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 emin 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
·
·
P1
T1
T2
P2
·
P3
P5
T3
P4
T4
P6
·
T1
P5
P3
P1
·
P3
T2
P2
T4
·
P1
P5
T3
P4
T4
P6
·
T1
P5
P6
T3
P3
P1
·
P3
T2
P2
T4
·
P1
P5
T3
P4
T4
P6
·
·
P1
T1
T2
P5
P4
T4
P6
P5
T3
P3
P4
T2
P1
P5
T3
P2
T4
·
P3
T4
P6
·
·
P1
T1
T2
P5
T4
P4
P6
T4
P6
P5
T3
P3
P4
P3
T2
P1
P5
T3
P2
T4
·
P3
P2
T1
·
·
P1
T1
T2
P5
T4
P4
T4
P6
P5
P6
P6
T3
P3
T3
P4
P3
T2
P1
P5
T3
P2
T4
·
P3
P2
T1
·
·
P1
T1
T2
P5
T4
P4
T4
P6
P5
P6
P6
T3
P3
P5
T3
P2
T4
·
P3
T3
P4
P3
T2
P1
P2
T1
P1
·
·
P1
T1
T2
P5
T4
P4
T4
P6
P5
P6
T4
P6
T3
P3
P5
T3
P2
T4
·
P3
P5
T3
P4
P3
P4
T2
P1
P2
T1
P1
·
·
P1
T1
T2
P5
T4
P4
T4
P6
P5
P6
T4
P6
T3
P3
P5
T3
P2
T4
·
P3
P5
T3
P4
P3
P4
T2
P1
P2
T1
P1
P6
The ERV unfolding algorithm
Unf  places from M0
pe  transitions enabled by M0
cut-off  
while pe  
extract emin 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
T6
P6
T5
P5
T4
P4
T3
P3
T2
P2
T1
P1
P7
T6
P6
T5
P5
T4
P4
T3
P3
T2
P2
T1 {P1}
P1
P7
T6
P6
T5
P5
T4
P4
T3
P3
T2
P2
T1 {P1}
P1
P7
T6
P6
T5
P5
T4
P4
T3 
P3
T2
P2
T1 {P1}
P1
P7
T6
P6
T5
P5
T4 {P5}
P4
T3 
P3
T2
P2
T1 {P1}
P1
P7
T6
P6
T5 {P5,P6}
P5
T4 {P5}
P4
T3 
P3
T2
P2
T1 {P1}
P1
P7
T6 {P5,P7}
P6
T5 {P5,P6}
P5
T4 {P5}
P4
T3 
P3
T2
P2
T1 {P1}
P1
Preset Trees
T1:{P1}
T3:
T4:{P5}
T5:{P6}
T5:{P5,P6}
T6:{P7}
T6:{P5,P7}
T1:{P1}
T4:{P5}
T3:
Weight = || + |{P1}| + |{P5}| + |{P6}| + |{P7}| = 4
Preset Trees
T1:{P1}
T3:
T4:{P5}
T5:{P5,P6}
T6:{P5,P7}
T1:{P1}
T4:{P5}
T5:{P5,P6} T6:{P5,P7}
T3:
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
P1
P4
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.
T2
P2
P3
T5
T1
T3
P1
P4
T4
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.
T2
P2
P3
T5
T1
T3
P1
P4
T4
T1:{P1,P2}
T2:{P2,P3}
T3:{P3,P4}
T4:{P1,P4}
T5:{P2,P4}
T1:{P1} T2:{P3} T5:{P4} T4:{P1}
{P2}
T3:{P3}
{P4}
{P1,P2,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}
{P4}
{P1,P2}
{P5}
{P1,P3,P4} {P1,P4} 
{P3}
{P4}
{P1,P2}
{P5}
{P1,P3,P4} {P1,P4} 
{P1,P4}
{P3}
{P4}
{P1,P2}
{P5}
{P3}
{P1,P4}

{P3}
{P4}
{P5}
{P1,P2}
{P3}
{P1,P4}
{P1}

{P3}
{P4}
{P5}
{P2}
{P3}
{P4}
{P1}

{P3}
{P4}
{P5}
{P2}
{P3}
{P4}
{P1}


{P3}
{P4}
{P5}
{P2}
{P3}
{P4}
{P1}

Building Preset Trees
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
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