Pre-nets, (read arcs) and unfolding: A functorial presentation Paolo Baldan (Venezia) Roberto Bruni (Pisa/Illinois) Ugo Montanari (Pisa) WADT Frauenchiemsee, Germany – 24/27 Sept Research supported by IST Project AGILE Italian MIUR Project COMETA CNR Fellowship on Information Sciences and Technologies

Roadmap Motivation P/T Petri Nets –Overall picture Processes / unfolding / algebraic approaches –Missing “tokens” Pre-Nets –Enlarged picture –A missing “token” (Read Arcs) Conclusions Ongoing Work!

Motivation P/T Petri nets (1962) –Basic model of concurrency –Widely used in different fields (graphical presentation, tools, …) –Enriched flavors (contexts, time, probability,…) Have 40 years been sufficient to completely understand P/T nets? –Many different semantics proposed over the years –Conceptual clarification advocated since the ’90s Techniques from category theory In the small/large, functoriality, universality The picture is still incomplete! –Limit of P/T nets, not of the applied techniques

P/T Petri Nets places transitions tokens a b c ts r

P/T Petri Nets places transitions tokens a b c ts r

P/T Petri Nets places transitions tokens a b c ts r

Processes Non-sequential behavior of P/T Petri nets –Causality and concurrency within a run of the net a b c ts r a b c s r t a t a

Unfolding All possible runs in a single structure –Causality (  ), concurrency (co), conflict (#) between events a b c ts r a b c t s r a t a r r #  co

Algebraic “Petri nets are monoids” –Algebra of (concurrent) computations via the lifting of the state structures to computations sequential composition “;” (of computations) plus identities (idle steps) plus parallel composition “  ” (from states) plus functoriality of  (concurrency) lead to a monoidal category of computations Collective Token Philosophy (CTPh) –T (_) (commutative processes) Individual Token Philosophy (ITPh) –P (_) (concatenable processes) –DP (_) (decorated concatenable processes) –Q (_) (strong concatenable processes)

(Part of) The ITPh Story So Far SafeOccPESDom Winskel’s chain of coreflections U (_) E (_) N (_) L (_) Pr (_) PTNetsDecOcc U (_) (_) + D (_) F (_) Sassone’s chain of adjunctions Petri * SsMonCat * PreOrd DP (N) (_) 

(Part of) The ITPh Story So Far Non functorial!Petri * SsMonCat * P (N) Objects: S  (commutative monoid) Arrows: Processes + ordering on minimal and maximal tokens in the same place –  a,b = id a  b if a  b a b t c d s e 2 f(a)=f(b)=d f(c)=e f(t)=s P (f)(  a,b )=  d,d  id 2d

(Part of) The ITPh Story So Far Suffers the same problem as P Petri * SsMonCat * DP (N) Objects: S  (strings, not multiset) Arrows: Processes + total ordering on minimal and maximal tokens –t:u  v implemented by {t p,q :p  q} p,q  S ,m(p)=u,m(q)=v Pseudo functorial Petri * SsMonCat * Q (_)

Pre-Nets Under the CTPh, the construction T (_) is completely satisfactory –T (_) is left adjoint to the forgetful functor from CMonCat  to Petri –T (_) can be conveniently expressed at the level of (suitable) theories (e.g. in PMEqtl) We argue that, under the ITPh, all the difficulties are due to the multiset view of states Pre-nets were proposed as the natural implementation of P/T nets under the ITPh –pre-sets and post-sets are strings, not multisets! –for each transition t: u  v, just one implementation t p,q : p  q is considered

Pre-Nets, Algebraically Under the ITPh, the construction Z (_) is completely satisfactory –Z (_) is left adjoint to the forgetful functor from SsMonCat  to PreNets –Z (_) can be conveniently expressed at the level of (suitable) theories (e.g. in PMEqtl) –All the pre-nets implementations R of the same P/T net N have the same semantics –Q (N) can be recovered from (any) Z (R) PreNetsSsMonCat  Z (_) G (_)

Pre-Net Processes? Deterministic Occurrence Pre-nets  –Finite conflict-free acyclic pre-net –(Like for P/T nets, but pre- and post-sets are strings of places) Processes of R –  :   R Concatenable processes –Total order on minimal and maximal places –Form a symmetric monoidal category PP (R) PP (R)  Z (R)

Pre-Net Unfolding? Non-Deterministic Occurrence Pre-nets  –Well-founded, finite-causes pre-net without forward conflicts –(Like for P/T nets, but pre- and post-sets are strings of places) Unfolding of R –Inductively defined non-deterministic occurrence pre-net U (R) PreNetsPreOcc U (_)

The Pre-Net Picture PreNetsPreOccPESDom U (_) E (_) ? L (_) Pr (_) PreOrd (_)  SsMonCat  Z (_) G (_) Functorial diagram reconciling all views Algebraic semantics via adjunction A missing link in the unfolding PTNets A (_) 

On The Missing Link The “most general” occurrence pre-net is hard to find … e1e1 e2e2  e1e1 e2e2 e1e1 e2e2

Read Arcs Read arcs model multiple concurrent accesses in reading to resources –t 1 and t 2 above can fire concurrently –(not possible if the situation is rendered with self- loops) Overall picture suffering of the same pathology as P/T nets… and more t1t1 t2t2

What Is There… Processes [Montanari, Rossi] Unfolding [Baldan, Corradini, Montanari] –Chain of coreflections Algebraic –Match-share categories –Non-free monoids of objects [Bruni, Sassone] CNets (semiweighted) OCNAESDom

What Is There… Processes [Montanari, Rossi] Unfolding [Baldan, Corradini, Montanari] –Chain of coreflections Algebraic –Match-share categories –Non-free monoids of objects [Bruni, Sassone] CNets (semiweighted) OCNAESDom

What Is There… Processes [Montanari, Rossi] Unfolding [Baldan, Corradini, Montanari] –Chain of coreflections Algebraic –Match-share categories –Non-free monoids of objects [Bruni, Sassone] CNets (semiweighted) OCNAESDom

What Is There… Processes [Montanari, Rossi] Unfolding [Baldan, Corradini, Montanari] –Chain of coreflections Algebraic –Match-share categories –Non-free monoids of objects [Bruni, Sassone] CNets (semiweighted) OCNAESDom

What Is There… Processes [Montanari, Rossi] Unfolding [Baldan, Corradini, Montanari] –Chain of coreflections Algebraic –Match-share categories –Non-free monoids of objects [Bruni, Sassone] CNets (semiweighted) OCNAESDom ==== … (atoms and electrons)

… And What Is Not Functoriality and universality of the algebraic approach? Reconciliation between the three views? Contextual pre-nets Algebraic (ok) –Based on (but slightly more complicated than) match-share categories Unfolding –Analogous to pre-nets (details to be worked out)

Conclusions Pre-nets are suitable for ITPh –A unique, straightforward algebraic construction –All views (algebraic, processes, unfolding) are satisfactorily reconciled To investigate: –From PES to PreOcc (hard) –Extension to read arcs (feasible) But also: –Algebraic approach for graphs (DPO / SPO) –Semantics of coloured / reconfigurable / dynamic nets

Pre-nets, read arcs and unfolding: A functorial presentation a paper byPaolo Baldan Roberto Bruni Ugo Montanari a WADT presentation byRoberto Bruni Research supported by IST Project AGILE Italian MIUR Project COMETA CNR Fellowship on Inf. Sci. and Techn. Electronic watercolor by Roberto Bruni