Modal Open Petri Nets Vitali Schneider Walter Vogler

Modal Open Petri Nets Vitali Schneider Walter Vogler Universität Augsburg PN 2019

Concurrency Semantics / Refinements are • … numerous for Labelled Transition Systems (LTS) • … easily transferred to Petri nets labelled with actions a, … or invisible internal action ¿: a – m m’ : some a-labelled t fires from m to m’ a – m m’ : sequence with additional ¿ -transitions visible actions: interface for synchronous communication a a a || = a PN 2019 a 2

Compositional Stepwise Refinement • If N 1 refines N 2, then N 1 || N refines N 2 || N (precongruence) • Compositionality results can be inherited from LTS, e. g. weak bisimilarity as refinement: • A weak bisimulation is a relation R that relates markings of N 1 and N 2 with (m 01, m 02) 2 R and (m 1, m 2) 2 R: i) if m 1 ii) if m 2 a a m 1‘ then m 2‘ then m 1 a a m 2‘ with (m 1’, m 2’) 2 R m 1‘ with (m 1’, m 2’) 2 R • pretty strict equivalence: same actions, same choices boring as a refinement; better: PN 2019 3

Modal Transition Systems (MTS) -> MPN a • m m b may-transition a m‘, m a m‘ b m‘ must-transition a || a b || b a || a a b a • MPN-relation R: relates markings of N 1 and N 2 with (m 01, m 02) 2 R and (m 1, m 2) 2 R: i) if m 1 ii) if m 2 a b m 1‘ then m 2‘ then m 1 • N 1 modally refines N 2, a b m 2‘ with (m 1’, m 2’) 2 R m 1‘ with (m 1’, m 2’) 2 R N 1 v MPN N 2 • Compositionality results can be inherited from MTS. PN 2019 4

Asynchronous Communication • modelled with a Petri Net place as channel • So far: same story in – Elhog-Benzina, D. , Haddad, S. , Hennicker, R. Refinement and asynchronous composition of modal Petri nets (2012) – Haddad, S. , Hennicker, R. , Møller, M. H. : Specification of asynchronous component systems with modal I/O-Petri nets (2014) • here: different net representation & refinement PN 2019 5

Modal Open Nets (MON) • • Unlabelled (¿ -labelled) modal nets with input / output places as interface studied in a common framework with MPN: l. MON parallel composition b b d a c PN 2019 6

MON-Refinement – the Idea • canonical idea how to inherit from synchronous setting • from [LNCS 625, 1992], worked out in [Stahl. V, Acta Inf 2012] • Add actions that describe sending/receiving by the environment. b a a b N wrap(N) c c • modalities of the new transitions? PN 2019 must ! 7

MON-Refinement – Idea Applied b N wrap(N) a c • Def. : N 1 v l. MON N 2 if wrap(N 1 ) v MPN wrap(N 2 ) • Idea works also well for modalities / modal refinement, i. e. • using also compositionality of v MPN, we can prove: v l. MON is compositional • We prove in the l. MON -framework, that v l. MON on MON is just right to preserve modal refinement on MPN. PN 2019 8

MAP – Haddad. Hennicker Representation of MON b b+ HHreduction a b+ a- c c+ • MAP only exists if each t is attached to ≤ 1 interface place. • For MAP (special MPN ): N 1 v HH N 2 if N 1 v MPN N 2 v HH (unnecessarily) stricter than v l. MON • • refuses typical asynchronous behaviour equivalence PN 2019 9

MAP – Parallel Composition b+ b- b+ a- c+ d+ • parallel composition HH • ad hoc (particular) proof for compositionality of v HH • We give a conceptually easy proof from compositionality of v MPN ; works for all compositional refinements. PN 2019 10

Example of Equivalent Asynchronous Behaviour N 1 a b N 2 a b • Both, wrap(N 1) and wrap(N 2), can perform ab and ba. • HH-reduction: a+ b+ vs b+ a+ • After firing … in N 1, environment can see “a first” with inhibitor arc to b? • Artefact: place is good model for channel in standard Petri net, but channel is really “long” (sequence of places). PN 2019 11

Take Home Points Ø Modal Open Nets: - interface of places for asynchronous communication - may- and must-transitions Ø MON-refinement: – compositional – exactly „reflects“ modal refinement on MPN • less restrictive than MAP-refinement: – does not refute some intuitive equivalence of asynchronous behaviour PN 2019 12

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MON-Refinement is Just Right b b d c a MON Composition: MPN a observer l. MON v MPN v l. MON b Composition a PN 2019 14

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