partial order semantics of types of nets robert lorenz, gabriel juhás, sebastian mauser sofsem...
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Partial Order Semantics of Types of Nets
Robert Lorenz, Gabriel Juhás, Sebastian Mauser
SOFSEM 2009,Špindlerův Mlýn, CZ, 27th February 2009
Road plan:
Present types of (Petri) nets
Road plan:
Present types of (Petri) nets
Present semantics of types of (Petri) nets
Road plan:
Present types of (Petri) nets
Present semantics of types of (Petri) nets:
step semantics
enabled labelled partial orders (pomsets)
token flow labelled partial orders (pomsets)
To unify different types of Petri nets Badouel and Darondeau defined the notion formally
A type of (Petri) nets defined by them is given by
Set of local states – LS
Commutative monoid of local events (LE, +, 0)
Transition relation given by triples (s e s‘) from LS x LE x LS satisfying for all states s,s‘,s‘‘ and all local events e:
s e s‘ s e s‘‘ s‘ = s‘‘ and s0 s
To unify different types of Petri nets Badouel and Darondeau defined the notion formally
A type of (Petri) nets defined by them is given by
Set of local states – LS
Commutative monoid of local events (LE, +, 0)
Transition relation given by triples (s e s‘) from LS x LE x LS satisfying for all states s,s‘,s‘‘ and all local events e:
s e s‘ s e s‘‘ s‘ = s‘‘ and s0 s
A net of type is given by
Set of places – P
Set of transitions – T
Weight function – W: P x T LE
Places are holders for local states
Markings – m: P LS
Transitions occur, changing the local states of places according to the weight function:
A transition t is enabled to occur in a marking m of the net, if each place p satisfies:
there exists a local state sp for which
m(p) W(p,t) sp
For each place p occurence of the enabled transition t in m changes the local state m(p) to the state sp
22
2
a b
c
Example - Place/transition nets:
p1 p2
p3
p4
22
2
a b
c
p1 p2
p3
p4
Example - Place/transition nets:
22
2
a b
c
p1 p2
p3
p4
Example - Place/transition nets:
22
2
a b
c
Example - Place/transition nets:
p1 p2
p3
p4
Local states: nonnegative integers N (with addition)
Example - Place/transition nets:
Local events: pairs of nonnegative integers (i,o) N x N with componetwise addition
22
2
a b
c
p1 p2
p3
p4
Local states: nonnegative integers N (with addition)
Example - Place/transition nets:
Transition relation: s(i,o) s’ s i s’ = s – i + o 22
2
a b
c
p1 p2
p3
p4
Local events: pairs of nonnegative integers (i,o) N x N with componetwise addition
Local states: nonnegative integers N (with addition)
Example - Place/transition nets:
Weight function: W(p1,a) = (1,0),W(p2,a) = (0,0),W(p3,a) = (0,2),W(p4,a) = (0,0),
…
22
2
a b
c
p1 p2
p3
p4
Transition relation: s(i,o) s’ s i s’ = s – i + o
Local events: pairs of nonnegative integers (i,o) N x N with componetwise addition
Local states: nonnegative integers N (with addition)
Weight function: W(p1,a) = (1,0,),W(p2,a) = (0,0,),W(p3,a) = (0,2,),W(p4,a) = (0,0,0),
…
22
2
a b
c
p1 p2
p3
p4
Transition relation: s(i,o) s’ s i s’ = s – i + o s + o h
Local events: triples (i,o,h) N x N x (N ) with componetwise addition for i and o and minimum for h
and neutral element (0,0,)
Local states: nonnegative integers N (with addition)
Example - Place/transition nets with inhibitor arcs and
a posteriori semantics:
22
2
a b
c
p1 p2
p3
p4
Step semantics: step sequences
Steps – mulitsets of transitions occur, changing the local states of places according to the weight function:
A step x: T N is enabled to occur in a state of the net, if each place p satisfies:
there exists a local state sp‘ for which
m(p) sp‘
For each place p occurence of the enabled transition t changes the local state sp to
the state sp‘
tTx(t)W(p,t)
22
2
a b
c
p1 p2
p3
p4
Place/transition nets:
Step semantics: step sequences
22
2
a b
c
p1 p2
p3
p4
Place/transition nets:
Step semantics: step sequences
(a+b) a b
22
2
a b
c
p1 p2
p3
p4
Place/transition nets:
Step semantics: step sequences
(a+b)
2x
a b
22
2
a b
c
p1 p2
p3
p4
Place/transition nets:
Step semantics: step sequences
(a+b) (2c) a b
c c
An LPO is enabled iff
Each step linearization of the LPO is enabled step sequence,
a b
c c
22
2
a b
c
p1 p2
p3
p4
abcc, bacc, acbc, (a+b)cc, a(b+c)c, ab(2c), ba(2c),
(a+b)(2c)
For p/t nets we have:
If a step linearization of an enabled step sequence is enabled then they both lead to the same marking
(WISP), e.g.
ab(2c) and (a+b)(2c) leads to the same marking
22
2
a b
c
p1 p2
p3
p4
For p/t nets we have:
If a step linearization of an enabled step sequence is enabled then they both lead to the same marking
(WISP), e.g.
ab(2c) and (a+b)(2c) leads to the same marking
If two enabled step sequences have equal multisets of transitions, then they lead to the same marking
(PIP, PIP implies WISP), e.g.
a(b+c)c and (a+b)(2c) leads to the same marking22
2
a b
c
p1 p2
p3
p4
For p/t nets we have:
If a step linearization of an enabled step sequence is enabled then they both lead to the same marking
(WISP), e.g.
ab(2c) and (a+b)(2c) leads to the same marking
If two enabled step sequences have equal multisets of transitions, then they lead to the same marking
(PIP, PIP implies WISP), e.g.
a(b+c)c and (a+b)(2c) leads to the same marking
If a step sequence is enabled and leads to a marking m then each its step linearization is enabled and lead to
the same marking m
(ISP, ISP implies WISP but is incomparable with PIP), e.g.
(a+b)(2c) is enabled implies that ab(2c) is enabled too and leads to the same marking
22
2
a b
c
p1 p2
p3
p4
We show that:
WISP is necessary and sufficient for unique final marking of enabled LPOs
PIP is necessary and sufficient for the same final markings of enabled LPOs differing only by causality
ISP is necessary and sufficient for checking enabledness by cuts = maximal co-sets:
An LPO is enabled iff for each cut of the LPO holds:
When all events „before the cut“ have fired, then the step of transitions given by the cut is enabled
22
2
a b
c
p1 p2
p3
p4
22
2
a b
c
Place/transition nets:
Causal Structures: processes
p1 p2
p3
p4
p1 p2 p1 p2
Process 1 Process 2
22
2
a b
c
p1 p2
p3
p4
a a
p1 p2 p1 p2
Process 1 Process 2
p3 p3 p3p3
Place/transition nets:
Causal Structures: processes
22
2
a b
c
p1 p2
p3
p4
a a
p1 p2 p1 p2
p3 p3 p3 p3
Process 1 Process 2
Place/transition nets:
Causal Structures: processes
22
2
a b
c
p1 p2
p3
p4
a b a b
p1 p2 p1 p2
p3 p3 p3 p3 p3 p3 p3 p3
Process 1 Process 2
Place/transition nets:
Causal Structures: processes
22
2
a b
c
p1 p2
p3
p4
a b a b
p1 p2 p1 p2
p3 p3 p3 p3 p3 p3 p3 p3
Process 1 Process 2
Place/transition nets:
Causal Structures: processes
22
2
a b
c
p1 p2
p3
p4
a b
c
a b
c
p1 p2 p1 p2
p3 p3 p3 p3 p3 p3 p3 p3
p4 p4
Process 1 Process 2
Place/transition nets:
Causal Structures: processes
22
2
a b
c
p1 p2
p3
p4
a b
c
a b
c
p1 p2 p1 p2
p3 p3 p3 p3 p3 p3 p3 p3
p4 p4
Process 1 Process 2
Place/transition nets:
Causal Structures: processes
22
2
a b
c
p1 p2
p3
p4
a b
c c
a b
c c
p1 p2 p1 p2
p3 p3 p3 p3 p3 p3 p3 p3
p4 p4 p4 p4
Process 1 Process 2
Place/transition nets:
Causal Structures: processes
22
2
a b
c
abcc
p1 p2
p3
p4
a b
c c
a b
c c
p1 p2 p1 p2
p3 p3 p3 p3 p3 p3 p3 p3
p4 p4 p4 p4
Process 1 Process 2
a b
c c
Run 2
a b
c c
Run 1
acbc
Characterization of enabledness by extended LPOs with labelled arcs
for „flow“ types of nets
Flow types of nets:LS is a free commutative monoid
There are morphisms i and o from LE to LS satisfyingse s’ s i(e) s’ = s – i(e) + o(e)
Strict flow types of nets:se s’ s i(e) s’ = s – i(e) + o(e)
Examples for strict flow types of nets: p/t nets, coloured Petri netsExample for non strict flow types of nets: p/t nets with inhibitor arcs
a b
c c
Characterization of enabledness by extended LPOs with labelled arcs
Extension of LPOs by adding an initial node:
a b
c c
Characterization of enabledness by extended LPOs with labelled arcs
Extension of LPOs by adding an initial node:
a b
c c
v0
a b
c c
v0
Characterization of enabledness by extended LPOs with labelled arcs
Token flow function:
Labelling of arcs in the LPO by multisets of places
a b
c c
v0
a b
c c
v0
a b
c c
Characterization of enabledness by extended LPOs with labelled arcs
Token flow function:
Labelling of arcs in the LPO by multisets of places
v0
a b
c c
p1 p2
p3 p3 p3 p3
p4 p4
0
a b
c c
v0
a b
c c
Characterization of enabledness by extended LPOs with labelled arcs
Token flow function:
Labelling of arcs in the LPO by multisets of places
v0
a b
c c
p1 p2
p3 p3 p3 p3
p4 p4
1p1
a b
c c
v00
a b
c c
Characterization of enabledness by extended LPOs with labelled arcs
Token flow function:
Labelling of arcs in the LPO by multisets of places
v0
a b
c c
p1 p2
p3 p3 p3 p3
p4 p4
1p21p1
a b
c c
v00
a b
c c
Characterization of enabledness by extended LPOs with labelled arcs
Token flow function:
Labelling of arcs in the LPO by multisets of places
v0
a b
c c
p1 p2
p3 p3 p3 p3
p4 p4
a b
c c
v0
1p21p1
0 0
a b
c c
Characterization of enabledness by extended LPOs with labelled arcs
Token flow function:
Labelling of arcs in the LPO by multisets of places
v0
a b
c c
p1 p2
p3 p3 p3 p3
p4 p4
2p3
a b
c c
v0
1p21p1
0 0
Characterization of enabledness by extended LPOs with labelled arcs
Token flow function:
Labelling of arcs in the LPO by multisets of places
a b
c c
p1 p2
p3 p3 p3 p3
p4 p4
a b
c c
v0
a b
c c
v0
2p3
1p21p1
0 0
2p3
Characterization of enabledness by extended LPOs with labelled arcs
Token flow function:
Labelling of arcs in the LPO by multisets of places
a b
c c
p1 p2
p3 p3 p3 p3
p4 p4
a b
c c
v0
a b
c c
v0
2p3
1p21p1
0 0
2p3
Characterization of enabledness by extended LPOs with labelled arcs
Token flow function:
Labelling of arcs in the LPO by multisets of places
a b
c c
p1 p2
p3 p3 p3 p3
p4 p4
a b
c c
p1 p2
p3
p3 p3
p4 p4
p3
a b
c c
v0
a b
c c
v0
2p3
1p21p1
0 0
2p3
a b
c c
Characterization of enabledness by extended LPOs with labelled arcs
Token flow function:
Labelling of arcs in the LPO by multisets of places
v0
a b
c c
p1 p2
p3 p3 p3 p3
p4 p4
a b
c c
p1 p2
p3p3 p3
p4 p4
1p31p3
1p3
1p3
p3
a b
c c
v0
2p3
1p21p1
0 0
2p3
1p21p1
0 0
Characterization of enabledness by extended LPOs with labelled arcs
Token flow function:
Labelling of arcs in the LPO by multisets of places
a b
c c
p1 p2
p3 p3 p3 p3
p4 p4
a b
c c
p1 p2
p3p3 p3
p4 p4
p3
In-token-flow of a node =Sum of ingoing labels
a b
c c
v0
1p31p3
1p3
1p3
1p21p1
0 0
a b
c c
v0
2p3
1p21p1
0 0
2p3
Characterization of enabledness by extended LPOs with labelled arcs
Token flow function:
Labelling of arcs in the LPO by multisets of places
a b
c c
p1 p2
p3 p3 p3 p3
p4 p4
a b
c c
p1 p2
p3p3 p3
p4 p4
p3
Out-token-flow of a node =Sum of outgoing labels
a b
c c
v0
1p31p3
1p3
1p3
1p21p1
0 0
a b
c c
v0
2p3
1p21p1
0 0
2p3
Characterization of enabledness by extended LPOs with labelled arcs
Token flow property: There is a token flow function s.t.
Out-token-flow of a node the number of tokens produced by the transition labelling the node
a b
c c
p1 p2
p3p3 p3
p4 p4
p3
Out-token-flow of a node =Sum of outgoing labels
a b
c c
v0
1p31p3
1p3
1p3
1p21p1
0 0
22
2
a b
c
p1 p2
p3
p4
2
Characterization of executability by extended LPOs with labelled arcs
Token flow property: There is a token flow function s.t.
Out-token-flow of a node the number of tokens produced by the transition labelling the node
In-token-flow of a node = the number of tokens consumed by the transition labelling the node
a b
c c
p1 p2
p3p3 p3
p4 p4
p3
22
a b
c
p1 p2
p3
p4
a b
c c
v0
1p31p3
1p3
1p3
1p21p1
0 0
In-token-flow of a node =Sum of ingoing labels
2
Characterization of executability by extended LPOs with labelled arcs
Token flow property: There is a token flow function s.t.
Transitions cannot comsume more than is produced by the precessors
but still have to consume the number of tokens prescribed by the net
a b
c c
p1 p2
p3p3 p3
p4 p4
p3
22
a b
c
p1 p2
p3
p4
a b
c c
v0
1p31p3
1p3
1p3
1p21p1
0 0
In-token-flow of a node =Sum of ingoing labels
Characterization of enabledness by extended LPOs with labelled arcs
Theorem (for strict flow types of nets, e.g. p/t nets or finite coloured Petri nets):
An LPO is enabled if and only if it fulfills the token flow property
(token flow property can be verified in (quasi) polynomial time!!!)
Theorem (for „non strict“ flow types of nets):
If an LPO is enabled then it fulfills the token flow property,
i.e. token flow property is necessary for enabledness
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