewritepart,someofthemdependingontherewritespecication).
Butrsttheadvancedmatching constructs s areintroducedinthis section,before theyare
elaboratedoninalatersection: theyallowtorequestmorenegrainormoredynamically
what types to match,aswellas allowingto specify amatch h froma storage. . Followedby
theadvancedrewrite constructswhicharehandledinthesameway(introductionhere,then
elaboration in a later r section); ; they y enable the e specication of f retyping g (relabeling) and
copying,aswellasnodemergingandedgeredirection.
113
114
AdvancedMatchingandRewriting
AdvancedNodeTypeConstructs
-
NodeType
-
TypeConstraint
-
StorageAccess
-
Retyping
-
Merging
-
CopyOperator
-
SpeciesanodeoftypeNodeType,constrainedintypewithaTypeConstraint (seeSec-
tion10.2,TypeConstraint),orboundbyastorageaccess(see13.6,StorageAccess),orretyped
withaReytping (seeSection10.4,Retyping),or mergedwithaMerging g (see e Section10.6,
Merging). Alternatively y itmay dene anodehavingthe sametypeandbearingthesame
attributesasanothermatchednode(seeSection10.5,CopyOperator). Typeconstraintsare
allowedinthepatternpartonly. TheCopyOperatorandtheMergingclauseareallowedin
thereplace/modifypartonly. TheRetypingclauseisachimerawhichrestrictsthetypeof
analreadymatchednodewhenusedontheLHS,andcaststothetargettypewhenusedon
theRHS,whichisitsprimaryuse.
AdvancedEdgeTypeConstructs
-
EdgeType
-
TypeConstraint
-
StorageAccess
-
Retyping
-
CopyOperator
-
TheAdvancedEdgeTypeConstructsspecifyanedgeoftypeEdgeTypeoracopyofanedge.
Typeconstraintsareallowedinthepatternpartonly(seeSection10.2,TypeConstraint);the
same holds s for the e storage access (see 13.6, StorageAccess). . The e CopyOperator r and the
Redirectclauseareallowedinthereplace/modifypartonly(seeSection10.5,CopyOperator,
seeSection10.7,Redirect). TheRetypingclauseisachimerawhichrestrictsthetypeofan
alreadymatchededgewhenusedontheLHS,andcaststothetargettypewhenusedonthe
RHS(itsprimaryuse). Furthermoreedgesmayberedirected,thisisshowninSection10.7,
Redirection.
10.1 RuleandPatternModiers
TestModier
-
exact
-
induced
-
10.2 StaticTypeConstraint
115
RuleModier
-
TestModier
-
dpo
-
dangling
-
identification
-
By default GrGen.NET performs rewritingaccording toSPOsemantics as explained
inSection5.4.1. Thisbehaviourcanbechangedwithpatternmodiersandrule e modiers
(andtheotheradvancedrewriteconstructsintroducedinthefollowingsectionswhichspoil
the theoretical foundation but are highly useful in practice). . Suchmodiers s add certain
conditionstotheapplicabilityofapattern.Theideaistomatchonlypartsofthehostgraph
that look k more e or less exactly like the pattern. . The e levelof\exactness" depends onthe
chosenmodier. Apatternmodierinfrontoftherule/test-keywordisequivalenttoone
modier-statementinsidethepatterncontainingallthespeciednodes(includinganonymous
nodes). Table10.1liststhepatternmodiers s withtheirsemantics,table10.2liststherule
onlymodierswiththeirsemantics. Example59explainsthemodiersbysmalltoy-graphs.
Modier
Meaning
exact
Switchestothemostrestrictivemode. Anexactly-matchednodeismatched,
ifallitsincidentedgesinthehostgrapharespeciedinthepattern.
induced
Switches to the induced-mode,wherenodescontainedinthesame induced
statement requiretheirinducedsubgraphwithinthehostgraphtobespec-
ied, in n order r to be matched. . In n particular this means that t in n general
induced(a,b,c)diersfrominduced(a,b); induced(b,c).
Table10.1:Semanticsofpatternmodiers
NOTE(35)
Internallyallthemodier-annotatedrulesareresolvedintoequivalentrulesinstandardSPO
semantics. Thesemantics s of the modiers ismostly implementedby NACs. . Inparticular
youmightwant tousesuchmodiersinordertoavoidwritingabunchofNACs yourself.
ThenumberofinternallycreatedNACsisboundedbyO(n)forexactanddpoandbyO(n
2
)
forinducedrespectively,wherenisthenumberofspeciednodes.
10.2 StaticTypeConstraint
Astatic typeconstraintgivenat anodeoredgedeclarationlimits the types onwhichthe
patternelementwillmatch(byexcludingforbiddentypes).
TypeConstraint
-
\
-
(
-
TypeExpr
+
-
)
-
TypeExpr
-
116
AdvancedMatchingandRewriting
Modier
Meaning
dpo
SwitchestoDPOsemantics.Thismodieraectsonlynodesthatare
to be deleted during the rewrite. . DPO O says|roughly spoken|that
implicit deletions must t not occur by y all means. . To o ensure this s the
danglingcondition(seedanglingbelow)andtheidenticationcondi-
tion(see identification below)get enforced(i.e. . dpo o = dangling
+ identification). . In n contrast to exact t and d induced this s modi-
er applies neither toapatternas such(can’t be usedwitha test)
nortoasinglestatementbutonlytoanentirerule. SeeCorradiniet
al.[CMR
+
99]foraDPOreference.
dangling
Ensuresthedanglingcondition.Thismodieraectsonlynodesthat
aretobe deletedduringtherewrite. . Nodesgoingtobedeleteddue
totherewriteparthavetobespeciedexactly(withalltheirincident
edges, exact t semantics) ) inorder r to be matched. . As s with h dpo, this
modierappliesonlytorules.
identification
Ensurestheidenticationcondition. Thismodieraectsonlynodes
thataretobe deletedduringtherewrite. . Ifyouspecify y twopattern
graphelementstobehomomorphicallymatchedbutonlyoneofthem
issubjecttodeletionduringrewrite,thosepatterngraphelementswill
neveractually be matchedtothe samehost graphelement. . As s with
dpo,thismodierappliesonlytorules.
Table10.2:Semanticsofruleonlymodiers
EXAMPLE(59)
HostGraph
Pattern/Rule
Eect
f . --> .; ; g
Produces no o match h for exact nor
induced
f x:node --> y:node; g
Produces
three
matches
for
induced(x,y)
but no
match
for
exact(x,y)
f x:node; induced(x); ; g
Producesnomatch
e
f --> > x:node e --> > ; ; g
modifyf delete(x); g
Produces no match in n DPO-mode e be-
causeofedgee
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