dependability engineering & Petri nets May 2020 dependability engineering & Petri nets May 2020 QUALITATIVE ANALYSIS METHODS , OVERVIEW NET REDUCTION STRUCTURAL PROPERTIES LINEAR PROGRAMMING static STRUCTURAL analysis place / transition invariants PETRI NET state equation ANALYSIS trap equation REACHABILITY ANALYSIS (complete) reachability graph compressed state spaces BDDs, NDDs, ..., XDDs dynamic Kronecker products analysis reduced state spaces (model checking) coverability graph symmetry stubborn sets branching process Y:\Documents\teaching\course-concurrency\skript-sources\nl_skript_fm\nl09_structuralProperties.sld.fm 6 - 1 / 24 monika.heiner@b-tu.de 6 - 2 / 24
dependability engineering & Petri nets May 2020 dependability engineering & Petri nets May 2020 QUALITATIVE PROPERTIES BEHAVIOURAL NET PROPERTIES MARKABILITY of places behavioural properties markable (place liveness) k-bounded (safe) ❑ general semantic properties boundedness LIVENESS of transitions liveness zero times firing (m 0 -dead) reversibility finite times firing (dead, non-live) general infinite times (probably) firing (live) semantic ❑ special semantic properties properties safety properties infinite times (definitely) firing (livelock free) progress properties REACHABILITY of states dead states reproducibility structural properties reversibility (m 0 - home state) bad states (facts) ❑ especially valuable: local(ly decidable) structural properties; user-specified states special ❑ certain combinations of structural properties NET INVARIANTS semantic allow conclusions to transition (sub/sur) invariants properties behavioural properties; place (sub/sur) invariants temporal relationship of logic formulae monika.heiner@b-tu.de 6 - 3 / 24 monika.heiner@b-tu.de 6 - 4 / 24
dependability engineering & Petri nets May 2020 dependability engineering & Petri nets May 2020 P ETRI NET PROPERTIES , M ORE E XPENSIVE O VERVIEW / C HARLIE S TRUCTURAL P ROPERTIES 1. S IMPLE S TRUCTURAL P ROPERTIES 2. S TRUCTURAL P ROPERTIES pure (no side conditions) RKTH rank theorem PUR ordinary (1-multiplicity of all arcs) STP siphon trap property ORD homogeneous (all output arcs of a given place SMC state machine coverable HOM (covered with SM components) have the same multiplicity) non-blocking multiplicity (for each place applies: SMD state machine decomposable NBM (covered with SCSM components) MIN multiplicity of input arcs >= MAX multiplicity of output arcs) SMA state machine allocatable conservative (any firing preserves token amount) CSV CPI covered with place invariants SCF static conflict free CTI covered with transition invariants Ft0 every transition has a pre-place SCTI strongly covered by transition invariants tF0 every transition has a post-place SB structurally bounded FP0 every place has a pre-transition 3. B EHAVIOURAL P ROPERTIES pF0 every place has a post-transition CON connected k-B k-bounded SC strongly connected DCF dynamically conflict free marked graph (synchronization graph) dead states (a state where no transition is enabled) MG DSt dead transitions (at the initial state) SM state machine DTr FC free choice net LIV live reversible (the initial state m 0 can be reached EFC extended free choice net REV again from all reachable states: home state) ES extended simple net monika.heiner@b-tu.de 6 - 5 / 24 monika.heiner@b-tu.de 6 - 6 / 24
dependability engineering & Petri nets May 2020 dependability engineering & Petri nets May 2020 CONCLUSIONS (1) CONCLUSIONS (2) STRUCTURAL -> STRUCTURAL -> BEHAVIOURAL P ROPERTIES BEHAVIOURAL P ROPERTIES not Ft0 not tF0 ⇒ ❑ CSV BND ⇒ ❑ CPI BND p t p t ⇔ ❑ covered by (structural) BND Sub-P-invariants (yC <= 0) t live t not live OR ⇐ p unbounded p unbounded ❑ SC LIVE & BND ⇒ ( not SC not LIVE or not BND ) not pF0 not Fp0 ⇐ ❑ CTI LIVE & BND ⇒ ( not CTI not LIVE or not BND) ⇒ ( not CTI & BND not LIVE ) p p t t ⇐ ❑ CTI REV ⇒ ( not CTI not REV) t not live t not live OR p bounded p unbounded input nodes allow net reduction: conclude properties -> delete nodes -> conclude properties -> monika.heiner@b-tu.de 6 - 7 / 24 monika.heiner@b-tu.de 6 - 8 / 24
dependability engineering & Petri nets May 2020 dependability engineering & Petri nets May 2020 NET CLASSES , OVERVIEW RELATIONSHIP OF NET CLASSES allowed not allowed State Machines ES Marked Graphs EFC FC nets FC SM MG EFC nets ES nets MG - synchronization graph, T-nets, SM - S-nets monika.heiner@b-tu.de 6 - 9 / 24 monika.heiner@b-tu.de 6 - 10 / 24
dependability engineering & Petri nets May 2020 dependability engineering & Petri nets May 2020 NET CLASS : NET CLASS : STATE MACHINE MARKED GRAPH ❑ ❑ no forward/backward branching of transitions no forward/backward branching of places -> finite (state) automaton -> precedence graph p1 t1 ❑ ❑ no concurrency, no conflicts, p1 p2 t1 t2 but conflicts but concurrency -> DCF (persistent) p5 p2 t2 t5 ❑ any static conflict = dynamic conflict p3 p4 ❑ prototype of a persistent pn t4 t3 ❑ no production/consumption p3 t3 ❑ number of tokens on each circle of tokens is invariant -> conservative (CSV) -> P-invariant -> bounded (BND) t1 ❑ elementary circle p1 ❑ prototype of a bounded pn p1 p2 -> no node in the circle t1 t2 appears twice SM ⇒ [ SC & ’at least one token’ ❑ p5 t2 ⇔ LIVE & BND & REV ] t5 p2 MG ⇒ [ SC & ’each elementary ❑ p4 p3 circle contains a token’ t3 t4 SM ⇒ [ SC & ’exactly one token’ ❑ ⇔ LIVE & BND & REV ] ⇔ LIVE & SAFE & REV ] t3 p3 MG ⇒ [ SC & ’each elementary ❑ ❑ SM and MG circle contains exactly one token’ ⇔ LIVE & SAFE & REV ] -> duality monika.heiner@b-tu.de 6 - 11 / 24 monika.heiner@b-tu.de 6 - 12 / 24
dependability engineering & Petri nets May 2020 dependability engineering & Petri nets May 2020 NET CLASS : NET CLASS : FREE - CHOICE NET EXTENDED FREE - CHOICE NET ❑ every shared place is the only pre-place of its post-transitions ❑ transitions in conflict have the same set of pre-places ❑ free choice for any conflict resolution ❑ free choice for any conflict resolution ❑ conflict & concurrency EFC ⇒ [ STP ( & HOM & NBM ) ⇔ LIVE ] ❑ ❑ sc not sufficient any more p1 p1 t1 t2 t1 t2 t4 p3 t4 p2 p3 p2 FC1 FC2 t3 t3 no bounded marking no live marking (except empty marking) p4 FC3 FC4 p4 STP -> live not STP -> not live ❑ all theorems for FCN hold also for EFCN -> tranformation FCN <-> EFCN ❑ EFC ( & HOM & NBM) ⇒ reduction p p monotonicity of liveness u u expansion q q v v monika.heiner@b-tu.de 6 - 13 / 24 monika.heiner@b-tu.de 6 - 14 / 24
dependability engineering & Petri nets May 2020 dependability engineering & Petri nets May 2020 NET CLASS : S IPHON -T RAP -P ROPERTY EXTENDED SIMPLE NET (STP) Siphon D Trap Q FD ⊆ DF QF ⊆ FQ IF two places p, q share post-transitions ❑ THEN post-transitions of q are also q post-transitions of p, or vice versa p ENDIF D -> one of the two places may have Q other post-transitions ❑ transitive conflict relation # -> t1 # t2 and t2 # t3 -> t1 # t3 any transition any transition putting token into the set taking tokens from the set t1 t1 also takes token from it also puts token into it p t2 t2 a marked trap will an empty siphon will t3 t3 q never again be empty never again carry a token ES not ES SIPHON ❑ ES & STP ( & HOM & NBM ) ⇒ LIVE TRAP ❑ ES & LIVE ⇒ time-independently LIVE, but not necessarily monotonically live STP: each siphon contains a (sufficiently) marked trap (at m 0 ) monika.heiner@b-tu.de 6 - 15 / 24 monika.heiner@b-tu.de 6 - 16 / 24
dependability engineering & Petri nets May 2020 dependability engineering & Petri nets May 2020 STP, C ONCLUSIONS (3) S TRUCTURAL -> EXAMPLES B EHAVIOURAL P ROPERTIES : ❑ example1 p1 SM ⇒ [ SC & ’at least one token’ ❑ -> siphon1: {p1, p2} ⇔ LIVE & BND & REV ] F(p1, p2) = {t1, t3} t1 t2 (p1, p2)F = {t1, t2, t3) SM ⇒ [ SC & ’exactly one token’ ❑ p2 p3 ⇔ LIVE & SAVE & REV ] -> siphon2: {p1, p3} F(p1, p3) = {t2, t3} t3 FC1 MG ⇒ [ SC & ’each elementary circle (p1, p3)F = {t1, t2, t3} ❑ contains a token’ -> trap: {p1, p2, p3} ⇔ LIVE & BND & REV ] (p1, p2, p3)F = T F(p1, p2, p3) = T MG ⇒ [ SC & ’each elementary circle ❑ => not STP -> not live contains exactly one token’ ⇔ LIVE & SAFE & REV ] ❑ example2 t1 EFC ⇒ [ STP ( & HOM & NBM ) ⇔ LIVE ] ❑ -> siphon: {p1, p2, p3} F(p1, p2, p3) = T p1 p2 ES & STP ( & HOM & NBM ) ⇒ LIVE ❑ (p1, p2, p3)F = T -> trap: {p1, p2, p3} t2 t3 STP ( & HOM & NBM ) ⇒ not DSt ❑ (p1, p2, p3)F = T FC2 p3 F(p1, p2, p3) = T ORD & ’there are no siphons’ ⇒ LIVE ❑ => STP -> live note: not bounded ORD & SC & SMA ⇒ structural LIVE ❑ monika.heiner@b-tu.de 6 - 17 / 24 monika.heiner@b-tu.de 6 - 18 / 24
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