A trichotomy for regular simple path queries on graphs

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A trichotomy for regular simple path queries on graphs

Guillaume Bagan, Angela Bonifati, Benoit Groz

To cite this version:

Guillaume Bagan, Angela Bonifati, Benoit Groz. A trichotomy for regular simple path queries on graphs. Journal of Computer and System Sciences, Elsevier, 2020, 108, pp.29-48.

�10.1016/j.jcss.2019.08.006�. �hal-02435355�

Contents lists available atScienceDirect

Journal of Computer and System Sciences

www.elsevier.com/locate/jcss

A trichotomy for regular simple path queries on graphs ^✩

Guillaume Bagan

^a

^,

^∗

, Angela Bonifati

^a

, Benoit Groz

^b

aUniversitéLyon1,LIRISUMRCNRS5205,F-69622,Lyon,France bUniversitéParisSud,LRIUMRCNRS8623,F-91405,Orsay,France

a r t i c l e i n f o a b s t r a c t

Articlehistory:

Received3June2018

Receivedinrevisedform 23June2019 Accepted19August2019

Availableonline20September2019 Keywords:

Graphs Paths

Regularsimplepaths Complexity Regularlanguages Automata

We focus onthe computational complexity ofregular simple pathqueries (RSPQs). We consider thefollowingproblemRSPQ(L) foraregularlanguage L:givenanedge-labeled digraphG andtwonodesxand y,isthereasimplepathfromxto ythatformsaword belonging toL?We fullycharacterize thefrontierbetweentractability andintractability forRSPQ(L).Moreprecisely,weproveRSPQ(L)iseitherAC⁰,NL-completeorNP-complete dependingonthe languageL.Wealsoprovideasimplecharacterizationofthetractable fragment in terms of regular expressions. Finally, we also discuss the complexity of deciding whether a language L belongs to the fragment above. We consider several alternative representations of L:DFAs, NFAsorregular expressions,and provethat this problemisNL-completeforthefirstrepresentationandPSpace-completefortheothertwo.

©^{2019ElsevierInc.Allrightsreserved.}

1. Introduction

Graphdatabaseshave beeninvestigatedstarting fromthelate 80s andarenow againinvogue duetotheir wide ap- plicationscenarios, rangingfromsocial networksto biologicalandscientificdatabases(see [2] fora survey).Regularpath queries (RPQs) are one ofthe mostnotable classesof queries on graphdatabases. Theyallow to retrievepairs ofnodes connectedbyapath,wherethepathisdescribedthrougharegularexpression.Suchregularpathqueriesarecomputablein timepolynomialinbothqueryanddatacomplexity(combinedcomplexity).Inthispaper,weinvestigatethecomputational complexityofregularsimplepathqueries(RSPQs),avariantofRPQinwhichthepathconnectingthepairhastobesimple, i.e., does not haverepeated vertices.Given an edge-labeled graph G anda regular language L, an RSPQ selects pairs of verticesconnectedbyasimplepathwhoseedgelabelsformawordinL.

TheevaluationofRSPQsisNP-completeevenforfixedbasiclanguagessuchas

(

aa

)

^∗ora^∗ba^∗[20],insharpcontrastwith RPQs.RSPQs are desirablein manyapplicationscenarios [17,23,6,15,13,30], such astransportation problems,VLSIdesign, metabolicnetworks,DNAmatchingandroutinginwirelessnetworks.Additionally,regularsimplepathshavebeenrecently considered in SPARQL 1.1 queries exhibiting property paths. In particular, recent studies on the complexity of property paths inSPARQL [3,18] havehighlighted the hardness ofthe semanticsproposed by W3Cto evaluate such paths inRDF graphs.Roughly speaking,accordingtothe semantics consideredin [18],the evaluationofexpressions underKleene-star closure should return a simple path, whereas the evaluationof the remaining expressions allows to traverse the same vertexmultipletimes.Assuch,thesemanticsstudiedin [18] isanhybridbetweenregularpaths andregularsimplepaths semantics.RPQshavebeenrecentlyfoundinpracticewithinreal-worldSPARQLquerylogs,suchasDBPediaandWikidata

✩ Thisarticleisanextendedversionof[5].

*

Correspondingauthor.

E-mailaddresses:guillaume.bagan@liris.cnrs.fr(G. Bagan),angela.bonifati@univ-lyon1.fr(A. Bonifati),benoit.groz@lri.fr(B. Groz).

https://doi.org/10.1016/j.jcss.2019.08.006

0022-0000/©^{2019ElsevierInc.Allrightsreserved.}

querylogs [8,7,9],showingthustheincreasinginterestofusersmanipulatingreal-worldgraphdatasetswiththesequeries.

In particular,the majorityofRPQsanalyzed in large corpuses [8,9] belongto thetractable fragment Ctract studiedinthis paper,furtherconfirmingtheapplicabilityandimpactofourtheoreticalinvestigation.

Contributions.Inthispaper,weaddressthelongstanding openquestion [20,6] ofexactlycharacterizingthemaximal class ofregular languagesforwhichRSPQsare tractable.By“tractable”wemeancomputableintime polynomialinthe sizeof thegraph.Precisely,we establishacomprehensiveclassificationofthecomplexityofRSPQsforafixedregularlanguage L.

Afirst steptowardsthisimportantissuehasbeenmadein [20].Theyexhibit atractablefragment: theclassoflanguages closedundertakingsubword.However,theirfragmentisnotmaximal.

Ourcontributions canbe detailedasfollows.Weintroducea classoflanguages,namedCtract,forwhichRSPQscan be evaluated inpolynomial time (data complexity),andevenin NL. Wethen show thatRSPQ evaluationisNP-completefor every regularlanguagethatdoesnotbelongtoCtract.Consequently,themaximaltractablefragment Ctract characterizesthe frontierbetweentractabilityandintractabilityforthisproblem,underthehypothesisNL

=

^{NP.Wefurtherrefineourresults} toshowthefollowingtrichotomy:theevaluationofRSPQproblemiseitherAC⁰,NL-completeorNP-complete.Wenotethat weconsiderthelanguage Lasafixedparameter,thusourresultscharacterizethedatacomplexityofRSPQevaluation.

We also discuss the complexity of deciding, given a language L, whether the RSPQ problem for L is tractable. We considerseveralalternativerepresentationsofL:DFAs,NFAsorregularexpressions.Weprovethatthisproblemofdeciding tractabilityisNL-completeforthefirstrepresentationandPSpace-completeforthetwoothers.

Next,wegiveacharacterizationofthetractablefragmentCtract foredge-labeledgraphsintermsofregularexpressions.

We also show that Ctract isclosed byunion andintersectionand show thatlanguages inCtract are aperiodic, i.e., can be expressedbyfirst-orderformulas [29].

Weconcludewithsome minorresultsthatidentifyfurthercaseswhereRSPQsadmitefficientsolutions.Wethusprove thatRSPQsare FPTfortheclassoffinitelanguages.Furthermore,weprovethattheproblemisalsoFPTfortheclassofall regular languageswhentheparameteristhesizeofthepath.Finally,weprovethattheproblemRSPQispolynomialw.r.t.

combinedcomplexityongraphsofboundeddirectedtreewidth.Thisisactuallyastraightforwardgeneralizationofaresult of[14].

A preliminary version ofthe presentarticlehas appearedin [5] without proofsofthe main results.Here we provide detailedproofs.

Relatedwork.Afew papersdealwithRSPQsorarerelatedtothem.Lapaughetal. [16] provethatfinding simplepathsof evenlengthispolynomialfornondirectedgraphsandNP-completefordirectedgraphs.Thisstudyhasbeenextendedin [4]

byconsideringpathsoflengthi modk.Similarly,findingk disjointpathswithextremitiesgivenasinputispolynomialfor non directedgraphs [26] andNP-completefordirected graphs [10]. Usingtheseresults,MendelzonandWoodprovethat evaluating anRSPQ onanedge-labeled directedgraphisNP-hardevenforfixed languages [20].However,they show that the problemcanbe decidedinpolynomialtime forsubword-closedlanguages.Theyalsoshowthattheproblembecomes polynomialundersomerestrictionsonthesizeofcyclesofbothgraphandautomaton.Asubsequentpaper [22] provesthe polynomialityfortheclassofouterplanargraphs.Barrettetal. [6] extendthisresult,provingthat theregularsimplepath problem ispolynomial w.r.t.combined complexity forgraphs ofboundedtreewidth. Barrett etal. [6] alsoshow that the problemisNP-completefortheclassofgridgraphsevenwhenthelanguageisfixed.

WealreadymentionedthatthecomplexityofevaluatingSPARQLpropertypathshasbeeninvestigatedinpreviousstud- ies [18,3]. As highlighted above, the semantics of SPARQL property paths blends the arbitrary paths and simple paths semantics. Losemann andMartens [18] andArenaset al. [3] focus onthe complexity of evaluating such property paths.

TheyshowthattheevaluationisNP-completeinseveralcases,andexhibitcasesinwhichitispolynomial.Moreprecisely, LosemannandMartens [18] classifyseveralfragmentsofpropertypathswithrespecttotheircomplexity.Boththeseman- ticsandthequeryfragmentsaredifferentfromtheonesinourpaper.Countingthenumberofpathsthatmatcharegular expression(whichispermittedforinstanceinSPARQL1.1)ishardinmanycases [18,3].Recently,MartensandTrautner [19]

studied thedecision- and enumerationproblemsconcerning theevaluationofRPQsby consideringseveralsemantics: ar- bitrary paths,shortestpaths,andsimplepaths.Whilewe proveherea trichotomyforthedatacomplexityofthedecision problem,theyfocusonthedatacomplexityofthepolynomialdelayenumerationproblem.

Section 2 introduces and illustrates the problem. We then establish our trichotomy. We first show insection 3 why languages outside our tractable fragment have NP-hard complexity.Section 4 discusses some properties ofthe tractable fragment, andshowshow thoseproperties(about stronglyconnectedcomponents) lead toa polynomial evaluationalgo- rithm.Andfinally,Section5detailsouralgorithmtodealwiththetractablelanguages.

2. Preliminaries

Let

[

ⁿ

]

^{denotethesetofintegers}

{

¹

, . . . ,

n

}

^and

[

ⁿ

,

m

]

^{denotethesetofintegers}

{

ⁿ

, . . . ,

m

}

^. 2.1. Complexity

A TM refers to a Turing Machine anda NDTM refers to a non deterministic TuringMachine. AC⁰

,

^L

,

^NL

,

^P

,

^NP

,

^PSpace refer to theclassical classesofcomplexity [24]. Therelations AC⁰

⊆

^L

⊂

^NL

⊆

^P

⊆

^NP

⊆

^{PSpacebetweentheseclasses are} wellknown.

FL is the class of functions computable by a deterministic log-space transducer. The class L^NL is the set of decision problemscomputable by a deterministic log-space algorithmwith an oraclein NL.The class FL^NL isthe set offunctions computablebyadeterministiclog-spacetransducerwithanoracleinNL.TheclassFL^NL

(

^NL

)

isthesetofproblemsreducible toaprobleminNLbyafunctioninFL^NL.

Lemma1.[24,11]NL

=

^LNL

=

^FLNL

(

^NL

)

.

NL^NL istheclassofdecisionproblemscomputablebyanondeterministiclog-spacealgorithmwithanoracleinNL.The nextlemma istrueonlyifwemakesomerestrictions ontheoraclemachinemodel(see[27]):theTMmustwriteonthe oracletapedeterministicallyi.e.itworksdeterministicallyassoonasitstartstowrite onthetapeuntilitcallstheoracle.

Theoracletapeiserasedattheendofeachcall.

Lemma2.[11]NL

=

^NLNL. 2.2. Graphs

Inourpaper,weessentiallyconsiderdb-graphs.Adb-graphisatupleG

= (

V

, ,

E

)

whereV isasetofvertices,

isa setoflabelsandE

⊆

^V

× ×

^{V isasetofedgeslabeledbysymbolsof}

.Givenaset S

⊆

^V,G

[

^S

]

^{theinducedsubgraph} ofG by S is

(

S

, ,

E

∩

^S

× ×

^S

)

.Apath pofadb-graphG fromxto y isasequence

(

v1

=

^x

,

a1

, . . . ,

vm

,

am

,

vm+¹

=

^y

)

suchthatforeachi

∈ [

^m

+

¹

]

^,viisavertexinGandforeachi

∈ [

^m

]

^,

(

v_i

,

a_i

,

v_i+1

)

isanedgeinG.Apath pissimpleifall verticesv_i inparedistinct.GivenalanguageL

⊆

^∗,pisL-labeledifa₁

. . .

a_m

∈

^{L.Givenasubset S}

⊆

^V,pis S-restricted ifeveryintermediate vertexof p belongsto S.Givena simplepath pandtwo verticesxand y in p,p

[

^x

,

y

]

^denotesthe subpathofpfromxto y.

2.3. Languagesandautomata

Let L be a regular language. Given a word w and a language L, w⁻¹L

= {

^w

:

^{w w}

∈

^L

}

^{. We denote by A}L

= (

Q_L

,

i_L

,

F_L

,

L

)

the minimal DFA for L, and by M_L the number of states M_L

= |

^QL

|

^{in A}L. Whenever the language is obviousfromcontext,wedropthesubscriptandwrite M insteadofML.Weassumethat AL iscompletei.e.

L isatotal function,so thatingeneral AL mayhavea sinkstate.Foranystate q

∈

^{Q andword w}

∈

^∗,

L

(

q

,

w

)

denotes thestate obtainedwhenreading wfromq.Foranystateq

∈

^{Q andsetofwords S}

⊆

^∗,

L

(

q

,

S

)

denotesthesetofstatesq such thatthereexists w

∈

^{S withq}

=

L

(

q

,

w

)

.Finally,Lq denotesthesetofallwordsacceptedfromq.Foreverystateq we denotebyLoop

(

q

)

thesetofallnonemptywordsthatallowtolooponq: Loop

(

q

) = {

^w

∈

⁺

|

L

(

q

,

w

) =

^q

}

^.Wesaythat astateq isreachablefromastate qifq

∈

L

(

q

,

^∗

)

.A(stronglyconnected)componentof AL isamaximalsetofstates thatarepairwisereachable.

TherunofL(or AL)overapathp

= (

v1

,

a1

, . . . ,

am

,

vm+¹

)

isthemapping

ρ : {

^v1

, . . . ,

vm+¹

} →

^QL suchthat:

ρ (

v1

) =

ⁱL and

ρ (

v_i+1

) =

L

(

i_L

,

a₁

. . .

a_i

)

forevery i

∈ [

^m

]

^{.Therearemany}characterizationsofaperiodiclanguages [29].Alanguage L isaperiodicifandonlyifitsatisfies

L

(

q

,

w^M+1

) =

L

(

q

,

w^M

)

foreverystate qandwordw.Intuitively,thatmeansthat foranystate q0 andaword w,intheinfinitesequence

(

q0

,

q1

,

q2

, . . .)

withqi+¹

=

L

(

q

,

w

)

foranyintegeri,thereisan integerMsuchthatq^M+1

=

^qM.

2.4. Regularsimplepaths

GivenaregularlanguageL,wedefinethefollowingproblem:

RSPQ

(

L

)

Input: Adb-graphG

= (

V

, ,

E

)

,andtwoverticesx

,

y

∈

^V Question:IsthereasimpleL-labeledpathfromxtoy?

Forthisproblem, L isfixed,sowefocuson datacomplexity.Notice thattherepresentationofL doesnot matterhere.

AlthoughweconsidertheBooleanversionoftheproblem,namelydecidingtheexistenceofapath,ouralgorithmsactually alsoreturnasimpleL-labeledpath.

The main problem that we address in thispaper is to distinguish cases when RSPQ

(

L

)

is tractable (i.e. decidablein polynomialtime)andwhenitisnot(i.e.NP-hard).

2.5. Theclassoftractablelanguages

WerecallthatM referstothesizeof Q_L,hereandhenceforth.WenextintroducetheclassCtract oflanguages.Wewill provethatitisexactlytheclassofregularlanguagesforwhichRSPQ

(

L

)

istractable.

Definition1.A regular language L belongs tothe class Ctract if the followingproperty is satisfied: forall pairs of states q1

,

q2

∈

^QL andallwordswwithLoop

(

q1

) = ∅

^,Loop

(

q2

) = ∅

^,q2

∈

L

(

q1

,

^∗

)

andw

∈

^Loop

(

q2

)

,itholdsthatw^MLq₂

⊆

Lq₁. ThisdefinitionismerelyatechnicaldefinitionforCtract,butwewillprovideinTheorem6amoreintuitivecharacteriza- tionsoftheclass.

Example1.Asan introductory example,considerthe language L

=

^a∗

(

bb⁺

+ )

c^∗.Weobserve thatthislanguage belongs toourclassCtract.WewishtodecideRSPQ(L),i.e.,whetherthereexistsasimplepathfromxto ylabeledby L,giventwo verticesx

,

y ofadb-graphG.ItisnotabsolutelytrivialthatRSPQ(L)canbesolvedefficiently:RSPQ(a^∗bc^∗)hasindeedbeen provedNP-complete.YetweoutlinebelowapolynomialalgorithmforL.

Wedistinguishtwocases:thereisasimpleL-labeledpathfromxto yifandonlyifoneofthefollowingcasesholds:

1: thereexistsasimplea^∗b^kc^∗-labeledpathfromxto yforsomek

∈ {

⁰

,

2

,

3

}

2: case 1doesnotholdandthereexistsasimplea^∗b⁴b^∗c^∗-labeledpathfromxto y.

The first caseis theeasiest to check. We firstcheck whether y canbe reachedfrom x by a(non-necessarily simple) a^∗c^∗-labeledpath.Ifwefindone,we obtainasimplea^∗c^∗-labeledpathby eliminatingits loops.Assumenowthere isno a^∗c^∗-labeledpathfromxto y.Wethencheckasfollowsifthereexistsasimplea^∗b^kc^∗-labeledpathfromxto y forsome k

∈ {

²

,

3

}

^{:we tryeverypossibleassignmentforthekmiddleb-edges.Foreach} combinationofk b-edges, wecheckifthe initialb-edgecanbereachedfromxthroughana^∗-labeledpath(avoidingtheverticesoftheotherbedges),andcheckifthe finalb-edgecanleadto ythroughsomec^∗-labeledpath(avoidingtheverticesoftheotherbedges).Intheresultinga^∗b^kc^∗- labeledpaththea^∗-labeledprefixandc^∗-labeledsuffixcannotintersect(weassumedthereisnoa^∗c^∗ path).Consequently weobtainasimplea^∗b^kc^∗-labeledpathbyeliminatingitsloops.Asthenumberofpossibleassignmentsforkedges(k

≤

³⁾ ispolynomial,wehaveprovedthatwecanfindoutinpolynomialtimewhethercase 1holds.

Letusnowassumew.l.o.g.that thereisnoa^∗b^kc^∗-labeledpathfromxto y fork

∈ {

⁰

,

2

,

3

}

^{.Wecanshow thatinthis} secondcasethereexistsasimpleL-labeledpathfromxto y ifandonlyifthereexistsixverticesv1

,

v2

,

v3

,

v4

,

v5

,

v6,two integersla

,

lb andtwosetsSa,Sbsatisfyingallfollowingconditions:

•

^theverticesv1

, . . . ,

v6 arealldistinctexceptthat v3 mayequalv4.

•

^{thereisab-labelededgefromv}1 tov₂,fromv₂to v₃,fromv₄ tov₅,andfromv₅to v₆.

•

^{thereisana∗-labeledpathfromxtov}1 avoidingallother vis

(

i

>

1

)

.Theshortestpossiblesuchpathhaslengthla.

•

^Saisthesetofallverticesreachablefromxthroughana^∗-labeledpathoflengthatmostlathatavoidsallvis

(

i

>

1

)

.

•

^{thereisab∗-labeledpathfromv}3 tov4 ofwhichallvertices(butthefirstandlast)avoidSa andthev_is.Theshortest possiblesuchpathhaslengthlb.

•

^Sb isthesetofall verticesreachablefrom v₃ throughanyb^∗-labeledpathoflengthatmostl_b that avoids S_a andall other vis

(

i

=

⁴

)

.

•

^{thereisac∗-labeledpathfromv}6 to yofwhichallvertices(butthefirst)avoidS_aandS_bandallother v_is

(

i

<

6

)

. Thefigurebelowsummarizesalltheseconditions.

Theseconditionscanclearlybeverifiedintimepolynomialin G.Itisrelativelyclearalsothatthepathconstructedabove isan L-labeledsimplepathfromx to y,sotheconditionsaresufficienttoobtain an L-labeledsimplepath.Toprovethat our procedureiscorrect, we onlyhavetoprove thatreciprocally,ifthereexists asimple L-labeledpathwe canfind one satisfyingourrestrictions(theconditionsaboveinvolvingthevi,Sa, Sb).

ForeveryshortestL-labeledsimplepathp fromxto y,letv₁

, . . . ,

v₆ denotetheverticesthatdelimitthefirstandlast two b-edges of p.We nextshow that thoseverticessatisfy theconditionsabove. Thelast vertexof pthat belongsto Sa cannotoccurafterv₃ inp.Otherwise,wecouldobtainasimplepath pbyreplacingtheprefixofpuptov withashorter path through Sa.Thisresultingpath p wouldstill be L-labeledby definitionof the vi,whichcontradicts theminimality of p.Asimilarargumentshowsthat thelastoccurrenceofavertexfrom S_b cannotoccurafter v₆.We concludethat the paths connectingv3 tov4 in p(resp.v6 to y)excluderespectivelyall verticesfromSa (resp.Sa

∪

^Sb).Asaconsequence, the pathfromxto v1 willonlyfeature verticesfromSa byminimalityof p,whichproves thatvertices v1

, . . . ,

v6 satisfy theconditionsabove.

The crux ofour approachis to constructthe a^∗, b^∗ andc^∗ subpathsindependently, lestwe enumerate exponentially manypaths.Thisiswhywerequirethattheb^∗ subpath avoids Sa: thiscondition isstrongerthannecessarytoguarantee thefirsttwosubpathsdonotintersect,butthestrongerrequirementallowsustobuildthetwosubpathsindependently,as Saisasupersetoftheverticesonthesubpathfromxtov1.Ouralgorithmfortractableinstanceswillgeneralizethisidea.

3. HardlanguagesforRSPQ

Thissectionisdevotedtotheproofofahardnessresult:RSPQ

(

L

)

isNP-hardforeveryregularlanguage Lthatdoesnot belongtoCtract.ThefirststeptowardthatproofliesinthefollowingcharacterizationofCtract.

Definition2(Witnessofhardness).Let L be aregular language. Awitnessforhardnessof L isa tuple

(

w_l

,

w_m

,

w_r

,

w₁

,

w₂

)

wherew_l

,

wr

∈

^∗ andwm

,

w1

,

w2

∈

⁺satisfying

•

^wlw^∗₁w_mw^∗₂w_r

⊆

^L

•

^wl

(

w1

+

^w2

)

^∗wr

∩

^L

= ∅

^.

Lemma3.LetL bearegularlanguagethatdoesnotbelongtoCtract.Then,L admitsawitnessforhardness.

Proof. Let L be a regular language that does not belong to Ctract. For commodity, we distinguish two cases, depending on whether L satisfies or not the following property: Lq2

⊆

Lq1 for every q₁

,

q₂

∈

^QL such that q₂

∈

L

(

q₁

,

^∗

)

and Loop

(

q1

) ∩

^Loop

(

q2

) = ∅

^(PropertyP^).

LetLbealanguagethatdoesnotsatisfyPropertyP,thereexistq

,

q2

,

wm

,

w

,

wrsuchthat

L

(

q

,

wm

) =

^q2,w

∈

^Loop

(

q

) ∩

Loop

(

q₂

)

,andw_r

∈

Lq2

\

Lq.Letw_l suchthat

(

i_L

,

w_l

) =

^{q.Then w}l

,

w_m

,

w_r

,

w₁

=

^w2

=

^{wisawitnessforhardness.}

Wenextplantoexhibitawitness forhardnessforthecasewhereL satisfiesPropertyP^{,butwefirstprovethatevery} language satisfyingproperty P ^(whetherin Ctract ornot)is aperiodic.Let L be a language satisfyingProperty P^{, q}

∈

^QL andw awordin

⁺.Letalsoqdenotethestateq

=

L

(

q

,

w^M

)

.Wedenotebyq thestate

L

(

q

,

w

)

.Wewanttoprove thatq

=

^{q.Bythepigeonhole principlethereexistssome k}0

<

k1

≤

^{M suchthat}

L

(

q

,

w^k0

) =

L

(

q

,

w^k1

)

.Wethen have

L

(

q

,

w^k

) =

^{q fork}

=

^k1

−

^k0.Then q andq both loopon w^k,sothat Lq

=

Lq bydefinitionofP^,henceq

=

^{q by} minimality.Consequently,Lisaperiodic.

Let L be a language that satisfies Property P ^{(and so in particular is} aperiodic), butthat does not belong to Ctract. Bydefinition ofCtract there existstates q

,

q2 andwords wl

,

w1

,

w2

,

wm

,

w_r such that

(

iL

,

wl

) =

^{q, w}1

∈

^Loop

(

q

)

, w2

∈

Loop

(

q2

)

,

L

(

q

,

wm

) =

^q2, w_r

∈

Lq2 and w^M₂ w_r

∈ /

Lq. W.l.o.g.we can supposethat w1

= (

w₁

)

^M forsome word w₁.We then claimthat Lq

⊆

Lq forevery q in

L

(

q

,

^∗w₁

)

.Indeed,forevery q

∈

L

(

q

,

^∗w₁

)

, thereexists somek

>

0 such that

L

(

q

,

w^k₁

) =

^{q, hence q loops over w}1 by aperiodicity of L. We thus have w1

∈

^Loop

(

q

) ∩

^Loop

(

q

)

and therefore Lq

⊆

LqduetoPropertyP^.

Letwr

=

^wM₂ ^wr.Bydefinition,wmw^∗₂wr

⊆

Lq becausewr

∈

Lq₂.Wenowprovethat

(

w1

+

^w2

)

^∗wr

∩

Lq

= ∅

^,because any wordin

(

w1

+

^w2

)

^∗wr can be decomposed into uv withu

∈ ₊ (

w1

+

^w2

)

^∗w1 and v

∈ (

w2

)

^∗wr. We recall that w_r

=

^w₂^Mwr

∈ /

LqandLisaperiodic,sothatv

∈ /

Lq.Furthermore,wehavejustprovedthatq

=

L

(

q

,

u

)

satisfiesLq

⊆

Lq. Consequently, v

∈ /

Lq anduv

∈ /

Lq.Thus, w_l, wm, wr, w1,and w2 providea witnessforhardness, whichconcludesthe proofofLemma3.

We cannow prove ourhardness result, by reduction fromVertex-Disjoint-Path, a problemalsoused in [20] toprove hardnessintheparticularcaseofa^∗ba^∗.

Vertex-Disjoint-Path

Input: AdirectedgraphG

= (

V

,

E

)

,fourverticesx₁

,

y₁

,

x₂

,

y₂

∈

^V

Question:Aretheretwodisjointpaths,onefromx1 toy1 andtheotherfromx2to y2?

Lemma4.LetL bearegularlanguagethatdoesnotbelongtoCtract.Then,RSPQ

(

L

)

isNP-hard.

Proof. LetL

∈ /

Ctract. Weexhibit areduction fromtheVertex-Disjoint-PathproblemtoRSPQ

(

L

)

. AccordingtoLemma3, L admitsawitnessforhardness wl

,

wm

,

wr

,

w1

,

w2.Bydefinitionwegetwl

(

w1

+

^w2

)

^∗wr

∩

^L

= ∅

^andwlw^∗₁wmw^∗₂wr

⊆

^L.

Webuild fromG adb-graph Gwhoseedges arelabeled bynon emptywords.Thisisactuallya generalizationofdb- graphs. Nevertheless,by addingintermediate vertices,an edge labeledby a word w canbe replaced witha pathwhose edgesformthewordw.

Gisconstructedasfollows.TheverticesofGarethesameastheverticesofG.Foreachedge

(

v₁

,

v₂

)

inG,weaddtwo edges

(

v1

,

w1

,

v2

)

and

(

v1

,

w2

,

v2

)

.Moreover,we addtwo newvertices x

,

y andthreeedges

(

x

,

w_l

,

x1

)

,

(

y1

,

wm

,

x2

)

and

(

y2

,

wr

,

y

)

.WenextprovethatRSPQ

(

L

)

returnsTruefor

(

G

,

x

,

y

)

iffVertex-Disjoint-PathreturnsTruefor

(

G

,

x1

,

y1

,

x2

,

y2

)

. Assume thereis a simple L-labeled path p fromx to y in G.By definitionof G, thispathnecessarily goesthrough the edge

(

y₁

,

w_m

,

x₂

)

since w_l

(

w₁

+

^w2

)

^∗w_r

∩

^L

= ∅

^{.Since p is simple, the subpaths from x}1 to y₁ andx₂ to y₂ are disjoint,henceVertex-Disjoint-PathreturnsTruefor

(

G

,

x1

,

y1

,

x2

,

y2

)

.Reciprocally,ifVertex-Disjoint-PathreturnsTruefor

(

G

,

x1

,

y1

,

x2

,

y2

)

,thereexistdisjointpaths fromx1 to y1 andfromx2 to y2.Bydefinitionthesetwopathsmatchaword in

(

w₁

+

^w2

)

^∗.We canthen obtain twodisjointsimple paths,one from x₁ to y₁ matchingaword in w^∗₁ andone from x2 to y2 matchinga wordin w^∗₂.Toobtain thosepathswe keepthevertices astheoriginal paths,eliminate theloopsif

Fig. 1.Reduction forL=^a∗b(cc)^∗d.

there areany,andswitch w1 andw2 edges whereneeded:wecanalways replacea w1 edgewitha w2 by construction of G sinceeverypairofverticesisconnectedby bothtypesofedges ornone.Concatenatingtheedge

(

x

,

w_l

,

x₁

)

withthe firstpath,theedge

(

y1

,

wm

,

x2

)

,thesecondpathandtheedge

(

y2

,

wr

,

y

)

providesasimpleL-labeledpath pfromxto y, whichconcludesourproof.WeillustrateinFig.1thereductionforL

=

^a∗b

(

cc

)

^∗d,onaninstance

(

G

,

x1

,

y1

,

x2

,

y2

)

,choosing w_l

=

^w1

=

^a,wm

=

^{b, w}2

=

^cc,andwr

=

^d.

This concludesour proof that languagesoutsideCtract are intractable. After thisnegative result, we now focuson the positiveresult,namelythatlanguagesinCtract admitefficientalgorithms.

4. PropertiesoflanguagesinCtract

The main result ofthispaper is that forevery L

∈

Ctract,RSPQ

(

L

) ∈

^{NL. The algorithm toevaluate efficiently RSPQ}

(

L

)

exploitsaparticularkindofpumpingargumentbetweenstronglyconnectedcomponentsoftheautomaton.Thispumping argument proves that if we build carefully a path using the usual reachability algorithm inside the strongly connected components,thenweneednotcareaboutpossibleintersectionsbetweensubpathscorrespondingtodifferentcomponents.

Inthissection,weintroduceandprovethispumpingargumentinLemma11throughaserieoftechnicallemmasaboutthe structureofautomatathatrecognizeCtractlanguages.

4.1. AlternativecharacterizationofCtract

Tobeginwith,weprovethateverylanguagefromCtractisaperiodicanddeduceanalternativecharacterizationofCtract. Lemma5.LetL bearegularlanguageinCtract.ThenL isaperiodic.

Proof. IntheproofofLemma3wedefinedapropertyP^{andshowedthatlanguagessatisfyingproperty}P^{areaperiodic.We} show thatevery L

∈

Ctract satisfiespropertyP^.LetL

∈

Ctract,q1

,

q2

∈

^QL andw satisfyq2

∈

L

(

q1

,

^∗

)

andw

∈

^Loop

(

q1

) ∩

Loop

(

q₂

)

.BydefinitionofCtract, w^MLq2

⊆

Lq1,henceLq2

⊆

Lq1 becausew

∈

^Loop

(

q₁

)

.

We then exploit thisaperiodicity property to establish the followingcharacterization of Ctract, which strengthens the requirementsfromDefinition1ontheloopsof AL.

Lemma6.LetL bea regularlanguage.Then, L belongstoCtractiffforeverypairofstatesq₁

,

q₂

∈

^QL suchthatLoop

(

q₁

) = ∅

^, Loop

(

q2

) = ∅

^andq2

∈

L

(

q1

,

^∗

)

,thefollowingstatementholds:

(

Loop

(

q2

))

^MLq2

⊆

Lq1.

Proof. The (if) implication is immediate by Definition 1. Let us now prove the (only if) implication. Assume L

∈

Ctract. Let q₁

,

q₂

∈

^QL satisfy Loop

(

q₁

) = ∅

^{, Loop}

(

q₂

) = ∅

^{, q}₂

∈

L

(

q₁

,

^∗

)

, and let w

∈

^Loop

(

q₂

)

. Let also q3 denote the state

L

(

q₁

,

w^M

)

.ThenDefinition1implies w^MLq₂

⊆

Lq₁.Thus,Lq₂

⊆

Lq3.Thecruxoftheproofistochoosecarefullyq₁,q₂ andw toexploittheconstraintsonLq₃.

Let q1

,

q2 be two statessuch that Loop

(

q1

) = ∅

^,Loop

(

q2

) = ∅

^andq2

∈

L

(

q1

,

^∗

)

. Let

(

v1

, . . . ,

vM

)

be a sequence of wordsin

(

Loop

(

q₂

))

^M andq₃

=

L

(

q₁

,

v₁

. . .

v_M

)

.WewishtoproveLq2

⊆

Lq3.

Forsome i

,

j,0

≤

ⁱ

<

j

≤

^{M, weget}

L

(

q1

,

v1

. . .

vi

) =

L

(

q1

,

v1

. . .

vj

)

,usingtheconvention

L

(

q1

,

v1

. . .

vi

) =

^q1 for i

=

^0.Letu1

=

^v1

. . .

vi,u2

=

^vi+¹

. . .

vjandu3

=

^vj+¹

. . .

vM.Letq4

=

L

(

q1

,

u1

)

.WeclaimthatLq2

⊆

Lq4.Theresultthen follows fromLq2

=

^u−3¹Lq2

⊆

^u−3¹Lq4

=

Lq3.Toprove theclaim, let w

=

^u1u^M₂ andq5

=

L

(

q1

,

w^M

)

.As

L

(

q1

,

w^M

) =

q₅ and w

∈

^Loop

(

q₂

)

, we get Lq2

⊆

Lq5 through Definition 1 with q₁

,

q₂ and w. Furthermore, u₂ belongs to Loop

(

q₅

)

because Lisaperiodic.Toconcludetheproof,weobservethatLq5

⊆

Lq₄,byDefinition1withq5

,

q4 andu2,andbecause

L

(

q4

,

u₂^M

) =

^q4 andu2

∈

^Loop

(

q5

)

.¹

1 ThislastapplicationofDefinition1correspondsactuallytoobservingthateverylanguageinC^{tractsatisfiesproperty}P^fromLemma3.

4.2. TechnicallemmasonthecomponentsofAL

In thissection, we show propertiesabout the componentsof Ctract languages. Notice that states in a componentare mutually reachable, but not reachable from states in other components that they can reach themselves. From now on, anduntilthe endofthe section,we fix alanguage L

∈

Ctract.We introduce inLemmas9 and11thepumping argument thatwe exploitinthealgorithm tocomputea simplepath.Intheotherlemmaswe proveauxiliaryresults,basedonthe decompositionofthe automatoninstronglyconnectedcomponents.We provethat componentsoflanguagesinCtract are veryparticular,inthesensethateverywordstayinglongenoughinthecomponentissynchronizing.Apreliminarylemma showsthattwodistinctstatesq₁ andq₂inthesamecomponentcannotlooponthesameword.

Lemma7.Letq1andq2betwostatesbelongingtothesamecomponentofAL.IfLoop

(

q1

) ∩

^Loop

(

q2

) = ∅

^,thenq1

=

^q2.

Proof. Letq1

,

q2asabove,andletw awordinLoop

(

q1

) ∩

^Loop

(

q2

)

.AccordingtoDefinition1,w^MLq2

⊆

Lq1,henceLq2

⊆

Lq1 since w

∈

^Loop

(

q1

)

.Bysymmetry,Lq2

=

Lq1,whichimpliesq2

=

^q1.

Thenexttwolemmascharacterizetheinternallanguageofacomponent.

Lemma8.LetC beacomponentofAL,q1

,

q2

∈

^{C anda}

∈

.Then

L

(

q1

,

a

) ∈

^{C iff}

L

(

q2

,

a

) ∈

^{C .}

Proof. Let q1

=

^q2 two states in the same component C. Let a satisfy

L

(

q1

,

a

) ∈

^{C. Let also w}

∈

^Loop

(

q1

) ∩

^a

^∗ and q₃

=

L

(

q₂

,

w^M

)

:a andw necessarilyexistbecauseC isthestronglyconnectedcomponentofq₁ andq₂.Wenextprove thatq3 belongstoC:byourdefinitionofC,thisimplies

L

(

q2

,

a

) ∈

^{C.As Lisaperiodic, w}

∈

^Loop

(

q3

)

,andconsequently, w^MLq3

⊆

Lq1 byDefinition1.Furthermore,w^MLq1

⊆

Lq2 alsobyDefinition1.HenceLq3

⊆

Lq1andLq1

⊆ (

w^M

)

⁻¹Lq2

=

Lq3.Thus,Lq1

=

Lq3 and,byminimalityofAL,q1

=

^q3,sothatq3

∈

^C.

Notation1.WedenotetheinternalalphabetofacomponentC ofA_L by

C

= {

^a

∈ : ∃

^q1

,

q₂

∈

^C

.

L

(

q₁

,

a

) =

^q2

}

^. AsadirectconsequenceofLemma8weget:

Lemma9.LetC beacomponentofAL,q

∈

^{C andw}

∈

^∗.Then

L

(

q

,

w

) ∈

^{C iffw}

∈ (

C

)

^∗.

Finally,weprovethatinsidea component,everywordwithlength atleast M² issynchronizing. Thisresultisthecore ofourpumpingargumentbetweenstronglyconnectedcomponentsasexposedinLemma11.

Lemma10.LetC beacomponentof AL,

C be theinternalalphabetofC , q1

,

q2 be twostatesof C and w

∈ (

C

)

^M2.Then,

L

(

q1

,

w

) =

L

(

q2

,

w

)

.

Proof. Assumethat w

=

^a1

. . .

a_M2. Foreach i from0 to M² and

α =

¹

,

2, let qα,i =

L

(

qα

,

a₁

. . .

a_i

)

. Since there are at mostM² distinctpairs

(

q1,i

,

q2,i

)

,thereexisti

,

j,withi

<

j suchthatq1,i

=

^q1,j andq2,i

=

^q2,j.ByLemma9,q1,i

,

q2,i

∈

^C. Let w

=

^ai+¹

. . .

aj. We have w

∈

^Loop

(

q1,i

) ∩

^Loop

(

q2,i

)

, hence q1,i

=

^q2,i by Lemma 7. As a consequence,

L

(

q1

,

w

) =

L

(

q2

,

w

)

.

Noticethattheabovelemmastillholdsforw

∈ (

C

)

^M2

^∗_C.Hereandthereafter,wefixtheconstant N

=

^2M2.

Lemma11.Letq1

,

q2betwostatessuchthatLoop

(

q1

) = ∅

^,Loop

(

q2

) = ∅

^,andq2

∈

L

(

q1

,

^∗

)

.LetC bethecomponentthatcontains q2and

CbetheinternalalphabetofC .Then,Lq2

∩ (

C

)

^N

^∗

⊆

Lq1.

Proof. Let w

∈

Lq2

∩ (

C

)

^N

^∗. There are some words u

,

v

∈ (

C

)

^M2, w

∈

^∗ such that w

=

^{uv w. By Lemma 9 and} the PigeonholePrinciple,there exista state q3

∈

^{C and M}

+

1 non-empty words v1

, . . . ,

vM+¹ such that v

=

^v1

. . .

vM+¹ and

L

(

q₂

,

uv₁

. . .

v_i

) =

^q3 for every i

∈ [

^M

]

^{. Therefore, w}

∈

^uv1

(

Loop

(

q₃

))

^M−1v_M+1w. By Lemma 10,

L

(

q₃

,

uv₁

) =

L

(

q2

,

uv1

) =

^q3.Thus, wbelongstoboth

(

Loop

(

q3

))

^MvM+¹wandLq₃.ByLemma6,w

∈

Lq₁.

Ormainresultfocusesondatacomplexityandthereforeassumesthelanguage(henceN)isconstant.Yetthecomplexity willbeexponentialinN thereforewenextprove,forthesakeofefficiency,thatwecantake N

=

^MinLemma11.

Lemma12.Letq₁

,

q₂betwostatessuchthatLoop

(

q₁

) = ∅

^,Loop

(

q₂

) = ∅

^,andq2

∈

L

(

q₁

,

^∗

)

.LetC bethecomponentthatcontains q2and

CbetheinternalalphabetofC .Then,Lq₂

∩ (

C

)

^M

^∗

⊆

Lq1.