Petri net analysis techniques Juli2000 dependability engineering & Petri nets Juli2000 BrandenburgTechnical Q UALITATIVE University at Cottbus, PROPERTIES : Computer Science Institute STRUCTURAL PROPERTIES P ETRI N ET ❑ especially valuable: A NALYSIS T ECHNIQUES local(ly decidable) structural properties; ❑ certain combinations of structural properties allow conclusions to behavioural properties; M ONIKA H EINER BEHAVIOURAL PROPERTIES mh@informatik.tu-cottbus.de ❑ general semantic properties boundedness http://www.informatik.tu-cottbus.de liveness reversibility ❑ special semantic properties safety properties progress properties D:\home\mh\docs\lv\pn\slides\pn_analysis.sld 1 / 20 mh@informatik.tu-cottbus.de 2 / 20
Petri net analysis techniques Juli2000 Petri net analysis techniques Juli2000 P ETRI N ET P ROPERTIES - O VERVIEW : 2. M ORE E XPENSIVE S TRUCTURAL P ROPERTIES DTP deadlock trap property 1. S IMPLE S TRUCTURAL P ROPERTIES SMC state machine coverable (covered with SM components) ORD ordinary (1-multiplicity of all arcs) SMD state machine decomposable HOM homogeneous (all output arcs of a given place (covered with SCSM components) have the same multiplicity) SMA state machine allocatable NBM non-blocking multiplicity (for each place applies: MIN multiplicity of input arcs >= CPI covered with place invariants MAX multiplicity of output arcs) CTI covered with transition invariants PUR pure (no side conditions) SB structurally bounded CSV conservative (any firing preserves token amount) SCF static conflict free 3. B EHAVIOURAL P ROPERTIES CON connected B bounded SC strongly connected REV reversible (the initial state m 0 can be reached Ft0 there is a transition without pre-place again from all reachable states: home state) tF0 there is a transition without post-place DSt dead states (a state where no transition is enabled) Fp0 there is a place without pre-transition BSt bad states (a state where a fact is enabled) pF0 there is a place without post-transition DTr dead transitions (at the initial state) MG marked graph (synchronization graph) DCF dynamically conflict free SM state machine L live FC free choice net LV live, excepted transitions dead at the initial marking (live, excepted implicit facts) EFC extended free choice net L&S live & safe (1-bounded) ES extended simple net mh@informatik.tu-cottbus.de 3 / 20 mh@informatik.tu-cottbus.de 4 / 20
Petri net analysis techniques Juli2000 Petri net analysis techniques Juli2000 S OFTWARE - ORIENTED INTERPRETATION B EHAVIOURAL NET PROPERTIES : OF NET PROPERTIES : MARKABILITY of places ❑ Dead code: markable (place liveness) statements which will never be executed; k-bounded (safe) pn: the corresponding transition never fires (dead at the initial marking); LIVENESS of transitions rg: transition does not appear at any edge; zero times firing (m 0 -dead) finite times firing (dead, non-live) general ❑ Total deadlock: semantic infinite times (probably) firing (live) system state from which there is no exit; properties infinite times (definitely) firing pn: dead marking; (livelock free) rg: final nodes (sheets); REACHABILITY of states dead states ❑ Partial deadlock : reproducibility not all parts of the system are available for all times; reversibility (m 0 - home state) pn: there are no dead markings, bad states (facts) but dead transition(s); user-specified states rg: not all final strongly connected components contain all transitions; special NET INVARIANTS semantic ❑ Well-structuredness: transition invariants properties all parts of the system may be executed for ever; place invariants pn: the net ist live; temporal relationship of logic formulae rg: all final strongly connected components contain all transitions; mh@informatik.tu-cottbus.de 5 / 20 mh@informatik.tu-cottbus.de 6 / 20
Petri net analysis techniques Juli2000 Petri net analysis techniques Juli2000 S OFTWARE - ORIENTED INTERPRETATION Q UALITATIVE ANALYSIS METHODS : OF NET PROPERTIES ( CONT .): NET REDUCTION ❑ Livelock : STRUCTURAL PROPERTIES parts of the system may be blocked for ever (due to the scheduler‘s strategy or something else not contai- ned in the model); LINEAR PROGRAMMING static pn: live, but not livelock-free; analysis place / transition invariants rg: not all circles contain all transitions; state equation trap equation ❑ Fault tolerance and self-synchronization : after a failure or from any abnormal state, the soft- REACHABILITY ANALYSIS ware will return to normal execution (recovery from failure) within finite time; (complete) reachability graph pn: reproducibility / reversibility; rg: from any state, the home state (initial state) compressed state spaces is reachable again; OBDDs, ONDDS dynamic Kronecker products analysis reduced state spaces (model checking) coverability graph symmetry stubborn sets sleep sets branching process mh@informatik.tu-cottbus.de 7 / 20 mh@informatik.tu-cottbus.de 8 / 20
Petri net analysis techniques Juli2000 Petri net analysis techniques Juli2000 S TATE E XPLOSION : N ET C LASSES : allowed not allowed EXAMPLE : p11 p21 pn1 State Machines Marked Graphs p12 p22 pn2 2 n n system components => system states (markings) FC nets GENERAL BEHAVIOUR : 50000 "ina.5.dat" "ina.5.dat" 45000 "ina.10.dat" "ina.10.dat" "ina.20.dat" 40000 "ina.20.dat" EFC nets 35000 30000 25000 20000 15000 10000 ES nets 5000 0 0 10000 20000 30000 40000 50000 60000 70000 80000 90000 100000 mh@informatik.tu-cottbus.de 9 / 20 mh@informatik.tu-cottbus.de 10 / 20
Petri net analysis techniques Juli2000 Petri net analysis techniques Juli2000 R ELATIONSHIP OF NET CLASSES : D EADLOCK -T RAP -P ROPERTY (DTP) Deadlock D Trap Q FD ⊆ DF QF ⊆ FQ ES D Q EFC FC ein leerer Deadlock kann ein markierter Trap wird nie wieder markiert werden nie wieder sauber; SM MG DEADLOCK TRAP DTP: Jeder Deadlock hat eine bei m 0 (ausreichend) markierte Falle. mh@informatik.tu-cottbus.de 11 / 20 mh@informatik.tu-cottbus.de 12 / 20
Petri net analysis techniques Juli2000 Petri net analysis techniques Juli2000 S TRUCTURAL P ROPERTIES : B INARY D ESICION D IAGRAMS , E XAMPLE 1: ( , , ) ( ∨ ) ∧ f x 1 x 2 x 3 = x 1 x 2 x 3 ❑ x1 x2 x3 f MG &SC & ’each elementary circle contains at least one token’ 0 0 0 0 ⇔ L & B 0 0 1 0 0 1 0 0 ❑ MG & SC & 0 1 1 1 ’each elementary circle contains exactly one token’ 1 0 0 0 ⇔ L & S 1 0 1 1 DESICION TABLE 1 1 0 0 ❑ SM & SC & ’at least one token’ 1 1 1 1 ⇔ L & B x1 DESISION TREE ❑ SM & SC & ’exactly one token’ ⇔ L & S x2 x2 ❑ EFC & DTP ( & HOM & NBM ) x3 x3 x3 x3 ⇔ L 0 0 0 1 0 1 0 1 ❑ ES & DTP ( & HOM & NBM ) ⇒ L x1 ROBDD OBDD x1 ❑ DTP ( & HOM & NBM) ⇒ not DSt x2 x2 x2 ❑ ORD & SC & SMA x3 x3 x3 ⇒ structural L 0 1 0 1 mh@informatik.tu-cottbus.de 13 / 20 mh@informatik.tu-cottbus.de 14 / 20
Petri net analysis techniques Juli2000 Petri net analysis techniques Juli2000 B INARY D ESICION D IAGRAMS , E XAMPLE 2: C OMPARISON RG - PREFIX : ( , , , , , ) ∧ ∨ ∧ ∨ ∧ f a 1 b 1 a 2 b 2 a 3 b 3 = a 1 b 1 a 2 b 2 a 3 b 3 PN RG PREFIX a a 1 a 1 x x x z b z 2 b b 1 BDD representations y y y of a single function a 2 for two different c c 3 variable orderings: b 2 z a 3 a b 3 a 1 a x y y a 1 z x z x y 0 1 1 c c a 2 a 2 2 c z a 3 a 3 a 3 a 3 a b 1 b 1 b 1 b 1 1 a c y2 x2 b 2 b 2 c a x1 y1 x1 y1 4 2 b 3 b d x2 y2 y1 x1 y1 x1 x2 y2 d b 0 1 y2 x2 a c 3 mh@informatik.tu-cottbus.de 15 / 20 mh@informatik.tu-cottbus.de 16 / 20
Petri net analysis techniques Juli2000 Petri net analysis techniques Juli2000 C ONCURRENT AUTOMATON , C ONCURRENT AUTOMATON : REDUCTION PRINCIPLE : Petri net reachability graph concurrent automaton ❑ combination of a reachability graph & finite prefix ...,a,... r ...,a,... b ❑ maintaining the reachability graph’s analysis power ...,b,... { r; s; t } -> deadlocks, liveness, livelock, home states s ...,c,... c ...,d,... ❑ separation of conflict <-> concurrency t ...,d,... -> data basis for evaluation of partial order properties d ❑ condensation of pure sequences and concurrencies a b c a,b,c -> larger state spaces manageable { r | s | t } r s t a,b,c ❑ restricted to 1-bounded nets d e f d,e,f s r t -> extension to k-bounded nets ? d,b,c a,b,f a,e,c s t r r t s d,e,c d,b,f a,e,f s t r d,e,f mh@informatik.tu-cottbus.de 17 / 20 mh@informatik.tu-cottbus.de 18 / 20
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