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On the Importance of the
Deadlock Trap Property for Monotonic Liveness
Monika Heiner1, Cristian Mahulea2, Manuel Silva2
1Department of Computer Science, Brandenburg University of Technology
Postbox 10 13 44, 03013 Cottbus, Germany
monika.heiner@tu-cottbus.de
2Instituto de Investigaci´on en Ingenier´ıa de Arag´on (I3A),
Universidad de Zaragoza, Maria de Luna 1, E-50018 Zaragoza, Spain
cmahulea@unizar.es, silva@unizar.es
Abstract. In Petri net systems, liveness is an important property cap-
turing the idea of no transition (action) becoming non-fireable (unattain-
able). Additionally, in some situations it is particularly interesting to
check if the net system is (marking) monotonically live, i.e., it remains
live for any marking greater than the initial one. In this paper, we dis-
cuss structural conditions preserving liveness under arbitrary marking
increase. It is proved that the deadlock trap property (DTP) is a neces-
sary condition for liveness monotonicity of ordinary nets, and necessary
and sufficient for some subclasses. We illustrate also how the result can
be used to study liveness monotonicity for non-ordinary nets using a sim-
ulation preserving the firing language. Finally, we apply these conditions
to several case studies of biomolecular networks.
1 Motivation
Petri nets are a natural choice to represent biomolecular networks. Various types
of Petri nets may be useful – qualitative, deterministically timed, stochastic, con-
tinuous or hybrid ones, depending on the available information and the kind of
properties to be analysed. Accordingly, the integrative framework demonstrated
by several case studies in [GHR+08], [HGD08], [HDG10] applies a family of re-
lated Petri net models, sharing structure, but differing in their kind of kinetic
information.
A key notion of the promoted strategy of biomodel engineering is the level
concept, which has been introduced in the Petri net framework in [GHL07]. Here,
a token stands for a specific amount of mass, defined by the total mass divided
by the number of levels. Thus, increasing the token number to represent a certain
amount of mass means to increase the resolution of accuracy.
This procedure silently assumes some kind of behaviour preservation while
the marking is increased (typically multiplied by a factor) to represent a finer
granularity of the mass flowing through the network. However, as it is well-known
in Petri net theory, liveness is not monotonic with respect to (w.r.t.) the initial
marking for general Petri nets. Thus, there is no reason to generally assume that
Int. Workshop on Biological Processes & Petri Nets (BioPPN),
satellite event of Petri Nets 2010, Braga, Portugal, pages 39-54, June 2010.
2 M. Heiner, C. Mahulea, and M. Silva
there is no significant change in the possible behaviour by marking increase.
Contrary, under liveness monotonicity w.r.t. the initial marking we can expect
continuization (fluidization) to be reasonable. However, only a particular kind
of monotonicity seems to be needed for continuization: homothetic liveness, i.e.,
liveness preservation while multiplying the initial marking by k[RTS99], [SR02].
At structural level, (monotonic) liveness can be considered using transforma-
tion (reduction) rules [Ber86], [Sil85], [Mur89], [Sta90], the classical analysis for
ordinary nets based on the Deadlock Trap Property (DTP) [Mur89], [Sta90], or
the results of Rank Theorems, which are directly applicable to non-ordinary nets
[TS96], [RTS98]. In this paper, we concentrate on the DTP, which will initially
be used for ordinary net models, and later extended to non-ordinary ones.
This paper is organized as follows. We start off with recalling relevant notions
and results of Petri net theory. Afterwards we introduce the considered subject
by looking briefly at two examples, before turning to our main result yielding
a necessary condition for monotonic liveness. We demonstrate the usefulness of
our results for the analysis of biomolecular networks by a variety of case studies.
We conclude with an outlook on open issues.
2 Preliminaries
We assume basic knowledge of the standard notions of place/transition Petri
nets, see e.g. [DHP+93], [HGD08], [DA10]. To be self-contained we recall the
fundamental notions relevant for our paper.
Definition 1 (Petri net, syntax).
APetri net is a tuple N=hP, T , P re,P osti, and a Petri net system is a
tuple Σ=hN ,m0i, where
–Pand Tare finite, non-empty, and disjoint sets. Pis the set of places.T
is the set of transitions.
–P re,P ost ∈N|P|×|T|are the pre- and post-matrices, where |·| is the car-
dinality of a set, i.e., its number of elements. For a place pi∈Pand a
transition tj∈T,P re(pi, tj)is the weight of the arc connecting pito tj(0
if there is no arc), while P ost(pi, tj)is the weight of the arc connecting tjto
pi.
–m0∈N|P|
≥0gives the initial marking.
–m(p)yields the number of tokens on place pin the marking m. A place
pwith m(p)=0is called empty (unmarked) in m, otherwise it is called
marked (non-empty). A set of places is called empty if all its places are
empty, otherwise marked.
–The preset and postset of a node x∈P∪Tare denoted by •xand x•.
They represent the input and output transitions of a place x, or the input
and output places of a transition x. More specifically, if tj∈T,•tj={pi∈
P|P re(pi, tj)>0}and tj•={pi∈P|P ost(pi, tj)>0}. Similarly, if pi∈P,
•pi={tj∈T|P ost(pi, tj)>0}and pi•={tj∈T|P re(pi, tj)>0}.
We extend both notions to a set of nodes X⊆P∪Tand define the set of
all prenodes •X:= Sx∈X
•x, and the set of all postnodes X•:= Sx∈Xx•.
DTP and Monotonic Liveness 3
–A node x∈P∪Tis called source node, if •x=∅, and sink node if x•=∅.
Aboundary node is either a sink or a source node (but not both, because we
assume a connected net).
Definition 2 (Petri net, behaviour). Let hN ,m0ibe a net system.
–A transition tis enabled at marking m, written as m[ti, if
∀p∈•t:m(p)≥P re(p, t), else disabled.
–A transition t, enabled in m, may fire (occur), leading to a new marking m0,
written as m[tim0, with ∀p∈P:m0(p) = m(p)−P re(p, t) + P ost(p, t).
–The set of all markings reachable from a marking m0, written as [m0i, is
the smallest set such that m0∈[m0i,m∈[m0i ∧ m[tim0⇒m0∈[m0i.
–The reachability graph (RG) is a directed graph with [m0ias set of nodes,
and the labelled arcs denote the reachability relation m[tim0.
Definition 3 (Behavioural properties). Let hN ,m0ibe a net system.
–A place pis k-bounded (bounded for short) if there is a positive integer
number k, serving as an upper bound for the number of tokens on this place
in all reachable markings of the Petri net: ∃k∈N0:∀m∈[m0i:m(p)≤k .
–A Petri net system is k-bounded (bounded for short) if all its places are
k-bounded.
–A transition tis dead at marking mif it is not enabled in any marking m0
reachable from m:6 ∃ m0∈[mi:m0[ti.
–A transition tis live if it is not dead in any marking reachable from m0.
–A marking mis dead if there is no transition which is enabled in m.
–A Petri net system is deadlock-free (weakly live) if there are no reachable
dead markings.
–A Petri net system is live (strongly live) if each transition is live.
Definition 4 (Net structures). Let N=hP, T , P re,P ostibe a Petri net.
Nis
–Homogeneous (HOM) if ∀p∈P:t, t0∈p•⇒P re(p, t) = P r e(p, t0);
–Ordinary (ORD) if ∀p∈Pand ∀t∈T,P re(p, t)≤1and P ost(p, t)≤1;
–Extended Simple (ES) (sometimes also called asymmetric choice) if it is
ORD and ∀p, q ∈P:p•∩q•=∅ ∨ p•⊆q•∨q•⊆p•;
–Extended Free Choice (EFC) if it is ORD and ∀p, q ∈P:p•∩q•=∅ ∨ p•=
q•.
Definition 5 (DTP). Let N=hP, T, P r e,P ostibe a Petri net.
–Asiphon (structural deadlock, co-trap) is a non-empty set of places D⊆P
with •D⊆D•.
–Atrap is a non-empty set of places Q⊆Pwith Q•⊆•Q.
–Aminimal siphon (trap) is a siphon (trap) not including a siphon (trap) as
a proper subset.
–Abad siphon is a siphon, which does not include a trap.
4 M. Heiner, C. Mahulea, and M. Silva
–An empty siphon (trap) is a siphon (trap), not containing a token.
–The Deadlock Trap Property (DTP) asks for every siphon to include an
initially marked trap, i.e., marked at m0.
The DTP can be reformulated as: minimal siphons are not bad and the
maximal traps included are initially marked.
Definition 6 (Semiflows). Let N=hP, T, P re,P ostibe a net.
–The token flow matrix (or incidence matrix if the net is pure, i.e., self-loop
free) is a matrix C=P ost −P re.
–Aplace vector is a vector y∈Z|P|; a transition vector is a vector x∈Z|T|.
–AP-semiflow is a place vector ywith y·C=0,y≥0,y6=0;
aT-semiflow is a transition vector xwith C·x=0,x≥0,x6=0.
–The support of a semiflow x, written as supp(x), is the set of nodes corre-
sponding to the non-zero entries of x.
–A net is conservative if every place belongs to the support of a P-semiflow.
–A net is consistent if every transition belongs to the support of a T-semiflow.
–In a minimal semiflow x,supp (x)does not contain the support of any other
semiflow z, i.e., 6 ∃ semiflow z:supp (z)⊂supp (x), and the greatest com-
mon divisor of xis 1.
–Amono-T-semiflow net (MTS net) is a consistent and conservative net that
has exactly one minimal T-semiflow.
For convenience, we give vectors (markings, semiflows) in a short-hand nota-
tion by enumerating only the non-zero entries. Finally, we recall some well-known
related propositions (see for example [Mur89], [Sta90]), which might be useful
for the reasoning we pursue in this paper.
Proposition 1 (Basics).
1. An empty siphon remains empty forever. A marked trap remains marked for
ever.
2. If Rand R0are siphons (traps), then R∪R0is also a siphon (trap).
3. A minimal siphon (trap) is a P-strongly-connected component, i.e., its places
are strongly connected.
4. A deadlocked Petri net system has an empty siphon.
5. Each siphon of a live net system is initially marked.
6. If there is a bad siphon, the DTP does not hold.
7. A source place pestablishes a bad siphon D={p}on its own, and a sink
place qa trap Q={q}.
8. If each transition has a pre-place, then P•=T, and if each transition has
a post-place, then •P=T. Thus, in a net without boundary transitions, the
whole set of places is a siphon as well as a trap (however, not necessarily
minimal ones).
9. For a P-semiflow xit holds •supp(x) = supp(x)•. Thus, the support of a
P-semiflow is siphon and trap as well (however, generally not vice versa).
DTP and Monotonic Liveness 5
Proposition 2 (DTP and behavioural properties).
1. An ordinary Petri net without siphons is live.
2. If Nis ordinary and the DTP holds for m0, then hN ,m0iis deadlock-free.
3. If Nis ES and the DTP holds for m0, then hN ,m0iis live.
4. Let Nbe an EFC net. hN ,m0iis live iff the DTP holds.
We conclude this section with a proposition from [CCS91], which might be
less known.
Proposition 3 (MTS net and behavioural properties). Liveness and
deadlock-freeness coincide in mono-T-semiflow net systems.
3 Monotonic Liveness
If a property holds for a Petri net Nwith the marking m0, and it also holds in
Nfor any m≥m0, then it is said to be monotonic in the system hN ,m0i. In
this paper we are especially interested in monotonic liveness.
Definition 7 (Monotonic liveness).
Let hN ,m0ibe a Petri net system. It is called monotonically live, if being
live for m0, it remains live for any m≥m0.
We are looking for conditions, at best structural conditions, preserving live-
ness under arbitrary marking increase. To illustrate the problem, let’s consider
a classical example [Sta90], [SR02].
Example 1. The net Nin Figure 1 is ES, conservative, consistent, and covered by
one T-semiflow. It is live for the given initial marking m1= (2p1, p4). Adding
a token to place p5yields the initial marking m2= (2p1, p4, p5) and the net
system remains live for m2≥m1. However, adding a token to p4yields the
initial marking m3= (2p1,2p4) and the net behaviour now contains finite firing
sequences, i.e., it can run into a deadlock (dead state). Thus, the net system is
not live for m3≥m1. It is not monotonically live.
How to distinguish both cases? The net has two (minimal) bad siphons
D1={p1, p2}and D2={p1, p3}. There is no chance to prevent these siphons
from getting empty for arbitrary markings. D1can potentially be emptied
by firing t2∈D1•\•D1, and D2by firing t1∈D2•\•D2. The latter case
destroyed the liveness for m3as it will equally occur for all initial markings
allowing transition sequences containing one of the troublemakers, in this
example t1and t2, sufficiently often.
One lesson learnt from the previous example is, a net does not have to make
use of the additional tokens. Thus, all behaviour (set of transition sequences),
which is possible for mis still possible for m0, with m≤m0. However, new
tokens may allow for additional system behaviour, which is actually well-known
in Petri net theory, see Proposition 4.
6 M. Heiner, C. Mahulea, and M. Silva
p1
p4
p5
p2
p3
t1
t2
t3
2
ORD PUR HOM NBM CSV SCF CON SC FT0 TF0 FP0 PF0 NC
N Y Y Y Y N Y Y N N N N ES
DTP CPI CTI SCTI SB k-b 1-b DCF DSt DTr LIV REV
N Y Y Y Y Y N Y 0 N Y Y
minimal deadlock, not containing a trap:
D1={p1, p2}; D2={p1,p3}
t1 may clean D2, t2 may clean D1;
2p1,p4
p1,p2,p5
p1,p3,2p4
p2,p3,p4
2p2,2p5
t1
t2
t3
p2,p3,p4,p5
p1,p2,2p5
2p1,p4,p5
t3
t2
t1
t2
t1
p2,p3,2p4
p1,p2,p4,p5
2p1,2p4
t3
t2
t1
t1
Fig. 1. A mono-T-semiflow and ES Petri net Nand its reachability graph
for the marking m1= (2p1, p4), generating the language LN(m1) =
(t1t2t3)∗{ε, t1, t1t2}. The siphon {p1, p3}does not contain a trap, i.e., it is a
bad siphon. If the initial marking is increased, it can potentially become empty
by firing of t1.
2p1,p4
p1,p2,p5
p1,p3,2p4
p2,p3,p4
2p2,2p5
t1
t2
t3
p2,p3,p4,p5
p1,p2,2p5
2p1,p4,p5
t3
t2
t1
t2
t1
p2,p3,2p4
p1,p2,p4,p5
2p1,2p4
t3
t2
t1
t1
Fig. 2. Two other reachability graphs for the net Nin Figure 1 for the initial
markings m2= (2p1, p4, p5) and m3= (2p1,2p4); both are greater than m1.
Obviously, LN(m1)⊂ LN(m2), LN(m1)⊂ LN(m3), but hN ,m3iis not live
while hN ,m1iis live.
Proposition 4. For any net Nand two markings mand m0, with m≤m0,
it holds LN(m)⊆ LN(m0)[BRA83]; nevertheless, hN ,mimay be live while
hN ,m0inot.
Example 1 is a mono-T-semiflow net, i.e., a net, where liveness and deadlock-
freeness coincide (see Proposition 3). We look briefly at Example 2 to understand
that this does not generally hold if there are several T-semiflows breathing life
into the net.
Example 2. The net Nin Figure 3 is a slight extension of Example 1. It is ES,
conservative, consistent and covered by two T-semiflows: x1= (t1, t2, t3),x2=
(t4, t5). It is live for the initial marking m1= (2p1, p4).
DTP and Monotonic Liveness 7
p1
p4
p5
p2
p3
t1
t2
t3
PUR ORD HOM NBM CSV SCF FT0 TF0 FP0 PF0 CON SC NC
Y N Y Y Y N N N N N Y Y ES
DTP CPI CTI SB k-B 1-B DCF DSt DTr LIV REV
N Y Y Y Y N N 0 N N N
minimal deadlock, not containing a trap:
D1={p1, p2}, D2={p1,p3};
t2 may clean D1, t1 may clean D2;
2
t4
t5
p6
transitions t1, t2, t3 are non-live.
Fig. 3. An ES Petri net which is not mono-T-semiflow. It is live for the initial
marking m1= (2p1, p4). The siphon {p1, p2}is bad. So it can potentially become
empty by firing t2sufficiently often. This happens for the initial marking m2=
(2p1, p4,2p5), making the net non-live, however keeping it deadlock-free (observe
that {p3, p6}behaves as a trap if the firing of t3is blocked forever).
The net has the following minimal siphons D1={p1, p2},D2={p1, p3, p6},
and D3={p4, p5}; the first two are bad siphons. With the initial marking
m2= (2p1, p4,2p5), D1can become empty by firing twice t2∈D1•\•D1,
which destroys the liveness, without causing a dead state. The transitions t4,t5
are live, the others not. Thus, the net system is not live, but deadlock-free.
The loss of liveness is not necessarily monotonic itself; i.e., a net may be live
for m1, non-live for a marking m2with m2≥m1, and live again for a marking
m3with m3≥m2(which works for all examples in this paper). Liveness may
also be lost by marking multiples (homothetic markings). Examples 1 and 2 are
homothetically live, Example 3 in Section 5 not.
4 Monotonic Liveness of Ordinary Nets
Let us turn to liveness criteria suitable for our objective looking at ordinary nets
first. Liveness criteria not relying on the marking obviously ensure monotonic
liveness. Unfortunately, there are only a few.
First of all, there are some structural reduction rules, see, e.g., [Sil85], [Ber86],
[Mur89], [SR99]. To give a sample, the following reduction rule is easy to accept:
a source transition is live, and all its post-places are unbounded. The transition
and its post-places can be deleted (for analysis purposes); the reduction can be
iterated as many times it is applicable. Sometimes, this kind of reasoning allows
to decide liveness (for examples, see Section 6).
Besides structural reduction we have the DTP, which in most cases does
depend on the marking, but it is obviously monotonic w.r.t. the marking: if each
siphon contains a marked trap at m, then – of course – it contains a marked
8 M. Heiner, C. Mahulea, and M. Silva
trap at m0≥m. Thus, the DTP-related conclusions on behavioural properties
in Proposition 2 are monotonic as well:
Proposition 5 (Monotonic DTP).
1. An ordinary net without siphons is monotonically live.
2. An ordinary net system which holds the DTP is monotonically deadlock-free.
3. A live ES net system which holds the DTP is monotonically live.
4. An EFC net system is monotonically live iff the DTP holds.
Proposition 5.1 can be considered as a special case of the DTP. Then, there
must be source transitions (see Proposition 1.8), and the net is not strongly
connected and not bounded.
Lemma 1. Let be Nan ordinary Petri net. If Nis monotonically live, then
there are no bad siphons.
Proof. We will prove its reverse – if there exist a bad siphon, then the net system
is not monotonically live – by contradiction. Let PSbe a bad siphon. Then there
exist troublemaking transitions Θi∈PS•\•PS. There must be such transitions,
because otherwise PS•=•PS, and then the siphon PSwould be a trap as well.
Since the net system is monotonically live, the marking of the places P\
PScan be increased in such a way that it will never restrict the firing of the
transitions PS•, i.e., the transitions depending on the siphon. Therefore, we can
consider the subnet restricted to PSin isolation.
We will show that the subsystem restricted to PScan be emptied eventually
by increasing the marking, hence cannot be monotonically live. Obviously we
can assume that PSis a minimal siphon. We consider two cases.
(1) The siphon has no forks (tjis a fork if |tj•|>1). Based on the P-strongly-
conectedness (see Proposition 1.3), there exists at least one path from each place
p∈PSto one of the troublemakers Θi. Moving a token from pto •Θidoes not
increase the marking of any other place of PSnot belonging to the considered
path. Obviously, this path can contain joins (tjis a join if |•t|>1), but we can
add any tokens that are missing in the input places of the join. Firing the join,
the marking of the places in the siphon is not increased. Using this process we
can move the tokens from any p∈PSto some •Θi, and by firing Θiwhen it
is enabled, PScan be emptied. Thus, the net system can not be monotonically
live.
(2) On the contrary, let us assume that there exists at least one fork tjand
let p1, p2∈tj•be its output places. For the same reason as discussed in case (1),
there exists a directed path from both places to one or several troublemakers.
If all paths from p1to any troublemaker Θicontain tj, then they form a trap.
This is impossible because siphons are assumed to be bad. By symmetry, in the
case in which the paths from p2to troublemakers contain tj, there exists a trap
as well.
Finally, let us assume that there exists a path from p1to a troublemaker
Θiand one path from p2to a troublemakers Θk, none of them containing tj.
DTP and Monotonic Liveness 9
On both paths the same kind of reasoning can be applied (in an iterative way
if several forks appear). Therefore, the siphon can be emptied even if firing tj
increases the tokens in PS.
Lemma 1 helps to preclude monotonic liveness for Examples 1 and 2 as well
as for all other non-monotonically live examples we are aware of.
Theorem 1. Let be Nan ordinary Petri net. If hN ,m0iis monotonically live,
then the DTP holds.
Proof. The structural check of the DTP can have three possible outcomes.
1. If there are no siphons, then the DTP holds trivially and the net is mono-
tonically live (see Proposition 5.1).
2. If there are bad siphons, then the DTP does not hold for any initial marking
and the net is not monotonically live (see Lemma 1).
3. If each siphon includes a trap, then the maximal trap PTin every minimal
siphon PShas to be initially marked to fulfill the DTP. Because we assume
liveness of the net system, there has to be at least one token in each minimal
siphon (see Proposition 1.5). Let us assume that a token is not in PT, but
in a place p∈PS\PT. If there exists at least one path without forks from p
to a troublemaking transition Θi∈PS•\•PSnot containing any transition
belonging to the trap, •PT, then pcan be emptied using the same reasoning
as used in the proof of in Lemma 1, case (1). Therefore the net can not
be live. If the path from pto a troublemaking transition Θi∈PS•\•PS
contains a fork, then the output places of the fork will be marked when p
is emptied, and the paths from the output places of the forks to the output
should be considered separately.
Finally, if all paths from pto the troublemaking transitions contain at least
one transition •PT, then the trap PTis not maximal since PTtogether with
all places belonging to the above mentioned paths (including all non-minimal
ones) from pto transitions •PTare also a trap.
According to Theorem 1, the DTP establishes a necessary condition for mono-
tonic liveness, which complements Proposition 5.3.
Corollary 1. A live ES net system is monotonically live iff the DTP holds.
Moreover, for those systems for which deadlock-freeness is equivalent to live-
ness, the DTP is a sufficient criteria for liveness monotonicity. This leads, for
example, to the following theorem:
Theorem 2. Let be Nan ordinary mono-T-semiflow Petri net which for m0
fulfills the DTP. Then the system hN ,miis live for any m≥m0.
Proof. It follows from Proposition 5.2 (DTP and deadlock-freeness mono-
tonicity) and Proposition 3 (equivalence of liveness and deadlock freeness in
10 M. Heiner, C. Mahulea, and M. Silva
mono-T-semiflow net systems).
Therefore, the DTP is a sufficient criterion for monotonic liveness of ordinary
mono-T-semiflow net systems as well. In summary, while the DTP is in general
neither necessary nor sufficient for liveness, it turns out to be the case to keep
alive ordinary ES nets or ordinary mono-T-semiflow nets under any marking
increase.
5 Monotonic Liveness of Non-ordinary Nets
It is well-known that non-ordinary nets can be simulated under interleaving
semantics by ordinary ones [Sil85] (see Figure 4 for an example). Let us look
on the net structures we get by this simulation to learn how far the results for
ordinary nets of Section 4 can be uplifted to non-ordinary nets.
p1
p1
a
b
c
d
a
b
d
c
2
5
2
3
unfolding of non-ordinary nets to ordinary nets
Fig. 4. A general principle to simulate a non-ordinary net system by an ordinary
net system (here, the firing language of the second net projected on {a, b, c, d}
is always equal to that of the first) [Sil85].
Example 3. We take a non-ordinary net from [SR02] and consider its simula-
tion by an ordinary net, which we construct according to the general principle
demonstrated in Figure 4.
The two net systems in Figure 5 are conservative, consistent, and live for the
given initial marking. The ordinary net on the right hand side is not ES, and it
DTP and Monotonic Liveness 11
has two minimal bad siphons {q1, p1, p1b, p1c},{p2, p1, p1b, p1c}. Thus, according
to Lemma 1, it is not monotonically live. Because our simulation preserves
the projection of the firing language, in particular, preserves monotonicity
of liveness. Thus, we conclude that the model on the left hand side is not
monotonically live. Indeed, both nets are not live for any initial marking with
an even number of tokens in p1, but live for infinitely many other markings
greater than or equal to (1,1).
As a consequence of firing simulation by the ordinary net systems of the
non-ordinary ones (preserving always the markings of the places involved in the
head of the tail and complement, here q1and q1co), liveness monotonicity can
be studied on the ordinary simulation.
q1co
q1
p1c
p1b
p2
p1
t2
t1
p1
p2
t1
t2
2
3
Fig. 5. A non-ordinary Petri net system and its simulation by an ordinary one.
Both systems are non-live for any initial marking with an even number of tokens
in p1, and live for any other odd marking. Note that the markings of q1and q1co
should not be increased in order to keep the language simulation in the right
hand model. The net system on the right has a bad siphon {q1, p1, p1b, p1c}that
can potentially become empty by firing t2sufficiently often.
6 Applications
We consider a variety of test cases of our benchmark repository to demonstrate
the helpfulness of the DTP for biomolecular networks. The following list sketches
some basic characteristics. The essential analysis results are summarized in Ta-
ble 1. All models hold the DTP, they are consistent and (supposed to be) live.
For non-ordinary nets, the DTP refers to its simulation by an ordinary one.
1. Apoptosis (size: 37 places, 45 transitions, 89 arcs) is a signal transduc-
tion network, which governs complex mechanisms to control and execute
genetically programmed cell death in mammalian cells. Disturbances in the
apoptotic processes may lead to various diseases. This essential part of nor-
mal physiology for most metazoan species is not really well understood; thus
12 M. Heiner, C. Mahulea, and M. Silva
there exist many model versions. The validation by Petri net invariants of
the model considered here is discussed in [HKW04], [HK04].
2. RKIP (size: 11 places, 11 transitions, 14 arcs) models the core of the in-
fluence of the Raf-1 Kinase Inhibitor Protein (RKIP) on the Extracellular
signal Regulated Kinase (ERK) signalling pathway. It is one of the standard
examples used in the systems biology community. It has been introduced in
[CSK+03]; the corresponding qualitative, stochastic, continuous Petri nets
are scrutinized in [GH06], [HDG10].
3. Biosensor (size: 6 places, 10 transitions, 21 arcs) is a gene expression net-
work extended by metabolic activity. The model is a general template of a
biosensor, which can be instantiated to be adapted to specic pollutants. It is
considered as qualitative, stochastic, and continuous Petri net in [GHR+08]
to demonstrate a model-driven design of a self-powering electrochemical
biosensor.
4. Hypoxia (size: 14 places, 19 transitions, 56 arcs) is one of the well-
studied molecular pathways activated under hypoxia condition. It mod-
els the Hypoxia Induced Factor (HIF) pathway responsible for regulating
oxygen-sensitive gene expression. The version considered here is discussed in
[YWS+07]; the corresponding qualitative and continuous Petri nets are used
in [HS10] to determine the core network.
5. Lac operon (size: 11 places, 17 transitions, 41 arcs) is a classical example
of prokaryotic gene regulation. We re-use the simplified model discussed in
[Wil06]. Its corresponding stochastic Petri net is considered in [HLGM09].
6. G/PPP (size: 26 places, 32 transitions, 76 arcs) is a simplified model of the
combined glycolysis (G) and pentose phosphate pathway (PPP) in erythro-
cytes (red blood cells). It belongs to the classical examples of biochemistry
textbooks, see e.g. [BTS02], and thus of systems biology as well. The model
was first discussed using Petri net technologies in [Red94]. Its validation by
Petri net invariants is shown in [HK04], and a more exhaustive qualitative
analysis in [KH08].
7. MAPK (size: 22 places, 30 transitions, 90 arcs) models the signalling
pathway of the mitogen-activated protein kinase cascade, published in
[LBS00]. It is a three-stage double phosphorylation cascade; each phosphory-
lation/dephosphorylation step applies the mass action kinetics pattern. The
corresponding qualitative, stochastic, and continuous Petri net are scruti-
nized in [GHL07], [HGD08].
8. CC – Circadian clock (size: 14 places, 16 transitions, 58 arcs) refers to the
central time signals of a roughly 24-hour cycle in living entities. Circadian
rhythms are used by a wide range of organisms to anticipate daily changes
in the environment. The model published in [BL00] demonstrates that cir-
cadian network can oscillate reliably in the presence of stochastic biomolec-
ular noise and when cellular conditions are altered. It is also available as
PRISM model on the PRISM website (http://www.prismmodelchecker.org).
Its corresponding stochastic Petri net belongs to the benchmark suite used in
[SH09]. We consider here a version with inhibitor arcs modelled by co-places.
DTP and Monotonic Liveness 13
9. Halo (size: 37 places, 38 transitions, 138 arcs) is a cellular signaling and reg-
ulation network, describing the phototaxis in the halobacterium salinarum
[NMOG03]. It models the sophisticated survival strategy, which the halobac-
terium developed for harsh conditions (high temperature, high salt). A light
sensing system and flagellar motor switching allows the cells to swim to
those places of their habitat where the best light conditions are available.
The model is the result of prolonged investigations by experimentally work-
ing scientists [Mar10].
10. Pheromone (size: 42 places, 48 transitions, 119 arcs) is a signal transduc-
tion network of the well understood mating pheromone response pathway in
Saccharomyces cerevisiae. The qualitative Petri net in [SHK06] extends a for-
mer ODE model [KK04]. The Petri net was validated by Petri net invariants
and a partitioning of the transition set.
11. Potato (size: 17 places, 25 transitions, 78 arcs) describes the main car-
bon metabolism, the sucrose-to-starch breakdown in Solanum tuberosum
(potato) tubers. The qualitative Petri net model was developed in co-
operation with experimentally working scientists, experienced in ODE mod-
elling. Its validation by Petri net invariants is discussed in [HK04], and a
more detailed pathway exploration in [KJH05].
Table 1. Some biomolecular case studies; all of them hold the DTP, are con-
sistent and live.
# case study multiplicities net class bounded liveness shown by
1 apoptosis ORD ES no Proposition 2.1
2 RKIP ORD ES yes Proposition 2.3
3 biosensor ORD ES no Proposition 2.3
4 hypoxia ORD not ES no structural reduction
5 lac operon HOM not ES no structural reduction
6 G/PPP HOM not ES no structural reduction
7 MAPK ORD not ES yes dynamic analysis (RG)
8 CC HOM not ES yes dynamic analysis (RG)
9 halo not HOM not ES yes dynamic analysis (RG)
10 pheromone HOM not ES no by reasoning
11 potato not HOM not ES no by reasoning
Contrary, the model of signal transduction events involved in the angiogenesis
processes, which is discussed in [NMC+09] as a stochastic and continuous Petri
net model (size: 39 places, 64 transitions, 185 arcs) is to a large extent covered by
a (non-minimal) bad siphon. Thus, even if the net is live for a certain marking m,
there is always a larger marking m0, which will allow to remove all tokens from
the bad siphon. Consequently, an arbitrary marking increase will not preserve
liveness.
14 M. Heiner, C. Mahulea, and M. Silva
7 Tools
The Petri nets for the case studies have been constructed using Snoopy [RMH10],
a tool to design and animate or simulate hierarchical graphs, among them qual-
itative, stochastic and continuous Petri nets as used in the case studies in Sec-
tion 6. Snoopy provides export to various analysis tools as well as import and
export of the Systems Biology Markup Language (SBML).
The qualitative analyses have been made with the Petri net analysis tool
Charlie [Fra09], complemented by the structural reduction rules supported by
the Integrated Net Analyser INA [SR99].
8 Conclusions
We have discussed the problem of monotonic liveness, with one of the motivations
originating from bio-model engineering. We have presented a new result showing
the necessity of the DTP for monotonic liveness.
Moreover, we immediately know – thanks to the well-known propositions of
the DTP – that ordinary ES nets are monotonically iff the DTP holds. Further-
more, we know – because the DTP monotonically ensures deadlock freeness –
that for any net class, in which liveness and deadlock freeness coincide, mono-
tonic liveness is characterized by the DTP. We have shown one instance for this
case: the mono-T-semiflow nets (MTS).
We have demonstrated the usefulness of our results by applying them to a
variety of biomolecular networks.
One of the remaining open issues is: what are sufficient conditions for mono-
tonic liveness for more general net structures? While none of our test cases is an
MTS net, this line might be worth being explored more carefully, e.g. by looking
at FRT nets (Freely Related T-Semiflows) [CS92] and extensions.
Acknowledgements. This work has been partially supported by CICYT
- FEDER grant DPI2006-15390 and by the European Community’s Seventh
Framework Programme under project DISC (Grant Agreement n. INFSO-ICT-
224498). The work of M. Heiner was supported in part by the grant DPI2006-
15390 for a stay with the Group of Discrete Event Systems Engineering (GISED);
the main ideas of this paper have been conceived during this period.
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