Labeling Workflow Views with Fine-Grained Dependencies

Transcription

1 Labeling Workflow Views with Fine-Graine Depenenies Zhuowei Bao Department of omputer an Information iene University of Pennsylvania Philaelphia, P 1914, U zhuowei@is.upenn.eu usan B. Davison Department of omputer an Information iene University of Pennsylvania Philaelphia, P 1914, U susan@is.upenn.eu Tova Milo hool of omputer iene Tel viv University Tel viv, Israel milo@s.tau.a.il BTRT This paper onsiers the problem of effiiently answering reahability queries over views of provenane graphs, erive from exeutions of workflows that may inlue reursion. uh views inlue omposite moules an moel fine-graine epenenies between moule inputs an outputs. novel view-aaptive ynami labeling sheme is evelope for effiient query evaluation, in whih view speifiations are labele statially (i.e. as they are reate) an ata items are labele ynamially as they are proue uring a workflow exeution. lthough the ombination of fine-graine epenenies an reursive workflows entail, in general, long (linear-size) ata labels, we show that for a large natural lass of workflows an views, labels are ompat (logarithmi-size) an reahability queries an be evaluate in onstant time. Experimental results emonstrate the benefit of this approah over the state-of-the-art tehnique when applie for labeling multiple views. 1. INTRODUTION The ability to manage workflow provenane is inreasingly important for sientifi as well as business appliations. For example, if an input to a workflow exeution is isovere to be inorret, we may wish to etermine whether a partiular workflow output epens on it an is thus also potentially inorret. Fining effiient tehniques to answer suh reahability queries is thus of partiular interest. However, provenane information an be extremely large, so we may wish to provie ifferent views of this information. For example, users may wish to speify abstration views whih fous user attention on relevant provenane information an abstrat away irrelevant etails, an iea propose in [8]. Workflow owners may also wish to speify seurity views whih an be use to hie private information from ertain user groups (e.g., sensitive intermeiate ata an moule funtionality [1]). Provenane views onsist of a set of omposite moules whih enapsulate subworkflows. 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 istribute for profit or ommerial avantage an that opies bear this notie an the full itation on the first page. To opy otherwise, to republish, to post on servers or to reistribute to lists, requires prior speifi permission an/or a fee. rtiles from this volume were invite to present their results at The 38th International onferene on Very Large Data Bases, ugust 27th - 31st 212, Istanbul, Turkey. Proeeings of the VLDB Enowment, Vol. 5, No. 11 opyright 212 VLDB Enowment /12/7... $ 1.. 1: Input DF 2: Input ignature M1: hek Moleules Example 1. Figure 1 shows an abstration of a real-life sientifi workflow ollete from the myexperiment reposi3: DF M2: Generate tom ignatures 4: Output DF 5: tom ignature Figure 1: Views with Fine-Graine Depenenies tory [19]. It generates atom signatures for iniviual ompouns given a trutural Data File (DF) as input (ignore for now the ashe eges insie moules M 1 an M 2). In a high-level view of this workflow, users see only one omposite moule, iniate as the big ashe box, with two inputs ( 1 an 2) an two outputs ( 4 an 5), while moules M 1 an M 2 an intermeiate ata 3 are hien. n important thing to keep in min is that Workflow provenane not only reors the orer of moule exeutions but also the epenenies between inputs an outputs of moules. Therefore, workflow views shoul expliitly speify the input-output epenenies for moules that are expose to users. Previous researh [13, 21, 4, 5] has aopte a simplifie provenane moel whih assumes that every output of a moule epens on every input, terme blak-box epenenies. However, a more fine-graine provenane moel aptures the fat that the output of a moule may epen on only a subset of its inputs. To unerstan why fine-graine epenenies are useful, onsier the two types of views mentione earlier. In abstration views, although irrelevant workflow etails are hien insie omposite moules, users shoul still be able to see the true epenenies between inputs an outputs of omposite moules (white-box epenenes). In seurity views, however, one may want to hie the true epenenies between inputs an outputs of ertain omposite moules in orer to preserve strutural or moule privay [1]. To this en, one may move to somewhere on the spetrum between white-box an blak-box epenenies (grey-box epenenies). With grey-box epenenies, aitional (false) epenenies between inputs an outputs may be ae. Example 2. Returning to Figure 1, fine-graine epenenies between the inputs an outputs of moules M 1 an M 2 are iniate as ashe eges insie the moules. In an abstration view, the omposite moule woul be assoiate with white-box epenenies, in whih 4 epens on 1 but not on 2. However, in a seurity view, the omposite moule oul be assoiate with a grey-box epeneny matrix in whih every output epens on every input. Hene, the answer to the reahability query Does 4 epen on 2? is ifferent in the two views. 128

2 This paper onsiers the problem of effiiently answering reahability queries over views of provenane graphs, of the types illustrate above. ommon approah for proessing reahability queries is to label ata items so that the reahability between any two items an be answere effiiently by omparing their labels. Moreover, ata items must be labele ynamially as soon as they are proue uring the exeution, sine sientifi workflows an take a long time to exeute an users may wish to query partial exeutions. In ontrast to previous work, we stuy effetive ynami labeling in the ontext of (1) fine-graine epenenies between inputs an outputs of moules; an (2) views with grey-box epenenies. This ontext introues several new hallenges. First, none of the existing ynami labeling shemes applies to fine-graine epenenies, sine they all rely on a simplifie provenane moel with blak-box epenenies. eon, ue to grey-box epenenies, the answer to a reahability query may alter in ifferent views. brutefore approah to hanling multiple views is to label ata items for eah view repeately an separately. This has two rawbaks: (i) large inex: for eah ata item, we must maintain one label for eah view; an (ii) expensive inex maintenane: when a new view is ae, all existing ata items must be re-labele. To aress the hallenges, more effetive labeling tehniques must be evelope. The main ontributions of this paper are summarize as follows. We propose a formal moel base on graph grammars whih apture a rih lass of (possibly reursive) workflows with fine-graine epenenies between the inputs an outputs of moules. We then use the moel to formalize the notion of views. They are efine over the workflow speifiation an then naturally projete onto its runs(etion 2). To get a hanle on the iffiulty introue by fine-graine epenenies to the ynami labeling problem, we prove that in general, long (linear-size) labels are require. We further show that ommon restritions on the workflow speifiation, that suffie to reue the label length for blakbox epenenies [5], are no longer helpful. Nevertheless, we ientify a large natural lass of safe views over stritly linear-reursive workflows for whih ynami, yet ompat (logarithmi-size) labeling is possible (etion 3). Base on this founation we propose a novel labeling approah whereby view speifiations are labele statially (i.e. as they are reate), whereas ata items are labele ynamially as they are proue uring a workflow exeution. t query time, the labeling of the view over whih the reahability query is aske is use to augment the ata labels to provie the orret answer in onstant time. We all this a view-aaptive ynami labeling sheme. It has the great avantage that, sine ata labels are unrelate to any view, views an be ae/elete/moifie without having to touh the ata. It is both spae-effiient an time-effiient relative to the brute-fore approah (etion 4). Finally, we evaluate the propose view-aaptive labeling sheme over both real-life an syntheti workflows. The experimental stuy emonstrates the superiority of our viewaaptive labeling approah over the state-of-the-art tehnique [5] when applie to label multiple views (etion 5). Relate Work. Before presenting our results, we briefly review relate work. The problem of reahability labeling has been stuie for ifferent lasses of graphs in both stati an ynami settings. Ieally, one woul like to buil ompat (logarithmi-size) labels whih enable effiient (onstant) query proessing. While ompat an effiient labeling is shown to be feasible for stati trees [2], when labeling general irete ayl graphs (DGs), any possible sheme requires linear-size labels even if arbitrary query time is allowe [4]. On the other han, ynami labeling is also muh harer than stati labeling. [9] shows that even labeling ynami trees requires linear-size labels. Fortunately, although workflow runs an have arbitrarily more omplex DG strutures than trees, [4, 5] show that knowlege of the speifiation an be exploite to obtain ompat an effiient labeling shemes for both stati an ynami runs erive from a given speifiation. more etaile omparison between existing stati an ynami labeling shemes for XML trees [2, 1, 9, 18, 23], for DGs [15, 24, 22, 16, 11] an for workflow runs [13, 4, 5] is summarize in [5]. However, as mentione above, none of the existing ynami labeling shemes is appliable to our problem as they neither support fine-graine epenenies nor hanle views. 2. MODEL ND PROBLEM TTEMENT We present a fine-graine workflow moel with white-box epenenies in etion 2.1. Base on this moel, we efine views with grey-box epenenies in etion 2.2. etion 2.3 formulates the view-aaptive ynami labeling problem. 2.1 Fine-Graine Workflow Moel Our workflow moel is built upon two onepts: workflow speifiation, whih esribes the esign of a workflow, an workflow run, whih esribes a partiular workflow exeution. We moel the struture of a speifiation as a ontextfree workflow grammar whose language orrespons to exatly the set of all possible runs of this speifiation. The grammar that we use is similar to [5, 7]. However, previous work[17, 13, 21, 5, 7]aopteasimplifieprovenanemoel whih impliitly assumes blak-box epenenies every output of a moule epens on every input. In ontrast, this paper proposes a more fine-graine provenane moel whih aptures the fat that an output of a moule may epen on only a subset of inputs. We all this white-box epenenies. In partiular, our moel assoiates the grammar with a epeneny assignment that expliitly speifies the epenenies between inputs an outputs of atomi moules. The basi builing bloks of our moel are moules an simple workflows. moule has a set of input ports an a set of output ports; an a simple workflow is built up from a set of moules by onneting their input an output ports. Definition 1. (Moule) moule is M = (I,O), where I is a set of input ports an O is a set of output ports. Definition 2. (imple Workflow) simple workflow is W = (V,E), where V is a multiset of moules an E is a set of ata eges from an output port of one moule to an input port of another moule. Eah ata ege arries a unique ata item that is proue by the former an then onsume by the latter. Input ports with no inoming ata eges are alle initial input ports; an output ports with no outgoing ata eges are alle final output ports. To simplify the presentation, we assume that (1) pairwise non-ajaent ata eges: any pair of ata eges are not inient to the same port; an (2) ayli simple workflow: ata eges o not form yles among the moules. Note that the above two restritions o not limit the expressive power of our moel. For (1), ajaent ata eges an be resolve by introuing ummy moules that istribute or aggregate multiple ata items. For(2), we will see that loops an be impliitly apture by reursive proutions. 129

3 Example 3. The top left orner of Figure 2 shows a moule with two input ports an three output ports, whih are enote by soli an empty yles, respetively. The top right orner of Figure 2 shows a simple workflow W 1 with six moules an ten ata eges (soli eges, ignore the ashe eges insie moules for now). W 1 has two initial input ports an three final output ports, whih are highlighte by soli an empty thik arrows, respetively. To buil a new workflow, an existing (simple) workflow may be reuse as a omposite moule. This is moele by a workflow proution. Definition 3. (Workflow Proution) workflow proution is of form M f W, where M is a omposite moule, W is a simple workflow an f is a bijetion that maps input ports an output ports of M to initial input ports an final output ports of W, respetively. When f is lear from the ontext, we simply enote a proution by M W. Example 4. In Figure 2, eah row efines one or two proutions. For example, the first row efines W 1, an the seon row efines W 2 an W 3. Note that also appears as a omposite moule in both W 1 an W 4. For simpliity, we assume that for eah proution M W, the (initial) input ports an (final) output ports of M an W are mappe by f from top to bottom as shown in the figure. The ontext-free workflow grammar is a natural extension of the well-known ontext-free string grammar, where moules orrespon to haraters, an simple workflows that are built up from moules orrespon to strings that are sequenes of haraters. In partiular, atomi an omposite moules orrespon to terminals an variables, respetively. We also efine a start moule an a finite set of workflow proutions. By Definition 3, eah proution M f W replaes a omposite moule M with a simple workflow W. The ata eges ajaent to M are onnete to W base on the bijetion f. The language of a ontext-free workflow grammar onsists of all simple workflows that an be erive from the start moule an ontain only atomi moules. Following the stanar notations for string grammars, given a finite set Σ of moules, let Σ enote the set of all simple workflows that are built up from a multiset of moules in Σ. Given two simple workflows W 1 an W 2, let W 1 f W 2 enote that W 2 an be erive from W 1 by applying a sequene of zero or more proutions, an f is a bijetion that maps initial input ports an final output ports from W 1 to W 2. gain, f may be omitte for simpliity. Definition 4. (ontext-free Workflow Grammar) ontext-free workflow grammar (abbr. workflow grammar) is G = (Σ,,,P), where Σ is a finite set of moules, Σ is a set of omposite moules (then Σ\ is the set of atomi moules), Σ is a start moule, an P = {M W M,W Σ } is a finite set of workflow proutions. The language of G is L(G) = {R (Σ\ ) R}. Example 5. Our running example of a workflow grammar G is shown in Figure 2. omposite moules are iniate by upperase letters an atomi moules by lowerase letters. Formally, G = (Σ,,,P), where Σ = {,, B,..., E, a, b,..., f}, = {,, B,..., E}, an P = {p 1 = W 1, p 2 = W 2, p 3 = W 3, p 4 = B W 4, p 5 = W 5, p 6 = D W 6, p 7 = D W 7, p 8 = E W 8}. Note that p 2 an p 4 form a reursion between an B. p 6 forms a self-reursion over D, an along with p 7, iniates a loop (sequential exeution) over f. B D E W 1 W 2 W 4 W 5 a b b B D W 6 f W 7 f D W 8 e f E W 3 Figure 2: Workflow peifiation One possible simple workflow run R L(G) is shown in Figure 3, where the atomi moules in R are enote by soli boxes, an the omposite moules that are reate uring the erivation of R are enote by ashe boxes. We reate a unique i for eah atomi an omposite moule in R by appening a istint number to the moule name. 1, 2,..., 41 are unique is for ata items (ata eges) in R. For sake of illustration, we omit etails of :1, :2 an :3, an show etails of :4 in Figure 4. Observe that R an be erive from by applying a sequene of proutions p 1, p 2, p 4, p 2, p 4, p 3, p 5, p 6, p 6, p 7, p 8,... o far we onsier only workflow struture the way in whih moules are onnete to onstrut workflows. Next, we enrih the moel by efining fine-graine epenenies between inputs an outputs of atomi moules. Naturally, we assume that every input ontributes to at least one output; an every output epens on at least one input. Definition 5. (Depeneny ssignment) Given a finite set Σ of moules, a epeneny assignment to Σ is a funtion λ that, for eah moule M = (I,O) Σ, efines a set λ(m) of epeneny eges from I to O, suh that i I, o O, (i,o) λ(m); an o O, i I, (i,o) λ(m). Finally, ombining all the above omponents, our finegraine workflow moel is formalize as follows. Definition 6. (Fine-Graine Workflow Moel) workflow speifiation is G λ, where G = (Σ,,,P) is a workflow grammar an λ is a epeneny assignment to Σ \. The set of all workflow runs w.r.t. G λ is L(G λ ) = {R λ R L(G)}, where R λ is obtaine from R by aing to eah moule M in R a set λ(m) of epeneny eges. Example 6. For the grammar G in Figure 2, we efine a epeneny assignment λ to all atomi moules (i.e., a, b,..., f). The epeneny eges introue by λ are shown in Figure 2 as ashe eges from input ports to output ports of atomi moules. With both ata (soli) an epeneny (ashe) eges, Figures 3 an 4 represent a run R λ L(G λ ). e 121

4 1 :1 a:1 4 :1 3 :2 8 B:1 e:1 : : B:2 :3 e: e: : : :2 35 : :1 Figure 3: Workflow Run b:1 2 :4 D:1 23 D:2 25 D: f:1 24 f:2 26 f:3 b: E:1 29 f: :3 : W 1 W 2 a b B W 3 e : Figure 4: Details of omposite Moule :4 In etion 3, we will ompare our fine-graine moel (i.e., with white-box epenenies) to the existing oarse-graine moel (i.e., with blak-box epenenies) [5, 7]. Both are grammar-base, but the oarse-graine moel is less expressive, an aptures only a sublass of fine-graine workflows. Definition 7. (oarse-graine Workflows) workflow speifiation G λ is sai to be oarse-graine if (1) λ is efine suh that for any atomi moule, every output epens on every input; an (2) every simple workflow use by G has a single soure moule an a single sink moule Views with Grey-Box Depenenies workflow view is onstrute over a speifiation an then projete onto its runs. uh approah is ommon in workflows [8, 21, 1] (unlike typial atabase views that are efineviaqueries), butourworkisthefirsttobebaseona fine-graine moel. Formally, a view is efine by two omponents. One esribes the struture of a view by restriting the possible expansions of workflow hierarhy to a subset of omposite moules. The other speifies the pereive fine-graine epenenies between inputs an outputs of all unexpanable moules in this view. s mentione in etion 1, for abstration views, the pereive epenenies always reflet the true epenenies, whih we all white-box epenenies. In ontrast, for seurity views, false epenenies may be introue in orer to hie private provenane information, whih we all grey-box epenenies. Definition 8. (Workflow View) Give a workflow speifiation G λ = (Σ,,,P) λ, a view over G λ is efine by a pair (,λ ), where is a subset of omposite moules an λ is a new epeneny assignment for Σ\. In partiular, (,λ) is sai to be the efault view over G λ. Remark 1. s will be seen in etion 3.1, from the inputoutput epenenies of atomi moules, we an ompute those of omposite moules. We thus say that a view (,λ ) has white-box epenenies, if λ efines the same epenenies as λ oes, otherwise, it has grey-box epenenies. 1 (2) ensures blak-box epenenies for omposite moules. B W 4 e Figure 5: View of Workflow peifiation view U = (,λ ) efine over a speifiation G λ proues a new grammar, enote G, by restriting G to the subset of proutions for omposite moules in. Together with λ, it efines a new speifiation, enote G U = (G ) λ, whih we all a view of this speifiation. imilarly, given a run R λ L(G λ ), by restriting the erivation of R to only proutions for omposite moules in an using λ, we obtain a view of this run, enote R U = (R ) λ. Example 7. Using the speifiation G λ in Figure 2, we efine a view U = (,λ ), where = {,,B}. The new grammar G is shown in Figure 5, whih ontains only the proutions for, an B. Note that is treate as an atomi moule in this view, whih makes D, E an f unerivable. Therefore, λ nees to be efine for only atomi moules a, b,,, e an. The epeneny eges introue by λ are shown in Figure 5 as ashe eges. omparing with λ efine in Figure 2, we observe that λ () is newly efine, λ (e) is hange, an others are unhange. Hene, this view introues grey-box epenenies. We projet this view onto the run R λ in Figures 3 an 4. ine is treate as atomi, etails of : 1, : 2, : 3 an : 4 (Figure 4) are hien an R has exatly the struture in Figure 3. However, all the epeneny eges for R shoul be given aoring to λ as in Figure 5. In the rest of this paper, we may simply enote a speifiation by G an a run by R, sine the original epeneny assignment λ is irrelevant to views (i.e., overwritten by λ ). 2.3 View-aptive Dynami Labeling We start with the basi ynami labeling problem. The goal is to assign eah ata item a reahability label as soon as it is proue (ynamially) suh that using only the labels of any two ata items, we an quikly eie if one epens on the other. Two ifferent but relate ynami labeling problems were formulate in [5]. In the exeution-base 1211

5 problem, atomi moules of a run are generate one-by-one aoring to some topologial orering. In the erivationbase problem, a run is erive from the start moule by applying a sequene of proutions. s observe in [5], any solution for the former also provies a solution for the latter. We thus fous only on the erivation-base problem. Definition 9. [5] (Dynami Labeling) ynami labeling sheme for a given speifiation G λ is (φ,π), where φ is a labeling funtion an π is a binary preiate. φ takes as input a erivation of a run R λ L(G λ ), that is, a sequene of proutions that transform the start moule to R. Initially, φ assigns a label φ() to eah input an output of. In the ith step of the erivation, φ assigns a label φ() to eah new ata item introue by the ith proution. Note that we o not know the proution sequene in avane, but reeive them online. The assigne labels annot be moifie subsequently. φ an π are suh that for any erivation of a run R λ L(G λ ) an any two ata items 1 an 2 in R λ, π(φ( 1),φ( 2)) = true iff 2 epens on 1. In ontrast to the previous work [5], this paper stuies the ynami labeling problem in more general an useful workflow settings. peifially, we onsier (1) fine-graine input-output epenenes an (2) views with grey-box epenenies. Both ingreients entail new hallenges, whih will be aresse in etions 3 an 4, respetively. To hanle views, we propose in etion 4 a novel viewaaptive labeling approah whereby view speifiations are labele statially (i.e., as they are reate), whereas ata items are labele ynamially as they are proue uring a workflow exeution. t query time, the label of the view over whih the query is aske is ombine with the labels of relevant ata items to provie the orret answer. In this framework, sine ata labels are unrelate to any view (view-aaptive), views an be ae/elete/moifie without having to touh the ata. It is both spae-effiient an time-effiient relative to the alternative approah where ata items are labele repeately an separately for eah view. Definition 1. (View-aptive Dynami Labeling) view-aaptive ynami labeling sheme for a given speifiation G is (φ r,φ v,π), where φ r is a labeling funtion for runs, φ v is a labeling funtion for view speifiations, an π isaternarypreiate. GivenaerivationofarunR L(G), φ r as before assigns a label φ r() (alle ata label) to eah ata item as soon as it is proue uring the erivation of R. Given a view U over G, φ v treats U as one objet an assigns a label φ v(u) (alle view label). φ r, φ v an π are suh that for any erivation of a run R L(G), any view U over G an any two ata items 1 an 2 in R U, π(φ r( 1), φ r( 2),φ v(u)) = true iff 2 epens on 1 w.r.t. U. (view-aaptive) ynami labeling sheme is sai to be ompat if for any erivation of a run with n ata items, it reates ata labels of O(log n) bits. learly, it provies shortest possible ata labels up to a onstant fator. 3. FEIBILITY OF DYNMI LBELING To aress the hallenges brought by fine-graine epenenies, we first onsier the basi ynami labeling problem (see Definition 9), where there is only one efault view efine over the speifiation. Note that the labels reate for the efault view also work for other views with white-box epenenies, but not those with grey-box epenenies. s a formal analysis, we present in this setion a lassifiation of fine-graine workflows base on the feasibility of eveloping (ompat) ynami labeling shemes. In etion 3.1, we first ientify a lass of safe workflows, an show that they are the largest set of workflows that allow ynami labeling shemes. In etion 3.2, we further ientify a lass of stritly linear-reursive workflow strutures for whih ynami, yet ompat labeling shemes are possible. Polynomial-time algorithms are also given to eie if a workflow is safe or if its struture is stritly linear-reursive. Interestingly, our results show that the ommon restrition on the workflow struture, whih suffie to reue the label length for blak-box epenenies [5], are no longer helpful. This formally proves the iffiulty introue by fine-graine epenenies to the ynami labeling problem. 3.1 afe Workflows ome workflows annot be labele on-the-fly even if arbitrary label size is allowe. We illustrate by an example. 1 2 a b Figure 6: Unsafe Workflow Example 8. onsier the speifiation in Figure 6 with two proutions a an b. 1 an 2 are an input an an output of, respetively. Observe that if a is applie, then 2 epens on 1; otherwise (if b is applie), 2 oes not epen on 1. Reall from Definition 9 that the labels for 1 an 2 must be assigne before we see the proution, an annot be moifie subsequently. Therefore, no ynami labeling shemes exist for this example. In general, if two simple workflows with only atomi moules an be erive from the same omposite moule, an they are inonsistent, in the sense that they have ifferent epenenies between initial inputs an final outputs, then ynami labeling is impossible for this speifiation. uh workflows are sai to be unsafe, an the others are safe. Definition 11. (afe Workflow) workflow speifiation G λ = (Σ,,,P) λ is sai to be safe if M an W 1,W 2 (Σ\ ) suh that M W 1 an M W 2, W 1 is onsistent with W 2 w.r.t. λ. lso, λ is sai to be safe if G λ is safe; an a view U is sai to be safe if G U is safe. Remark 2. afety is a natural restrition on fine-graine workflows. It essentially says that for any moule, either atomi or omposite, the epenenes between inputs an outputs are eterministi, in the sense that they an be preite from the speifiation, an are onsistent among all possible exeutions. In partiular, by Definition 7, any oarsegraine workflow (i.e., with blak-box epenenies) is always safe. Moreover, it is important to notie that from the perspetive of ata provenane, the output of an aggregate funtion epens on eah of its inputs [3], even though the output may take the value from only one of its inputs (e.g., max or min funtions). Therefore, a workflow that use those aggregate funtions as moules is still safe. Our first result shows that safety haraterizes the feasibility of ynami labeling for fine-graine workflows. Theorem 1. Given any workflow speifiation G λ, there is a ynami labeling sheme for G λ iff G λ is safe. Proof. (keth) By Definition 11, unsafe workflows o not allow any ynami labeling shemes. On the other han, the view-aaptive ynami labeling sheme, whih we will present in etion 4, an be moifie to label arbitrary safe workflows, though it may reate linear-size ata labels. 1212

6 Itispossibletotestin polynomialtimeifagivenspeifiation G λ is safe. Our algorithm base on Lemma 1 is briefly esribe as follows. We start by efining λ = λ for eah atomi moule, an then ompute λ for omposite moules by verifying all the proutions. proution M W is sai to be verifiable, if λ is alreay efine for all the moules in W, so that λ (M) an be ompute. The algorithm reports that G λ is safe, if λ is onsistently efine for all omposite moules, an outputs λ as a by-prout. Lemma 1. (Full ssignment) workflow speifiation G λ = (Σ,,,P) λ is safe iff there is a unique epeneny assignment λ to Σ (alle the full epeneny assignment) suh that (1) M Σ\, λ (M) = λ(m); an (2) M W P, M is onsistent with W w.r.t. λ. B D E B Figure 7: Full Depeneny ssignment Example 9. We illustrate the above algorithm using the speifiation G λ in Figure 2. Initially, both p 7 = D W 7 an p 8 = E W 8 are verifiable. We ompute λ (D) an λ (E) by p 7 an p 8. One λ (D) an λ (E) are efine, p 5 = W 5 an p 6 = D W 6 beome verifiable. We ompute λ () by p 6, an verify that λ (D) ompute by p 6 is onsistent with the one ompute before by p 7. We ontinue this proess until all the proutions are verifie. Hene, G λ is safe, an λ is shown on the top of Figure 7. imilarly, one an verify that the view U = (,λ ) efine in Example 7 is safe using Figure 5. The full epeneny assignment for U is shown on the bottom of Figure 7. omparing the two full assignments in Figure 7, while B gets the same epenenies, the ones for an are ifferent. 3.2 Linear-Reursive Workflow trutures For safe workflows, we further examine the feasibility of eveloping ompat ynami labeling shemes. First of all, a negative result in [5] shows that there is a oarse-graine workflow that oes not allow any ompat ynami labeling sheme. By Definition 7 an Lemma 1, we know that any oarse-graine workflow is safe. o the negative result also applies to the fine-graine moel: there is a safe workflow that oes not allow any ompat ynami labeling sheme. Given this, our next goal is to ientify safe workflows that enable ompat ynami labeling. n elegant haraterization for oarse-graine workflows is prove in [5]: given any oarse-graine workflow speifiation G λ, there is a ompat ynami labeling sheme for G λ iff G is a linear-reursive workflow grammar whih is formally efine as follows. Definition 12. [5] (Linear-Reursive Workflow Grammar) workflow grammar G = (Σ,,,P) is sai to be linear-reursive if M an W Σ suh that M W, W has at most one instane of M. Note that oarse-graine workflows are only a restrite lass of (fine-graine) safe workflows. We show here that, in the fine-graine moel, linear-reursiveness is not enough to enable ompat ynami labeling for safe workflows. Theorem 2. There is a linear-reursive grammar G an a safe epeneny assignment λ suh that any ynami labeling sheme for G λ requires linear-size ata labels. W a a a b b a Figure 8: ounterexample in Proof of Theorem 2 Proof. (keth)figure8givesaounterexampleg λ with three proutions p a = W a, p b = W b an p = W, where G is linear-reursive an λ is safe. Observe that a run R λ L(G λ ) is erive from the start moule by applying an arbitrary sequene of p a an p b, followe by one p. Both p a an p b proue three new ata items (ata eges). We fous only on the epeneny eges between the first two ata items. Observe from Figure 8 that they form a binary tree that is reate ynamially from left to right: if p a is applie, then the first ata item is expane, otherwise (if p b is applie), the seon ata item is expane. Using a similar tehnique to [9], we an prove that labeling suh a ynami tree requires linear-size ata labels. Theorem 2 tells us that while fine-graine epenenies inrease the expressive power of the moel, they limit the reursive workflow struture that allows ompat ynami labeling. We thus ientify a natural lass of stritly linearreursive workflow grammars for whih ynami, yet ompat labeling is feasible for any safe epeneny assignment. To efine them, we introue a proution graph that esribes the erivation relationship between moules. Definition 13. (Proution Graph) Given a workflow grammar G = (Σ,,,P), the proution graph of G is a irete multigraph P(G) in whih eah vertex enotes a unique moule in Σ. For eah proution M W in P an eah moule M in W, there is an ege from M to M in P(G). Note that if W has multiple instanes of a moule M, then P(G) has multiple parallel eges from M to M. Intuitively, every yle in P(G) orrespons to a reursion in G. G is sai to be reursive if P(G) is yli. moule in G is sai to be reursive, if it belongs to a yle in P(G). Definition 14. (tritly Linear-Reursive Workflow Grammar) workflow grammar G is sai to be stritly linear-reursive if all the yles in P(G) are vertex-isjoint. Remark 3. tritly linear reursion is able to apture ommon reursive patterns that we observe from the myexperiment workflow repository [19]. In partiular, onsier two ommon forms of reursion that we enounter in reallife sientifi workflows. The first is alle the loop exeution for whih a sub-workflow is repeate sequentially a number of times until ertain onition is met. The seon is alle the fork exeution for whih multiple opies of a sub-workflow are exeute in parallel. In sientifi workflow systems, suh as Taverna [14] an Kepler [2], fork exeutions are ommonly use to moel operations over omplex ata (e.g., maps over sets). Both loop an fork exeutions belong to a simple form of stritly linear reursion. It is easy to show that every stritly linear-reursive workflow grammar is also linear-reursive, but not vie versa. W b b W 1213

7 e (3,1) (4,1) (2,1) a (2,2) (1,6) (1,3) B (1,1) (4,2) (2,3) (3,2) (1,4) (1,2) (1,5) (5,2) (5,1) D b (5,3) (6,2) (6,1) (5,4) (7,1) E f (8,2) (8,1) Figure 9: Proution Graphs Example 1. Figure 9 (left) shows the proution graph P(G) for the grammar G in Figure 2 (ignore number pairs on the eges). Observe that P(G) has two yles: one between an B an the other (self-loop) over D. ine they are vertex-isjoint, G is stritly linear-reursive. Figure 9 (right) shows the proution graph P(G ) for the grammar G in Figure 8. ine P(G ) has two self-loops that share, G is linear-reursive but not stritly linear-reursive. It is possible to test in polynomial time if a given grammar G is stritly linear-reursive. The algorithm starts by builing the proution graph P(G), then aoring to Definition 14, heks if any two yles in P(G) share a vertex. The main result of this paper is to show that ynami, yet ompat labeling is feasible for stritly linear-reursive grammars with any safe epeneny assignment. Theorem 3. Given any stritly linear-reursive workflow grammar G, for any safe epeneny assignment λ, there is a ompat ynami labeling sheme for G λ. a The following setion esribes our labeling sheme. 4. VIEW-DPTIVE DYNMI LBELING This setion presents a ompat view-aaptive ynami labeling sheme for stritly linear-reursive workflows with safe views. The rationale behin our label esign is explaine as follows. Both ata labels an view labels enoe only partial(but orthogonal) reahability information. More preisely, a ata label enoes only a subsequene of the run erivation that reates this ata item, while a view label enoes only the fine-graine epenenies that are efine in this view. However, a ombination of two ata labels an a view label provies the omplete information to infer the reahability between the two ata items over this view. We start with a preproessing step in etion 4.1. Two inepenent tasks for labeling ynami runs an labeling safe views are esribe in etions 4.2 an 4.3, respetively. etion 4.4 presents how to effiiently answer queries using a ombination of ata labels an view labels. Finally, etion 4.5 analyzes the quality of our labeling sheme. 4.1 Preproessing s a preproessing step, we assign a pair of numbers to eah ege in the proution graph. These pairs serve as unique is for the eges, an will be use later to label runs anviews. LetG = (Σ,,,P)beastritlylinear-reursive grammar an P(G) be its proution graph. First of all, we fix an arbitrary orering among the proutions in P, an for eah proution M W, fix an arbitrary topologial orering among the moules in W. Let p k = M W be the kth proution in P, an M i be the ith moule in W, then we assign the ege from M to M i in P(G) a pair (k,i). Hereafter, we simply refer to this ege as (k,i). In b aition, we also fix an arbitrary orering among all the (vertex-isjoint) yles in P(G), an for eah yle, fix an arbitrary ege as the first ege of the yle. We enote by (s) the sth yle in P(G) ontaining a list of number pairs. Example 11. For the grammar G in Figure 2, the pairs of numbers assigne to the eges in P(G) are shown in Figure 9. Note that the proutions p 1, p 2,..., p 8 are simply sorte by their subsripts. In Figure 2, all the moules in W 1 are sorte topologially as a b. Therefore, the ege from to in Figure 9 is assigne (1,5) beause p 1 = W 1 is the first proution, an is the fifth moule in W 1. Moreover, the two yles in P(G) are enote by (1) = {(2,2),(4,2)} an (2) = {(6,2)}. 4.2 Labeling Dynami Runs Given a erivation of a run R L(G), our goal is to assign a ata label φ r() to eah ata item in R as soon as it is proue. The labeling is base on a tree representation for runs, alle the ompresse parse tree. In ontrast to the traitional parse tree use for ontext-free grammars whose epth may be proportional to the size of the run, the epth of a ompresse parse tree is always boune by the size of the speifiation. We will see later that this property is ritial to enable ompat (logarithmi-size) ata labels. Definition 15. (ompresse Parse Tree) The ompresse parse tree for a run R is an orere tree T (R), where eah leaf noe enotes an atomi moule, an eah non-leaf noe enotes either a omposite moule (alle the omposite noe), or a linear reursion (alle the reursive noe). The hilren of a omposite noe enote all the moules of a simple workflow proue by a proution, an are orere by a fixe topologial orering; an the hilren of a reursive noe enote a sequene of neste omposite moules obtaine by unfoling a yle in the proution graph. (2,1) (1,1,1) :1 a:1 b:1 R:1 :1 :1 :1 :1 B:1 :2 B:2 :3 :2 :2 (1,1) (1,6) (1,2) (1,3) (1,4) (1,5) (1,1,2) (2,3) (4,1) (2,1) (1,1,3) (1,1,4) (1,1,5) (2,3) (4,1) (3,1) e:1 :3 :3 e:2 e:3 (2,1,1) b:2 R:2 E:1 :2 D:1 D:2 D:3 (6,1) (6,1) (7,1) (3,2) f:1 f:2 f:3 Figure 1: ompresse Parse Tree :4 (5,1) (5,4) (5,3) (5,2) (2,1,2) (2,1,3) (8,1) (8,2) f:4 :3 Example 12. The ompresse parse tree T (R) for the run R in Figures 3 an 4 is shown in Figure 1 (ignore the ege labels), where R:1 an R:2 are reursive noes. Note that :1, B:1, :2, B:2, :3 (hilren of R:1) are obtaine by unfoling the yle between an B in Figure 9. In a stanar parse tree, they woul be onnete in a path. Lemma 2. Given a stritly linear-reursive workflow grammar G, for any erivation of a run R L(G), the epth of the ompresse parse tree T (R) is no greater than 2, where is the number of omposite moules in G. 1214

8 We now esribe the ynami labeling algorithm. Given a erivation of a run R, we buil T (R) in a top-own manner. During this proess, we label eah new ege an use the ege labels to onstrut labels for new ata items on-the-fly. We next explain the esign of ata labels step by step. Firstly, we esribe the label for an ege in T (R). Let e be an ege from u to v in T (R). We enote by φ r(e) the label of e. (1) If u is a omposite noe, then e an be mappe to an ege e in P(G). Reall from etion 4.1 that eah ege in P(G) is uniquely ientifie by a pair of numbers. Let e = (k,i), then φ r(e) = (k,i); an (2) otherwise (if u is a reursive noe), let u enote the sth yle in P(G) starting from the tth ege. This an be etermine by the first hil of u. Let v be the ith hil of u, then φ r(e) = (s,t,i). eonly, we use a sequene of ege labels to onstrut the label for an input port i in R. We enote by φ r(i) the label of i. uppose i is first reate as the xth input port of a moule M uring the erivation of R, an M is enote by a noe v in T (R). Let e 1, e 2,..., e l be the path from the root noe to v in T (R), then φ r(i) = {φ r(e 1), φ r(e 2),..., φ r(e l ), x}. For an output port o, φ r(o) is efine similarly. Finally, we use a pair of input an output port labels to onstrut the label for a ata item (ata ege) = (o,i) in R. We enote by φ r() the label of, then φ r() = (φ r(o),φ r(i)). ine o an i must be reate by the same proution, φ r(o) an φ r(i) iffer only in the last one or two ege labels. The size of φ r() an be reue almost by half by fatoring out the ommon prefix of φ r(o) an φ r(i). Example 13. The ege labels for the ompresse parse tree T (R) are shown in Figure 1. E.g., the ege from R:1 to :3 is labele by (1,1,5), beause R:1 enotes the first yle in the proution graph starting from the first ege (see Example 11), an :3 is the fifth hil of R:1. Next, we label the ata items. E.g., onsier 21 = (o,i) in Figure 4, where o is the first output port of b:2, an i is first reate as the seon input port of D:1 (note that i is also the seon input port of f:1). Then, φ r( 21) = (φ r(o),φ r(i)), where φ r(o) = {(1,3),(1,1,5),(3,2),(5,1),1} φ r(i) = {(1,3),(1,1,5),(3,2),(5,2),(2,1,1),2} 4.3 Labeling afe Views Given a safe view U = (,λ) over G, our goal is to reate a view label φ v(u) whih an be ombine with above ata labels to infer reahability over U. Using the algorithm in etion 3.1, we first ompute the full epeneny assignment λ by extening λ to all the omposite moules in. Next, we efine three funtions, I, O an Z. Reall from etion 2.2 that G enotes the grammar obtaine by restriting G to. Let P(G ) be the proution graph of G, then P(G ) is a subgraph of P(G). Reall from etion 4.1 that eah ege in P(G) is uniquely ientifie by a pair of numbers (k,i). The input of I an O is an ege in P(G ), enote by a pair (k,i). The input of Z is a pair of eges in P(G ) of form (k,i) an (k,j). For simpliity, we also enote them by a triple (k,i,j). The output of all three funtions is a reahability matrix, whih is efine next. Funtions I an O. Given an ege (k,i) in P(G ), let p k = M W be the kth proution in P, an M i be the ith moule in W, then (1) I(k,i) is efine as a reahability matrix from the inputs of M to the inputs of M i (w.r.t. λ ); an (2) O(k, i) is efine as a (reverse) reahability matrix from the outputs of M to the outputs of M i (w.r.t. λ ). Funtion Z. Given a pair of eges (k,i) an (k,j) in P(G ), let p k = M W be the kth proution in P, an M i an M j be the ith an jth moule in W, respetively, then Z(k,i,j) is efine as a reahability matrix from the outputs of M i to the inputs of M j (w.r.t. λ ). Note that Z(k,i,j) is an empty matrix (with only false values) if i j, sine M i an M j are sorte in topologial orering. Finally, φ v(u) onsists of all the above three funtions, along with λ () for the start moule. That is, φ v(u) = {λ (),I,O,Z} Basially, the above view label enoes all the fine-graine epeneny information that is speifi to this view an is neessary for our eoing algorithm given in etion 4.4. Example 14. For the running example, we first label the efault view U 1 = (,λ) for whih λ is ompute in Example 9, an is shown on the top of Figure 7. Using λ, we an ompute the funtions I, O an Z. E.g., onsier the ege (1,5) from to in Figure 9. The first proution p 1 = W 1 is shown in Figure 2. I(1,5) enotes the reahability from the inputs of (i.e., the initial inputs of W 1) to the inputs of (i.e., the fifth moule in W 1); similarly, O(1,2) enotes the (reverse) reahability from the outputs of (i.e., the final outputs of W 1) to the outputs of b (i.e., the seon moule in W 1); an Z(1,2,5) enotes the reahability from the outputs of b to the inputs of in W 1. [ ] 1 1 I(1,5) = O(1,2) = [ ] 1 Z(1,2,5) = 1 imilarly, we an label the other view U 2 = (,λ ) efine in Example 7, whose full epeneny assignment is shown on the bottom of Figure 7. Using Figure 5, we have [ ] 1 1 I(1,5) = O(1,2) = 1 [ ] Z(1,2,5) = s we an see above, the funtions enoe by the view labels φ v(u 1) an φ v(u 2) may evaluate to ifferent values for the same input. Moreover, they are efine over ifferent omains. E.g., I(5,1) is efine for U 1 but not for U 2. pae-effiient View Labeling. By efault, we preompute all the reahability matries for I, O an Z, an materialize them in the view label. lternatively, one an ompute them on-the-fly by performing a graph searh over the view of a speifiation uring the query time. In general, more sophistiate approahes (e.g., [15, 24, 22]) an be use to label the view, in orer to fin a better balane between the overhea of labeling views an query effiieny. We will further explore this traeoff in the experiments. 4.4 Deoing Data Labels with View Labels Using only two ata labels φ r( 1) an φ r( 2) an a view label φ v(u), one an eie if 2 epens on 1 w.r.t. U by a eoing preiate π. We first efine in etion two proeures use by π, namely, Inputs an Outputs, an then esribe π in etion etion presents fast matrix multipliation use to ahieve onstant query time PreeuresInputs anoutputs Let e be an ege from u to v in the ompresse parse tree T (R). Given the ege label φ r(e) (efine in etion 4.2) an a view label φ v(u), our proeure Inputs omputes a reahability matrix Inputs(φ r(e),φ v(u)) by lgorithm 1. ase 1. [Line 1 to Line 2] If φ r(e) = (k,i), that is, if u is a omposite noe, then Inputs omputes a reahability matrix from the inputs of the moule enote by u to the inputs of the moule enote by v, simply given by I(k,i). 1215

9 ase 2. [Line 3 to Line 8] If φ r(e) = (s,t,i), that is, if u is a reursive noe, then v is the ith hil of u. Let M 1, M 2,..., M i be the moules enote by the first i hilren of u. They are a sequene of neste omposite moules in R U obtaine by unfoling the sth yle in P(G) starting from the tth ege. Inputs finally omputes a reahability matrix from the inputs of M 1 to the inputs of M i in R U by multiplying all i 1 intermeiate reahability matries. lgorithm 1 Proeure Inputs Input: φ r(e) = (k,i) or (s,t,i) φ v(u) = {λ (),I,O,Z} Output: Inputs(φ r(e),φ v(u)) 1: if φ r(e) = (k,i) then 2: return I(k, i) 3: else {φ r(e) = (s,t,i)} 4: let (s) = {(k 1,i 1),(k 2,i 2),...,(k l,i l )} 5: // (s) enotes the sth yle in P(G) of length l 6: let a 1, k a+l = k a an i a+l = i a 7: return i 1 a=1 I(kt+a 1,it+a 1) 8: en if The other proeure Outputs is efine similarly, whih omputes a (reverse) reahability matrix for output ports. Example 15. Let e be the ege from R : 1 to : 3 in Figure 1 an U 1 be the efault view. φ r(e) = (1,1,5) an φ v(u 1) are explaine in Examples 13 an 14. For this pair of labels, lgorithm 1 omputes the reahability matrix from the inputs of :1 to the inputs of :3 in R U1. By Example 11, the first yle is (1) = {(2,2),(4,2)}. Therefore, Inputs(φ r(e),φ v(u 1)) = I(2,2) I(4,2) I(2,2) I(4,2) Deoing Preiate Given a pair of ata labels φ r( 1) an φ r( 2) an a view label φ v(u) = {λ (),I,O,Z}, our goal is to evaluate π to true iff 2 epens on 1 w.r.t. U. Due to spae onstraints, we sketh only the main ases, where both 1 an 2 are intermeiate ata items of R. The omplete esription an be foun in the full version of this paper [6]. Let φ r( 1) = (φ r(o 1),φ r(i 1)) an φ r( 2) = (φ r(o 2), φ r(i 2)), then 2 epens on 1 w.r.t. U iff i 2 is reahable from o 1 in R U. Let φ r(o 1) = {l 1,x} an φ r(i 2) = {l 2,y}, where l 1 an l 2 are two lists of ege labels. uppose uring the erivation of R, o 1 is first reate as the xth output port of some moule M 1 an i 2 is first reate as the yth input port of some moule M 2. uppose M 1 an M 2 are enote by two noes v 1 an v 2 in the ompresse parse tree T (R). ase 1. If l 1 = l 2 or one is a prefix of the other, that is, v 1 = v 2 or one is an anestor of the other in T (R), then M 1 = M 2 or one is erive from the other. Thus, i 2 is not reahable from o 1 in R U, an π evaluates to false. ase 2. Otherwise, suppose l 1 an l 2 agree on the first l 1 ege labels, but iffer on the lth ege label. Moreover, let the length of l 1 an l 2 be p an q, respetively. That is, l 1 = {φ r(e 1),...,φ r(e l 1 ),φ r(e l ),...,φ r(e p)} l 2 = {φ r(e 1),...,φ r(e l 1 ),φ r(e l),...,φ r(e q)} where φ r(e l ) φ r(e l). We enote by v = L(v 1,v 2) the least ommon anestor of v 1 an v 2 in T (R). Let e l be an ege from v to v 1 an e l be an ege from v to v 2. Let M 1 an M 2 be the moule enote by v 1 an v 2, respetively. ase 2a ase 2b v / M (k,i) (k,j) v 1 / M 1 v 1 / M 1 (s,t,i) v 2 / M 2 v 2 / M 2 v / R (s,t,i+1) (s,t,j) M v 1 / M 1 v 2 / M 2 v 2 / M 2 (k,j ) (k,i ) v 1 / M 1 v 2 / M 2 v 1 / M 1 M 1 M 2 M 1 x y M 2 o 1 M 1 M 1 O T M 1 x o 1 Figure 11: Main ases of Deoing Preiate O T Z Z i 2 I I i 2 I y M 2 M 2 ase 2a. If φ r(e l ) = (k,i) an φ r(e l) = (k,j), that is, v = L(v 1,v 2) is not a reursive noe, then we ompute O = Π p a=l+1 Outputs(φr(ea),φv(U)) I = Π q a=l+1 Inputs(φr(e a),φ v(u)) an Z = Z(k,i,j). s illustrate by the top right orner of Figure 11, O is the (reverse) reahability matrix from the outputs of M 1 to the outputs of M 1, Z is the reahability matrix from the outputs of M 1 to the inputs of M 2, an I is the reahability matrix from the inputs of M 2 to the inputs of M 2. Thus, O T Z I gives the reahability matrix from the outputs of M 1 to the inputs of M 2, where O T enotes the transpose of O. o π evaluates to (O T Z I)[x,y]. ase 2b. If φ r(e l ) = (s,t,i) an φ r(e l) = (s,t,j), that is, v = L(v 1,v 2) is a reursive noe, we onsier the ase where i < j. The other ase where i > j an be hanle in a similar manner. First of all, if p = l, that is, v 1 = v 1 an M 1 = M 1, then M 2 is erive from M 1. By ase 1, we know that i 2 is not reahable from o 1 in R U. o π evaluates to false. Otherwise, as illustrate by the bottom right orner of Figure 11, using a similar eoing proess to ase 2a, π evaluates to (O T Z I I)[x,y] (see [6] for etails) Fast Matrix Multipliation To ahieve onstant query time, we nee to show that Inputs an Outputs an be implemente in onstant time. Lemma 3. Given a fixe stritly linear-reursive grammar G, for any ege label φ r(e) an any ata label φ v(u), Inputs an Outputs an be ompute in onstant time. Proof. onsier ase 2 in lgorithm 1. First observe the repeate pattern of length l in the i 1 intermeiate reahability matries. Let X be the multipliation of the first l matries. o we only nee to effiiently ompute X i 1/l. Further observe the repeate pattern in the sequene X, X 2,..., X i 1/l. uppose any moule has at most input or output ports. Note that is a onstant for a fixe G. ine eah matrix has at most 2 possible boolean values, we an fin in onstant time a an b suh that a < b <= an X a = X b. One a an b are foun, X i 1/l an be ompute in onstant time. Query-Effiient View Labeling. To spee up the query proessing, one an also pre-ompute a an b for eah reursion in the view, an materialize a an b (as well as X 1,X 2,...,X b ) in the view label. In ontrast to spaeeffiient view labeling (etion 4.3), this is the other extreme alternative that will be ompare in the experiments. M