Uniform Parsing for Hyperedge Replacement Grammars

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Journal of Computer and System Sciences

www.elsevier.com/locate/jcss

Uniform parsing for hyperedge replacement grammars

Henrik Björklund^a, Frank Drewes^a,∗, Petter Ericson^b,a,Florian Starke^c

aDepartmentofComputingScience,UmeåUniversity,Sweden

bDigitalandCognitiveMusicologyLab,ÉcolePolytechniqueFédéraledeLausanne,Switzerland cFacultyofComputerScience,TUDresden,Germany

a rt i c l e i nf o a b s t ra c t

Articlehistory:

Received24April2018 Accepted26October2020 Availableonline27November2020

Keywords:

Graphgrammar Hyperedgereplacement Uniformparsing Complexity

NaturalLanguageProcessing Meaningrepresentation

It iswellknownthathyperedge-replacement grammarscangenerateNP-completegraph languagesevenunderseeminglyharshrestrictions.Thismeansthattheparsingproblemis difficulteveninthenon-uniformsetting,inwhichthegrammarisconsideredtobefixed rather thanbeingpartoftheinput.Littleisknown aboutrestrictionsunderwhichtruly uniformpolynomialparsingispossible.Inthispaperweproposealow-degreepolynomial- timealgorithmthatsolvestheuniformparsingproblemforarestrictedtypeofhyperedge- replacementgrammarswhichweexpecttobeofinterestforpracticalapplications.

©^{2020TheAuthor(s).PublishedbyElsevierInc.Thisisanopenaccessarticleunderthe} CCBYlicense(http://creativecommons.org/licenses/by/4.0/).

1. Introduction

Hyperedge-replacement grammars (HR grammars, for short) are context-free graph grammars that were introduced in [3,18], see also [17,11]. They represent one of the two most successful formal models for the description of graph languages (the other being confluent node-replacement grammars), becauseof their favorable algorithmic andlanguage- theoretic propertieswhichcloselyresemble thoseofcontext-freestringgrammars.Unfortunately,thesimilarities between the stringandgraphcases failto extendto oneof themostimportantcomputational problemsin thecontext offormal languages: theparsing problem. Ithas beenknownfora long time that eventhe non-uniformmembership problemfor context-freegraphlanguagesisintractable(unlessP=^NP).Inparticular,therearehyperedgereplacementgraphlanguages whichareNP-complete [1,19].Severerestrictionsmustbeplacedonthegrammarsinordertomake atleastnon-uniform polynomial parsingpossible.Early resultsinthisregard can befound in [20,21,10].In [20] thedegree ofthepolynomial thatboundstherunningtimevarieswiththelanguage.Thealgorithmin [21],whichconsidersonlyedgereplacement,and its generalizationtohyperedgereplacementby [10] arecubicinthesize oftheinputgraph,butdependexponentially on the grammarifconsidered ina uniformsetting.Moreover, therestrictions [21] and [10] placed onthe considered graph languagesareverystrong,anditwasshownin [9] thatevenaslightrelaxationresultsinNP-completenessagain.Forthese reasons,theseparsingalgorithmsaremainlyoftheoreticalinterest.

Inrecentyearsthequestionofefficientlyparsinghyperedgereplacementlanguagesreceived renewedinterest,because hyperedgereplacementwasproposed asasuitable mechanismfordescribingsentencesemanticsinnaturallanguage pro- cessing, and in particular the abstract meaning representation (AMR) proposed in [2]. Regarding the use of hyperedge replacement in thisapplication area, see [7]. The same paper described a generalrecognition algorithm together witha

*Correspondingauthor.

E-mailaddresses:henrikb@cs.umu.se(H. Björklund),drewes@cs.umu.se(F. Drewes),petter.ericson@epfl.ch(P. Ericson),Florian.Starke@tu-dresden.de (F. Starke).

https://doi.org/10.1016/j.jcss.2020.10.002

0022-0000/©^{2020TheAuthor(s).PublishedbyElsevierInc.ThisisanopenaccessarticleundertheCCBYlicense} (http://creativecommons.org/licenses/by/4.0/).

detailedcomplexityanalysis.Unsurprisingly,therunningtimeofthealgorithmisexponentialeveninthenon-uniformcase, one oftheexponents beingthe maximumdegreeofnodesinthe inputgraph.The sameistruefortherecentalgorithm by [16] whichimplementsparsingforso-calledregulargraphgrammars.

Unfortunately, the node degree is one of the parameters one would ideally not wish to limit, since meaning repre- sentations do not have boundednode degree. Moreover, naturallanguage processingoften has to deal withalgorithmic learning situationsinwhichlargecorporamustbe parsedandgrammarsadjusted inan iterativeprocess. Thus,trulyuni- formpolynomial-time solutionswouldbe valuable, providedthat thepolynomials haveareasonably lowdegree andthe restrictionsonthegrammarsare“natural”.

Parsing agraphG withrespecttoagivenHRgrammarG ^{meanstocheckwhetherthereisaderivationtreein}G ^that yields G.Thus, thetaskistodecompose G recursivelyinto subgraphsthat canbe generatedfromthe nonterminalsof G^. Intuitively,theNP-completenessoftheproblemcomesfromthefactthatagraphhasexponentiallymanysubgraphs.Thisis themaindifferencebetweengraphandstringparsing.Inthelattercase,thewell-knowndynamic programmingapproach by Cocke, Kasami, andYounger isefficient because a stringhas only quadraticallymany substrings. One wayto achieve polynomial parsing in thegraph caseas well is to makesure that only polynomially manydecompositions are possible candidates forwell-formedderivation trees.Inthispaperweachieve thisby imposingrestrictions onG ^{which guarantee} that the overall shape of a suitable decomposition of G can be “read off” G itself. Intuitively,what remains is to check whetherappropriate rulesofG ^{can beassignedtotheverticesofthis}decompositioninordertoturnitintoaderivation tree.

Anattemptata setofrestrictionsserving thispurposewasmadein [5].Motivatedbythefactthat meaningrepresen- tationssuchasAMRaretypicallyacyclic,HRgrammarswereconsideredthatgeneratedirectedacyclicgraphs.However,as acyclicityalonedoesnotmakeparsinganyeasieradditionalconditionswereplacedontheformoftherules.Inthepresent paper,wegeneralizetheapproach:thegeneratedgraphsmayhavecycles,theallowedrulesareconsiderablymoregeneral, andtherestrictions arefewer andformulatedinan axiomaticwaywhichallows fordifferentconcretizations.Weimpose twoconditionsonourgrammars,calledreentrancypreservation andorderpreservation.Thelatterisrelativetoanorderingof thenodesofinputgraphsthatcanbeinstantiatedindifferentways.

WeexpectourparsingalgorithmtobeusefulfordescribingandprocessinglanguagesofsemanticgraphssuchasAMR.In fact,typicalstructuresoccurringinsuchgraphsprovidedthestartingpointforthedevelopmentoftherestrictionsproposed in thispaper.Along withtheformal development ofthe notions andresultsleading to ourparsing algorithm,we tryto illustratethepotential usefulnessofourgrammarsby meansofasmallrunningexamplethat stretchesfromSection 2to Section6.ItshowshowtogeneratealanguageofAMR-likesemanticgraphsbyanHRgrammarthatsatisfiesourconditions foruniformpolynomialparsing(Examples2.4,4.3and5.4,aswellasSection6.1).

Letusbrieflydescribe theideabehind therestrictions weimposeon HRgrammarstomake themefficientlyparsable.

Whenworkingwithhyperedgereplacement,anonterminalhyperedgeisaplaceholderattachedtoasequenceofnodes.This placeholder willeventually be replacedby a subgraph thatshares the attachednodesofthe hyperedge(andonly those) withtherestofthegeneratedgraph.Onedifficultyparsinghastofaceisthat,afterthereplacementofahyperedge,itmay not bevisible intheresultinggraphwhich nodesthereplacedhyperedgehadbeenattachedto.Reentrancy preservation, whichisthefirstconditionwedescribeinthispaper,makesitpossibletorecoverthissetofnodessolelyfromthestructure ofthegeneratedgraph.

One difficultyremains:even iftheattachednodesofa nonterminalhyperedgecan uniquelybe recovered,it maystill be unclear inwhich order they had been attachedto the hyperedge.This is what is avoided by the condition oforder preservation.Itensures,forexample,thatarulecannotreplaceanonterminalhyperedgebyanothernonterminalhyperedge attachedtothesamenodesbutinadifferentorder.

Thankstothetworestrictions,weobtainauniformparsingalgorithmwhichisroughlyquadraticinboththesizeofthe grammarandthatoftheinputgraph.¹

Asafinalnoteonrelatedwork,wementionherethatanotherrecentapproachtoefficientparsingforHRgrammarswas presentedin [13,14],wherepredictivetop-downandbottom-upparsersareproposed,generalizingtechniquesfromcompiler construction tothe graphcase. Theapproach thusdiffers fromoursinthat ityields a parsergeneratorwhich,withonly thegrammarasinput,constructsaquadraticparserforthespecificlanguagegeneratedbythatgrammar.Providedthatthe grammaranalysiscanbeperformedinpolynomialtime(whichdependsontheexactvariantoftheparsergeneratorused), thisapproachisthusuniformlypolynomialaswell.

The nextsection compiles thebasicnotions relevanttohyperedgereplacementgrammars.Section 3and4define and study reentrancyandorderpreservation,respectively.Section5presentsonepossibleconcretizationofourabstractnotion of preservedorders. The parsing algorithm and the main resultof thispaper are presented in Section 6, andSection 7 concludesthepaper.

1 Theexactrunningtimedependsonhowefficientlythechosenordercanbecomputed.

2. Preliminaries

The setofnon-negativeintegersisdenotedby N.Forn∈ N^, [ⁿ]^denotes {¹,. . . ,n}^{.Givena set S, S∗ denotesthe set} of all finite sequences over S,and S^ denotes the set ofnon-repeating sequences in S^∗, i.e.those sequences in which no element of S occurs twice. The empty sequence is denoted by ε, S⁺=^S∗\ {ε}^{,and S⊕}=^S\ {ε}^{. The length ofa} sequence w∈^{S∗isdenotedby}|^w|^,and[^w]^{denotesthesmallestsubset A ofS suchthat w}∈^{A∗.Thecanonicalextensions} of a mapping f:^S→^{T to S∗ andto the powerset of S are denoted by f as well, i.e., f}(a1· · ·^ak)= ^f(a1)· · ·^f(ak) for a₁,. . . ,a_k∈^{S,and f}(S^)= {^f(a)|^a∈^S}^forS⊆^{S.Asequencesw}∈^S∗withs∈^{S mayalsobedenotedby}(s,w). 2.1. Orderingsubsetsofaset

As mentionedin theintroduction,one oftheprerequisitesof ourparsingalgorithm isa waytoordervarious subsets ofthenodesofan inputgraph.Consideranarbitrarybinaryrelation≺^{onaset S.Givenasequence w}=^s1. . .s_k∈^S∗,we saythat w isorderedby≺^ifsi≺^si+¹foralli∈ [^k−¹]^{,andmoreover,s}i≺^sjimpliesi<j foralli,j∈ [^k]^.Wefurthermore say that≺ ^{orders asubset A}⊆^{S ifthe elements of A canbe arrangedin asequence w whichis orderedby} ≺^.Clearly, if≺^{orders A,thissequence w isuniquely}determined.Inthefollowing,wedenotethissequenceby ^A≺ (providedthat

≺ ^{indeed orders A). Note that, for the sake of} generality, we place no further restrictions on ≺^{, andit is thus neither} necessarilyanorderon A (whereitmaylacktransitivity)noronS (whereitmaybeotherwiseentirelyarbitrary).

2.2. Hypergraphs

Throughoutthispaper,we fixa countablyinfinitesupply LAB ofsymbolscalledlabels,such thatevery σ∈^{LAB hasa} uniquerank rank(σ)∈ N^{.Similarly,wefixcountablyinfinitesupplies}V ^andE ^{ofverticesand}hyperedges,respectively.

Definition2.1(hypergraph).A(directedhyperedge-labeled)hypergraph over⊆^{LAB isatupleG}= (^V,E,att,lab,ext)with thefollowingcomponents:

• ^V⊆V^andE⊆E ^{aredisjointfinitesetsofnodes and}hyperedges,respectively.

• ^Theattachment att:^E→^{V⊕ assigns toeachhyperedgee asequenceofattachednodes.Fore}∈^{E with att}(e)= (^v,w) wealsodenotev bysrc(e)andw bytar(e),callingthemthesource andthesequenceoftargets ofe,respectively.

• ^Thelabeling lab:^E→ ^{assignsalabeltoeachhyperedge,subjecttotheconditionthatrank}(lab(e))= |^tar(e)|^forevery e∈^E.

• ^{The sequence ext}∈^{V⊕ is thesequence ofexternalnodes. Ifext}G= (^v,w), then we denote thenode v by G and the sequencew ofnodesbyG ,respectively,andweimposetheadditionalrequirementthatsrc(e)∈ [/ ^G ]^foralle∈^E.2 Thesize|^G|^{ofG is}

e∈^E|^att(e)|^.3

Notethatweforbidatt(e)(fore∈^{E)tocontainanynode}repeatedly.Inthefollowing,wesimplycallhyperedgesedges andhypergraphsgraphs.Ourdivisionoftheattachmentofeveryedgeintoasinglesourcenodeandanynumberoftarget nodesissimilartothatusedintheliteratureonterm(hyper)graphs.Itmakesitmeaningfultospeakaboutdirectedpaths (definedbelow).Ourgraphsare,however,moregeneralthantermgraphsinthatwe,forthemoment,donotimposefurther structuralconditionsonthem.

Throughoutthepaper,ifthecomponentsofagraph G arenotexplicitlynamed,wedenotethemby V_G, E_G,att_G,etc.

Ifthe componentsof G are given explicitnames (and thusthe subscript isdropped) weextend thisinthe obviousway toderived notations,dropping thesubscripteventhere.Wefurthermoreusethenotation outG(v) todenotethesetofall outgoingedgesofanode v∈VG,i.e.,outG(v)= {e∈EG|srcG(e)=v}.

Anisomorphism h:^G→^{H isapairofbijectivemappings}(h_V:^VG→^VH,h_E:^EG→^EH)suchthatatt_H◦^hE=^hV◦^attG, labH◦^hE=^labG,andextH=^hV(extG).IfsuchanisomorphismexistswewriteG≡^{H andsaythatthegraphsare}isomorphic.

Apath oflengthk∈ N ^fromu∈^{V toe}∈^{E inG isasequence p}=^e1· · ·^ek∈^E+wheresrc(e1)=^{u, src}(ei+¹)∈ [^tar(ei)] forall i∈ [^k−¹]^,andek=^{e. If}furthermore v isanode in[^tar(ek)]^{then pv is apathfrom u to v.Both p and pv pass} the nodes src(e₂),. . . ,src(e_k),and we saythat p contains e₁,. . . ,e_k aswell assrc(e₁),. . . ,src(e_k),while pv additionally contains v.Ifsrc(e₁)∈ [^tar(e_k)]^{,thepathisacycle.Wesaythatthepathisasourcepath ifu}=^G.

Anode v oran edgee isreachable fromanode u ifu=^{v orthereisapathfromu tov orfromu toe,} respectively.

Wesimplysaythatv ande arereachableinG iftheyarereachablefromG.IfG isclearfromthecontextwemayjustwrite

2 Recallthat[^G ]^{denotesthesetofnodesoccurringinG .}

3 Thissimpledefinitionofsizeissufficientandappropriateforourpurposesastheclassesofgrammarsconsideredinthepaperonlygenerateconnected hypergraphs,andbythedefinitionofhypergraphsitholdsthatexternalnodesarepairwisedistinctand1≤ |^att(e)|≤ |^V|^{for allhyperedgese.Thus,}

|^V|≤ |^G|^,|^E|≤ |^G|^,and|^ext|≤ |^G|^.

s

u

v w

v^ w^

A e

c f

gb a h

Fig. 1. Example drawing of a graph G.

“reachable” instead of“reachable inG”. Notethat, by definition,paths arealways directed, andthus all ofthesenotions refertodirectedpaths.

Therank ofG= (^V,E,att,lab,ext)isrank(G)= |^G |^andthatofe∈^{E isrank}G(e)=^rank(lab(e)).Thein-degree ofanode u∈^{V is}|{^e∈^E|^u∈ [^tar(e)]}| ^anditsout-degree is |{^e∈^E|^src(e)=^u}|^.Anodeofout-degree 0isa leaf,andanode v of in-degree 0, suchthateveryother nodeinV isreachablefromv,isaroot.Thus,therootofagraphisuniqueifitexists.

Ifitdoes,we saythat G isrooted.Notethat, iftherootisG,thenthewholegraphG isalsoreachable.Notefurthermore that,byourgeneralconditiononthesourcesofedges,allnodesinG areleaves.Thereadershouldkeepthisfactinmind becausewewilloccasionallymakeuseofitwithoutexplicitlymentioningit.

Foralabel A ofrankk,welet A^• denotethegraph({⁰,. . . ,k},{^e},^att,lab,0· · ·^k)suchthatatt(e)=⁰· · ·^k,andlab(e)= A.

2.3. Drawingconventions

WedrawgraphsasshowninFig.1: externalnodesaredepictedasbulletsandnon-externalonesascircles.Thenode G isalways thetopmostbullet.An edgee∈^EG isdepictedasaboxwiththeedgelabelinscribed,whichcanbedropped if itisnotrelevant.TheattachmentattG(e)isindicatedbyalinedrawnfromsrcG(e)to(theboxrepresenting)e,andarrows pointingfrome tothenodesintarG(e).ThearrowsleavetheboxintheorderinwhichtheyappearintarG(e),fromleftto right.Similarly,thenodesinG arearrangedfromlefttoright.Forexample,inthefigurewehavetarG(e)=^uv,G=^s,and G =^{v w.}

2.4. Hyperedgereplacement

LetH andF begraphsande∈^EH suchthat VH∩^VF= [^extF]^,EH∩^EF = ∅^,andattH(e)=^extF.Theresultofsubstituting e by F inH isthegraphG=^H[^e:^F]^suchthatG= (^VH∪^VF,(E_H∪^EF)\ {^e},^attG,lab_G,ext_H)with

attG(f)=



att_H(f) if f∈^EH\ {^e} att_F(f) if f∈^EF

labG(f)=



lab_H(f) if f∈^EH\ {^e} lab_F(f) if f∈^EF.

For graphs H and F and an edge e∈^EH withrank_H(e)=^rank(F) it should be clearthat we mayalways choose an isomorphiccopy F^ ofF suchthat H[^e:^F]^{isdefined.Toavoidthecumbersome}technicalitiesofconstantlyhavingtodeal with explicitisomorphisms, we shall thereforealways assume that F itself fulfills the requirements.If it does not, it is assumedthat F issilently replacedby anappropriate isomorphic copy.Notethat thisispossibleby ourassumptionthat neitherattachmentsofedgesnorthesequencesofexternalnodesofgraphscontainrepetitions.

For the remainder of the paper, we assume that LAB is partitioned into two disjoint subsets LABN and LABT, both countably infinite,whoseelements are callednonterminals and terminals,respectively.Naturally, a terminal(nonterminal) edgeisanedgelabeledbyaterminal(nonterminal,respectively).Wesometimesjustcallthemterminalsandnonterminals ifthereisnodanger ofconfusion.Byconvention, weusecapitalletterstodenotenonterminals,andlowercaselettersfor terminalsymbols.Furthermore,werefertothegraphH above,inwhichthereplacementtakesplace,asthehostgraph.

Definition2.2(hyperedgereplacementgrammar).Ahyperedgereplacementgrammar (HRgrammar,forshort)isasystemG = (,N,S,R) where⊆^LABT, N⊆^LABN, S∈^{N is theinitial}nonterminal,and R isasetofrules, alsocalledHR rules.Each ruleisoftheformA→^{F where A}∈^{N andF isagraphover}∪^{N withrank}(F)=^rank(A).

Thesize ofG is|G|=

(A→F)∈R|^F|^.

Forgraphs G,H , we let H⇒RG if there exista rule A→^F∈^{R and an edge e}∈^EH withlab(e)=^{A such that G}= H[^e:^F]^.Asusual, ⇒^∗_R ^{denotesthereflexivetransitiveclosureof}⇒R.Ifthereisnodangerofconfusionwe oftenwrite ⇒