Bidirectionalizing Graph Transformations

Transcription

1 Biiretionalizing Graph Transformations Soihiro Hiaka Kazuhiro Inaa Zhenjiang Hu Hiroyuki Kato National Institute of Informatis, Japan Kazutaka Matsua Tohoku University, Japan Keisuke Nakano The University of Eletro-Communiations, Japan Astrat Biiretional transformations provie a novel mehanism for synhronizing an maintaining the onsisteny of information etween input an output. Despite many promising results on iiretional transformations, these have een limite to the ontext of relational or XML (tree-like) ataases. We hallenge the prolem of iiretional transformations within the ontext of graphs, y proposing a formal efinition of a well-ehave iiretional semantis for UnCAL, i.e., a graph algera for the known UnQL graph query language. The key to our suessful formalization is full utilization of oth the reursive an ulk semantis of strutural reursion on graphs. We arefully refine the existing forwar evaluation of strutural reursion so that it an proue suffiient trae information for later akwar evaluation. We use the trae information for akwar evaluation to reflet in-plae upates an eletions on the view to the soure, an aopt the universal resolving algorithm for inverse omputation an the narrowing tehnique to takle the iffiult prolem with insertion. We prove our iiretional evaluation is well-ehave. Our urrent implementation is availale online an onfirms the usefulness of our approah with nontrivial appliations. Categories an Sujet Desriptors D.3.2 [Programming Languages]: Language Classifiations Speialize appliation languages; E. [Data Strutures]: Graphs an networks General Terms Design, Languages Keywors iiretional transformation, view upating, graph query an transformation, strutural reursion. Introution Biiretional transformations (Czarneki et al. 2009; Foster et al. 2005) provie a novel mehanism for synhronizing an maintaining the onsisteny of information etween input an output. They onsist of a pair of well-ehave transformations: forwar transformation is use to proue a target view from a soure, while the akwar transformation is use to reflet moifiation on the view to the soure. This pair of forwar an akwar transformations shoul satisfy ertain iiretional properties. Biiretional transformations are inee pervasive an an e seen in many interesting Permission to make igital or har opies of all or part of this work for personal or lassroom use is grante without fee provie that opies are not mae or istriute for profit or ommerial avantage an that opies ear this notie an the full itation on the first page. To opy otherwise, to repulish, to post on servers or to reistriute to lists, requires prior speifi permission an/or a fee. ICFP 0, Septemer 27 29, 200, Baltimore, Marylan, USA. Copyright 200 ACM /0/09... $0.00 appliations, inluing the synhronization of repliate ata in ifferent formats (Foster et al. 2005), presentation-oriente struture oument evelopment (Hu et al. 2008), interative user interfae esign (Meertens 998), ouple software transformation (Lämmel 2004), an the well-known view upating mehanism whih has een intensively stuie in the ataase ommunity (Banilhon an Spyratos 98; Dayal an Bernstein 982; Gottlo et al. 988; Hegner 990; Lehtenörger an Vossen 2003). Despite many promising results on iiretional transformations, they are limite to the ontext of relational or XML (tree-like) ataases. It remains unresolve (Czarneki et al. 2009) whether iiretional transformations an e aresse within the ontext of graphs ontaining noe sharing an yles. It woul e remarkaly useful in pratie if iiretional transformation oul e applie to graph ata strutures, eause graphs play an irreplaeale role in naturally representing more omplex ata strutures suh as those in iologial information, WWW, UML iagrams in software engineering (Stevens 2007), an the Ojet Exhange Moel (OEM) use for exhanging aritrary ataase strutures (Papakonstantinou et al. 995). There are many hallenges in aressing iiretional transformation on graphs. First, unlike relational or XML ataases, there is no unique way of representing, onstruting, or eomposing a general graph, an this requires a more preise efinition of equivalene etween two graphs. Seon, graphs have share noes an yles, whih makes oth forwar an akwar omputation muh more ompliate than that on trees; naïve omputation on graphs woul visit the same noes many times an possily infinitely. It is partiularly iffiult to hanle insertion in akwar transformation eause it requires a suitale sugraph to e reate an inserte into a proper plae in the soure. This paper reports our first solution to the prolem of iiretional graph transformation. We approah this prolem y proviing a iiretional semantis for UnCAL, whih is a graph algera for the known graph query language UnQL (Buneman et al. 2000); forwar semantis (forwar evaluation) orrespons to forwar transformation an akwar semantis (akwar evaluation) orrespons to akwar transformation. We hoose UnQL/UnCAL as the asis of our iiretional graph transformation for two main reasons. First, UnQL/UnCAL is a graph query language that has een well stuie in the ataase ommunity with a soli founation an effiient implementation. It has a onise an pratial surfae syntax ase on selet-where lauses like SQL, an an e easily use to esrie many interesting graph transformations. Seon, an more importantly, graph transformations in UnQL an e automatially mappe to those in terms of strutural reursion in UnCAL, whih an e evaluate in a ulk manner (Buneman et al. 2000); a strutural reursion is evaluate y first proessing in parallel on all eges of the input graph an

2 then omining the results. This ulk semantis signifiantly ontriutes to our iiretionalization, proviing a smart way of treating share noes an yles in graphs an of traing ak from the view to the soure. Our main tehnial ontriutions are summarize as follows. We are, as far as we are aware, the first to have reognize the importane of strutural reursion an its ulk semantis in aressing the hallenging prolem of iiretional graph transformation, an sueee in a novel two-stage framework of iiretional graph transformation ase on strutural reursion. We emonstrate that graph transformations efine in terms of strutural reursions (eing suitale for optimization as have een intensively stuie thus far (Buneman et al. 2000)) make akwar evaluation easier. We give a formal efinition of iiretional semantis for Un- CAL y () refining the existing forwar evaluation so that it an proue useful trae information for later akwar evaluation (Setion 4), an (2) using the trae information to reflet in-plae upates an eletions on the view to the soure, an aopt the narrowing tehnique to takle the iffiult prolem with insertion (Setion 5). We prove our iiretional evaluation is well-ehave. We have fully implemente our iiretionalization presente in this paper an onfirme the effetiveness of our approah through many non-trivial examples, inluing all those presente in this paper an some typial iiretional graph transformations in ataase management an software engineering. More examples an emos are availale on our BiG projet We site. We onsier an operation ase approah, whih means that the user expliitly provies eiting operations in terms of rename, elete, an insert. Currently these operations are treate aoring to the orer speifie y users. It might e hallenging to proue these operation sequenes automatially from the states efore an after user s moifiations on the view, ut it is eyon the sope of this paper. The forwar transformations we onsier is ase on UnCAL, whih is isimulation generi, meaning that the transformation an t istinguish etween graphs that are isimilar. For example, it an t extrat first hil of a noe. Extening our moel to ope with orer is inlue in our future work. Also note that akwar transformation is not isimulation generi, meaning that two results of upates that are isimilar o not always lea to isimilar soure. However, this is not neessarily a limitation introue y our iiretionalization, sine this asymmetry omes from the expressiveness of onitional expression in the original UnCAL graph algera. Similar argument apply for isomorphi upates. Outline We start with a rief review of the asi onept of a graph ata moel an the strutural reursion of UnCAL in Setion 2. Then, we larify the iiretional properties within our ontext an give an overview of our two-stage framework for iiretionalizing graph transformations in Setion 3. After explaining how to exten the forwar evaluation of UnCAL with trae information in Setion 4, we give a formal efinition of iiretional semantis for UnCAL an prove that it is well-ehave in Setion 5. We isuss implementation issues in Setion 6 an relate work in Setion 7. We onlue the paper in Setion a 5 6 a 3 a 4 (a) A Simple Graph a 0 3 a a () An Equivalent Graph Figure. Graph Equivalene Base on Bisimulation 2. UnCAL: A Graph Algera We aopte UnCAL (Buneman et al. 2000), a well-stuie graph algera, as the asis of our iiretional graph transformation. We will riefly review its graph ata moel an the ore of UnCAL. 2. Graph Data Moel We eal with roote, irete, an ege-laele graphs with no orer on outgoing eges. They are ege-laele in the sense that all information is store on laels of eges while laels of noes serve only as a unique ientifier without a partiular meaning. UnCAL graph ata moel has two prominent features, markers an ε-eges. Noes may e marke with input an output markers, whih are use as an interfae to onnet them to other graphs. An ε-ege represents a shortut of two noes, working like the ε-transition in an automaton. We use Lael to enote the set of laels an M to enote the set of markers. Formally, a graph G, sometimes enote y G (V,E,I,O), is a quaruple (V, E, I, O), where V is a set of noes, E V (Lael {ε}) V is a set of eges, I M V is a set of pairs of an input marker an the orresponing input noe, an O V M is a set of pairs of output noes an assoiate output markers. For eah marker &x M, there is at most one noe v suh that (&x, v) I. The noe v is alle an input noe with marker &x an is enote y I(&x). Unlike input markers, more than one noe an e marke with an iential output marker. They are alle output noes. Intuitively, input noes are root noes of the graph (we allow a graph to have multiple root noes, an for singly roote graphs, we often use efault marker & to iniate the root), while an output noe an e seen as a ontext-hole of graphs where an input noe with the same marker will e plugge later. We write inmarker(g) to enote the set of input markers an outmarker(g) to enote the set of output markers in a graph G. In aition, we write lael(ζ) to enote the lael of the ege ζ. Note that multiple-marker graphs are meant to e an internal ata struture for graph omposition. In fat, the initial soure graphs of our transformation have one input marker (single-roote) an no output markers (no holes). For instane, the graph in Figure (a) is enote y (V, E, I, O) where V = {, 2, 3, 4, 5, 6}, E = {(, a, 2), (,, 3), (,, 4), (2, a, 5), (3, a, 5), (4,, 4), (5,, 6)}, I = {(&, )}, an O = {}. Value Equivalene etween Graphs Two graphs are value equivalent if they are isimilar. Please refer to (Buneman et al. 2000) for the omplete efinition. Informally, graph G is isimilar to graph G 2 if every noe x in G has at least a isimilar ounterpart x 2 This analogy woul hoose NFA rather than DFA, sine we allow multiple outgoing eges with iential laels from a noe. 4 4

3 e ::= {} {l : e} e e &x := e &y () e e e yle(e) { onstrutor } $g { graph variale } if l = l then e else e { onitional } re(λ($l, $g).e)(e) { strutural reursion appliation } Figure 2. Graph Construtors in G 2 an vie versa, an if there is an ege from x to y in G, then there is a orresponing ege from x 2 to y 2 in G 2 that is a isimilar ounterpart of y, an vie versa. Therefore, unfoling a yle or upliating share noes oes not really hange a graph. This notion of isimulation is extene to ope with ε-eges. For instane, the graph in Figure () is value equivalent to the graph in Figure (a); the new graph has an aitional ε-ege (enote y the otte line), upliates the graph roote at noe 5, an unfols an splits the yle at noe 4. Unreahale parts are also isregare, i.e., two isimilar graphs are still isimilar if one as sugraphs unreahale from input noes. Graph Construtors Figure 2 summarizes the nine graph onstrutors that are powerful enough to esrie aritrary (irete, ege-laele, an roote) graphs (Buneman et al. 2000): G ::= {} { single noe graph } {l : G} { an ege pointing to a graph } G G 2 { graph union } &x := G { lael the root noe with an input marker } &y { a noe graph with an output marker } () { empty graph } G G 2 { isjoint graph union } G 2 { appen of two graphs } yle(g) { graph with yles } Here, {} onstruts a root-only graph, {l : G} onstruts a graph y aing ege l pointing to the root of graph G, an G G 2 as two ε-eges from the new root to the roots of G an G 2. Also, &x := G assoiates an input marker, &x, to the root noe of G, &y onstruts a graph with a single noe marke with one output marker &y, an () onstruts an empty graph that has neither a noe nor an ege. Further, G G 2 onstruts a graph y using a omponentwise (V, E, I an O) union. iffers from in that unifies input noes while oes not. requires input markers of operans to e isjoint, while requires them to e iential. G 2 omposes two graphs vertially y onneting the output noes of G with the orresponing input noes of G 2 with ε-eges, an yle(g) onnets the output noes with the input noes of G to form yles. Newly reate noes have unique ientifiers. We will give this reation rule extene for our iiretionalization in Setion 4.. The efinition here is ase on graph isomorphism (iential graph onstrution expressions results in iential graphs up to isomorphism), an they are, together with other operators, also isimulation generi (Buneman et al. 2000), i.e., isimilar result is otaine for isimilar inputs. Figure 3. Core UnCAL Language Example. The graph equivalent to that in Figure (a) an e onstrute as follows (though not uniquely). yle((&z := {a : {a : &z }} { : {a : &z }} { : &z 2}) (&z := { : {}}) (&z 2 := { : &z 2 })) For simpliity, we often write {l : G,..., l n : G n} to enote {l : G } {l n : G n}. 2.2 The Core UnCAL UnCAL (Unstruture Calulus) is an internal graph algera for the graph query language UnQL, an its ore syntax is epite in Figure 3. It onsists of the graph onstrutors, variales, onitionals, an strutural reursion. We have alreay etaile the ata onstrutors, while variales an onitionals are self explanatory. Therefore, we will fous on strutural reursion, whih is a powerful mehanism in UnCAL to esrie graph transformations. A funtion f on graphs is alle a strutural reursion if it is efine y the following equations f({}) = {} f({$l : $g}) = f($g) f($g $g 2 ) = f($g ) f($g 2 ), where the expression e may ontain referenes to variales $l an $g (ut no reursive alls to f). Sine the first an the thir equations are ommon in all strutural reursions, we write the strutural reursion in UnCAL simply as f($) = re(λ($l, $g).e)($). Despite its simpliity, the ore UnCAL is powerful enough to esrie interesting graph transformation inluing all graph queries (in UnQL) (Buneman et al. 2000), an nontrivial moel transformations (Hiaka et al. 2009). Some simple examples are given elow. Example 2. The following strutural reursion a2 replaes ege lael a with an leaves other laels unhange. a2($) = re(λ($l, $g). if $l = a then { : & } 2 else {$l : & 3 } 4 ) ($) 5 (The supersripts are for ientifying oe positions, whih will e important in Setion 4; they an simply e ignore for now.) Here is an instane of an exeution: a2 a = where enotes the root of the graph. Informally, the meaning of this efinition an e onsiere to e a fixe point (though may not neessarily unique) over the graph, whih is again efine y a set of equations using the three onstrutors {}, :, an. For instane, the graph in Figure (a) an e onsiere to e the fixe point of the following equations: G root = {a : {a : G 5 }, : {a : G 5 }, : G 4 } G 5 = { : {}} G 4 = { : G 4 }.

4 (a) Before Removing ε- eges S2 S3 S4 E2 2 S25 E25 5 S56 E56 E3 3 S35 E35 E4 4 S44 E44 6 () Removing ε-eges () Removing ε-eges After Figure 4. Bulk Semantis of Strutural Reursion in UnCAL Example 3. The following strutural reursion a2 x replaes all laels a with an removes eges laele. a2 x($) = re(λ($l, $g). if $l =a then { : & } 2 else if $l = then {ε : & 3 } 4 else {$l : & 5 } 6 ) ($) 7 Applying the funtion a2 x to the graph in Figure (a) yiels the graph in Figure 4(). Example 4. The following strutural reursion onseutive extrats sugraphs that an e aessile y traversing two onnete eges of the same lael. onseutive($) = re(λ($l, $g). re(λ($l, $g ). if $l = $l then {result : $g } else {} 2 )($g) 3 )($) 4 For example, we have onseutive! a a X a Y = result X Note that the strutural reursive efinition of onseutive uses graph parameter $g to ahieve the transformation. Also note that strutural reursions are allowe to e neste, an inner reursion an refer to outer variales (as $l in the example). This enales us to express the join of multiple queries. Example 5. Although the examples given so far are self-reursive, it is possile to simulate mutual reursion y returning graphs with multiple markers. For instane, the following funtion aa aa($) = re(λ($l, $g). &z :={a : &z 2} &z 2 :={ : &z })($) hanges all eges of even istanes from the root noe to a, an o istane eges to. We may onsier the markers &z i as a mutually reursive all, an aa to onsist of two mutual reursive funtions. The first is &z, whih, at eah ege in the original graph, generates a new a ege pointing to the result of &z 2 at the original estination noe. The seon is &z 2 that generates eges pointing to the result of &z from its estination. The result of the whole expression is efine to e the result of the &z at the root noe of the argument graph. The following figure shoul e helpful. The ashe eges enote the eges that are unreahale from the output root noe. aa( e ) = a a = a a a 2.3 Bulk Semantis of Strutural Reursion By allowing ε-eges, we an evaluate a strutural reursion in a ulk manner. Consier the strutural reursion, re(λ($l, $g). e) whih is to e applie to an input graph G. In ulk semantis, we apply oy e inepenently on every ege (l, g) in G where l is the lael of the ege an g is the graph that the ege is pointing to, then join the results with ε-eges (as in onstrutor). Reall the strutural reursion a2 x efine in Example 3. Applying it to the input graph in Figure (a) yiels the graph in Figure 4(a), where eah ege from i to j in the input graph leas to a sugraph ontaining a graph with an ege from Sij to Eij in the output graph (where the otte ege enotes an ε-ege), an these sugraphs are onnete with ε-eges aoring to the original shape of the graph. If we eliminate all ε-eges as explaine in Setion 3.2, we otain a stanar graph in Figure 4(). One istint feature of ulk semantis is that the shape of the input graph is rememere through aitional ε-eges, whih will e fully utilize in our later iiretionalization. 3. Overview: Biiretionalizing UnCAL It is more hallenging to iiretionalize transformations on graphs than trees, eause graphs may ontain share noes or yles. We shall emonstrate that the strutural reursion in UnCAL an serve as the asis to solve this prolem. Although strutural reursion was propose within the ontext of query optimization, we will show that it plays a ruial role in our iiretionalization. 3. Biiretional Properties Biiretionalization is use to erive akwar transformation from forwar transformation. We approah the prolem of iiretionalization in graph transformation y proviing a iiretional semantis for UnCAL; forwar semantis (forwar evaluation) orrespons to forwar transformation an akwar semantis (akwar evaluation) orrespons to akwar transformation. Before giving our iiretional semantis for UnCAL, let us larify the iiretional properties that the forwar an akwar evaluations shoul satisfy. Let F[[e]]ρ enote a forwar evaluation (get) of expression e uner environment ρ to proue a view, an B[[e]](ρ, G ) enote a akwar evaluation (put) of expression e uner environment ρ to reflet a possily moifie view G to the soure y omputing an upate environment. ρ is a set of mappings with form $x G with a graph (or lael) G. The following are two important properties: F[[e]]ρ = G B[[e]](ρ, G) = ρ (GETPUT) B[[e]](ρ, G ) = ρ G Range(F[[e]]) F[[e]]ρ = G (PUTGET) The (GETPUT) property states that unhange view G shoul give no hange on the environment ρ in the akwar evaluation, while the (PUTGET) property states that if a view is moifie to G whih is in the range of the forwar evaluation, then this moifiation an e reflete to the soure suh that a forwar evaluation will proue the same view G. These two properties are essentially the same as those in (Foster et al. 2005). One prolem with the (PUTGET) property is that it

5 v ε losure v ε losure Figure 5. General ε-ege Elimination Proeure nees to hek whether a graph is in the range of forwar evaluation, whih is iffiult to o in pratie. To avoi this range heking, we allow the moifie view an the view otaine y akwar evaluation followe y forwar evaluation to iffer, ut require oth views to have the same effet on the original soure if akwar evaluation is applie. B[[e]](ρ, G ) = ρ F [e]ρ = G B [e](ρ, G ) = ρ (WPUTGET) The get in our (WPUTGET) an e onsiere as an amenment of the moifie view G to G. Certainly, if the (PUTGET) property hols, so oes the (WPUTGET). We say that a pair of forwar an akwar evaluations is well-ehave if it satisfies (GETPUT) an (WPUTGET) properties. In the rest of this paper, we will give a iiretional evaluation (semantis) for UnCAL, an prove the following theorem, whih is a iret onsequene of Lemmas 2, 3, an 4 that will e isusse later. Theorem (Well-ehaveness). Our forwar an akwar evaluations are well-ehave, provie their evaluations suee. 3.2 Two-Stage Biiretionalization Reall a2 x, whih maps the soure graph in Figure (a) to the view graph in Figure 4(). The ig gap etween the soure an the view makes it har to reflet hanges on the view to the soure. Our iea to rige this gap was to ivie the forwar evaluation into two easily hanle stages: Stage : Forwar evaluation (in the ulk semantis) with suffiient ε-eges, so that the output graph will have a similar shape to the input graph, making the later akwar evaluation easier. Stage 2: Elimination of ε-eges to proue a usual view. For a2 x, Stage maps the soure graph to the intermeiate graph in Figure 4(a), an Stage 2 maps the intermeiate graph to the view graph (Figure 4()). By oing so, eah stage eomes easier to iiretionalize. First, let us onsier Stage 2. The ε-ege elimination proeure is simple: new eges are ae to skip the ε-losure (Figure 5). It is easy to efine a well-ehave akwar evaluation for this proeure. First, all noes in the result graph, G v, exist in the original graph, G s, so eah noe in G v an e trae to G s. Seon, although an ege in G s may e upliate in G v ((E25,, E56) an (E35,, E56) in Figure 4()), eah ege in G v shoul have a uniquely orresponing ege in G s. Therefore, aing a new noe to G v orrespons to aing a new noe to G s, an aing a new ege to G v orrespons to aing a new ege etween two orresponing noes in G s. Similar orresponene hols for eletions of noes an eges, an in-plae upates of eges. Next, let us onsier Stage. One fat worth noting is that after the akwar evaluation in Stage 2, the moifiation to the view in Note that Figure 4() oes not have this upliation eause for this partiular graph, it is safe to glue the soure an the estination noes of an ε-ege together. It is unsafe, if an only if, the soure has another outgoing ege an the estination has another inoming ege. Here, upliation is unavoiale. Stage satisfies the ε-marker preserving property: () No ε-eges are ae or elete, (2) Markers are not ae, elete, or hange an (3) Unreahale parts are not moifie. This property is very important in our iiretionalization, eause it not only enfores the nine graph onstrutors so that they are invertile, ut it also makes it easy to iiretionalize strutural reursion eause there is a lear orresponene etween the input an output graphs. In the rest of this paper, we will fous on iiretional graph transformation in Stage. 4. Traeale Forwar Evaluation An UnCAL expression usually speifies a forwar evaluation mapping a graph ataase (whih is just a graph) to a view graph (in Setion 2). The main purpose of the present paper is to give akwar evaluation (akwar semantis), whih speifies how to reflet view upates to the graph ataase. For this purpose, we have to etet how eah noe of the view is generate, partiularly when it is onstrute through onneting input/output markers an removing ε-eges, whih are no longer in the view. To make the view more informative, viz., traeale, we enrih the original semantis of UnCAL y emeing trae information (like provenane traes (Cheney et al. 2008)) in all noes of the view that possily inlues ε-eges. In this setion, we explain what kin of trae information is emee in the view, an exten the original semantis for UnCAL expressions to e evaluate into traeale views. 4. Traeale Views A view is otaine y evaluating an UnCAL expression with a ataase. Every noe of the view originates in either a noe of the ataase or a onstrut in the UnCAL expression, exept when the noe is generate through a strutural reursion with a re onstrut (in the ulk semantis). Reall that an expression re(λ($l, $g).e )(e 2 ) is evaluate y ining variales $l an $g in e to a part of the evaluation result of e 2. In this ase, a noe in the view may originate not only in the whole re expression ut also a su-expression in e 2. A traeale view is a view eah noe of whih has information for traing its origin. The information, alle trae ID, is efine y TraeID ::= SrID Coe Pos Marker ReN Pos TraeID Marker ReE Pos TraeID Ege, where SrID ranges over ientifiers uniquely assigne to all noes of the ataase, Pos ranges over oe positions in the UnCAL expression, Marker ranges over input/output markers, an Ege stans for TraeID Lael TraeID with a set of laels Lael. We now riefly explain the meaning of eah trae ID. Let i e a trae ID of a noe u in a traeale view. When i is a noe ientifier in SrID, noe u originates in the noe assigne y i in the ataase. When i is Coe p &m with oe position p an input marker &m, noe u originates in the suexpression at p in the UnCAL expression. The marker &m is only require when the suexpression is given y the or yle onstrut. This is eause these onstruts yiel as many ε-eges as input markers. When i is either ReN p i 0 &m or ReE p i 0 (i, a, i 2), noe u is generate through the re onstrut at the oe position p. ReN an ReE stan for what noe an ege, respetively, of the argument of the reursion, the noe originates in. Let us explain these ases through an example where the Un- CAL expression a2 x in Example 3 is applie to the ataase G sr in Figure (a). The traeale view we want an e otaine from the graph G view in Figure 4(a) y assigning trae IDs to all noes. The trae ID assigne to noe in G view is

6 (ReN 7 &) eause the noe originates in noe of G sr in SrID, whih is use as a part of the argument of the re onstrut at oe position 7 in a2 x. The trae ID assigne to noe S2 in G view is (ReE 7 (Coe 2) (, a, 2)) eause the noe originates in the a-laele ege from noe to 2 of G sr in Ege through the graph onstrutor { : } at oe position 2 in the re onstrut at 7 in a2 x. When the argument of the re onstrut is also a re expression, ReN an ReE in the trae ID are neste like (ReN p (ReE p... )... ) an (ReE p (ReE p... ) (ReN..., a, ReN... )). A traeale view is enote y a quaruple (V, E, I, O) just like an orinary UnCAL graph. The only ifferene is that in traeale views, trae IDs are assigne to all noes. 4.2 Enrihe Forwar Semantis Traeale views an e ompute y a simple extension of the original forwar semantis of UnCAL so that traing information is reore when a noe is reate. Let e p enote an UnCAL suexpression e at oe position p. We write ρ($x) for G when ($x G) ρ. ρ is naturally use as variale sustitution in UnCAL expressions, e.g., eρ for an expression e. We inutively efine the enrihe forwar semantis F [e p ]ρ for eah UnCAL onstrut of e. Graph Construtor Expressions. The semantis of graph onstrutor expressions is straightforwar aoring to the onstrution in Figure 2. For instane, we have F[[{} p ]]ρ = ({Coe p},, {(&, Coe p)}, ), whih reates a graph having a single noe with the trae ID of Coe p (iniating the noe is onstrute y the oe at position p), no eges, an input noe (the single noe itself), an no output noes. As another example, the semantis for the expression e e 2 is efine elow to unify two graphs y onneting their input noes with mathing markers using ε-eges: F[[(e e 2 ) p ]]ρ = F [e ]ρ p F [e 2 ]]ρ, where p is a union operator for two graphs onerning position p an is efine y G p G 2 = (V V V 2, E E E 2, I, O O 2 ) where (V, E, I, O ) = G (V 2, E 2, I 2, O 2 ) = G 2 M = inmarker(g ) = inmarker(g 2 ) V = {Coe p &m &m M} E = {(Coe p &m, ε, v) (&m, v) I I 2} I = {(&m, Coe p &m) &m M}. We omit efinitions for other onstrutor expressions. Variale. A variale looks up its ining from environment ρ. Conition. F [($v) p ]ρ = ρ($v) The forwar semantis of a onition is efine as F[[(if l j= l 2 then e else e 2) p ]ρ F [e ]ρ if l = ρ = l 2ρ F [e 2 ]ρ otherwise. It first evaluates the onitional expression l = l 2, an with the result it evaluates either the then ranh or the else ranh. Strutural Reursion. The semantis of a strutural reursion is given y ulk semantis as reviewe in Setion 2.3, whih an e formally efine y F[[(re(λ($l, $g). e )(e a )) p ]ρ = ompose p re (fw eahege(g a, ρ, e ), G a, M) where M = inmarker(e ) outmarker(e ) G a = F [e a ]ρ, where fw eahege an ompose re are efine in Figure 6. Intuitively, fw eahege evaluates the oy expression e at eah ege ζ of the argument graph G a otaine y evaluating e a an returns the set of result graphs. Then, ompose p re glues all the graphs together along the struture of G a onerning oe position p. Note that sugraph(g, ζ) enotes the sugraph to whih the ege ζ is pointing in the graph G. Example 6. We will now illustrate the semantis of re through an example: the strutural reursion a2 x, whih is efine with position information in Example 3, is applie to G sr in Figure (a), an the traeale view is a graph similar to G view in Figure 4(a). First, G sr is oun to a variale $. Then, fw eahege generates a set of pairs of an ege an a loal result for eah ege in G sr. The loal result is otaine y evaluating the oy of re uner ρ = {$ G sr } {$l L, $g G} with the lael L of the ege an a sugraph G reahale from the ege. For example, as the loal result for ege (3, a, 5) in G sr, ege (Coe 2,, Coe ) with input noe Coe 2 an output noe Coe is generate eause the suexpression { : & } 2 is use ue to $l = a. The funtion ompose p re glues all pairs of an ege an a loal result after aing ReN or ReE to their noes. For example, regaring a pair of ege ζ = (3, a, 5) an its loal result ontaining ege (Coe 2,, Coe ), the set E ReE ontains ege (ReE 7 (Coe 2) ζ,, ReE 7 (Coe ) ζ) where 7 is the oe position of the onerne re, while set E ReN ontains ege (ReN 7 3 &, ε, ReE 7 (Coe 2) ζ) an (ReE 7 (Coe ) ζ, ε, ReN 7 5 &) ue to (&, Coe 2) I an (Coe, &) O. The former orrespons to the ege from S35 to E35 of G view an the latter orrespons to two eges from 3 to S35 an from E35 to 5 of G view. In this example, E ε is an empty set sine G sr has no ε-eges. The sets I ReN an O ReN of input an output noes are otaine with I = {(&, )} an O =, respetively, whih are those of G sr. Hene, I ReN = {(&.&, ReN 7 &)} an O ReN = eause M = inmarker(e ) outmarker(e ) = {&}. Here,. enotes Skolem funtion (Buneman et al. 2000) that satisfies (&x.&y).&z = &x.(&y.&z) (assoiativity) an &.&x = &x.& = &x (left an right ientity). More onretely, if the soure graph is s = a2 x(s) gives the graph ReN 7 & ReE 7 (Coe 6) (,, 2) ReE 7 (Coe 5) (,, 2) ReN 7 2 & whih is isimilar to the graph 5. Bakwar Evaluation of UnCAL ReE 7 (Coe 2) (, a, 2) ReE 7 (Coe ) (, a, 2). a 2, With traeale views an the ε-marker preserving property (Setion 3) on the moifiation of suh views, akwar evaluation (in Stage ) turns out to e simpler for two reasons. First, the graph onstrutors eome invertile. For instane, if G = G G 2, G is moifie to G, ut the moifiation is ε-marker preserving; then, we an follow traing information, ε-eges, an marker information to uniquely eompose G to G an G 2 suh that G G 2 G hols. We will write this G (V,E,I,O ) G 2(V2,E 2,I 2,O 2 ), the exat equivalene of two graphs, is efine y V = V 2 E = E 2 I = I 2 O = O 2.

7 n o fw eahege(g (,E,, ), ρ, e) = (ζ, F[[e]]ρ ζ ) ζ E, lael(ζ) ε, ρ ζ = ρ {$l lael(ζ), $g sugraph(g, ζ)} ompose p re(g, (V, E, I, O), M) = (V ReE V ReN, E ReE E ReN E ε, I ReN, O ReN ) where V ReE = {ReE p v ζ (ζ, (V ζ,,, )) G, v V ζ } E ReE = {(ReE p u ζ, l, ReE p v ζ) (ζ, (, E ζ,, )) G, (u, l, v) E ζ } V ReN = {ReN p v &m v V, &m M} E ReN = {(ReN p v &m, ε, ReE p u ζ) &m M, (ζ = (v,, ), (,, I ζ, )) G, (&m, u) I ζ } {(ReE p u ζ, ε, ReN p v &m) &m M, (ζ = (,, v), (,,, O ζ )) G, (u, &m) O ζ } E ε = {(ReN p v &m, ε, ReN p u &m) (v, ε, u) E, &m M} I ReN = {(&n.&m, ReN p v &m) (&n, v) I, &m M} O ReN = {(ReN p v &m, &n.&m) (v, &n) O, &m M} Figure 6. Core of Forwar Semantis of re at Coe Position p eomposition as eomp G G 2, an applying it to G will give (G, G 2). Seon, akwar evaluation of a strutural reursion re(e) is reue to that of its oy e (followe y result gluing), eause of the ulk semantis of strutural reursion. Bakwar evaluation greatly epens on what upates are allowe on the view. We allow the following three general upates on our ege-laele graphs: () in-plae upates as moifiation of ege laels, (2) eletion of eges, an (3) insertion of eges or a sugraph roote at a noe. An we aept a sequene of these upates on the view an reflet them to the soure. In the rest of this setion, we shall explain the respetive akwar evaluation for these upates on views. 5. Refletion of In-plae Upates In this setion, we formally efine akwar semantis for UnCAL, where only in-plae upates are onsiere. Reall that akwar semantis B[[e](ρ, G ) is use to ompute a new environment from the original input environment ρ an the moifie view G. Like forwar semantis, akwar semantis an e efine inutively over the onstrution of expression. 5.. Bakwar Evaluation of Simple Expressions Graph Construtor Expressions. Sine eah onstrutor is revertile an is assoiate with a eomposition funtion, we an eompose the views of onstrutor expressions so as to efine the akwar semantis inutively. For example, we have B[[(e e 2) p ]](ρ, G ) = B [e ]](ρ, G ) ρ B [e 2 ](ρ, G 2) where G = F [e ]]ρ G 2 = F [e 2 ]]ρ (G, G 2) = eomp G G 2 (G ) Unlike Foster et al. (2005), we have variale ining, an therefore multiple environments proue y akwar evaluation of the operans are merge y ρ efine elow, using an approah similar to that in Liu et al. (2007), whih eals with variale inings. (ρ 8 ρ ρ 2 ) 9 < = : ($v mg(g, G ($v G ) ρ, =, G 2 ) ($v G) ρ, ($v G 2) ρ ; 8 2 < G if G 2 = G G = G 2 where mg(g, G, G 2 ) = G 2 if G = G : FAIL otherwise ρ unifies eah ining y mg. If only the ining on the left han sie is moifie (G 2 = G), or oth are onsistently upate It woul e more preise to write it as eomp G,,G 2 in that the eomposition epens on three arguments. (G = G 2 ), then the ining on the left is aopte, an vie versa. If oth are upate to ifferent values, it fails, leaing to the failure of the entire akwar evaluation. Lael variale inings are treate similarly. We have omite the efinitions for other onstrutor expressions, whih an e efine similarly. Variale. A variale simply upates its ining as B[[$v]](ρ, G ) = ρ[$v G ]. Here, ρ[$v G ] is an areviation for (ρ \ {$v G }. Conition. }) {$v The akwar evaluation of a onition is efine y B[[if l 8= l 2 then e else e 2 ]](ρ, G ) < ρ if l ρ = l 2ρ l ρ = l 2ρ = ρ 2 if l ρ l 2ρ l ρ 2 l 2ρ 2 : FAIL otherwise where ρ = B[[e ]](ρ, G ) ρ 2 = B[[e 2]](ρ, G ), whih is reue to the akwar evaluation of e if l = l 2 hols, an to the akwar evaluation of e 2 otherwise. To guarantee well-ehaveness, we ensure that l = l 2 oes not hange after akwar evaluation Bakwar Evaluation of Strutural Reursion Due to the traeale ulk forwar evaluation of strutural reursion re an the ε-marker preserving property that retains similarity in shape etween input an output graphs, akwar semantis an easily e efine as B[[re(λ($l, $g). e )(e a)]](ρ, G ) = merge(ρ, e a, E a, w eahege(g a, ρ, e, eomp re (G, E a))) where G a = (, E a,, ) = F[[e a ]]ρ This efinition is easy to unerstan if we note uality with the efinition of its forwar semantis. Bakwar semantis first eomposes through eomp re the moifie result graph G into piees of graphs, whih is intuitively an inverse operation of ompose re. For every non-ε ege ζ E a in the soure argument graph, the eomposition extrats (possily moifie) supart G ζ of G, whih originates at the result G ζ of the forwar omputation on the ege. Then, in w eahege, we arry out akwar omputation of the oy expression e on eah ege an ompute the upate environment ρ ζ. Finally, these environments are merge into the upate environment ρ of the whole expression. The merge funtion oes two piees of work. First, y omining the information ρ ζ($l) an ρ ζ($g) from the upate environments (an ε-eges existing in the eges E a of the soure argument graph), it omputes the moifie argument graph G a. Then, we inutively arry out akwar

8 8 9 ζ E a, lael(ζ) ε, >< V eomp re ((V, E, I, O ), E a ) = (ζ, (V ζ, E ζ, I ζ, O ζ )) ζ = {w (ReE p w ζ) V }, >= E ζ = {(w, l, w 2 ) (ReE p w ζ, l, ReE p w 2 ζ) E }, >: I ζ = {(&m, w) (ReN p v &m, ε, ReE p w ζ) E }, O ζ = {(w, &m) (ReE p w ζ, ε, ReN p v &m) >; E } n o w eahege(g, ρ, e, G ) = (ζ, B[[e]](ρ ζ, G ζ )) (ζ, G ζ ) G, ρ ζ = ρ {$l lael(ζ), $g sugraph(g, ζ)} merge(ρ, e a, E a, R) = B[[e a]](ρ, G U n o a) ρ ρ ζ \ {$l } \ {$g } (ζ, ρ ζ ) R S where G a = V ζ, E eps S E ζ, I a, O a E eps = {(u, ε, v) (u, ε, v) E a } (V ζ, E ζ ) = V ζ {u}, E ζ {(u, ρ ζ ($l), I ζ (&))} for eah (ζ, ρ ζ ) R, letting (u,, ) = ζ an (V ζ, E ζ, I ζ, O ζ ) = ρ ζ ($g) Figure 7. Core of Bakwar Semantis of re at Coe Position p evaluation on the argument expression e a to otain another upate environment ρ a. This ρ a an all ρ ζs are merge into ρ. Let us explain in more etail the efinition of eomp re, whih is the key point of the akwar evaluation. The funtion first extrats from result graph G noes V ζ an eges E ζ that elong to eah ege ζ y mathing trae ID ReE p ζ. Note that if there are noes that have een freshly inserte into the view, we also require these noes to have this struture, so that these noes are also passe to the akwar evaluation of the reursion oy. Input an output noes with marker &m are reovere y seleting those pointe from/to hu noes having struture ReN &m. Top-level onstrutors of trae ID are erase so that we an inutively ompute the akwar image from the oy expression. Example 7. Reall the simple example in Example 3 where the soure is s = a 2, an a2 x(s) gives the graph G. If the graph G is moifie to G where the ege lael is upate to X, then B[[a2 x]]({$ s}, G ) returns ining {$ s } X where s = a 2. Therefore, the in-plae upate of the hange on the view graph is reflete to the soure. Lemma 2 (Well-ehaveness for In-plae Upates). If output graphs are moifie y in-plae upates on eges, then for any expression e, the two evaluations F[[e] an B [e](, ) form a well-ehave iiretional transformation, if they suee. Proof. This statement an e prove y inution on the struture of e. For the ase ase where e is a variale, it learly hols. Consiering the inutive ase, () if e is a onstrutor expression, it hols eause eah onstrutor is revertile within our ontext, (2) if e is a onition, its akwar evaluation is reue to that on either its true ranh or its false ranh, so the statement hols y inution, an (3) if e is a strutural reursion, y ulk semantis, its akwar omputation is reue to its oy expression, so the statement hols y inution. 5.2 Refletion of Deletion Deletion in a view is reflete as eletion of the orresponing part in the soure y using trae IDs. Suppose we want to elete the ege laele in the view of Example 7. Sine oth enpoints of the ege have trae IDs of the form ReE 7 (, a, 2), we an see that the selete ege has een generate ue to the existene of the soure ege (, a, 2), whih is the orresponing part to e elete in the soure. In general, for a laele ege ζ = (u, l, v) with l ε, its orresponing ege orr(ζ) is efine as: orr((u, l, v)) = (u, l, v) if u, v SrID orr((ree p u ζ j, l, ReE p v ζ )) orr((u, l, v)) if orr((u, l, v)) FAIL = orr(ζ ) if orr((u, l, v)) = FAIL orr(ζ) = FAIL otherwise. Here, FAIL means failure on fining the orresponing ege. The first ase means that if the ege ζ is a opy of an ege in the soure, then ζ itself is the orresponing ege. The seon an the thir ases are for when ζ is a result of some strutural reursion. Aoring to the forwar semantis of re in Figure 6, the non-ε ege ζ must have the form (ReE p u ζ, l, ReE p v ζ ) for some p, u, v, an another non-ε ege ζ. This means that ζ onsists of an ege (u, l, v) originating from an evaluation of a reursion-oy at ζ. Hene, for this ase, we first reursively trae the orresponing soure of (u, l, v), an if this fails, then try that of ζ. In other ases, orr fails to fin the orresponing soure, eause it must e the ase that u has a trae ID of the form Coe, meaning that the ege is not erive from the soure ut from an UnCAL expression. Let $ e the soure graph, G view e the view proue y F[[e]]ρ from a forwar omputation of expression e with environment ρ, an G view e a graph from G view with a set of eges D out = {ζ,..., ζ n} remove. Our akwar evaluation B[[e]](ρ, G view) onsists of the following three steps.. Compute the set of soure eges D in = {orr(ζ i ) ζ i is not an ε-ege}. 2. If FAIL D in, akwar evaluation fails. If it is otaine suessfully without failure, ompute G sr = ρ($) D in, where G E enotes removal of the eges in the set E from graph G. 3. Return ρ = ρ[$ G sr] as the result if F[[e]]ρ = G view, an fail otherwise. Lemma 3 (Well-ehaveness for Deletion). If output graphs are moifie y ege eletion, then for any expression e, the two evaluations F[[e]] an B[[e]](, ) form a well-ehave iiretional transformation, if they suee. Proof. The (GETPUT) property is lear eause of the fat that D in = if D out =. For the (WPUTGET) property, it hols eause the thir step atually oes this hek.

9 5.3 Refletion of Insertion Refletion of insertion is muh more ompliate than that of inplae-upating an eletion. This is eause there are no orresponing eges in the soure for the freshly inserte eges in the view, whih requires us not only to reate new information ut also to a it to a proper loation in the soure graph. Our iea was to use the Universal Resolving Algorithm (URA) (Aramov an Glük 2002), a powerful metho of inversion omputation, to erive a right inverse of the forwar evaluation, an use the istriutive property of strutural reursion re(e)($g $g 2 ) = re(e)($g ) re(e)($g 2 ) to properly reflet insertion to the soure. In this setion, we shall give our algorithm for this refletion, efore we highlight how URA an e use to erive the right inverse Insertion Refletion with Right Inverse We assume the monotoniity of insertion in that an insertion on the view is translate to an insertion on the soure rather than other upating operations. The monotoniity omes from the asene of isempty (Buneman et al. 2000) in our ore UnCAL. We only onsier insertion on the view graph proue y forwar omputation of a variale expression or a strutural reursion. For the ase of a variale, this refletion is one in the same way as in Setion 5... Insertion for strutural reursion, the asi omputation unit in Un- CAL, nees to e arefully esigne. In the following, we will fous on strutural reursion, omitting other ases for simpliity. Before giving our refletion algorithm, we shoul larify the meaning of right inverse. In general, a funtion h is sai to e a right inverse of f if for any x in the range of f, f(h(x)) = x hols. Within our ontext, for an expression e an a graph G, F [[e]](g) is sai to e a right inverse omputation if it returns ρ suh that F[[e]]ρ = G. Now, we will return to our refletion algorithm. Let G sr e the soure graph, G view = F [re(e)($)]ρ, where ρ = {$ G sr }, an G view e a graph from G with new eges inserte. Notie that it is suffiient to onsier $ as the argument of re, eause $ an e oun to other expression. Our akwar evaluation B[[re(e)($)](ρ, G view) returns ρ as the result if there are no new eges inserte in G view ; otherwise, it oes the following:. Extrat the inserte sugraph G from G view suh that G view = G view G. 2. Compute with right inverse omputation: ρ = F [[re(e)($)]](g ). 3. Return ρ 2 = {$ G sr ρ ($)} as the result. The first step of extration is possile provie that insertion happens at the root noe. The seon step of right inverse omputation will e explaine in Setion The last step is to upate the ining of $ an return this environment as our result. The following lemma shows the orretness of the algorithm. Lemma 4 (Well-ehaveness for Insertion). If output graphs are moifie y ege insertion, then for a strutural reursion of the form re(e)($) where e ontains no free variales, then two evaluations F[[e]] an B[[e](, ) form a well-ehave iiretional transformation, if they suee. Insertions to non-root positions are possile ue to ulk semantis that allows similar treatment for every noe. Proof. First, the (GETPUT) property learly hols eause ρ is returne when no insertions our. Next, we prove the (WPUTGET) property y using the following alulation. F[[re(e)($)]]ρ 2 = { partial appliation } F[[re(e)(ρ 2($))]]ρ 2 = { ef. of ρ 2 } F[[re(e)(G sr ρ ($))]]ρ 2 = { strutural reursion property } F[[re(e)(G sr ) re(e)(ρ ($)))]]ρ 2 = { forwar evaluation } F[[re(e)(G sr )]]ρ 2 F[[re(e)(ρ ($)))]ρ 2 = { e oes not ontain free variale } G view F[[re(e)($)]]ρ = { right inversion } G view G It is worth noting that we have simplifie our isussion in oth the aove algorithm an lemma y making it a requirement that e in re(e)($) oes not ontain any free variales. With this requirement, our forwar an akwar evaluation satisfies the stronger (PUTGET) property. In fat, it is aeptale to relax this onition y allowing e to ontain other free variales an the initial ρ ontains ining of other variales. Then, right inversion will proue ρ that will e use to upate all variale inings in aition to $. In this ase, F[[re(e)(G sr )]]ρ may proue a graph that is ifferent from the original view G view. In any ase, this ifferent graph will not have an aitional effet on the soure when we apply akwar evaluation to this new graph. Therefore, (WPUTGET) always hols. With this iea, we shall propose an algorithm in whih (PUTGET) property is satisfie without any aitional requirements. The iea is to utilize the Trae ID information, as will e isusse later Improving Insertion Refletion The metho aove satisfies the (PUTGET) property only if the variales of e are isjoint from the variales oun in the initial environment ρ. However, in general, sine a transformation may have multiple variale referenes, more effort is require to ahieve the (PUTGET) property. We takle the prolem y first loating where we insert a graph y using trae IDs, an then applying the URA algorithm (to e esrie later) to fin what graph shoul e inserte. Consier the transformation a2 x an the view in Example 6. Suppose we want to insert a graph G vins roote at the view noe v = ReN 7 2 &. Where shoul some graph e inserte into the soure to reflet this insertion? The answer is that we must insert a graph roote at the soure noe 2 eause there woul e no ege from v in the view unless there were an ege from 2 in the soure aoring to the ulk semantis of strutural reursion. Now, our next task is to fin what graph shoul e inserte uner the soure noe 2. That is, we hope to fin G sins suh that the following hols.! a2 x a 2 G sins ReN 7 & ReE 7 (Coe 4) (,, 2) = ReE 7 (Coe 3) (,, 2) ReN 7 2 & ReE 7 (Coe 2) (, a, 2) ReE 7 (Coe ) (, a, 2) G vins

10 tr(srid) = SrID tr(ren ti ) = tr(ti) tr(coe ) = FAIL tr(ree ti ) = tr(ti) Figure 8. Traing Noe ID URA an help us to fin suh G sins for G vins. For example, if G vins is { : {}}, then URA returns G sins = { : {}}. If G vins is { : {}}, then URA returns one of the possiilities, G sins = {a : {}} or G sins = { : {}}, epening on the searh metho use in URA. Aoring to the sounness an the ompleteness of URA, the refletion y URA is always orret in the sense that (PUTGET) hols, an moreover URA always returns a G sins if suh G sins exists. Of these, sounness is the key to insertion refletion satisfying (PUTGET) for general UnCAL transformations. In summary, our insertion-refletion algorithm is as follows.. Let v e a noe uner whih we want to insert a graph G vins. 2. By using the tr funtion in Figure 8, we fin the soure noe u = tr(v) uner whih we insert a graph to reflet the insertion. 3. Let G view e a graph otaine from the view y aing ε-ege from v to G vins. 4. We fin a graph G sins onnete from u y an ε-ege, y applying URA for G view. 5. We return a graph G sr otaine from the soure y aing an ε-ege from u to G sins. The sounness of the insertion-refletion algorithm is iretly erive from the sounness of URA. Lemma 5 (Sounness of Insertion). Our insertion-refletion algorithm satisfies (PUTGET). Note that we use URA for G view instea of G vins. Thus, URA rejets any insertion of G sins that violates (PUTGET). In aition, our insertion-refletion algorithm is omplete in the sense that, if there exist some soure insertions to reflet the view insertion uner some onitions, the algorithm will fin one of them. Lemma 6 (Completeness of Insertion). Let v e a noe suh that tr(v) FAIL. For any soure graph G, we an insert any graph into its view if there exists a soure insertion that reflets the view insertion an v still ours in the view of the insertion-reflete soure. Reall that we only onsier insertion on the view graph proue y forwar omputation of a variale expression or a strutural reursion, whih is expresse y tr(v) FAIL. This lemma an e prove using the property of trae IDs stating that, to insert a graph roote at view noe v, we must insert a graph roote at soure noe tr(v). By inution on the trae ID of v, we an show that, if there is an ege from v, it must e the ase that there is an ege from tr(v), whih is implie y the property of trae IDs. Note that G vins has no ege to the original view. However, this is not a restrition sine if there is a rossing ege pointing to a sugraph of the original view, we an upliate the sugraph an integrate it to G vins so that the ege an e eliminate Right Inverse Computation y URA Reall that the right inverse omputation of an expression e is to take a graph G view an return a ρ suh that F [e]ρ = G view. We aopt the universal resolving algorithm (URA) (Aramov an Glük 2002), a powerful an general inversion mehanism, to ompute ρ. The asi iea ehin URA is to searh on a perfet proess tree (Glük an Klimov 993), whih represents all possile omputations of an expression, an to fin a omputation path that proue the result. Our right inverse omputation onsists of three steps.. It lazily enumerates possile evaluation paths y symoli omputation alle neee narrowing (Antoy et al. 994). 2. From the generate evaluation paths, it onstruts a tale of input/output pairs of omputations. 3. If there is a pair in the tale whose output is G view, it generates a sustitution (environment) from the path an returns it as the result. Example 8. As a simple example, let us see how we fin ρ suh that F[[a2 x($x)]]ρ = G view where G view = { : {}}. We searh ρ y symoli evaluation of a2 x($x). To evaluate a2 x($x), we unfol $x an reursively evaluate a2 x, i.e., a strutural reursion. There are many ways to instantiate $x suh as $x {}, $x {$l : $x }, $x {$l : $x, $l 2 : $x 2}. If we hoose $x {}, the omputation finishes, yieling a tale onsisting of an input/output pair ({}, {}). Sine this tale oes not ontain a pair whose output is G view, we ontinue searhing. Assume that we hoose $x {$l : $x }. Then a2 x($x) is unfole to (if $l = a then { : &} else (if $l = then {ε : &} else {$l : a2 x($x ). As evaluation gets stuk here eause of a free variale $l in the if onition, we fin a suitale $l to resume the evaluation. If we hoose $l a, then the expression is reue to { : a2 x($x ) an input/output pair ({a : {}}, { : {}}) is otaine y hoosing $x {}. Sine G view = { : {}}, we gather all inings along this omputation an return the following environment as the result. {$x {a : {}}} Figure 9 shows part of a perfet proess tree in our right-inverse omputation: the left is the tree an the right is a tale of a pair of input/output graph templates (it is more general than a pair of input/output graph instanes, as we isusse aove). Note this tree is a variant of SLD-resolution trees (Glük an Sørensen 994). To use URA effetively for our right inverse omputation of UnCAL, we efine a small-step semantis for UnCAL suh that a perfet proess tree an e onstrute though these small steps. The only non-stanar feature of this semantis is that we use memoization to avoi infinite loops proaly ause y yles in the soure graph (See Appenix of (Hiaka et al. 200) for etails). In aition, we provie a Dijkstra-searhing strategy to enumerate all the possile evaluation paths so that a solution an always e foun if one exists. The two heuristis we use to esign the ost funtion are: We use a (weighte) size of graphs (to e inserte into the soure) as a ost funtion in the Dijkstra-searh. For the weighte size, the epth (the length of the path) has more weight than the with (the numer of paths). This strategy works niely for onseutive in Example 4. Moreover, we show that a suitale ining to ontinue evaluation of onitional expressions an easily e foun for our ore UnCAL, eause the onitional part of a onitional expression is in the simple form of l = l 2. The same notion is alle riving (Glük an Klimov 993; Glük an Sørensen 994) in (Aramov an Glük 2002).