Pre-nets, (read arcs) and unfolding:

A functorial presentation

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

WADT 2002 - Frauenchiemsee, Germany – 24/27 Sept. 2002

Research supported by• IST-2001-32747 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

t s

r

P/T Petri Nets

places

transitions

tokens

a

b

c

t s

r

P/T Petri Nets

places

transitions

tokens

a

b

c

t s

r

Processes• Non-sequential behavior of P/T Petri nets

– Causality and concurrency within a run of the net

a

b

c

t s

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

t s

r

a

b

ct

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

Safe Occ PES Dom

Winskel’s chain of coreflections

U(_) E(_)

N(_)

L(_)

Pr(_)

PTNets DecOccU(_)

(_)+

D(_) F(_)

Sassone’s chain of adjunctions

Petri* SsMonCat* PreOrdDP(N) <u_>

(_)

(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 = idab if a b

a bt

c

d

s

e

2f(a)=f(b)=d f(c)=e

f(t)=s

P(f)(a,b)=d,d id2d

(Part of) The ITPh Story So Far

Suffers the same problem as PPetri* SsMonCat*

DP(N)

• Objects: S (strings, not multiset)• Arrows: Processes + total ordering on

minimal and maximal tokens– t:u v implemented by

{tp,q:p q}p,qS

,m(p)=u,m(q)=v

Pseudo functorialPetri* 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

tp,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)

PreNets SsMonCatZ(_)

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)

PreNets PreOccU(_)

The Pre-Net Picture

PreNets PreOcc PES DomU(_) E(_)

?

L(_)

Pr(_)

PreOrd<p_>

(_)

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 …

e1

e2

e1

e2

e1

e2

Read Arcs

• 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

t1 t2

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

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

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

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

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)

… 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 by Paolo BaldanRoberto Bruni

Ugo Montanari

a WADT presentation by Roberto Bruni

Research supported byIST-2001-32747 Project AGILEItalian MIUR Project COMETACNR Fellowship on Inf. Sci. and Techn.

Electronic watercolor by Roberto Bruni