Analyzing Workflows implied by Instance-Dependent Access Rules

Transcription

1 Analyzing Wokflows implid by Instan-Dpndnt Ass Ruls Toon Calds 1 Stijn Dkys 2 Jan Hidds 1 Jan Padans 1 1 Univsity of Antwp (Blgium) 2 Univsity of Southn Qunsland (Austalia) toon.alds@ua.a.b dkys@usq.du.au jan.hidds@ua.a.b jan.padans@ua.a.b ABSTRACT Rntly poposd fom-basd wb infomation systms libat th aptu and us of data in oganizations by substituting th dvlopmnt of thnial implmntations of ltoni foms fo th onptual modlling of foms t-stutud shmas and thi data ass uls. Signifiantly, ths instan-dpndnt uls also imply a wokflow poss assoiatd to a fom, liminating th nd fo a ostly wokflow dsign phas. Instad, th wokflows thus atd in an ad ho mann by unsophistiatd nd-uss an b automatially analyzd, and inot foms jtd. This pap xamins fundamntal otnss poptis of wokflows that a implid by instan-dpndnt ass uls. Spifially, w study th didability of th fom ompltability popty and th smi-soundnss of a fom s wokflow. Ths poblms a afftd by a hoi of onstaints on th path languag usd to xpss ass uls and ompltion fomulas, and on th dpth of th fom s shma t. Hn, w study ths poblms by xamining thm in th ontxt of sval diffnt fagmnts dtmind by suh onstaints. 1. INTRODUCTION In pvious wok [4] w psntd a pliminay dsiption of a novl wb infomation systm basd on foms to solv a patial poblm that many oganizations dal with. Th main goal of suh a systm is to nabl staff to asily and suly aptu data using wb foms, and us data that oths (possibly onntd via oth ps) hav aptud, povidd suh us dos not violat suity o pivay uls. Uss should only onntat on dvloping th shma and ass uls of thi foms, and lt th systm handl stoag, suity and oth latd issus. As suh, th poposd systm alady omplmnts th apabilitis of ommial softwa suh as Miosoft s InfoPath [1] and th nw XFoms [2] standad bing dfind by th W3C to pla th oiginal HTML-basd wb foms. Mo signifiantly, fom a thotial point of viw th Pmission to mak digital o had opis of all o pat of this wok fo psonal o lassoom us is gantd without f povidd that opis a not mad o distibutd fo pofit o ommial advantag and that opis ba this noti and th full itation on th fist pag. To opy othwis, to publish, to post on svs o to distibut to lists, quis pio spifi pmission and/o a f. PODS 06, Jun 26 28, 2006, Chiago, Illinois, USA. Copyight 2006 ACM /06/ $5.00. a many diffnt, taditionally databas-ointd fomal aspts assoiatd to ths fom-basd wb infomation systms (fb-wis), inluding data and shma intgation [3] and wokflow analysis [6, 10]. This pap studis th latt: ass uls inludd in a fom s dfinition imply a wokflow poss that ath simply put dsibs th od in whih data an b ntd in a fom. Hn, in this typ of wokflow systms, th data-flow implis th ontol-flow. In ontast to xisting wokflow systms and sah, an fb-wis dos not qui omplx initial dsign and subsqunt modifiation phass fo wokflows. This stablishs th motivation fo ou pap: wokflow posss implid by foms that w atd in an ad ho mann by unsophistiatd uss nd to b analyzd automatially suh that foms with an inot wokflow will b jtd by th fb-wis and uss an b told how thy should modify thi fom s dfinition. Impotantly, th poblms w study in this pap do not only ou in th fom-basd wis stting. Indd, vn lational databas managmnt systms suitably nhand with instan-dpndnt ass uls imply wokflows. Whil th poblm aa is thfo a gnal on, w will us th fb-wis stting as th patial appliation in whih to study it, sin that stting is patiulaly wll-suitd to solv - govnmnt issus. An xampl of a omplx fom usd in this ontxt is th ltoni vsion of a tax dlaation: vaious pats of th -fom may only b ompltd by tain psons and thn only dpnding on infomation that has alady bn ntd. Th assoiatd wokflow poss an b omplx, spially whn th fom allows intation btwn itizns and vaious lvls of th administativ buauay that posss th data aptud though th fom. Modl and Fagmnts. Th gnal modl w us fo th fb-wis mploys nstd lations as th shma fo foms, and a subst of XPath s abbviatd syntax fo ass uls and ompltion fomulas. Rstitions on ith (o both) th shma and th fomula languags onstitut vaious fagmnts that w xamin in dpth. Fo instan, w will at tims stit th dpth of a shma (as in th dpth-k fagmnt) o pohibit th us of ngation in ith ass o ompltion fomulas (as in th positiv fagmnts). Not that th oiginal modl w poposd fo p-to-p fb-wis [4] addd isa onstaints to at links btwn multipl foms shmas to nabl us of data aoss ps. W hav omittd this fatu in this pap sin w will onntat on a singl wokflow implid by a singl fom.

2 Poblms. Th stitions mployd in th vaious fagmnts failitat th study of som fats of th gnal poblm whih xamins th poptis of wokflows implid by instan-dpndnt ass uls. W onsid th didability of a fom s otnss, on aspt of whih w all th ompltability poblm baus it osponds to dtmining whth a fom an b ompltd (aoding to som fomula st by th fom s ato) givn its ass uls. A latd qustion, th smi-soundnss poblm 1, is to did whth ah ahabl instan is ompltabl. Running Exampl. Consid that a wb fom xists that nabls staff to apply fo ational lav and manags to did on all appliations. Th Lav Appliation fom has a shma and data ass uls that allow a pson to at nw instans and updat dtails suh as stat and nd dats. Whn th us submits th appliation (.g. by hking th appopiat box), thos dtails annot b hangd anymo, and th fom (o ath that instan of th fom) boms assibl by th manag. Th latt an appov o jt th lav appliation, possibly giving a motivation fo doing so. Th fom is ompltd by hking a final hk box, aft whih nothing an b hangd. Hn, th fom ompltion fomula simply hks th valu of this hk box. Stutu. Stion 2 disusss lvant sah, whil th maind of this pap is dividd in two pats. Stion 3 fomally dfins th fb-wis modl, th fagmnts w study and th ompltability and smi-soundnss poblms. Stions 4 and 5 xamin th omplxity of diding ths poptis fo foms with unstitd ass uls and fom ompltion fomulas, and with only positiv fomulas, sptivly. W wap up with a shot onlusion. 2. RELATED WORK 2.1 Patial Wok Th Wold Wid Wb Consotium s XFoms [2] ommndation is a spifiation fo a dlaativ languag usd to spify wb foms whih may inlud data onstaints and onditional fomatting logi basd on XML Shma and XPath. Compad to taditional HTML foms, XFoms povid sialization of aptud data to XML, spaat psntation fom ontnt, minimiz ound-tips to th sv, off dvi indpndn, and du th nd fo sipting. Th main diffns with fb-wis a that an fb-wis is an ativ sv-sid systm that (1) fs th fom ato fom having to xpliitly spify ass to a databas bak-nd; (2) inluds data ass uls in th fom dfinitions nabling a mo fin-gaind suity spifiation; and (3) implis and manags a wokflow poss automatially. As XFoms omplmnts th funtionality of an fb-wis, th latt will nomally mak us of XFoms fo lint-sid onstaint hking and sialization of data. Miosoft s InfoPath [1] softwa (msip) is an altnativ to XFoms but simplifis onditional fomatting and maks its us xpliit. Whil th fous is on psntation, ondi- 1 This popty is alld smi-soundnss baus it is a wak vsion of th usual notion of soundnss fo wokflow nts [9] whih also quis that ah vnt ous in at last on possibl un of th wokflow. tional fomatting uss a powful path xpssion languag to did whth o not a spifi fild in a fom should b mad visibl to th us. This is usually dtmind on th basis of data alady ntd in th fom. Hn, this thniqu is simila to th instan-dpndnt data ass uls w popos fo fb-wis. Th patial diffns btwn th systms a: (1) ou data ass uls a simpl path xpssions making th study of wokflow otnss fasibl; (2) msip s uls dtmin psntation issus in th lint, not data ass in th stoag bak-nd; and (3) fom atos in msip, as in XFoms, a taskd with xpliitly ating ass to data sous. Of intst to this pap, both msip and XFoms a popula tools that allow unsophistiatd uss to imply a wokflow by using onditional fomatting in thi foms, but do not analyz th otnss of ths wokflows. Hn, fom a thotial point of viw, both systms giv is to th sam st of wokflow-latd poblms as thos implid by an fbwis. Som of ths poblms a studid in this pap. 2.2 Thotial Wok Th ida to lt ass uls impliitly dfin a wokflow is patly inspid by Podut-Basd Wokflow Dsign as intodud by Hajo Rijs t al. [7] wh a mthod is poposd to dsign wokflows basd upon th final podut, simila to a fom, and dpndnis btwn th diffnt pis of infomation in th fom. Th has also alady bn a signifiant amount of sah on spifying ass ontol poliis fo XML doumnts in tms of XPath xpssions (s Fundulaki and Max [5] fo an ovviw) but to th bst of ou knowldg non of thm attmpt to analys th wokflow that is implid by ths ass ontol poliis. 3. PRELIMINARIES 3.1 Shmas Infomally a shma dfins th undlying data stutu of a fom. It is basially a shma of a nstd lation as dfind in th nstd lational modl [8]. Fo xampl, in th lav appliation, a possibl shma ould look lik th modl givn in Figu 1. nam appliation dision dpt piod appov jt bgin submit nd ason Figu 1: Th lav appliation shma. final W assum th xistn of a st L of nod labls, and a spial labl L that will b usd to labl th oots of ts. In th following of this pap w will abbviat th nod labls in Figu 1 to thi fist ltt, i.., appliation to a, nam to n, t ta.

3 Dfinition 3.1 (Shma and Instan). A ootd nod-lablld t is a fou-tupl M = (V, E,, λ) wh th gaph (V, E) is a t with oot V, and λ : V L is a lablling funtion on th nods. A shma is a ootd nod-lablld t in whih no two siblings hav th sam labl, and th oot has labl. A ootd nod-lablld t I = (V, E,, λ ) is alld an instan of th shma M = (V, E,, λ) if th xists a homomophism fom I to M. That is, th xists a funtion h : V V suh that (1) fo all v 1, v 2 V it holds that if (v 1, v 2) E, thn (h(v 1), h(v 2)) E, (2) h( ) =, and (3) fo all v V, λ (v ) = λ(h(v )). a n d p b (a) s b p n a d d p b s (b) Figu 2: Instans of th lav appliation shma. Exampl 3.2. In Figu 2 two xampls of instans of th shma in Figu 1 a givn. Th instan (a) shows a submittd appliation fo two piods, and (b) an appliation fo a singl piod that was jtd. Unlss xpliitly indiatd th filds in a fom an ontain zo o mo lmnts. It will b disussd lat how filds an b mad singl-valud by hoosing appopiat ass uls. Not that in pati th will b valus assoiatd with th filds suh as th atual nam, dpatmnt and th dats, but this is byond th sop of this pap. Poposition 3.3. If a fom I is an instan of a shma M, thn th homomophism fom I to M is uniqu. Hn, vy nod v and dg in an instan I of shma M a always assoiatd with a singl nod and dg in M. W dnot this uniqu nod and uniqu dg as ˆv and ê, sptivly. 3.2 Fomulas Dfinition 3.4 (Fomula). Fomulas a dfind by th following abstat syntax: F ::= P F (F F ) (F F ) P ::=.. L (P/P ) P [F ]. W will us fomulas to spify ass and updat uls and fom ompltion fomulas fo foms. Th abstat syntax losly smbls XPath s abbviatd syntax: th dsndnt-o-slf (//) and slf (.) axs a omittd, th fom baus th shma s dpth and hn that of its instans is fixd whn it is atd, and th latt baus it dos not add xpssiv pow in this ontxt. In [4] w also allowd any onditions in fomulas to inlud ompaisons to th systm vaiabls us-id f and dattim. In this pap w assum only on us; th data ass fomulas a valuatd fo this us and do not inlud tmpoal onditions. Dfinition 3.5 (Smantis of Fomulas). Th xpssion n = T φ indiats that th fomula φ is tu fo th ootd nod-lablld t T = (V, E,, λ) and a nod n in T. Th xpssion n p T n indiats that th path xpssion p is tu fo th ootd nod-lablld t T, a bgin nod n, and an nd nod n in T. W will omit th subsipt T in both ass if it is la fom th ontxt. Th following uls dfin both onpts: n = p th xists a n V suh that n p n n = φ n = φ dos not hold n = φ ψ (n = φ) and (n = ψ) n = φ ψ (n = φ) o (n = ψ) n.. n (n, n) E n l n (n, n ) E and λ(n ) = l n th xists a n V suh that n p n and n q n n p[φ] n n p n and n = φ n p/q Exampl 3.6. W giv som xampls of fomulas fo instans of th shma in Figu 1. Thy a assumd to b valuatd fo th oot : a/p[ b ] : All piods hav bgin and nd dats. f d[a ] : Th appliation annot b final unlss it was jtd o appovd. d[ (a )] : Th appliation annot b both jtd and appovd. 3.3 Canonial Instan W dfin a notion of fomula quivaln that an b infomally dsibd as bisimulation und th assumption that all dgs a biditional. Dfinition 3.7 (Fomula quivaln). A fomula quivaln btwn two fom instans I = (V, E,, λ) and I = (V, E,, λ ) is a lation R V V, suh that: (, ) R; fo all (v, v ) R: λ(v) = λ(v ); fo all (v, w) E: if (w, w ) R, thn th xists a v V suh that (v, v ) R, and (v, w ) E ; fo all (v, w ) E : if (w, w ) R, thn th xists a v V suh that (v, v ) R, and (v, w) E; fo all (v, w) E: if (v, v ) R, thn th xists a w V suh that (w, w ) R, and (v, w ) E ; and, fo all (v, w ) E : if (v, v ) R, thn th xists a w V suh that (w, w ) R, and (v, w) E. If th xists a fomula quivaln btwn I 1 and I 2, w all thm fomula quivalnt, and this is dnotd I 1 I 2. Two nods v, v of a fom instan I a alld fomula quivalnt nods if th xists a fomula quivaln R btwn I and itslf, suh that (v, v ) R.

4 It is wll-known that bisimilaity btwn instans, and btwn th nods of an instan a quivaln lations btwn sptivly th instans, and th nods of an instan, so this also holds fo fomula quivaln. Futhmo, if two instans a fomula quivalnt, all fomulas will valuat to th sam tuth valu on thm. Fo vy lass of fomula quivalnt instans, w an isolat a singl anonial (up to isomophism) instan as is indiatd in th following dfinition and lmma. Dfinition 3.8 (Canonial Instan). Lt I b an instan. Lt [v] dnot th quivaln lass of a nod v. Th anonial instan an(i) is th following fom instan: ([V ], [E], [], [λ]), with nods [V ] = {[v] v V }, dgs [E] = {([v], [w]) (v, w) E}, and [λ]([v]) = λ(v). Noti that [λ] is wll-dfind, baus vy pai of fomula quivalnt nods must hav th sam labl. Moov, th sult is a t baus if two nods in I a fomula quivalnt thn thy ith a both th oot o both hav pants whih a fomula quivalnt. Lmma 3.9. Lt I and J b two fom instans. If R is a fomula quivaln btwn I and J and (v, w) R thn fo all fomulas ϕ, v = I ϕ iff w = J ϕ. Futhmo, I an(i), and an(i) is isomophi to an(j). Exampl In Figu 3 w s an xampl of an instan (a) and th osponding anonial instan (b). a a a a b (a) d Figu 3: An instan and osponding anonial instan. 3.4 Ass and Updat Ruls W postulat th st of ass ights R = {add, dl} wh add and dl a th ight to at and dlt sptivly. Dfinition 3.11 (Guadd Fom). A guadd fom is a tupl (M, A, I 0, ϕ) wh M = (V, E,, λ) is a shma, A : R E F th ass-ul funtion that maps ah ass ight and dg in M to a fomula, I 0 th initial instan that is an instan of M, and ϕ a ompltion fomula that dfins whn th fom is omplt by bing tu fo th oot nod. Th only updats on instan ts that a onsidd a th additions and dltions of dgs that add and mov laf nods, sptivly. Th addition of an dg = (n, n ) is allowd iff th fomula A(add, ê) is tu fo n in th unt instan. Similaly, th dltion of an dg = (n, n ) is allowd iff th fomula A(dl, ê) is tu fo n in th unt instan. Givn a guadd fom (M, A, I 0, ϕ) w all a squn I 0,..., I n of instans of M a un of this guadd d a a (b) b d fom if fo vy 1 i n it holds that I i an b obtaind fom I i 1 by a singl dg addition o dltion whih is allowd by A fo I i 1, and a omplt un if I n satisfis ϕ. Not that w do not onsid ad o updat ights h baus to th xtnt that ths a lvant fo wokflow analysis ths an b simulatd in th psntd fomalism. Fo xampl, it might b that th us an only s (and thfo updat) a nod if h o sh has ad-ights on this nod and all its anstos, but this an b simulatd by adding th fomulas of th ad ights to th fomulas that dtmin th at and dlt-ights. Exampl To illustat th onpt of guadd fom tak th shma M in Figu 1, th initial instan I 0 that onsists only of th oot (w stat with an mpty fom), a ompltion fomula ϕ = f (th final fild has bn makd) and th ass-ul funtion A as follows (shma dgs a idntifid by th paths to thi nd nods): A(add, a) = a A(dl, a) = a A(add, a/n) =../s n A(dl, a/n) =../s A(add, a/d) =../s d A(dl, a/d) =../s A(add, a/p) =../s A(dl, a/p) =../s A(add, a/p/b) =../../s b A(dl, a/p/b) =../../s A(add, a/p/) =../../s A(dl, a/p/) =../../s A(add, s) = s a[n d p] a/p[ b ] A(dl, s) = s A(add, d) = s d A(dl, d) = f A(add, d/a) = (a ) A(dl, d/a) =../f A(add, d/) = (a ) A(dl, d/) =../f A(add, d//) = A(dl, d//) =../../f A(add, f) = d[a ] f A(dl, f) = f In Exampl 3.12 A(add, a) quis that th is not alady an a fild psnt and so th annot b two appliations in an instan. Th ul A(dl, a) quis that th is no a fild so w an nv dlt an appliation fild on it has bn addd. Fo th dg a/n th a simila uls xpt that it is also quid that th is no s fild yt, i., th appliation was not yt submittd. Not that sin th fomula is valuatd fo th a nod and not th oot it ads../s and not just s. Fo adding an a/p dg it is not quid p and so th an b mo than on piod in an appliation. This is fltd in A(add, s) wh it is quid that bfo submission th not only must b a nam, dpatmnt and at last on piod, but it must also hold that th is no piod without bgin o nd dat. 3.5 Compltability and Smi-soundnss To illustat th poblms that w will study onsid th guadd fom in Exampl 3.12 xpt that ϕ = f s. It an b obsvd that if w stat fom th initial instan th is no full un, i.., w an nv ah an instan that satisfis ϕ via a squn of allowd updats. Claly suh a guadd fom annot b onsidd ot, so w dfin th following poblm. Dfinition 3.13 (Compltability Poblm). Fo a guadd fom G = (M, A, I 0, ϕ) th fom ompltability poblm is to did if th xists at last on omplt un of G. Not that ompltability is not only intsting as a otnss quimnt but also impotant fo diding invaiants. Fo xampl, by hking ompltability fo ϕ = d[a ] w an hk if at any stag th an b a dision fild that ontains both apt and jt.

5 If a guadd fom is ompltabl it may still hav ompltion poblms. Consid again Exampl 3.12 but with ϕ = f d[a ], A(add, f) = d f, A(add, d/a) = (a )../f and A(add, d/) = (a )../f. In this as th guadd fom is still ompltabl but at th sam tim it is possibl to ah an instan wh th is a final fild but no appoval o jt fild. Fom that instan th fom annot b ompltd baus th final fild pvnts th addition of th quid appoval/jt fild. Obviously, if fom ompltability is a didabl poblm, a fom manag might disallow any updats that lad to suh an instan fom whih ompltion is not possibl, but in a wll-dsignd guadd fom th ass uls should pvnt this. This lads to th following poblm. Dfinition 3.14 (Smi-soundnss Poblm). Givn a guadd fom G = (M, A, I 0, ϕ) th fom smi-soundnss poblm is to did if fo vy un I 0,..., I n of G it holds that (M, A, I n, ϕ) is ompltabl. In th following stions, w study th omplxity of diding th ompltability and th smi-soundnss of guadd foms. Fo th gnal as, w show that ths poblms a both undidabl. Thfo, w also study mo stitd fagmnts of guadd foms, fo whih ths poptis a didabl. Th stitions a thfold: w study th ass in whih (a) th ass uls do not ontain ngation (A + ), (b) th ompltion fomula dos not ontain ngation (ϕ + ), and () th dpth is limitd to ith 1, o a fixd onstant k. W also study all possibl ombinations of ths stitions. W will dnot ths lasss of fagmnts by F (A, ϕ, d), with A ith A + o A, dnoting sptivly only positiv ass uls o any ass uls; ϕ is ith ϕ + o ϕ, dnoting sptivly a positiv ompltion fomula, o an unstitd ompltion fomula; and d bing 1, k, o, dnoting that th dpth of th shmas is sptivly 1 (.g., und th oot th is at most 1 lvl of nods), a fixd onstant k, o unstitd. Fo xampl, F (A +, ϕ, 1) dnots th lass of guadd foms wh all ass uls a positiv fomulas, th ompltion fomula an ontain ngation, and th dpth of th shma is at most 1. Th lass without stitions is dnotd F (A, ϕ, ). 4. UNRESTRICTED ACCESS RULES 4.1 Unstitd ompltion fomulas W will show that within th ontxt of th fom ompltability poblm w an simulat a two-ount mahin. It is wll-known that two-ount mahins a Tuing omplt, and that thi halting poblm is undidabl. Fom th simulation it is staightfowad, givn a two-ount mahin, to at an instan of th fom ompltability poblm that is ompltabl if and only if th two-ount mahin will vntually halt whn th mpty sting is givn as input. This popty of two-ount mahins is undidabl. A two-ount mahin is a dtministi stat mahin with two ounts. Th stat tansitions dpnd only on th input symbol ad, th unt stat, and on th ounts bing zo. Futhmo, duing a tansition th ounts an b inmntd o dmntd by 1, indpndntly, and th input uso an ith stay in pla, o advan on position. Baus w only onsid th poblm of diding whth o not th two-ount mahin halts on th mpty sting, w do not hav to tak a of th input-uso. Thus, a two-ount mahin without input an b modlld as a th-tupl (Q, F, δ), with Q a finit st of stats, F Q th st of apting stats, and δ th tansition funtion that maps Q {0, +} {0, +} to Q {, 0, +} {, 0, +}. Fo xampl, δ(q, +, +) = (p,, 0) dnots that if th twoount mahin is in stat q, and both ounts a nonzo, th stat will hang to p, th fist ount will b dmntd, and th sond mains unhangd. A onfiguation of a two-ount mahin without input will b dnotd as a th-tupl (q, n, m) with q th stat, and n and m positiv intgs that indiat th sptiv ontnts of th two ounts. Thom 4.1. Fo guadd foms in F (A, ϕ, ) th ompltability and smi-soundnss poblms a undidabl, vn if only foms of dpth 2 a onsidd. Poof. Th poof pods by showing that th onfiguation of an inputlss two-ount mahin an b psntd, and how w an mov fom on onfiguation to th nxt by disussing th podus fo dmnting and inmnting th ounts. Thfo, w an simulat th halting poblm of an inputlss two-ount mahin with th ompltability of a guadd fom. Baus this popty is undidabl, th ompltability poblm is undidabl as wll. Futhmo, th two-ount mahins w onsid a dtministi. As a onsqun, in vy ahabl instan of th guadd fom w onstut, only on oth instan (up to dupliations of nods) is ahabl. Thus, in this as, th ompltability poblm and th smisoundnss poblm a quivalnt. Noti inidntally that th dpth of th guadd fom onstutd blow is always 2, indpndnt of th simulatd two-ount mahin. Configuation: Lt (q, n, m) b a onfiguation of a twoount mahin. This onfiguation will in th simulation b psntd by th following instan Conf(q, n, m): blow th oot th is a nod lablld q, n nods lablld 1 and m nods lablld 2. Fo xampl, th following instan dnots th onfiguation (q, 5, 3). q Tansitions: Th most omplx pat of th dution is to giv an ass-ight funtion, suh that w an guaant that whn moving fom on onfiguation t to anoth, w us th tansition funtion δ. On w an nsu this guaant, th stopping ondition in th fom will simply b th disjuntion of all apting stats. Fo vy possibl tansition δ(q, s 1, s 2) = (p, a 1, a 2), a nw nod init(q, s 1, s 2) is intodud in th shma of th guadd fom. Th psn of this nod in an instan will dnot that th tansition init(q, s 1, s 2) is in pogss. This is nssay in od to avoid that th xution of on tansition alady stats bfo th pvious tansition is ompltly finishd. To ahiv this, th addition ondition fo init(q, s 1, s 2) will b: ^ q σ 1 σ 2 init(p, t 1, t 2), p Q,t 1 {0,+},t 2 {0,+} (p,t 1,t 2 ) (q,s 1,s 2 )

6 wh σ i is i (sp. i) if s i is + (sp. 0). That is, th tansition init(q, s 1, s 2) an only b initiatd if w a in stat q, th is a i iff s i is + (indiating that ount i is stitly positiv), and non of th oth tansitions is still in pogss. Inmnts: Fo th inmnt of ount 1, w nd to add xatly on 1 blow th oot. W hav to b aful in this as, baus w want to avoid that mo than on 1 an b addd. Thfo, w nd to b abl to mak a diffn btwn th situation bfo th addition, and aft. This an b don as follows: fist w mak all 1-nods, by adding a nod lablld d blow thm and adding a spial nod m on this is don fo all of thm. Thn, on 1 is addd. Noti that w an laly mak a distintion btwn th situation bfo th addition, and th situation aftwads; bfo th addition all 1 nods a makd; hn ( 1[ d]) is tu, and aft th addition, th is on unmakd nod; hn 1[ d]) is tu). Th ation is thn ompltd by moving all maks, and moving to th nxt stat. What follows illustats som of th osponding ass uls if δ onsists only of th tansition δ(q 0, 0, +) = (q 1, +, 0): A(add, init(q 0, 0, +)) = q init(q 0, 0, 0) init(q 0, +, 0) init(q 0, +, +) init(q 1, 0, 0) init(q 1, 0, +) init(q 1, +, 0) init(q 1, +, +) A(add, 1 /d) =../init(q 0, +, 0) A(add, m) = init(q 0, +, 0) 1 [ d] A(add, 1 ) = init(q 0, +, 0) m 1 [ d] A(dl, 1 /d) =../init(q 0, +, 0)../m A(dl, m) = init(q 0, +, 0) 1 [d] A(add, q 1 ) = init(q 0, +, 0) A(dl, q 0 ) = init(q 0, +, 0) A(dl, init(q 0, 0, +)) = init(q 0, +, 0) 1 [d] m q 1 q 0 Dmnts: Th dmnt is th most diffiult opation to tanslat. A fist attmpt ould b to mak xatly on 1-nod, and subsquntly mov th makd nod. This is, howv, impossibl, as w an only do opations on lavs of th instan. A sond attmpt ould b to mak all nods, xpt fo on. But, this is not at all staightfowad, baus at th momnt that w nd to did whth o not to add an additional mak, w annot distinguish btwn th situation wh on nod is unmakd, o wh two nods a unmakd. Thfo, th following, ath umbsom, podu is followd. Fist w mak xatly on 1-nod by adding a nod lablld d blow it. This is th nod that will b dltd lat on. Thn, w mak vy oth 1-nod with d. Subsquntly, w unmak th nod with mak d and thn mov th sol unmakd 1-nod (this is th nod that was makd with d bfo.) Finally, w mov all maks d. It might b susptd that th psn of dltions maks th poblm had but th following oollay shows othwis. Coollay 4.2. Fo th lass F (A, ϕ, ) th ompltability poblm and smi-soundnss poblm a undidabl, vn if only additions and foms of dpth 3 a onsidd. Poof. Th poof is simila to that of Thom 4.1 xpt that (1) vy dltion of an dg is plad with th addition of an dg und that dg that nds in a nod with a spial labl, say dltd, and (2) in all fomulas w pla path xpssions of th fom l with l[ dltd]. Lmma 4.3. Lt G b a guadd fom in F (A, ϕ, 1) and lt C b a anonial instan of th shma of G. Th xists an instan J, ahabl fom I, with an(j) = C, if and only if C is ahabl fom an(i). Futhmo, in a guadd fom of dpth 1, instan I an b ompltd if and only if an(i) an b ompltd. Poof. Evy addition that is allowd in I, is also allowd in an(i). As suh, vy opation on I, sulting in an instan J with a diffnt anonial fom, an b imitatd on an(i) to gt an(j), and vi vsa. Lmma 4.4. Lt ϕ b an abitay fomula, and lt T b a ootd t that satisfis ϕ, thn th xists an mbddd subt T of T that also satisfis ϕ and has a banhing fato that is lina in th lngth of ϕ. Poof. W stat with th obsvation that th following quivalnis hold, wh ψ 1 ψ 2 mans that fo vy instan I and nod n in I it holds that n = I ψ 1 n = I ψ 2: (p 1/p 2)[ψ] p 1[p 2[ψ]] (p 1[ψ 1])[ψ 2] p 1[ψ 1 ψ 2] (p 1/p 2)/p 3 p 1/(p 2/p 3) (p 1[ψ])/p 2 p 1[ψ p 2] l/p l[p]../p..[p] By applying ths uls fom lft to ight to ϕ and its subfomulas w an obtain a fomula ϕ that is lina in th siz of ϕ and blongs to th following syntax: F ::= P F F F F F P ::= L.. L[F ]..[F ] Givn a fomula ψ of th fom F w dfin a sltion of ψ as a st S of sub-fomulas and ngatd sub-fomulas of ψ suh that (1) S = {p} if ψ = p with p in P, (1) S = { p} if ψ = p with p in P, (3) S = S if ψ = (ψ 1 ψ 2) and S is a sltion of ψ 1 ψ 2, (4) S = S if ψ = (ψ 1 ψ 2) and S is a sltion of ψ 1 ψ 2, (5) S = S 1 S 2 wh ψ = ψ 1 ψ 2 and S 1 and S 2 a sltions of ψ 1 and ψ 2, sptivly and (6) S = S 1 o S = S 2 wh ψ = ψ 1 ψ 2 and S 1 and S 2 a sltions of ψ 1 and ψ 2, sptivly. Obsv that a nod n in an instan I satisfis a fomula ψ iff th is a sltion of ψ suh that all fomulas in this sltion a satisfid by n in I. If this holds thn w say that th sltion is satisifid by nod n in I. Fom ϕ and T w onstut T that satisfis ϕ as follows. W stat by ltting T b th nod n fo whih φ holds in T and assoiat with n a st Φ(n) whih is a sltion of ϕ that was satisfid by n in T. Thn w pod by applying th following uls: Fo ah fomula of th fom l[ψ ] in Φ(n ) w add to T a hild n of n with labl l that mad this fomula tu fo n in T, unlss suh a hild is alady psnt in T, and add to Φ(n ) a sltion of ψ that is satisfid by n in T, and fo ah fomula of th fom l in Φ(n ) w add to T a hild n with labl l.

7 Fo ah fomula of th fom l[ψ ] in Φ(n ) and in T ah hild n of n with labl l w add a sltion of ψ that was was satisfid fo n in T to Φ(n ), and if th is a fomula of th fom l in Φ(n ) and in T a hild n of n with labl l thn w abot th onstution. Fo ah fomula of th fom..[ψ ] in Φ(n ) w add to T th pant n of n, if it was not alady in T, and add to Φ(n ) a sltion of ψ that is satisfid by n in T, and fo ah fomula of th fom.. in Φ(n ) w add to T a pant n of n, if it was not alady in T. Fo ah fomula of th fom..[ψ ] in Φ(n ) and pant n of n w add n to T, if it was not alady in T, and add to Φ(n ) a sltion of ψ that is satisfid by n in T, and if th is a fomula of th fom.. in Φ(n ) and a pant n of n thn w abot th onstution. It is asy to s that if ϕ was satisfid by n in T thn th onstution of T will not abot and ϕ will b satisfid by n in T. Moov, baus of th way hild nods a addd th fan-out of ah nod is stitd by th numb of sub-fomulas of ϕ and hn lina in th siz of ϕ. Coollay 4.5. Tsting satisfiability of a fomula ϕ is NP-omplt if th dpth of th instans is limitd by a onstant k, and is PSPACE-omplt if th dpth is unlimitd. Poof. Th NP upp bound follows fom Lmma 4.4 sin it shows that th is a suint tifiat of siz O(n k ) fo th satisfiability of ϕ if n is th siz of ϕ Th NP-hadnss poof is a staightfowad dution fom SAT to satisfiability;.g., th satisfiability of th popositional fomula (x 1 x 2) x 3 osponds to th satisfiability of th fomula (a b). Th PSPACE upp bound an b shown as follows. W tansfom ϕ to ϕ and us th sam gnation uls as in th poof of Lmma 4.4. Thn w follow th sam podu xpt that (1) w do not gnat th whol t but only walk though it dpth-fist and (2) with ah nod n that is addd w also guss immdiatly th st Φ(n ) and hk if it onsistnt with that of n. Sin th siz of Φ(n ) is boundd by th siz of ϕ it follows that fo ah nod in th t w hav to do a polynomial amount of wok. Moov, sin th dpth of th t annot bom lag thn th siz of ϕ it follows that th stak that is quid fo th dpth-fist t-walk is polynomial in th siz of ϕ. Th PSPACE-hadnss an b shown with a dution fom QSAT to satisfiability. E.g., th satisfiability of th quantifid Boolan fomula x y z : (x y z) osponds to th satisfiability of th path xpssion ( a x/a y/a z[ (../../x) (../y) z]) (4.1) (a x/x ( a x[ x])) (4.2) ( (a x[ a y/y])) ( (a x[ a y[ y]])) (4.3) (a x/a y[a z/z ( a z[ z])]) (4.4) In this path xpssion, assignmnts fo x a nodd with an a x-nod. Th assignmnt making x tu is nodd as a nod a x with a subnod x. Similaly fo a y and a z. Th assignmnts a nstd to psnt th quantifi od. Futhmo, lin (4.2) maks su that th is only on tuth assignmnt fo x (possibly psntd by multipl idntial opis). Lin (4.3) nsus that within vy a x-nod, th is an assignmnt a y making y tu, and on making y fals. Lin (4.4) nsus again that a within vy a y-nod a uniqu hoi fo z is mad. Finally, lin (4.1) xpsss that on vy path, th givn fomula must b tu. Thom 4.6. Diding ompltability fo F (A, ϕ, 1) is PSPACE-omplt. Poof. Th upp bound follows fom Lmma 4.3 sin it implis that w an did ompltability in NPSPACE by gussing a squn of anonial instans of th shma suh that (1) th fist is an(i 0) wh I 0 is th initial instan, (2) ah nxt instan an b obtaind fom th pding on by applying on o mo tims a tain addition o dltion that is allowd and (3) th last instan satisfis th ompltion fomula. W show PSPACE-hadnss by duing th PSPACEomplt poblm ahabl dadlok to th ompltability poblm. Th ahabl dadlok poblm is th following. Th input onsists of a list of gaphs G 1 = (V 1, E 1),..., G k = (V k, E k ) with disjoint sts of vtis, a squn of vtis v 1,..., v k, with v i V i, i = 1... k, and a st T of pais of dgs ( i, j) with i and j in diffnt gaphs. A onfiguation is any st a 1,..., a k with a 1 V 1,..., a k V k. Th is a tansition of onfiguation a 1,..., a k to onfiguation b 1,..., b k, if th xist two indis 1 i < j k suh that fo vy l = 1... k, l i, j, a l = b l, and ((a i, a j), (b i, b j)) is in T. W thn all b 1,..., b k a susso of a 1,..., a k. Th ahabl dadlok poblm now is: dos th xist a onfiguation ahabl fom v 1,..., v k that dos not hav a susso? W du th ahabl dadlok poblm to th fom ompltability poblm fo dpth on. Fo vy vtx v in V 1... V k, and fo vy tansition t T, th is a nod n(v) sptivly n(t) in th shma. A onfiguation a 1,..., a k will b nodd by th fom instan that onsists of th oot nod and th nods n(a 1),..., n(a k ). W dnot this instan I(a 1,..., a k ). Hn, th initial stat will b th instan I(v 1,..., v k ). Th nods n(t), t T will b usd to ontol th tansitions. Fo as of notation, w intodu th notation onf fo th fomula ( W t T n(t)), whih is tu if th instan dos not ontain any of th ontol nods. An nd onfiguation is on that has no sussos. That is, if th dos not xist a tansition t T with t = ((a, b), (, d)) suh that a and a both in th onfiguation. Hn, w an dsib suh a dadlok situation with th following ompltion fomula: ^ φ = onf (a ). ((a,b),(,d)) T In od to pfom an atual tansition t = ((a, b), (, d)) in ou fom ompltability poblm, w will us th ontol nod n(t) to dnot that th tansition is taking pla. Th addition ondition of th ontol nod n(t) is thus (onf n(a) n()), and th dltion ondition ( n(a) n() n(b) n(d)). Futhmo, fo vy vtx v, th addition ondition will b: v ( n(t) n(t)), t=((x,v),(y,z)) T t=((x,y),(z,v)) T

8 and th dltion ondition: _ ( n(t) t=((v,x),(y,z)) T _ t=((x,y),(v,z)) T n(t)). Th a no oth ass ights. Th fom instan I(v 1,..., v k ) is ompltabl if and only if th is a ahabl dadlok. Coollay 4.7. Diding smi-soundnss fo th lass F (A, ϕ, 1) is PSPACE-omplt. Poof. Th poof of th upp bound is simila to that of Thom 4.6 xpt that w guss a squn of anonial instans suh that fom th last anonial instan th fom is not ompltabl. Not that by Thom 4.6 this last quimnt an indd b vifid in PSPACE. Th PSPACE-hadnss is shown by duing th ompltability poblm fo this fagmnt to th smi-soundnss poblm fo th sam fagmnt. Fom a guadd fom G w onstut a guadd fom G with th sam initial instan and a slightly lag shma. Th ass uls of G a simila to that of G but nsu that fo all ahabl instans I and I of G it holds that I is ahabl fom I in G iff I is ahabl fom I in G o I is th initial instan. It thn follows by Lmma 4.3 that G is smi-sound iff G is ompltabl. Th onstution of G is dtaild by th following points: In th shma w add th filds st and build und. Infomally ths will indiat th phas in whih th instan is dltd an in whih th initial instan is built, sptivly. All ass uls fom G a opid xpt that fo additions th fomula ψ is tansfomd to ψ st build and fo dltions it is tanslatd to ψ st. Th ompltion fomula ϕ is tansfomd to ϕ st build. A(add, st) = build st, A(dl, st) = build, A(add, build) = st build (l 1... l n) wh l 1,..., l n a th filds in th shma und th oot, and A(dl, build) is a fomula that tsts if th instan is an(i 0). Finally th ass uls fo additions a xtndd suh that whil build holds an(i 0) is built. 4.2 Positiv ompltion fomulas All th pding sults fo fagmnts with unstitd ass uls and unstitd ompltion fomulas also hold fo th fagmnts with only positiv ompltion fomulas. This is baus w an simulat th unstitd fagmnts as follows. W add in th shma a nw fild final und th oot, lt th ompltion fomula b final and add ass uls fo final suh that it an b addd if th old ompltion fomula holds. It is la that th sulting guadd fom has th ompltability and smi-soundnss poptis iff th oiginal guadd fom has thm. 5. POSITIVE ACCESS RULES 5.1 Unstitd ompltion fomulas Thom 5.1. Diding ompltability fo guadd foms in th lass F (A +, ϕ, k) is NP-had, and fo th lass F (A +, ϕ, ) is PSPACE-had. Poof. Lt ϕ b an abitay Boolan fomula. W du diding satisfiability of this fomula to th ompltability poblm of a guadd fom as follows: fo vy vaiabl x in ϕ, th is on nod lablld x in th shma of th guadd fom. All ass uls a st to tu. Th ompltion fomula is th givn fomula ϕ. It is asy to s that th guadd fom is ompltabl if and only if ϕ is satisfiabl, baus th ass uls allow any instan that satisfis th shma to b onstutd. Thom 5.2. Diding th ompltability of an instan I of a guadd fom in F (A +, ϕ, k) is in NP. Poof. (Skth) Suppos that th guadd fom is ompltabl by th instan J, and lt o 1,..., o n b th path fom th initial instan I to J. Thn, w an show that if a dltion is followd by an addition w an swap th two and again obtain a possibl path fom I to J. So w an assum that th is a path btwn I and J that onsists of a squn a 1,..., a k of additions, followd by a squn d 1,..., d n k of dltions. In what follows w show that with only a polynomially lag subsqun of this squn w an also ah an instan that satisfis th ompltion fomula ϕ. Baus of Lmma 4.4, within th instan J, w an idntify a subt T that satisfis ϕ, with banhing fato lina in th lngth of ϕ. Baus th dpth of th guadd fom is onstant, th siz of T is polynomial in th lngth of th ompltion fomula ϕ. Th whol onstution that follows is aimd at onstuting a minimal ahabl instan that ontains th t T as a subt. W all this t th witnss t. Duing its onstution w maintain two sts Add and Dl that ontain substs of th additions a 1... a k and dltions d 1,..., d n k, sptivly, and w stat with Add ontain th addition that onstut T and Dl mpty. Thn w add additions and dltions to Add and Dl if thy a quid by th ass uls thos that a alady in ths sts o by th ompltion fomula. W an do this in suh a way that in od to nsu that a tain fomula ψ holds fo a tain nod w add at most on addition that adds a hild und that nod. Sin all suh fomulas ψ will b subfomulas o ngations of subfomulas in th guadd fom it follows that th fan-out of th nods in th ts that a addd by th additions in th sulting Add st is lina in th siz of th guadd fom. Sin th hight of ths ts is at most k and thi numb is at most th numb of nods in T it follows that Add is polynomial in th siz of th guadd fom. Thom 5.3. Diding th smi-soundnss of guadd foms in F (A +, ϕ, k) is Π P 2k-had. Poof. W show Π P 2k-hadnss by duing th QSAT 2k - poblm to th omplmnt of smi-soundnss. That is, givn an instan of th QSAT 2k -poblm, w onstut a guadd fom of F (A +, ϕ, k), that is not smi-sound if and only if th QSAT 2k -instan valuats to tu. Consid th following QSAT 2k -instan: x x 1 n 1 y y 1 m 1 x x 2 n 2 y y 2 m 2... x k 1... x k n k y k 1... y k m k ψ.

9 W an assum without loss of gnality that all quantifi bloks hav th sam numb of vaiabls; that is, n 1 =... = n k = m 1 =... = m k = n. Th guadd fom w onstut fo this QSAT 2k -instan, will hav th following fom shma: x x 1 n 1 y1 k... yn k x x 2 n 2 y yn 1 x x 3 n 3 y yn 2 k 1 x k... 1 x k n y k yn k 1 Th ass uls a lativly simpl: fo all nods, xpt y1 k,..., yn k and u, th addition and dltion onditions a: /u; that is, thy an b addd and dltd, as long as th u-lmnt is psnt (u stands fo und onstution ). Th ass uls fo addition and dltion of y1 k,..., yn k a always tu. Th addition ondition fo u is u; that is, u an only b addd if it is alady psnt. Put othwis, if an instan no long ontains u, thn non of th instans ahabl fom this instan will ontain u. Th dltion ondition fo u is always tu. To summaiz: as long as u is psnt, all lmnts an fly b addd and dltd. On u is dltd, only manipulations of y1 k,..., yn k a allowd. Th initial instan of th guadd fom is th instan onsisting of only th oot and on nod lablld u. W will now onstut th ompltion fomula in suh a way that th xists a ahabl instan J that annot b ompltd if and only if th QSAT 2k -instan valuats to tu. Intuitivly, this instan J will psnt a situation wh fo th fist xistntial blok, a hoi has bn mad. Fo vy possibl assignmnt in th fist univsal blok, th is an lmnt lablld 1 that psnts this assignmnt. Within vy assignmnt blok 1, a hoi fo th sond xistntial blok is mad. Fo vy 1-lmnt, fo vy possibl assignmnt fo th vaiabls in th sond univsal blok, an 2-lmnt psnting this assignmnt is psnt, and so on, until th k-th xistntial blok. Noti that th a no lmnts with labl k to psnt th last univsal quantifi blok. Th ompltion fomula is thn dfind as: u Wk 1 i=1 1/... / i 1[ i[(ηi 1... ηn]] i 1/... / k 1 [ ψ ] wh ηj i = yj i /yj k and ψ is th following fomula: all vaiabls x i j a plad by th path xpssion: (k i) tims z } {../... /.. /x i j u all vaiabls y i j, i k, a plad by: and th vaiabls y k j (k 1 i) tims z } {../... /.. /y i j a plad by: (k 1) tims z } {../... /.. /y k j Th ompltion fomula is of th fom u... so a ahabl inompltabl instan dos not ontain u. Suppos that th xists an lmnt i 1, suh that th xists an assignmnt v that is not psntd by a hild lmnt i. Thn, w an omplt th instan J by making th following disjunt in th ompltion fomula tu: 1/... / i 1[ i[(y i 1 /y k 1 )... (y i n /y k n)]] Finally, suppos that th QSAT 2k fomula is tu. Thn w an find an assignmnt fo x 1 1,..., x 1 n, suh that fo all assignmnts fo y1, 1..., yn 1 th xists an assignmnt fo x 2 1,..., x 2 n, suh that fo all..., t ta, ψ holds. W onstut th instan that osponds to ths assignmnts. That is, w psnt th assignmnt fo x 1 1,..., x 1 n und, fo ah assignmnt fo y1, 1..., yn 1 w intodu an 1 nod that psnts this assignmnt. Und this 1 nod w also add th osponding assignmnt fo x 2 1,..., x 2 n and fo ah osponding assignmnt fo y1, 1..., yn 1 w intodu an 2 nod that psnts this assignmnt. Et ta, until w hav addd th nods fo th assignmnt fo x k 1,..., x k n und th k 1 nods. Not that if this instan is ahd only manipulations of y1 k,..., yn k a allowd sin th is no u fild. Moov, fom this instan th fom annot b ompltd sin (1) th u wil main absnt, (2) th sond pat of th ompltion fomula annot b satisfid baus th a nods fo all possibl assignmnts and (3) th thid pat annot b satisfid baus th QSAT 2k fomula is tu. On th oth hand, assum that th QSAT 2k fomula is not tu. Thn fom vy ahabl instan th fom an b ompltd sin at last on of th following holds: (1) u has not yt bn movd, o (2) not at all lvls i all assignmnts fo th x i 1,..., x i n hav osponding i nods that psnt thm, o (3) th is an assignmnt fo y1 k,..., yn k suh that whn nodd in th osponding filds nsus that th instan satisfis 1/... / k 1 [ ψ ]. Coollay 5.4. Diding th smi-soundnss of guadd foms in th lass F (A +, ϕ, ) is PSPACE-had. Poof. W an du any QSAT 2k -instan to diding smi-soundnss fo guadd foms of dpth k. Th poof in th pvious poof was onstutiv, and unifom fo diffnt k s. Hn, QSAT an b dud to diding smisoundnss fo F (A +, ϕ, ). 5.2 Positiv ompltion fomulas Thom 5.5. Diding th ompltability of a guadd fom in F (A +, ϕ +, ) is in P. Poof. (Skth) Th ompltability of a guadd fom in F (A +, ϕ +, ) an b vifid by adding as muh dg as possibl without adding a sibling with th sam labl as an alady xisting sibling. Th sult will satisfy ϕ iff th guadd fom is ompltabl. Th only-if pat is povd by

10 showing that a squn of additions that lads fom th initial instan to an instan that satisfis ϕ an b tansfomd to a squn additions that nv adds a sond sibling with th sam labl by moving suh additions and plaing subsqunt additions und th sond sibling with simila additions und th fist sibling. Sin th intmdiat instans annot bom lag than th podut of th siz of th initial instan and th siz shma this an b don in polynomial tim. Thom 5.6. Diding smi-soundnss of guadd foms in th lass F (A +, ϕ +, 1) is onp-had. Poof. (Skth) W giv a dution of th NP-omplt SAT poblm to th poblm of diding whth a guadd fom is not smi-sound. This shows that diding smisoundnss is onp-had. Lt ψ b a Boolan fomula in 3-onjuntiv nomal fom, and lt x 1,..., x k b th vaiabls in ψ. Fo vy vaiabl x i, i = 1... k, w intodu two nods, lablld sptivly x i and x i, in th shma of th guadd fom. Th psn of nods with ths labls psnt sptivly x i is tu and x i is fals. Th initial fom instan I in th smisoundnss poblm onsists of th oot-nod with all x i and x i, i = 1... k. Th ass uls a as follows: A(dl, x i) = x i, A(dl, x i) = x i, A(add, x i) = x i, and A(add, x i) = x i. Finally, th aptan ondition ϕ will flt th ngation of th SAT fomula ψ, and is dfind as ng(ψ), with ng(ψ) usivly dfind as follows: ng(x i) = x i, and ng( x i) = x i. Futhmo, fo vy onjunt of ψ, ng(l 1 l 2 l 3) is dfind as ng(l 1) ng(l 2) ng(l 3). Finally, ng(c 1 C 2... C m) is dfind as ng(c 1) ng(c 2)... ng(c m). It an thn b shown that ψ is satisfiabl iff th onstutd guadd fom is not smi-sound. Coollay 5.7. Diding smi-soundnss of a guadd fom in th lass F (A +, ϕ +, 1) is onp-omplt. Poof. Baus of Lmma 4.3, to show that a guadd fom in F (A +, ϕ +, 1) is not smi-sound, it suffis to giv a anonial instan J, suh that J annot b ahd fom an(i), and J is inompltabl. Thus, w an guss suh a anonial instan J. Baus of th dpth of 1, J is suint. Both hking th ahability of J fom an(i) and th inompltability of J an b don in polynomial tim. (Thom 5.5) Tabl 1: Summay of th omplxity sults. Fagmnt Compltability Smi- Soundnss A +, ϕ +, 1 P onp-ompl. A +, ϕ +, k P onp-had A +, ϕ +, P onp-had A +, ϕ, 1 NP-ompl. Π P 2 -ompl. A +, ϕ, k NP-ompl. Π P 2k -had A +, ϕ, PSPACE-had PSPACE-had A, ϕ, 1 PSPACE-ompl. PSPACE-ompl. A, ϕ, k undidabl undidabl A, ϕ, undidabl undidabl A, ϕ +, 1 PSPACE-ompl. PSPACE-ompl. A, ϕ +, k undidabl undidabl A, ϕ +, undidabl undidabl 6. SUMMARY AND CONCLUSION In this pap w hav analyzd th omplxity of diding otnss poptis, viz. ompltability and smisoundnss, fo wokflows that a impliitly dfind by a fom and instan-dpndnt ass uls that dtmin whih updats a allowd on th ontnts of th fom. Th sults in this pap a summaizd in Tabl 1. Th undlind sults indiat opn poblms sin th an upp bound of th poblms has not bn dtmind yt. In futu sah w intnd to invstigat th psntd fomalism as a mhanism fo xpssing wokflows and s und whih stitions tain wokflows an o annot b xpssd. Aknowldgmnt W would lik to thank Ron Addi, Hua Wang, and Rihad Watson fom th Univsity of Southn Qunsland fo patiipating in valuabl aly disussions about fom-basd wb infomation systms that inspid th wok disussd in this pap. 7. REFERENCES [1] Miosoft Offi InfoPath, 2003, podinfo/dfault.mspx. [2] J. Boy, D. Landwh, R. Mik, T. Raman, M. Dubinko, and L. Klotz, XFoms 1.0 (sond dition), W3C Rommndation, Mah 2006, [3] D. Calvans, G. D Giaomo, M. Lnzini, and R. Rosati, Logial foundations of p-to-p data intgation, PODS, 2004, pp [4] S. Dkys, J. Hidds, R. Watson, and R. Addi, P-to-p fom basd wb infomation systms, ADC 2006 (Hobat, Tasmania, Austalia) (Gill Dobbi and Jams Baily, ds.), Austalian Comput Soity, In., Januay [5] Iini Fundulaki and Maatn Max, Spifying ass ontol poliis fo XML doumnts with XPath, SACMAT 2004 (Yoktown Hights, Nw Yok, USA), ACM Pss, 2004, pp [6] J. Hidds, M. Dumas, W. van d Aalst, A. t Hofstd, and J. Vlst, Whn a two wokflows th sam?, CATS, 2005, pp [7] Hajo A. Rijs, Slma Limam, and Wil M. P. van d Aalst, Podut-basd wokflow dsign, Jounal of Managmnt Infomation Systms 20 (2003), no. 1, [8] Hans-Jög Shk and Ma H. Sholl, Th lational modl with lation-valud attibuts, Inf. Syst. 11 (1986), no. 2, [9] W. van d Aalst, Th appliation of pti nts to wokflow managmnt, Jounal of Ciuits, Systms, and Computs 8 (1998), no. 1, [10] W. van d Aalst, A. Hinshall, and H. Vbk, An altnativ way to analyz wokflow gaphs, CAiSE, 2002, pp