Characterisation of the State Spaces of Live and Bounded Marked - PowerPoint PPT Presentation

Characterisation of the State Spaces of Live and Bounded Marked Graph Petri Nets Eike Best and Raymond Devillers LATA’ 2014, Madrid – March 11, 2014

Analysis and synthesis of Petri nets • Analysis (Esparza et al.) Deduce behavioural properties of a Petri net • Synthesis (Rozenberg et al.) Derive a Petri net realising a labelled transition system A line of research (Darondeau et al.) Classes of Petri nets vs. classes of labelled transition systems This paper relates • marked graph Petri nets • to a subclass of labelled transition systems Marked graphs have been applied in manufacturing, in controller synthesis, and in asynchronous hardware design

A live and bounded marked graph M 0 a t b A marked graph Petri net and its initial marking M 0

A live and bounded marked graph M 0 b a t b after executing b

A live and bounded marked graph M 0 b a t b t after executing bt

A live and bounded marked graph M 0 a b a t b a t b a t b b a t b A marked graph Petri net a t b b a t and its reachability graph.. b a t b b a t b ..which has several nice properties: a t b a

It is finite M 0 a b a t b a t b a t b b a t b Finiteness a t b b ..due to the boundedness of the net a t b a t b b a t b a t b a

It is deterministic M 0 a b a t b a t b a t b b a t b Determinism If a state enables b and t , a t leading to different states, then b � = t b b a t b a .. true because the reachability graph t b b a comes from a Petri net t b a t b a

It is totally reachable M 0 a b a t b a t b a t b b a t b Total reachability Every state is re- a t achable from the initial state M 0 b b a t b a .. true by the definition of reachability t b b a graph t b a t b a

It is reversible M 0 a b a t b a t b a t b b a t b Reversibility The initial state is a t reachable from every reachable state b b a t b a .. true (for marked graphs) by t b b a liveness and boundedness t b a t b a

It is persistent M 0 a b a t b a t b a t b b a t b Persistency If a state enables b and t a t for b � = t , then it also enables bt and tb b b a t b a .. true by the marked graph property t b b a t b a also called strong confluence t b a

It is backward persistent M 0 a b a t b a t b a t b b a t b Backward persistency a t If a state backward enables b and t for b b a t b � = t , from two reachable states, then b a it also backward enables bt and tb t b b a t b .. true by the marked graph property a t b a

It satisfies the P1 property M 0 a b a t b a t b a t b b a t b The Parikh 1 property a t In a small cycle, every firable b b a t transition occurs exactly once b a t b b a .. true by the marked graph property t b a t Note: M 0 [ bbttaa � M 0 is not small b a

Properties of live and bounded marked graphs Definition A labelled transition system is nice if • it is finite • deterministic • totally reachable • reversible • persistent • backward persistent • and satisfies the P1 property of small cycles Theorem Commoner, Genrich, Holt, Even, Lautenbach, Pnueli (1968..) The reachability graph of a live and bounded marked graph Petri net is nice

Main result of this paper – A converse Theorem (LATA’ 2014) If a labelled transition system is nice, then it is the reachability graph of some live and bounded marked graph Petri net Moreover: There is a unique minimal marked graph realising it Moreover: Place bounds can be calculated from the lts Proof: Constructively

Reducing a labelled transition system TS Let x be a label and let TS - x be obtained from TS by erasing all x -labelled arrows TS - x TS 4 = r x 0 2 0 2 4 b b x a a d d a a a a b b x 5 1 3 = s x 1 3 5 Lemma: If TS is nice, then TS - x • is connected and acyclic • has a unique minimal element r x • has a unique maximal element s x

Computing a net from a labelled transition system TS The states of TS - x can be partitioned into • NE ( x ) = states not enabling x (including r x ) • EN ( x ) = states enabling x (including s x ) Lemma: Every maximal state in NE ( x ) equals s a , for some a � = x • Let the labels of TS be the transitions of the net • For a label x , pick a maximal state s = s a ∈ NE ( x ) • Create a place p with incoming transition a and outgoing transition x • Let the initial marking of p be the number of a ’s on a path from r x to s 0 Lemma: It doesn’t matter which path • Exhaustively perform this construction to obtain a net Theorem: This net is a live and bounded marked graph realising TS Moreover: it is side-condition-free, unique, and minimal Moreover: the bound of p is the number of a ’s on a path from s a to s x

A worked example with initial state s 0 = 0 TS TS - a TS - x 2 4 0 b x 0 0 b x b a d a a a d d b a a b b x 1 3 5 x Solution: TS - b TS - d 0 0 a b x b x a a d a a a a b x d x x States in EN ( x ) are drawn in gray Maximal states in NE ( x ) are drawn in red The places correspond to the red states

Necessity of niceness If all but one niceness properties are satisfied for some lts TS then no live and bounded marked graph has a reachability graph isomorphic to TS Also: The uniqueness of minimal solutions may fail

Niceness minus the P1 property The lts shown below satisfies all niceness properties except P1 c s 0 c 2 2 a c b a b a b c There are two different minimal non-marked graph solutions

Niceness minus backward persistency The lts shown below satisfies all niceness properties except backward persistency a p c a d d a s 0 2 b c b b b d a d a c b d There are two different minimal non-marked graph solutions

Concluding remarks Done: A characterisation of ‘nice’ labelled transition systems in terms of a structurally defined class of Petri nets Applications: E.g., a fast and direct synthesis algorithm Other possible uses: Help in proving open conjectures Extensions: Modify / relax lts properties or net classes E.g., what if backward persistency is omitted / relaxed? Answer: doesn’t work easily / we don’t know, respectively