Sesqui-pushout rewriting

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1 Sesqui-pushout ewiting Andea Coadini Dipatimento di Infomatica, Pisa, Italy IFIP WG La Roche en Adennes, June 6, Joint wok with Tobias Heindel Fank Hemann Babaa König Univesität Stuttgat, Gemany Technisce Univesität Belin, Gemany Univesität Duisbug-Essen, Gemany IFIP WG La Roche en Adennes, June 6, p.1/25

2 Outline What and why? Some shallow motivations... Algebaic gaph ewiting: DPO and SPO Sesqui-pushout (SqPO) ewiting An example of SqPO ewiting at wok On the existence of pushback complements Relating SqPO with DPO and SPO Some deepe motivations and futue pespectives IFIP WG La Roche en Adennes, June 6, p.2/25

3 Some shallow motivations Sesqui-pushout ewiting is a new categoical definition of ewiting in an abitay categoy, simila to double-pushout o single-pushout ewiting The name (sesqui means one and a half in Latin) indicates that conceptually it lies between the SPO and the DPO Technically, it is defined as DPO ewiting, whee the left pushout is eplaced by a suitable pullback It looks moe adequate than DPO/SPO in some cases, and it enjoys seveal nice popeties that will be detailed late IFIP WG La Roche en Adennes, June 6, p.3/25

4 Double-pushout ewiting in C A ule is a span q = L α K β R A match is an aow m : L G Diect deivation A === m,q B if the following double-pushout diagam can be constucted: L α K β R m i c A γ Note: The left squae is built as a pushout complement, not chaacteized as a univesal constuction: geneal DPO is ambiguous D δ B IFIP WG La Roche en Adennes, June 6, p.4/25

5 Pushout complements in Gaph Example of ambiguity: IFIP WG La Roche en Adennes, June 6, p.5/25

6 Pushout complements in Gaph Example of ambiguity: fo injective top-aows, if the POC exists, it is unique; in this case, it exists iff the identification and dangling conditions ae satisfied. IFIP WG La Roche en Adennes, June 6, p.5/25

7 Quasi-adhesive categoies A quasi-adhesive categoy: has pullbacks, has pushouts along egula monos pushouts along egula monos ae Van Kampen squaes A m g C D f n B a A g A g C m f d D D m n C n c f B B b IFIP WG La Roche en Adennes, June 6, p.6/25

8 DPO theoy in quasi-adhesive cats Paallel and Sequential Independence Paallel Poductions and Deivations Local Chuch-Rosse and Paallelism Theoem Shift Equivalence and Canonical Deivations Concuency Theoem Embedding and extensions Citical pai lemma... IFIP WG La Roche en Adennes, June 6, p.7/25

9 POC in quasi-adhesive categoies Pushout complements along egula mono ae unique. Examples of quasi-adhesive categoies: Categoy of tem gaphs egula monos ae monos eflecting vaiables Categoy of simple gaphs egula monos ae monos eflecting edges If the top aow is mono but not egula, the POC might not be unique. IFIP WG La Roche en Adennes, June 6, p.8/25

10 Single-Pushout ewiting on gaphs Poductions ae patial mophisms Match: total mophism No dangling and identification conditions deletion in unknown context deletion stonge than pesevation m L dom(p) p R G H p q m n n* IFIP WG La Roche en Adennes, June 6, p.9/25

11 Defining sesqui-pushout ewiting A === m,q B if L α K β R the ight squae is a pushout m i c D, i, γ is a pushback complement of A m L α K that is, a final pullback complement of m: A γ D δ B the squae is a pullback fo each othe pullback (ove m) and fo each f : K K such that α f = α, thee exists a unique f : D D making eveything commute m L A α γ α γ K i D f bf K i D IFIP WG La Roche en Adennes, June 6, p.10/25

12 A few popeties of SqPO ewiting In any categoy C, pushback complements ae unique: SqPO ewiting is not ambiguous! even if L α K is not mono cloning even if C is quasi-adhesive, L α K is mono but not egula. If L α K is mono, then A γ D is mono, and D is the lagest subobject of A making the squae a pullback Pushback complement along monic, non-egula mophism in categoy of simple gaphs. IFIP WG La Roche en Adennes, June 6, p.11/25

13 An example: Access Contol Modeling basic opeations of simple Access Contol system fom [Haison, Ruzzo, Ullman, CACM 1976] Simple gaphs including nodes epesenting subjects ( ) and objects ( ), and labeled edges epesenting ights ( ). Aleady modeled by [Koch, Mancini, Paisi-Pesicce, ESORICS 00] using DPO on (multi-)gaphs, with Negative Application Conditions. Basic Opeations: ceate subject X s ceate object X o destoy subject X s destoy object X o ente i into(x s, X o ) delete i into(x s, X o ) IFIP WG La Roche en Adennes, June 6, p.12/25

14 Some ules and thei effect 1 1 α β m 1 1 w Application of destoy subject X s : deletion in unknown context as fo SPO IFIP WG La Roche en Adennes, June 6, p.13/25

15 Some ules and thei effect 1 1 α β m 1 1 w Application of destoy subject X s : deletion in unknown context as fo SPO w α β m cp staffnuse 1 mh 2 w cp staffnuse 1 mh 2 cp staffnuse 1 mh 2 Application of delete w into(x s, X o ): not ambiguous. IFIP WG La Roche en Adennes, June 6, p.13/25

16 A new ule: clone subject Non-left-injective ules model cloning 1 2 α β staffnuse 1 2 m cp w mh staffnuse 1 newnuse 2 cp w w mh staffnuse 1 newnuse 2 w cp w mh IFIP WG La Roche en Adennes, June 6, p.14/25

17 A new ule: clone subject Non-left-injective ules model cloning 1 2 α β staffnuse 1 2 m cp w mh staffnuse 1 newnuse 2 cp w w mh staffnuse 1 newnuse 2 In Gaph, the pushback complement might not be a POC. cp w w mh Pushout complements Pushback complement IFIP WG La Roche en Adennes, June 6, p.14/25

18 n the existence of pushback complement In categoies like Set, Gaph, gaph stuctues in geneal: IFIP WG La Roche en Adennes, June 6, p.15/25

19 n the existence of pushback complement In categoies like Set, Gaph, gaph stuctues in geneal: α fo injective L K the pushback complement exists iff the match is conflict-fee, i.e., m(l \ K) m(k) =. { 1, 2 } m { 1 } { 0 } X IFIP WG La Roche en Adennes, June 6, p.15/25

20 n the existence of pushback complement In categoies like Set, Gaph, gaph stuctues in geneal: α fo injective L K the pushback complement exists iff the match is conflict-fee, i.e., m(l \ K) m(k) =. { 1, 2 } m { 1 } { 0 } X fo abitay L α K and injective matches: see Constuction 6 in the pape... IFIP WG La Roche en Adennes, June 6, p.15/25

21 n the existence of pushback complement In categoies like Set, Gaph, gaph stuctues in geneal: α fo injective L K the pushback complement exists iff the match is conflict-fee, i.e., m(l \ K) m(k) =. { 1, 2 } m { 1 } { 0 } X fo abitay L α K and injective matches: see Constuction 6 in the pape... fo abitay L α K and abitay matches: IFIP WG La Roche en Adennes, June 6, p.15/25

22 n the existence of pushback complement In categoies like Set, Gaph, gaph stuctues in geneal: α fo injective L K the pushback complement exists iff the match is conflict-fee, i.e., m(l \ K) m(k) =. { 1, 2 } m { 1 } { 0 } X fo abitay L α K and injective matches: see Constuction 6 in the pape... fo abitay L α K and abitay matches: conditions... athe involved...;... beyond the scope of the pape...; the inteested eade is encouaged to specialize the concepts that ae availabe fo evey topos [Goldblatt] and the esults in next section...;... the pushback constuction cannot be pefomed IFIP WG La Roche en Adennes, June 6, p.15/25 componentwise...

23 Existence in an abitay categoy C Given L m A conside the pullback functo m : C A C L along m. If its ight adjoint Π m : C L C A exists patially at α, it povides a pullback complement iff the co-unit ε α is an iso. m L A α m (δ) ε α Π m (α) K f m ( ) bf m (D) This povides a constuction of pushback complements in categoies whee the pullback functos have ight adjoints (like locally catesian closed cats). δ D m ( b f) IFIP WG La Roche en Adennes, June 6, p.16/25

24 squi-pushout vs double-pushout ewitin In quasi-adhesive categoies, fo a left-egula ule q and a match m, if the POC exists, then it is a PshBC. Thus 1. If A === m,q DPO B then also A === m,q B. 2. If A === m,q B and a pushout complement exists, then also A === m,q DPO B. IFIP WG La Roche en Adennes, June 6, p.17/25

25 squi-pushout vs double-pushout ewitin In quasi-adhesive categoies, fo a left-egula ule q and a match m, if the POC exists, then it is a PshBC. Thus 1. If A === m,q DPO B then also A === m,q B. 2. If A === m,q B and a pushout complement exists, then also A === m,q DPO B. Fo left-monic but non-left-egula ules, in some examples the PshBC is a POC. It is open if this is a geneal popety. IFIP WG La Roche en Adennes, June 6, p.17/25

26 squi-pushout vs double-pushout ewitin In quasi-adhesive categoies, fo a left-egula ule q and a match m, if the POC exists, then it is a PshBC. Thus 1. If A === m,q DPO B then also A === m,q B. 2. If A === m,q B and a pushout complement exists, then also A === m,q DPO B. Fo left-monic but non-left-egula ules, in some examples the PshBC is a POC. It is open if this is a geneal popety. Fo non-left-monic ules, the PshBC is not a POC, in geneal. IFIP WG La Roche en Adennes, June 6, p.17/25

27 esqui-pushout vs single-pushout ewiting In categoies of gaph stuctues, given a patial mophism q (seen also as left-injective span) and a match m, 1. If A === m,q B then A === m,q SPO B. 2. If A === m,q SPO B and m is conflict-fee then A === m,q B. Note that usually non-conflict-fee matches ae uled out in pactical uses o theoetical developments of the SPO theoy, esticting to d-injective o even to injective matches. IFIP WG La Roche en Adennes, June 6, p.18/25

28 Theoy of paallelism Some fist esults of the DPO/SPO theoy have been ecast fo SqPO ewiting Paallel Independence β 1 R 1 K 1 n 1 k 1 v α 1 α 2 L1 L 2 K 2 m 1 m 2 u k 2 β 2 R 2 n 2 H 1 D 1 δ 1 γ 1 G D 2 γ 2 δ 2 H 2 Local Chuch-Rosse Theoem G m 1,p 1 m 2,p 2 Given paallel independent G m 1,p 1 ====== H 1 and H 1 H 2 G m 2,p 2 ====== H 2, thee ae an object G and diect deivations H 1 m 2,p 2 ====== G and H 2 m 1,p 1 ====== G. m 2,p 2 G m 1,p 1 IFIP WG La Roche en Adennes, June 6, p.19/25

29 Back to motivations... Semantics of concuency fo GTS Rewiting of gaphs is intinsically concuent Peti nets ae a efeence model fo concuency (Place/Tansition) Peti nets can be seen as a degeneate case of Gaph Tansfomation A obust semantics of concuency fo GT should specialize to known semantics fo nets IFIP WG La Roche en Adennes, June 6, p.20/25

30 Some contibutions to the field We defined genealizations to othe classes of nets and to DPO o SPO ewiting in Gaph of deteministic and non-deteministic pocesses unfolding and event stuctue semantics functoial (coeflective) semantics IFIP WG La Roche en Adennes, June 6, p.21/25

31 Winskel s style semantics fo GTS Safe Nets U Occuence Nets N E Pime Event Stuctues P L Domains IFIP WG La Roche en Adennes, June 6, p.22/25

32 Winskel s style semantics fo GTS Safe Nets U Occuence Nets N E Pime Event Stuctues P L Domains [Woks with Paolo Baldan, Ugo Montanai, Leila Ribeio] DPO Gaph Gammas DPO Occ. Gammas Inhibito Event Stuctues Domains IFIP WG La Roche en Adennes, June 6, p.22/25

33 Winskel s style semantics fo GTS Safe Nets U Occuence Nets N E Pime Event Stuctues P L Domains [Woks with Paolo Baldan, Ugo Montanai, Leila Ribeio] DPO Gaph Gammas DPO Occ. Gammas Inhibito Event Stuctues Domains SPO Gaph Gammas SPO Occ. Gammas Asymmetic Event Stuctues Domains IFIP WG La Roche en Adennes, June 6, p.22/25

34 The next steps, quite obviously......genealizing the semantics developed fo concete models to ewiting systems in adhesive categoies... IFIP WG La Roche en Adennes, June 6, p.23/25

35 The next steps, quite obviously......genealizing the semantics developed fo concete models to ewiting systems in adhesive categoies... Fist esults: genealization of pocesses [with Paolo Baldan, Tobias Heindel, Babaa König, FoSSaCS 06] IFIP WG La Roche en Adennes, June 6, p.23/25

36 The next steps, quite obviously......genealizing the semantics developed fo concete models to ewiting systems in adhesive categoies... Fist esults: genealization of pocesses [with Paolo Baldan, Tobias Heindel, Babaa König, FoSSaCS 06] Befoe moving to unfolding semantics, we noted that: fo DPO ewiting, a coeflective semantics is impossible; SPO ewiting moe appealing, but genealization to abitay categoies is quite involved; no consensus on the way conflicts ae esolved; thus, need fo a notion of ewiting simila to DPO, but without application conditions IFIP WG La Roche en Adennes, June 6, p.23/25

37 Conclusions... I pesented the definition and a few popeties of Sesqui-pushout ewiting, elating it to DPO and SPO ewiting It is not-ambiguous, allows to model cloning, and coincides with DPO and SPO unde suitable assumptions Some basic esults about paallelism have been lifted to SqPO ewiting: a lot has to be done, still. Seveal esults of DPO/SPO theoy should lift easily. IFIP WG La Roche en Adennes, June 6, p.24/25

38 ... and Futue Wok The expessiveness of the appoach should be compaed with that of DPO/SPO ewiting on pactical case studies Genealizing the coeflective semantics of SPO ewiting to that if SqPO ewiting in quasi-adhesive theoies IFIP WG La Roche en Adennes, June 6, p.25/25

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