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Towards a n Efficient Algorithm for Unfolding Petri Nets. Victor Khomenko and Maciej Koutny Department of Computing Science University of Newcastle upon Tyne. Motivation. Partial order semantics of Petri nets Alleviate the state space explosion problem Efficient model checking algorithms. - PowerPoint PPT Presentation
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Towards an Efficient Algorithm for Unfolding Petri Nets
Victor Khomenko and Maciej Koutny
Department of Computing Science
University of Newcastle upon Tyne
Motivation
• Partial order semantics of Petri nets
• Alleviate the state space explosion problem
• Efficient model checking algorithms
The ERV unfolding algorithm
Unf places from M0
pe transitions enabled by M0
cut-off
while pe
extract emin pe
if e is a cut-off event
then cut-off cut-off {e}
else
add e and its postset into Unf
UpdatePotExt(pe,Unf,e)
add cut-off events and their postsets into Unf
·T1
·T2
·T3 T4
P1 P3 P5
P2 P4 P6
·T1
·T2
·T3 T4
P1 P3 P5
P2 P4 P6
T4
P1
P3
P5
·T1
·T2
·T3 T4
P1 P3 P5
P2 P4 P6
T4
T3
P1
P3
P5P6
·T1
·T2
·T3 T4
P1 P3 P5
P2 P4 P6
T4
T3
T4
T2
P1
P3
P5P6
P4
P5
·T1
·T2
·T3 T4
P1 P3 P5
P2 P4 P6
T4
T3
T4
T2
T1P1
P3
P5P6
P4
P5
P2
P3
·T1
·T2
·T3 T4
P1 P3 P5
P2 P4 P6
T4
T3
T4
T2
T3
T1P1
P3
P5P6
P4
P5
P2
P3
P6
·T1
·T2
·T3 T4
P1 P3 P5
P2 P4 P6
T4
T3
T4
T2
T3
T1P1
P3
P5P6
P4
P5
P2
P3
P6
P1
·T1
·T2
·T3 T4
P1 P3 P5
P2 P4 P6
T4
T3
T4
T2
T3
T1
T4
P1
P3
P5P6
P4
P5
P2
P3
P6
P1
P4
P5
·T1
·T2
·T3 T4
P1 P3 P5
P2 P4 P6
T4
T3
T4
T2
T3
T1
T4
P1
P3
P5P6
P4
P5
P2
P3
P6
P1
P4
P5 P6
The ERV unfolding algorithm
Unf places from M0
pe transitions enabled by M0
cut-off
while pe
extract emin pe
if e is a cut-off event
then cut-off cut-off {e}
else
add e and its postset into Unf
UpdatePotExt(pe,Unf,e)
add cut-off events and their postsets into Unf
P7
P1
P2
P3
P4
P5
P6
T6
T5
T4
T3
T2
T1
P7
P1
P2
P3
P4
P5
P6
T6
T5
T4
T3
T2
T1 {P1}
P7
P1
P2
P3
P4
P5
P6
T6
T5
T4
T3
T2
T1 {P1}
P7
P1
P2
P3
P4
P5
P6
T6
T5
T4
T3
T2
T1
{P1}
P7
P1
P2
P3
P4
P5
P6
T6
T5
T4
T3
T2
T1
{P1}
{P5}
P7
P1
P2
P3
P4
P5
P6
T6
T5
T4
T3
T2
T1
{P1}
{P5}
{P5,P6}
P7
P1
P2
P3
P4
P5
P6
T6
T5
T4
T3
T2
T1
{P1}
{P5}
{P5,P6}
{P5,P7}
T3:
T1:{P1}
T4:{P5}
T5:{P5,P6}
T6:{P5,P7}
Preset Trees
T3:
T1:{P1} T4:{P5}
T5:{P6} T6:{P7}
Weight = || + |{P1}| + |{P5}| + |{P6}| + |{P7}| = 4
T3:
T1:{P1}
T4:{P5}
T5:{P5,P6}
T6:{P5,P7}
Preset Trees
T3:
T1:{P1} T4:{P5} T5:{P5,P6} T6:{P5,P7}
Weight = || + |{P1}| + |{P5}| + |{P5,P6}| + |{P5,P7}| = 6
Proposition (P.Rossmanith). Building a minimal-weight preset tree is an NP-complete problem in the size of a Petri net, even if all transition presets have the size 3.
Proposition (P.Rossmanith). Building a minimal-weight preset tree is an NP-complete problem in the size of a Petri net, even if all transition presets have the size 3.
Proposition (P.Rossmanith). Building a minimal-weight preset tree is an NP-complete problem in the size of a Petri net, even if all transition presets have the size 3.
P2 P3
P4P1
Proposition (P.Rossmanith). Building a minimal-weight preset tree is an NP-complete problem in the size of a Petri net, even if all transition presets have the size 3.
P2 P3
P4P1
T2
T3T1
T4
T5
Proposition (P.Rossmanith). Building a minimal-weight preset tree is an NP-complete problem in the size of a Petri net, even if all transition presets have the size 3.
P2 P3
P4P1
T2
T3T1
T4
T5
T2:{P2,P3}
T1:{P1,P2}
T3:{P3,P4}
T4:{P1,P4}
T5:{P2,P4}
T1:{P1} T5:{P4}T2:{P3} T4:{P1}
{P2} {P4}
T3:{P3}
{P1,P2,P4} {P1,P2,P5} {P1,P3,P4} {P1,P4}{P1,P2,P3}
{P1,P2,P4} {P1,P2,P5} {P1,P3,P4} {P1,P4}{P1,P2,P3}
{P1,P2}
{P4} {P5} {P1,P3,P4} {P1,P4}{P3}
{P1,P2}
{P4} {P5} {P1,P3,P4} {P1,P4}{P3}
{P1,P2} {P1,P4}
{P4} {P5} {P3}{P3}
{P1,P2} {P1,P4}
{P4} {P5} {P3}{P3}
{P1,P2} {P1,P4}
{P1}
{P4} {P5} {P3}{P3}
{P2} {P4}
{P1}
{P4} {P5} {P3}{P3}
{P2} {P4}
{P1}
{P4} {P5} {P3}{P3}
{P2} {P4}
{P1}
function BuildTree({A1, ..., Ak})
TS {Tree(A1, ),…,Tree(Ak, )}
while |TS|>1
choose Tree(A', · ) TS and Tree(A'', · ) TS
such that A‘ A'' and |A'A''| is maximal
I A'A'‘
T {Tree(B \ I, ts) | Tree(B, ts) TS and I B}
T= {ts | Tree(I,ts) TS and ts }
TS TS\{Tree(B, ·) TS | I B}
TS TS {Tree(I, T T=)}
/* |TS|=1 */
return the remaining tree Tr TS
Building Preset Trees
Experimental Results
• Small presets (2-3 places): – improvements for some examples
• Medium-size presets (5-6 places):– improvements for all examples
• Large presets (7 and more places):– significant improvements for all examples
