Automatic Model View Controller Engineering. Kalle Burbeck Henrik Larsson

Transcription

1 Final Thsis Automatic Modl Viw Controllr Enginring by Kall Burbck Hnrik Larsson LiTH-IDA-Ex-02/

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3 Linköpings univrsitt Dpartmnt of Computr and Information Scinc Final Thsis Automatic Modl Viw Controllr Enginring by Kall Burbck Hnrik Larsson LiTH-IDA-Ex-02/ Suprvisor: Uw Assmann Examinr: Uw Assmann

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5 Abstract Abstract Invrtibl transformations can b usd for ralizing a systm whr dvlopmnt is don in diffrnt viws, kping th viws consistnt with a common modl. A mthod to raliz such a systm is calld automatic Modl-Viw-Controllr nginring (MVARE). In this thsis important proprtis of gnral MVARE ar ntif and discussd. Th invrtibility of a numbr of rwriting tchniqus is discussd. Invrtibility of doubl pushout graph rwriting systms is furthr invstigatd whn thy ar usd for modl-viw transformations. A mthod using tags is prsntd allowing th invrtibl transformations to bcom lss rstrictd. Th main rsult of this thsis is th thortical foundations for CODEX, an MVARE systm. Futur works lis in implmnting a CODEX systm, and by using th principls of CODEX to invstigat mor xprssiv rwriting tchniqus.

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7 Acknowldgmnts Acknowldgmnts This thsis was writtn as a part of our Mastr s dgr in Computr Scinc at th Univrsity of Linköping, Swdn. Our mployr was th Programming Environmnts Laboratory (PELAB) at th Dpartmnt of Computr Scinc (IDA), Univrsity of Linköping. Spcial thanks gos to our xaminr and suprvisor Uw Assmann, assoc. profssor at PELAB for his nvr nding nthusiasm and as. Thanks also to our opponnts, Malin Alzén and Karolina Eliasson, for valuabl suggstions and commnts.

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14

15 Chaptr 1: Introduction 1 Introduction 1.1 Purpos Automatic Roundtrip Enginring (ARE), Assmann (2002), is a nw paradigm of softwar nginring. Th ovrall purpos of this thsis is to invstigat automatic Modl-Viw-Controllr nginring (MVARE), a spcial cas of ARE. 1.2 Motivation MVARE consists of a modl with a collction of viws and th corrsponding modl-viw rlations. Th rlations ar spcif by transformations. Th transformation must b invrtibl to b abl to kp th modl and th diffrnt viws consistnt. Whn a viw updats th modl, th othr viws ar updatd using thir invrs transformations. This is furthr xplaind in sction 2.2. If MVARE is ralizabl to its full xtnt it could hav ths nic proprtis: homognous rprsntation for multipl viws (th Modl) asy maintainability asy dbugging asy porting to nw platforms asy xtnsion of xisting viws asy dclaration of nw viws Thr ar a lot of applications today, that in principl us th concpt of working in diffrnt viws. But rlations btwn th viws ar mad in an ad hoc fashion. Ths applications would bnfit from MVARE. Th following sctions giv som short xampls of using MVARE WYSIWYG In som wysiwyg (What You S Is What You Gt) nvironmnts, for xampl HTML or wysiwyg TX, it is dsirabl to b abl to also dit th txt rprsntation in addition to th formattd documnt. In such a systm th txt rprsntation would b ditd in on viw and th formattd documnt in anothr viw. Two applications that hav this functionality, in an ad hoc fashion, and would bnfit from MVARE ar Mozilla s Composr and Macromdia s Dramwavr. 15

16 Chaptr 1: Introduction Aspct-orintd programming Aspct-orintd programming (AOP) is a programming paradigm whr MVARE could b usd. In AOP a wavr distributs cod into diffrnt parts of a program. If th wavd aspcts ar to b rgaind a dwavr is ndd to collct th cod fragmnts throughout th program. If th waving is spcif as MVARE, th dwavr is automatically givn. Mor on AOP can b found in Czarncki, K., Eisnckr, U.W. (2000) and at th aspct-orintd softwar dvlopmnt hompag Roundtrip nginring with UML Fujaba and Rational Ros both hav th ability to gnrat UML from Java cod and vic vrsa. This could b ralizd with MVARE and th invrs transformation would b gaind automatically. This also holds for roundtrip nginring in gnral Programming languags With MVARE transformations mapping programming languags to a common modl it would b possibl to dvlop softwar in th programmr s prfrrd languag. It is also asir to xtnd languags and add nw faturs by using multi-tird MVARE systms. 1.3 Targt Group Th targt group of this thsis is popl with a good undrstanding of computr scinc and intrst in rwriting systms. 1.4 Problm Statmnt 1. Find proprtis that can b usd for classifying instancs of MVARE. 2. Dscrib rwriting tchniqus that may b suitabl for modlviw transformations. 3. Find rquirmnts for making a suitabl xisting rwriting tchniqu invrtibl or dfin a nw invrtibl tchniqu. 1.5 Mthod 1. To bgin with, in dpth studis was ndd to undrstand rwriting, its applications and th diffrnt tchniqus that ar usd. Th following aras among othrs had to b invstigatd. Trm rwriting 16

17 Chaptr 1: Introduction Catgory thory Graph rwriting Attribut grammars XSLT, XSL, XML, TXL, AGG 2. A slction of xisting rwriting tchniqus was invstigatd brifly and th critria xprssibility and invrtibility wr usd to choos th most suitabl on. Invrtibility was only consrd at rul lvl at this stag. Automatically gnrating th invrs for a larg and complicatd transformation is of cours vry hard. Starting with obtaining invrss for indivual ruls and aftrwards putting th invrs ruls togthr to th complt invrs transformation smd mor promising. This is th motivation for only consring indivual ruls to start with. 3. Invrtibility of th chosn tchniqu was invstigatd furthr. Whn invrtibl ruls ar usd togthr in a rwriting systm, how can th systm b mad invrtibl? Rstrictions and xtnsions ncssary to gt invrtibl transformations wr spcif. 4. Th plan was to us an xisting rwriting tool and try a largr xampl of MVARE using th invrtibl transformations. Sinc no tool was vry suitabl for CODEX a papr xampl was workd through instad. 1.6 Ovrviw Th structur of th rst of this thsis is as follows. Chaptr 2 dscribs xisting basic concpts, which form th foundation of th latr rsults. Us this chaptr mainly as a rfrnc, with th xcption of sction 2.1 and 2.2 dfining ARE and MVARE. Th rwriting tchniqus of chaptr 2 ar furthr stud in chaptr 3 whr th suitability of th diffrnt tchniqus with rgard to invrtibility is discussd. In chaptr 4 proprtis of MVARE systms ar dfind and discussd. Existing tools and libraris that at last to som xtnt could b usd for ralizing MVARE ar dscribd. Chaptr 5 invstigats how wll suitd doubl pushout is for a graph rwriting systm such as is ndd in MVARE. Th MVARE systm rsulting from this thsis, CODEX, is introducd in chaptr 6. Th DPO rwriting tchniqu is xtndd 17

18 Chaptr 1: Introduction with tags and diffrnt matching critria. Also th issu of usr diting is rsolvd. In chaptr 7 a brif ovrviw of rlatd works is givn. Th conclusions of th thsis ar givn in chaptr 8. Finally, chaptr 9 prsnts svral aras of rsarch, which would b intrsting to invstigat furthr but ar outs th scop of this thsis. 18

19 Chaptr 2: Basic Concpts 2 Basic Concpts This chaptr dscribs th fundamntals and notations of ARE, MVARE and th rwriting tchniqus consrd in latr chaptrs. Sinc som basic proprtis such as normal forms ar intrsting for all tchniqus, ths ar dfind for abstract rduction systms. 2.1 Automatic Roundtrip Enginring Th basic a of ARE is simpl. Givn a spcification of a transformation btwn two domains, how can th invrs transformation b drivd automatically? Figur 1 shows two instancs of ARE. For ARE itslf it dos not mattr which transformation tchniqus that ar usd but som ar mor suitabl than othrs. Diffrnt tchniqus also hav proprtis that mak thm mor or lss suitabl for diffrnt applications. For vry application of ARE it is thrfor important to dc which tchniqu to us. Domain D h -1 h Domain B Domain B Domain C f -1 f f -1 f g g -1 Domain A Domain A Figur 1: Exampls of ARE To th lft, th most basic instanc of ARE. To th right is an instanc with multipl tirs Dfinitions and notations Th following dfinition is from Assmann (2002): Lt A and B b two domains, and f: A B a transformation function from function spac F. If thr is a function i: F F 19

20 Chaptr 2: Basic Concpts which calculats for f its invrs f -1 F thn R = (A, B, f, i) is an automatic roundtrip nginring systm (ARE). 2.2 Automatic Modl-Viw-Controllr Enginring Using ARE with multipl viws is calld Automatic Modl-Viw- Controllr Enginring (MVARE), s Assmann Th Modl-Viw-Controllr concpt In Krasnr and Pop (1988) th Modl-Viw-Controllr (MVC) concpt is prsntd as a way of sparating th application (modl) from th visualization (viw) prsntd to th usr. Th controllr handls th usr s intraction with th modl and maks sur th viw is consistnt with th modl. Usr input snsors Controllr Usr input dvic intraction viw mssags Modl accss and diting mssags Viw Display layout and intraction viws Display output Dpndnts chang mssags Modl Application domain stat and bhavior Dpndnts chang mssags Figur 2: Modl Viw Controllr Th intraction btwn th modl-, viw- and controllrobjcts (Krasnr and Pop, 1998) Th MVARE concpt MVARE uss th concpts of MVC. A domain is from now on dnotd viw or modl according to th rol it plays. Th modl is th systm in dvlopmnt, and diting is don in viws. No diting is allowd dirctly on th modl, but of cours a viw could 20

21 Chaptr 2: Basic Concpts show th modl as-is. Th dvlopr may choos to work in whichvr viw sh finds most appropriat as shown in figur 3. E=mc 2 Linux Arithmtic viw = ( ) = Java viw for (int i = 0; i <= 5; i++) { x = x + 2 * i; } Modl Figur 3: Exampl of MVARE Dvlopmnt in an arithmtic languag basd viw and a Java basd viw. It is clar from th pictur that both th physicist to th lft, and th programmr to th right ar vry happy to b abl to dit th modl in thir own prfrrd viw. In this thsis focus will b on th modl viw rlations. In an implmntation, vry viw could hav a controllr (or on shard controllr), which is rsponsibl for kping th viw consistnt with th modl. A modl viw rlation is a spcification of how cntral parts of a controllr should b implmntd Dfinitions and notations An MVARE systm (M, V 1 V k, T 1 T k, f) is a modl M, a collction of viws V 1 V k and for vry viw a modl viw rlation MVR x, which kps th viw consistnt with th modl. MVR x consists of a transformation T x : V x M and th invrs transformation T x -1 : M V x for x = 1 k. T x -1 is drivd automatically 21

22 Chaptr 2: Basic Concpts by applying th function f: F F on T x, whr F is th function spac of th transformations (Assmann 2002). In this work a rul or transformation with a -1 suprscript (.g. r -1 ) is always a transformation from th modl to th viw. A rul or transformation without this suprscript is a transformation from th viw to th modl. This is motivatd by th most natural way of looking at an MVARE systm. That is, th diting is don in th viws and transformd to th modl. 2.3 Abstract Rduction Systms Th notations in this sction ar thos usd by Boy t al. (1999) Dfinitions and notations An abstract rduction systm (ARS) is a pair (A, { i } i I ), whr A is a st and ach i is a binary rlation on A, calld a rwrit rlation. If a A, b A and (a, b) i this is writtn a i b. Th rflxiv and transitiv closur of a rlation R is dnotd R* or R *. Th invrs rlations ar dnotd R -1 and R* -1. Givn an ARS (A, R ), a A is a normal form (NF) if thr is no b A such that a R b. Th st of normal forms in A is dnotd NF(A). If thr is a b NF(A) such that a R * b, thn a has a normal form. R is Church-Rossr (or conflunt) if for all a, b, c A such that a R * b and a R * c thr is a d A such that b R * d and c R * d. If R is conflunt thn th normal form, if it xists, of vry lmnt is uniqu. 2.4 Trm Rwriting Systms This sction givs a brif introduction to trm rwriting, and givs th basic notations and dfinitions, which will b usd in th latr discussion about invrtibility History According to Drshowitz and Jounnaud (1990) rwrit systms can b tracd back to Gorn (1967), whn thy mrgd as a form of computr program. Sinc thn a substantial amount of rsarch 22

23 Chaptr 2: Basic Concpts has bn don on rwriting systms. An xtnsiv list of rfrncs can b found on th rwriting hom pag of Drshowitz and Vignron, (2001) Dfinitions and notations W will adopt som of th notations of Boy t al. (1999) for trm rwriting systms. A signatur F is a finit st of function symbols. Associatd with vry f F is a natural numbr dnoting its arity, i.. numbr of argumnts. Functions with arity n ar dnotd f / n. Constants, dnotd c, ar functions with arity 0. Lt F b a signatur, and V a dnumrabl st of variabls. Th st Tr(F, V) of trms ovr F and V is dfind: if x V thn x Tr(F, V) if f / n F and t 1 t n Tr(F, V) thn f(t 1,, t n ) Tr(F, V) nothing ls is in Tr(F, V) Var(t) dnots all variabls in th trm t. Syntactical quality of th trms t and s is xprssd by t s. Th st O(t) of positions in a trm t is dfind as follows whr ε is th mpty squnc. if t V thn O(t) = {ε} if t f(t 1,, t n ) thn O(t) = {ε} {i.p 1 i n and p O(t i )} If p O(t) thn t p dnots th subtrm of t at position p dfind as if p = ε thn t p = t if t f(t 1,, t n ) and p = i.q thn t p = (t i ) q. If p O(t) thn t[s] p dnots th trm obtaind from t by rplacing th subtrm at position p with th trm s. if p = ε thn t[s] p = s if t f(t 1,, t n ) and p = i.q thn t[s] p = f(t 1,, t i [s] q,, t n ) A substitution is a function σ: Tr(F, V) Tr(F, V) such that σ(f(t 1,, t n )) f(σ(t 1 ),, σ(t n )) A trm rwrit systm (TRS) is a pair (F, R) whr F is a signatur and R Tr(F, V) Tr(F, V) is a st of rwrit ruls. Evry 23

24 Chaptr 2: Basic Concpts rwrit rul is a pair (l, r) of trms, dnotd l r such that l V and Var(r) Var(l). Lt R b a TRS. Its associatd rwriting rlation R is dfind as: s R t iff thr xist: a rwrit rul l r in R a position p O(s) whr s p V a substitution σ such that s p σ(l) and t s[σ(r)] p. 2.5 Attribut Grammars with Forwarding This sction dscribs attribut grammars and a mor rcnt xtnsion namd forwarding. Th notation is from Knuth (1968) History Attribut grammars (AG) wr originally invntd by Knuth (1968). Thy ar an xtnsion of contxt-fr grammars. Th diffrnc is that th nontrminals also hav attributs, which ar assignd valus whn applying a production. AGs hav bn xtnsivly usd to spcify smantic analysis of programming languags. Attribut grammars with forwarding (AGF), invntd by Van Wyk t al. (2002), is an xtnsion to attribut grammars. It cam into xistnc during dvlopmnt of intntional programming (Czarncki & Eisnckr 2000). Forwarding has similaritis to highr ordr attribut grammars (Vogt t al. 1989) and can b rwrittn as on (Van Wyk t. al. 2001) Dfinitions and notations A contxt fr grammar is a tupl G = N, T, P, D whr N is a finit st of nontrminal symbols, T is a finit st of trminal symbols, P is a finit st of productions (or ruls) and finally D N is th start symbol of th grammar. V = N T is a grammar symbol. Th productions ar mad up of a lft-hand s and a right-hand s and is of th form X α whr X is a nontrminal and α V *, i.. a string of grammar symbols. An attribut grammar is a tupl AG = G, A, R whr G is a contxt fr grammar. A(X) is a finit st of attributs associatd with ach grammar symbol and A is th union of all such sts, A(X). A(X) is also divd into two disjoint substs, I(X) and 24

25 Chaptr 2: Basic Concpts S(X). I(X) consists of th inhritd attributs and S(X) consists of th synthsizd attributs, that is th attributs that ar passd from th lft-hand s to th right-hand s and th attributs that ar passd from th right-hand s to th lft-hand s rspctivly. A production is writtn p: X 0 X 1 X n (n 0). An attribut occurrnc X i.a xists for a production if a A(X i ), 0 i n. Associatd with ach production p P is a finit st of smantic ruls R p, whr on and only on smantic rul is associatd with ach synthsizd, X 0.a, and ach inhritd, X i.a (1 i n), attribut occurrnc of th nontrminals in th production. Th smantic ruls of R p ar of th form X i.a = f(y 1,, y k ), k 0, whr th following conditions must b mt: If i = 0 thn a S(X i ) ls if i 1 thn a I(X i ). Evry y j, 1 j k, must b an attribut occurrnc in p. Th smantic function f maps th valus of y 1,,y k to th valu of X i.a, X i.a is sa to dpnd on th attributs y 1,,y k. Now R is th union of all such sts, that is R = R p. Whn introducing forwarding a fw modifications to attribut grammars ar ndd. As in highr ordr attribut grammars (Vogt t al., 1989) som nontrminals ar calld nontrminal attributs, which can hold abstract syntax trs gnratd by th smantic ruls. Van Wyk t al. (2001) rquir that all nontrminal attributs occur to th right of all standard nontrminals in a production. A nw optional nontrminal attribut, placd rightmost in a production, is introducd and is calld a forwards-to nontrminal. Thus vry production p P has th form: X 0 X 1,, X n,, X m, X f X f, as statd abov, is optional but if it xists it has to b th sam nontrminal as X 0. Th nontrminals X n,, X m ar th nontrminal attributs. Oftn production-valud attributs ar usd to assign valus to th nontrminal attributs. Ths attributs pass a production, 25

26 Chaptr 2: Basic Concpts which is appl to th appropriat trs and thn producs a tr, which is assignd to th nontrminal attribut. Th major diffrnc btwn highr ordr grammars and attribut grammars with forwarding is that in highr ordr AGs (and also standard AGs) it is rquird that thr, in vry production, is a smantic function for ach normal attribut, synthsizd and inhritd, and ach nontrminal attribut. Howvr, in forwarding it is only rquird that all nontrminal attributs hav smantic functions dfining thm. Synthsizd attributs that ar not xplicitly dfind instad rciv as thir valu th valu of X f.a. Inhritd attributs that ar not dfind xplicitly ar not ndd whn calculating th synthsizd attributs. Figur 4 and figur 5 shows an xampl takn from Van Wyk t al. (2001), which dmonstrats ovrloading. In figur 4 thr is a gnric plus production and a spcializd plus, calld add, for intgrs. 1. plus: Expr 1 ::= Expr 2 Expr 3 Typ 2. Typ = Expr 2.typ 3. forwardsto Typ.plusProd(Expr 2, Expr 3 ) intgr: Typ ::= ε 6. Typ.plusProd = add add: Expr 1 ::= Expr 2 Expr 3 9. Expr 1.cod = gn_add_cod(expr 2.cod, Expr 3.cod) 10. Expr 1.typ = intgr Figur 4: Ovrloading with AGF A gnric plus production and th ncssary productions, intgr and add, ndd to gt plus for intgrs. Now it is vry asy to xtnd this grammar to includ plus for strings. In figur 5 plus for strings is dfind as concatnating two strings in th production concat. No changs to th original plus production ar ndd which is vry nic. 1. string: Typ ::= ε 2. Typ.plusProd = concat 3. 26

27 Chaptr 2: Basic Concpts 4. concat: Expr 1 ::= Expr 2 Expr 3 5. Expr 1.cod = 6. gn_concat_cod(expr 2.cod, Expr 3.cod) 7. Expr 3.typ = string Figur 5: Ovrloading with AGF continud Two productions ar now vry asily addd to support th plus oprator for strings. 2.6 Doubl Pushout This sction dscribs a graph rwriting tchniqu calld doubl pushout (DPO). Th following dfinitions ar mainly from Ehrig t al. (1990) and Corradini t al. (1997) History Doubl pushout was first introducd by Hartmut Ehrig t al. (1973). It is a graph rwriting tchniqu basd on catgory thory Dfinitions and notations A labld, dirctd graph consists of a tupl G = N, E, s, t, lbl whr N is a st of nods, E is a st of dgs, s: E N and t : E N ar calld sourc and targt map and lbl is calld labl map. Th sourc map is a mapping from an dg to th nod it starts at. Th targt map is a mapping from an dg to th nod it nds at. Th labl map is a mapping lbl: N E L whr L is a st of labls. In this work undirctd graphs will b usd, which ar a spcial cas of dirctd graphs. A subgraph, G s, of a graph, G, is a graph whr E s E, N s N and all th mappings s s, t s and lbl s ar rstrictions of th corrsponding ons from G. A graph morphism is a mapping m: G G, which consists of a pair of mappings m = (m E : E E, m N : N N ) such that m N ο s = s ο m E, m N ο t = t ο m E, lbl ο m E = lbl and lbl ο m N = lbl. A graph morphism is injctiv if both m E and m N ar injctiv and it is surjctiv if both m E and m N ar surjctiv. Graphs and graph morphisms form a catgory in th sns of catgory thory (Fokkinga (1994)). 27

28 Chaptr 2: Basic Concpts A production (or rul) in DPO is givn by thr graphs, L, K and R, and two injctiv morphisms, l and r. In figur 6 a DPO production is shown. Th lftmost graph, L, is th pattrn to match against th graph. Th conditions to b abl to apply th production ar givn blow. Th graph in th mdl, K, is th subgraph that is part of both L and R. It is calld th intrfac of th production. Th rightmost graph, R, stats th nods/dgs that will b addd to th graph whn applying th production. Th morphisms l and r ar injctiv and maps K to L and R rspctivly. L K R l 3 2 r Figur 6: Exampl of a DPO production. Givn two graph morphisms K B and K D a graph togthr with th morphisms B G and D G is calld a pushout of K B and K D if th following conditions ar mt: Commutativity: K B G = K D G Univrsal proprty: For all graphs G and graph morphisms B G and D G satisfying K B G = K D G thr is a uniqu graph morphism G G such that B G G = B G and D G G = D G This is illustratd in figur 7. 28

29 Chaptr 2: Basic Concpts B K G G D Figur 7: A pushout Illustration of a pushout of th morphisms K B and K D. Th production may b appl to a graph if thr xists a morphism g: L G such that th pushout G of l and d in figur 8 xists. From this pushout and th production it is always possibl to gt th rsulting graph, H, of applying th production. H is uniquly dfind, up to isomorphism, by th pushout H of r and d. On application lik this is calld a dirct drivation from G to H with rgard to a production p and a morphism m and is writtn (p, m): G H (or brifly p: G H). L l K r R m d m * G D l * r * H Figur 8: Doubl pushout drivation Two pushouts forming a dirct drivation from G to H with rgard to p and m. L, R and K ar th production from DPO. G is th start graph to which th production is appl and H is th rsulting graph aftr th application. In DPO a production can only b appl if it fulfills th gluing condition. Th gluing condition consists of two parts, th dangling dgs condition and th ntification condition. Th dangling dgs condition spcifis that if a nod is dltd by a 29

30 Chaptr 2: Basic Concpts production thn also all dgs to that nod hav to b xplicitly dltd by th production. Th ntification condition rquirs that vry objct to b dltd has only on pr-imag in L. Two dirct drivations, p 1 and p 2, hav th Church-Rossr proprty if thy ar both paralll indpndnt and squntial indpndnt. Two dirct drivations (p 1, m 1 ): G H 1 and (p 2, m 2 ): G H 2 ar paralll indpndnt if m 1 (L 1 ) m 2 (L 2 ) m 1 (l 1 (K 1 )) m 2 (l 2 (K 2 )). This mans that th two matchs ar not allowd to ovrlap, s figur 9, on thir lft hand ss. A two-stp drivation (p 1, m 1 ): G H 1, (p 2, m 2 ) : H 1 H 2 is squntial indpndnt iff m * 1 (R 1 ) m 2 (L 2 ) m * 1 (r 1 (K 1 )) m 2 (l 2 (K 2 )). This mans that th scond match is not allowd to ovrlap with rwrittn part of th first drivation, s figur 9. Squntial ovrlap L R L R Paralll ovrlap R L L R Figur 9: Ovrlap btwn ruls 2.7 Singl Pushout Singl pushout (SPO) is a rlaxation of doubl pushout. In SPO it is not ncssary to mt th gluing condition whn applying a production. 30

31 Chaptr 2: Basic Concpts History Singl pushout is a lss rstrictd vrsion of doubl pushout. It was according to Ehrig t al. (1990) first introducd by Raoult (1984) and furthr rfind by Knnaway (1987) Dfinitions and notations As statd abov th SPO approach dos not rquir th gluing condition to hold whn applying a rul. This is achivd by consring a pair of injctiv morphisms (lik l and r in DPO), l : K L, r : K R as a singl partial morphism r : L R and prforming dirct drivations as a singl pushout construction in th catgory of graphs and partial graph morphisms. A partial graph morphism r : L R is a morphism from som subgraph L r of L to R. A singl pushout production (or rul) is an injctiv partial graph morphism. Th application of a production r : L R to a graph G is givn by a total morphism m : L G. Th dirct drivation from G to H is givn by th pushout H of r and m as sn in figur 10. L r R m m * G r * H Figur 10: Singl pushout drivation A dirct drivation is formd by on pushout in th singl pushout approach. 31

32 Chaptr 3: Ruls of Rwriting Tchniqus 3 Ruls of Rwriting Tchniqus Th purpos of this chaptr is to choos a rwriting tchniqu to invstigat furthr in th contxt of MVARE. Th invrtibility of a singl rul is stud whn it is appl onc without intrfrnc from othr ruls. As an xampl a whil loop is rwrittn to a BASIC-lik languag. 3.1 Lambda Calculus and Combinatory Logic Lambda calculus and combinatory logic (s Boy t al. 1999) ar two vry low lvl but powrful rwriting tchniqus. It would b intrsting to invstigat th invrtibility of thos tchniqus, but not in th contxt of this work. Lambda calculus and combinatory logic ar du to thir low lvl syntax not wll suitd to asily b usd by a human spcifying complx rwritings. Th primitiv syntax is illustratd by th xampl lambda trm of figur 11. λxy.x(λuxy.x(uxy)(y(λuxy.x(uxy))(λxy.y)) Figur 11: Exampl of lambda calculus A lambda trm dfining addition on Church numrals from Boy t al. (1999). 3.2 Trm Rwriting Systms Trm rwrit systms (TRS), s sction 2.4 for dfinitions, is on of th most wll-known and also oftn usd rwriting tchniqus. This may b du to its straightforward syntax and smantics. Rul r 1 : 0 * x 0 Rul application: 0 * (3 * 4) R 0 Figur 12: Exampl of trm rwriting A trm rwriting rul r 1, and a dirct drivation. Whn it coms to invrtibility on rul lvl, trm rwriting ruls ar too powrful and nd to b rstrictd. Th rul r 1 of figur 12 has a variabl on th lft hand s, which is not prsnt in th 32

33 Chaptr 3: Ruls of Rwriting Tchniqus right hand s. This mans an unspcif subtr is rmovd. Th invrs rul thn has to rconstruct this subtr, without knowing what it lookd lik from th bginning, which is of cours impossibl. By rstricting trm rwrit ruls to rquir vry variabl to b prsnt ithr at both ss of a rul or non th problm is rmd Whil Figur 13 shows a trm rwrit rul for a whil statmnt construct transformd to a lowr lvl languag. Th invrs is obtaind simply by xchanging th lft hand s of th rul with th right hand s and vic vrsa. Sourc: whil(x > 0){ x := x - 1; } Rwriting rul: sl(whil(x, y), sl(z, w)) Targt: labl start; if(not x > 0) goto nd; x := x - 1; goto start; labl nd; sl(labl(start), sl(if(not(x), goto(nd)), sl(y, sl(goto(start), sl(labl(nd), sl(z,w)))))) Figur 13: Trm rwriting rul for whil An invrtibl trm rwrit rul, which dscribs a transformation of a whil statmnt to lowr lvl statmnts. Rwriting is supposd to tak plac on an abstract syntax tr rprsntation of th sourc and targt cod. sl is short for statmnt list. 3.3 Attribut Grammars with Forwarding Attribut Grammars with Forwarding (AGF), s sction 2.5, is usd for dfining abstract concpts in trm of lowr lvl constructs (.g dfining a whil-loop in trms of gotos and labls). This motivatd a closr look on th tchniqu. 33

34 Chaptr 3: Ruls of Rwriting Tchniqus In this sction, th modl and th viw ar consrd to b in a txtual rprsntation, parsd by grammars to an abstract syntax tr and thn prtty printd to txtual form again Whil Figur 14 shows th AGF production for whil. Th prttyprint attribut assignmnt is thr only to show how th prttyprint attribut of th forwardsto nod can b ovrloadd. Whn valuating othr (non ovrloadd) attributs for th whil-nod th forwardsto nod s attributs ar usd. 1. whil: Stmt 1 whil Cond 1 StmtList 1 2. Stmt 1.prttyPrint = gnwhilprint( 3. Cond 1.prttyPrint, 4. StmtList 1.prttyPrint) 5. forwardsto pars( 6. labl _start; 7. if not Cond 1 thn goto _nd; 8. StmtList 1 ; 9. goto _start; 10. labl _nd; Figur 14: AGF production of whil. Th forwardsto nod is cratd using th function pars, which taks a string as input and producs an AST. Th string argumnt of pars can also contain rfrncs to othr ASTs such as th ons from th right hand s of th production in th xampl. Forwarding is a nic tchniqu for rducing abstractions into primitiv constructs. Whn parsing, an AG production matchs only on on lvl of th abstract syntax tr, thrfor, in th invrs w can not match on grandchildrn for xampl. Figur 15 shows that an invrs of th whil xampl can not b constructd. 1. whil -1 : StmtList Labl If StmtList 1 Goto Labl 2. forwardsto pars( 3. whil Cond 1 StmtList 1 ; ) Figur 15: Inadquacy of AGF Bcaus th production only matchs in on lvl of th tr, w hav no accss to th if-statmnt s Cond-xprssion.Th rfrnc to Cond 1 is undfind. 34