Prenets, read arcs and unfolding: A functorial presentation

1
Pre-nets, (read arcs) and unfoldingA functorial
presentation
WADT 2002 - Frauenchiemsee, Germany 24/27
Sept. 2002

• Paolo Baldan (Venezia)
• Roberto Bruni (Pisa/Illinois)
• Ugo Montanari (Pisa)

• Research supported by
• IST-2001-32747 Project AGILE
• Italian MIUR Project COMETA
• CNR Fellowship on Information Sciences and
  Technologies

2
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!
3
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

4
P/T Petri Nets
b
places
a
transitions
t
s
tokens
c
r
5
P/T Petri Nets
b
places
a
transitions
t
s
tokens
c
r
6
P/T Petri Nets
b
places
a
transitions
t
s
tokens
c
r
7
Processes

• Non-sequential behavior of P/T Petri nets
• Causality and concurrency within a run of the net

a
b
b
a
t
s
c
r
8
Unfolding

• All possible runs in a single structure
• Causality (?), concurrency (co), conflict ()
  between events

b
a
s
t
b
c
r
a
t
s
a
r
c
t
r
r
a
9
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)

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(Part of) The ITPh Story So Far
U(_)
E(_)
L(_)
Safe
Occ
PES
Dom
N(_)
Pr(_)
Winskels chain of coreflections
11
(Part of) The ITPh Story So Far
P(N)
Non functorial!
Petri
SsMonCat

• Objects S? (commutative monoid)
• Arrows Processes ordering on minimal and
  maximal tokens in the same place
• ?a,b ida?b if a ? b

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(Part of) The ITPh Story So Far
DP(N)
Petri
SsMonCat
Suffers the same problem as P
Q(_)
Petri
SsMonCat
Pseudo functorial

• Objects S? (strings, not multiset)
• Arrows Processes total ordering on minimal and
  maximal tokens
• tu ? v implemented by
• tp,qp ? qp,q?S?,m(p)u,m(q)v

13
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 tp,q p ? q is considered

14
Pre-Nets, Algebraically
Z(_)
PreNets
SsMonCat?
G(_)

• 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)

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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)

16
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)

U(_)
PreNets
PreOcc
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The Pre-Net Picture

• Functorial diagram reconciling all views
• Algebraic semantics via adjunction
• A missing link in the unfolding

U(_)
E(_)
L(_)
PreNets
PreOcc
PES
Dom
?
Pr(_)
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On The Missing Link
e1
e1
?
e2
e2

• The most general occurrence pre-net is hard to
  find

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Read Arcs
t1
t2

• Read arcs model multiple concurrent accesses in
  reading to resources
• t1 and t2 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

20
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)
OCN
AES
Dom
21
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)
OCN
AES
Dom
22
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)
OCN
AES
Dom
23
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)
OCN
AES
Dom
24
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)
OCN
AES
Dom



(atoms and electrons)
25
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)

26
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

27

• Pre-nets, read arcs and unfolding
• A functorial presentation
• a paper by Paolo Baldan
• Roberto Bruni
• Ugo Montanari
• a WADT presentation by Roberto Bruni
• Research supported by
• IST-2001-32747 Project AGILE
• Italian MIUR Project COMETA
• CNR Fellowship on Inf. Sci. and Techn.
• Electronic watercolor by Roberto Bruni