# Searching for Attractors: To infinity and beyond

Stefan Haar

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Stefan Haar. Searching for Attractors: To infinity and beyond. AUTOMATA 2025 - 31st International Workshop on Cellular Automata and Discrete Complex Systems, Jun 2025, Lille, France. hal-05115278

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# Searching for Attractors : To infinity and beyond

Stefan Haar¹

INRIA, France

stefan.haar@inria.fr

**Abstract.** Discrete and nondeterministic modelling of regulatory and signalling networks allows to characterize many of their crucial dynamical properties, with a reasonable computational effort. A dynamical feature of particular interest in its own right, as well as in terms of biological relevance, is the landscape of attractors and their attraction basins. In recent years, we have developed new approaches to discovery and global cartography of this basin landscape. We enlarge the set of models used, by moving to Petri nets on the one hand and an opening to continuous dynamics on the other. Surprisingly, these openings are rewarded not only by a sharpening of the analysis, but also the emergence of compact and readable data structures, and fast search algorithms. The talk will advocate the cross-fertilization between discrete and continuous approaches, and to not be afraid of pushing limits.

## 1 Boolean Networks and their attractors

Discrete models, and in particular Boolean Networks (BNs), are a rich and fertile object of study in their own right, with a powerful theory. A driving force in the development of this field lies in the massive and successful application of BNs as dynamical systems models for interaction systems in engineering and live sciences, especially in the area of gene regulation [1, 5, 11, 7].

While this application is not our main focus here, we highlight two desiderata that seem crucial to us in this context:

- The capacity of identifying the *attractors* of the system under study, that is, the absorbing strongly connected subspaces of the systems state space; these constitute the stable regimes of the network, and most notably the *phenotypes* of a cell whose regulatory network is captured by the BN model; and

- best possible coverage of the state space, i.e. a full account of the states reachable and pathways available in the network.

It is a well-known fact that the choice of *update mode* for applying the boolean function that constitutes the BN has a very strong impact on the way the systems dynamics are discovered. The following are the fundamental modes studied in the literature:

- asynchronous mode: in every round only one variable, chosen nondeterministically, updates its value according to its component function;
- synchronous: all variables update value simultaneously; and

- generalised asynchronous: any nondeterministically chosen subset of variables updates value simultaneously.

Further variants in the literature are obtained by e.g. imposing, for the asynchronous semantics, an order in which individual variables are considered in successive rounds, etc; we will not dwell on those variants here since they allow *fewer* actions than their counterpart in the above, and will not improve the coverage we are striving to achieve.

Note here that these semantics allow - with suitable adaptations according to the update mode chosen - to benefit from translation into other formal models, in particular into Petri nets. We have detailed some aspects of this transfer in [2]; the two-way translation allows to retrieve theoretical results from one domain into the other, and to benefit e.g. from the powerful *unfolding* method for Petri net to conduct efficient and exhaustive searches for attractors [1]. In continuation of the work in [2], we will advocate below to study *continuous* Petri nets and to move back and forth between continuous and discrete representations, obtaining better coverage of the dynamics and access to attractors, as in our desiderata above.

Returning to BNs, it had long been assumed that the generalised asynchronous semantics is the all-en-compassing updating mode of BNs and capable of expressing all possible behaviours. Doubts arose when, in particular, the authors of [11] present an element of gene regulation that exhibits behaviours that cannot be explained with the classical, generalized asynchronous BNs. One may think of it as a *pulse generator*, visualised below by the interaction graph and an approximate experimental expression profile of specie 3.

<figure><img src="image_2.png" /></figure>

Here, an incoming signal activating 1 causes a short term spike in activity of 3, thus generating a pulse. Note that in a continuous dynamical model, such as ODE, such a behaviour can be captured using precise timing parameters that reflect speed differences between the processes involved; or equivalently, in terms of activation or exhibition *thresholds*, admitting gradual continuous increases in variable values.

The issue touches upon the very essence and justification of using discrete models, who are hoped to provide sound *overapproximations* of the behaviour of the underlying natural systems. In fact, the above semantics are all grounded in abstracting away from such activation thresholds. The only behaviours possible with asynchronous BNs are those where 3 never activates due to the inhibition from 1, or 2 activates 3 in spite of that inhibition, resulting in a continuous activation, rather than a pulse.
In this context of growing evidence that discrete BNs may not be sufficient for coverage, our development of *Most permissive (MP) semantics* [13, 4, 2] can be seen to address this challenge, while remaining rooted in the discrete approach, with only a moderate increase of the size of the model. The development had been inspired by previous discoveries in the domain of Petri nets [3], showing how a weakening of synchronization limitations in the semantics allows for opening new behaviours which can in turn be captured by a suitable discrete refinement. In Boolean networks, the *most permissive* semantics we will detail below works in a similar way, adding nondeterministic choices in a slightly refined discrete structure. The keystone in this series of development will then be, further below, the connection with *continuous Petri nets*, capturing the rich behaviour of most permissive boolean networks, while eventually giving a compact and discrete picture of it. The following sections will discuss in greater depth, in this order,

- most permissive Boolean networks,

- Petri nets, both discrete and continuous, and finally

- the emerging picture of a connection between those models.

## 2 Beyond Limitations: Most Permissive Semantics for Boolean Networks

Recall that a BN of dimension $n \in \mathbb{N}$ (on $n$ variables) is a function $f: \mathbb{B}^n \to \mathbb{B}^n$, typically given component-wise as local Boolean functions $f_i: \mathbb{B}^n \to \mathbb{B}$ describing the evolution of the variables $i \in \{1, \dots, n\}$, $f = (f_1, \dots, f_n)$. An $n$-ary Boolean vector $\mathbf{x} \in \mathbb{B}^n$ is a configuration for $f$. The interaction graph of $f$ has the $n$ variables as vertex set, and links variable $i$ to $j$ by an oriented arc iff $i$ participates in the update function of $j$; a positive influence (promotion) is indicated by coloring the arc green, a negative one by red. An example BN network and its associated interaction graph are given in Figure 1.

<figure><img src="image_3.png" /><figcaption>**Fig. 1.** A running example BN on three variables (left) with its interaction graph (right). Positive interactions are depicted by pointed green arrows while negative interactions are depicted by flat red arrows.</figcaption></figure>

For a BN $f$ of dimension $n$ and any two configurations $\mathbf{x}, \mathbf{y} \in \mathbb{B}^n$, let $\Delta(\mathbf{x}, \mathbf{y}) = \{i \in \{1, \dots, n\} \mid \mathbf{x}_i \neq \mathbf{y}_i\}$ be the set of all variables whose value differs between $\mathbf{x}$ and $\mathbf{y}$. Then the fully asynchronous semantics of $f, fa(f)$,
synchronous semantics of $f, syn(f)$, and asynchronous semantics of $f, ga(f)$ are
defined as follows for any pair of configurations $\mathbf{x}, \mathbf{y} \in \mathbb{B}^n$:

$$ (\mathbf{x}, \mathbf{y}) \in fa(f) \iff \exists i \in \{1, \dots, n\}, \Delta(\mathbf{x}, \mathbf{y}) = \{i\} \wedge f_i(\mathbf{x}) = y_i $$

$$ (\mathbf{x}, \mathbf{y}) \in syn(f) \iff \forall i \in \{1, \dots, n\}, f_i(\mathbf{x}) = y_i $$

$$ (\mathbf{x}, \mathbf{y}) \in ga(f) \iff \forall i \in \Delta(\mathbf{x}, \mathbf{y}), f_i(\mathbf{x}) = y_i $$

It follows immediately that $fa(f) \subseteq ga(f)$ as well as $syn(f) \subseteq ga(f)$.

With the natural infix notation for the semantics relations we write, e.g.,
$\mathbf{x} \xrightarrow{fa} \mathbf{y}$. The associated reachability relations are obtained as the reflexive and

transitive closures of the semantics relations, $\mathbf{x} \xrightarrow{fa}^* \mathbf{y}$, etc.

Configurations of the example BN from Figure 1 reachable from (0,0,0)
under fully asynchronous or asynchronous semantics are visualised in Figure 2.
Synchronous semantics are omitted, as the only reachable configurations are
(1,1,0) and (0,0,0) itself (the cycle in asynchronous reachability).

<figure><img src="image_4.png" /><figcaption>**Fig. 2.** [9] All configurations of the example BN (Figure 1) reachable from (0,0,0)
under fully asynchronous semantics (left) and asynchronous semantics (right). The
variables are ordered from top to bottom and dark tiles represent value 0 while light
tiles represent value 1. Self-loops are omitted.</figcaption></figure>

Variants of definitions for MP semantics can be found in [13, 4]; to remain
conceptually close to state updates of PNs, we use here the original definition
from [13]. MP differs from other semantics by the introduction of two transient
(or intermediate) values, namely $\uparrow$ to signify that a variable is increasing from
0 to 1, and $\downarrow$ when the variable is decreasing from 1 to 0.

For a BN $f$ of dimension $n$ and any MP configuration $\hat{\mathbf{x}} \in \{0, 1, \uparrow, \downarrow\}^n$, let
$\beta(\hat{\mathbf{x}}) = \{\mathbf{x} \in \mathbb{B}^n | \forall i \in \{1, \dots, n\}, \hat{\mathbf{x}}_i \in \mathbb{B} \implies \mathbf{x}_i = \hat{\mathbf{x}}_i\}$ be the set of compatible
binarisations expressed as Boolean configurations. Then, the MP semantics of
a BN $f$, $\hat{m}_p(f)$, is defined as follows for any pair of MP configurations $\hat{\mathbf{x}}, \hat{\mathbf{y}} \in$
`{0, 1, ↗, ↘}^n`:

$$ (\hat{\mathbf{x}}, \hat{\mathbf{y}}) \in \hat{m}_p(f) \iff \exists i \in \{1, \dots, n\}, \Delta(\hat{\mathbf{x}}, \hat{\mathbf{y}}) = \{i\} \wedge [(\hat{\mathbf{x}}_i = ↗ \wedge \hat{\mathbf{y}}_i = 1) \vee \\ \phantom{(\hat{\mathbf{x}}_i = ↗ \wedge \hat{\mathbf{y}}_i = 1) \vee} (\hat{\mathbf{x}}_i = ↘ \wedge \hat{\mathbf{y}}_i = 0) \vee \\ \phantom{(\hat{\mathbf{x}}_i = ↗ \wedge \hat{\mathbf{y}}_i = 1) \vee (\hat{\mathbf{x}}_i = ↘ \wedge \hat{\mathbf{y}}_i = 0) \vee} (\hat{\mathbf{x}}_i \neq 1 \wedge \hat{\mathbf{y}}_i = ↗ \wedge \exists \mathbf{x} \in \beta(\hat{\mathbf{x}}), f_i(\mathbf{x}) = 1) \vee \\ \phantom{(\hat{\mathbf{x}}_i \neq 1 \wedge \hat{\mathbf{y}}_i = ↗ \wedge \exists \mathbf{x} \in \beta(\hat{\mathbf{x}}), f_i(\mathbf{x}) = 1) \vee} (\hat{\mathbf{x}}_i \neq 0 \wedge \hat{\mathbf{y}}_i = ↘ \wedge \exists \mathbf{x} \in \beta(\hat{\mathbf{x}}), f_i(\mathbf{x}) = 0)] $$

The top two lines of the disjunction enable a variable in a transient value to collapse into the target Boolean value at any time, allowing a nondeterministic choice between this 'unguarded' transition and idling in the intermediate state. The bottom two lines dictate that for a variable to change to a transient value, a binarisation must exist for which the local function returns the target Boolean value of said transient value.

Since Boolean configurations are the objects we are interested in, we will often only keep the induced reachability relations between such 'pure states', abstracting away the intermediate states that allowed MP semantics to obtain those paths in the first place, and note this relation as $(\mathbf{x}, \mathbf{y}) \in mp(f)$.

We have $ga(f) \subseteq mp(f)$, since any asynchronous update can be reproduced in MP by first putting all relevant variables into transient value and subsequently collapsing them. The inclusion is strict in general, as shown in Figure 3 of all Boolean configurations of the example BN from Figure 1 reachable under MP semantics.

<figure><img src="image_5.png" /><figcaption>**Fig. 3.** [9] All Boolean configurations of the example BN (Figure 1) reachable from (0, 0, 0) under MP semantics. While the fixed points are preserved, all other configurations become transiently reachable, as opposed to asynchronous semantics.</figcaption></figure>

MP BNs are known to have very favorable algorithmic properties [13, 4]:
- for any $(\mathbf{x}, \mathbf{y}) \in mp(f)^*$, there is a path of at most 3n transitions connecting $\mathbf{x}$ to $\mathbf{y}$;

- all attractors are hypercubes; in fact, minimal trap spaces and attractors coincide.

The question now is whether there are fast and compact ways of representing and analysing the semantics of MP, allowing to treat larger systems efficiently. The following will develop the lead along which this can be done

## 3 Beyond Infinity: Petri Nets, discrete and continuous

Nets are bipartite directed graphs with two sets of nodes, *places* and *transitions*. Places model resource types available in the system, and can hold a certain amount of *tokens* or *token mass* representing availability of said resource; once token mass has been assigned to places, the marked net is called a *Petri net*. Transitions allow removing (reducing) or adding tokens (token mass), according to the *arcs* connecting them to places.

Formally, a *net* is a tuple $N = (P, T, \mathbf{W})$, where $P$ is a finite set of places, $T$ is a finite set of transitions and $\mathbf{W}: (P \times T) \cup (T \times P) \rightarrow N_0$ is a weight function assigning each transition the amount of resources consumed or produced in each place. For $x, y \in P \cup T$, we say there is an arc from $x$ to $y$ if $\mathbf{W}(x, y) > 0$. A simple example net is visualised in Figure 4; it also shows a case of discrete transition firing.

<figure><img src="image_6.png" /><figcaption>**Fig. 4.** Left: An example net with two places (depicted as circles) and two transitions (depicted as squares). The arc weights are given as labels on the arrows. As per convention, arc weights equal to 1 are omitted. Right: An example of a transition firing in a DPN with the same net structure. The tokens assigned by the markings are depicted as black dots inside the places.</figcaption></figure>

For each place or transition $x \in P \cup T$, we call $\bullet x = \{y \in P \cup T \mid \mathbf{W}(y, x) > 0\}$ the *pre-set* of $x$. Similarly, $x^\bullet = \{y \in P \cup T \mid \mathbf{W}(x, y) > 0\}$ is the *post-set* of $x$.

*Discrete Petri nets.* A DPN $\mathcal{N} = (\mathbb{N}, M_0)$ is a net $N = (P, T, \mathbf{W})$, with $\mathbf{W}$ taking values in $\mathbb{Z}$, together with a (discrete) *marking* $M_0: P \to N_0$, called the *initial marking*. A discrete marking models the resource allocation in the system
by assigning a natural number of tokens to each place. A transition $t \in T$ is enabled at a marking $M$, denoted $M \xrightarrow{t}$, if and only if $\forall p \in \bullet t, M(p) \geq W(p, t)$. An enabled transition can fire, $M \xrightarrow{t} M'$, leading to a new marking $M'$ defined for each $p \in P$ as follows:

$$M'(p) \triangleq M(p) - W(p, t) + W(t, p)$$

Let us consider the example net from Figure 4 with the initial marking $M_0 = \{(p_1, 1), (p_2, 0)\}$. Transition $t_1$ is enabled at $M_0$, leading to marking $M_1 = \{(p_1, 0), (p_2, 1)\}$ as illustrated in Figure 4 on the right hand side. $M_1$, however, is a deadlock, i. e. a marking at which no transition is enabled.

A (finite) firing sequence is a word $w = (t_i)_{i<n \in \mathbb{N}_0}$ over $T$ for which there exists a sequence of markings $(M_i)_{i<n+1}$ such that $M_i \xrightarrow{t_i} M_{i+1}$ for all $i < n$. We denote the set of markings reachable from some marking $M$ in $\mathcal{N}$ by $RS_{\mathcal{N}}(M) = \{M' | \exists w \in T^*, M \xrightarrow{w} M'\}$, or simply $RS(M)$ when there is no ambiguity about $\mathcal{N}$.

A DPN all of whose reachable markings have at most one token per place, $\forall M \in RS(M_0), \forall p \in P, M(p) \leq 1$, is safe (also 1-bounded). The PNs that encode BNs are necessarily safe; therefore we will focus on bounded nets throughout.

We only describe discrete PNs in passing here, to prepare the discussion of continuous Petri nets; however, DPNs and their connection to BNs with classical asynchronous semantics are interesting in their own right, witness the results and discussion in [2] and the effective attractor search developed in [1].

**Beyond infinity: Continuous Petri nets.** A CPN $\mathcal{N} = (\mathbb{N}, \mathbf{m}_0)$ is a net $\mathbb{N} = (P, T, \mathbf{W})$ with a (continuous) marking $\mathbf{m}_0: P \to \mathbb{R}_0^+$, called the initial marking. Similarly to the discrete case, continuous markings also model resource allocation, but assign $\mathbb{R}_0^+$-valued token mass instead of discrete tokens. A transition $t \in T$ has an enabling degree at marking $\mathbf{m}$ given by $\text{enab}(t, \mathbf{m}) = \min_{p \in \bullet t} \frac{\mathbf{m}(p)}{W(p, t)}$ when $\bullet t \neq \emptyset$, and equal to $\infty$ otherwise. A transition with $\text{enab}(t, \mathbf{m}) > 0$ is enabled at $\mathbf{m}$ and can $\alpha$-fire for any $\alpha \in [0, \text{enab}(t, \mathbf{m})] \cap \mathbb{R}$, $\mathbf{m} \xrightarrow{\alpha t} \mathbf{m}'$, leading to a new marking $\mathbf{m}'$ defined for each $p \in P$ as follows:

$$\mathbf{m}'(p) \triangleq \mathbf{m}(p) - \alpha \mathbf{W}(p, t) + \alpha \mathbf{W}(t, p)$$

Again, a finite firing sequence is a finite word, but this time of the form $\sigma = (\alpha_i t_i)_{i<n \in \mathbb{N}_0}$ over $\mathbb{R}_0^+ \times T$ for which there exists a finite sequence of markings $(\mathbf{m}_i)_{i<n+1}$ such that $\mathbf{m}_i \xrightarrow{\alpha_i t_i} \mathbf{m}_{i+1}$ for all $i < n$. An infinite firing sequence is an infinite word $\sigma = (\alpha_i t_i)_{n \in \mathbb{N}_0}$ over $\mathbb{R}_0^+ \times T$ for which there exists an infinite family of markings $(\mathbf{m}_i)_{i \leq \omega}$ such that $\mathbf{m}_i \xrightarrow{\alpha_i t_i} \mathbf{m}_{i+1}$ for all $i \in \mathbb{N}_0$ and $\lim_{i \to \infty} \mathbf{m}_i = \mathbf{m}_\omega$.

We denote the set of markings reachable from some marking $\mathbf{m}$ in $\mathcal{N}$ by $RS_{\mathcal{N}}(\mathbf{m}) = \{\mathbf{m}' | \exists \sigma \in (\mathbb{R}_0^+ \times T)^*, \mathbf{m} \xrightarrow{\sigma} \mathbf{m}'\}$. We further denote the set of markings reachable in the limit (lim-reachable) from some marking $\mathbf{m}$ in $\mathcal{N}$ by $\lim RS_{\mathcal{N}}(\mathbf{m}) = \{\mathbf{m}' | \exists \sigma \in (\mathbb{R}_0^+ \times T)^*, \mathbf{m} \xrightarrow{\sigma} \mathbf{m}'\}$. Since an infinite firing
sequence can include null firings ($\alpha = 0$), we have $RS_N(\mathbf{m}) \subseteq \lim RS_N(\mathbf{m})$. We write simply $RS(\mathbf{m})$ and $\lim RS(\mathbf{m})$ when $\mathcal{N}$ is obvious.

Similar to DPNs, a CPN all of whose reachable markings have token mass at most 1 per place, $\forall \mathbf{m} \in RS(\mathbf{m}_0), \forall p \in P, \mathbf{m}(p) \le 1$, is safe. The notions of safeness and lim-safeness coincide [10].

The ability to fire a transition by varying degree means that there are infinitely many possible firings in any marking at which at least one transition is enabled. We therefore cannot visualise all the possible transition firings of a CPN in the same fashion as Figure 4 does for a DPN. Symbolic analysis methods have therefore been introduced for CPN dynamics, namely abstract reachability graphs (ARG) and symbolic reachability trees (SRT) [8].

The ARG constitutes an abstraction of the state space, in which markings that have the same *support*, i.e. the set of places which have non-zero token mass, $[[\mathbf{m}]] \triangleq \{p \in P \mid \mathbf{m}(p) > 0\}$, are grouped into equivalence classes. Supports are linked between one another by two types of edges, *anonymous* and *border*, both representing the possibility of the system evolving into a marking with different support. Anonymous edges indicate reversible changes; Border edges are of special interest, as they represent occurrences that *permanently disable* one or more transitions, thus shaping the reachability landscape.

The SRT provides a refined view at such a landscape in the form of a directed tree linked by border edges. The nodes of the tree are aggregations of markings by their *mode*, i.e. the set of transitions eventually fireable from the given marking: $T_{\mathbf{m}} \triangleq \{t \in T \mid \exists \mathbf{m} \xrightarrow{\sigma} t \in [[\sigma]]\}$, where $[[\sigma]]$ is the set of all transitions occurring along $\sigma$ with $\alpha > 0$. The branches of the tree then indicate refinements of those modes, as less and less transitions remain available. The depth of the SRT is naturally bounded by $|T|$.

We omit the formal definition of ARG and SRT due to space constraints. The reader is instead referred to [8]. Both the ARG and SRT of the example net are illustrated in Figure 5.

<figure><img src="image_7.png" /><figcaption>**Fig. 5.** [8, 9] The ARG (left) and the SRT (right) of the example net from Figure 4. The border edges are given as double line arrows in the ARG. The SRT for this example is trivial, as there is only one border edge leading from a marking whose support (set of places with positive token mass) is $\{p_1, p_2\}$, and which uses an infinite firing sequence of instances of $t_1$ and $t_2$ to reach the empty marking.</figcaption></figure>
Observe that as opposed to the DPN (Figure 4), $t_2$ can be fired in a CPN with the same initial marking $m_0 = M_0$. Firing of $t_2$ halves the token mass, every loop around the net thus results in a marking with less total token mass. This phenomenon is illustrative, as the two places of the net are a trap, that is $\{p_1, p_2\} \cdot \subseteq \cdot \{p_1, p_2\}$. It is a well known result for DPNs that an initially marked trap cannot be emptied [6]. This result extends to reachability in CPNs, indeed after any finite number of transition firings, there will always be some token mass left, however, the empty marking is lim-reachable.

**Transfinite Sequences.** Classical reachability, i.e. underwritten by some finite firing sequences, has been seen to be an insufficient measure of the CPN dynamics; the natural semantics here is indeed lim-reachability, as argued above. However, to capture the dynamics that sets in from some lim-reached state $m$, one would like to concatenate the new sequence of actions after the infinite sequence that led into $m$: it becomes thus natural to look at transfinite sequences $(\alpha_n t_n)_{n<\kappa}$ for $\kappa$ some transfinite ordinal. In fact, even small examples suffice to see that the exploration of CPN dynamics requires the passage to such transfinite sequences. In the CPN of Figure 6, consider the following transfinite firing sequence whose (countable) ordinal is $\omega + 1$:

$$ \sigma = (2^{-n}t_1)_{n \ge 1} \cdot 1t_3 $$

<figure><img src="image_8.png" /><figcaption>**Fig. 6.** An example CPN exhibiting the necessity of studying transfinite sequences</figcaption></figure>

The trajectory generated by $\sigma$ is: $T \cdot \{t_3\} \cdot \emptyset$; it cannot be generated by an infinite firing sequence. This means that in order to fully apprehend the state space and pathways in a CPN, transfinite sequences have to be systematically explored; note that results from [8] show that no sequences longer than $\omega^2$ are relevant.

# 4 Why CPNs are worth the effort

We will now state the main reasons why the use of CPNs is relevant, in our opinion, for the attractor search in boolean systems.
## 4.1 The SRT gives all attractors

Indeed :

**Theorem 1 ([8]).** Let ($\mathcal{N}$, $\mathbf{m}_0$) be a bounded marked CPN such that for all $\mathbf{m} \in \lim\mathbf{-RS}(\mathcal{N}, \mathbf{m}_0)$, $T_{\mathbf{m}} = T_{\mathbf{m}_0}$. Then for all $\mathbf{m} \in \lim\mathbf{-RS}(\mathcal{N}, \mathbf{m}_0)$, $\lim\mathbf{-RS}(\mathcal{N}, \mathbf{m}) = \lim\mathbf{-RS}(\mathcal{N}, \mathbf{m}_0)$.

In fact, this result implies that the leaves of the SRT (those are the modes from which no other mode is reachable) all correspond (in general not 1-1) to an attractor of the underlying continuous Petri net; conversely, every attractor is represented by one or more of those leaves.

<figure><img src="image_9.png" /><figcaption>**Fig. 7.** [9] The PN encoding of the example BN given in Figure 1. The loops are depicted as double headed arrows to avoid clutter.</figcaption></figure>

## 4.2 Encoding a BN as a CPN faithfully translates reachability, both ways

While we will not detail the construction that takes as input a BN and produces a CPN (those elements can be found in [9]), a comparison of Figure 7 with Figure 1 may suffice here: for every variable we create two Petri net places, whose token mass will always add up to 1 and represent the increase or decrease in value of the variable concerned. Transitions are created for all cases of increase or decrease that the boolean function imposes, depending on the marking of influencing variables that are connected by arc loops. If we denote the Petri net that is associated to boolean network $f$ by $N(f)$, we obtain the following result (not yet reviewed):
**Theorem 2 ([9] Continuous Petri Nets Simulate MP Boolean Networks).** Let $f$ be a Boolean network of dimension $n$. Then, for any configurations $\mathbf{x}, \mathbf{y} \in \mathbb{B}^n$, $\mathbf{x} \underset{f}{\overset{mp}{\longrightarrow}}^* \mathbf{y} \iff \langle \mathbf{y} \rangle \in \lim\nolimits_{\mathbf{S}_{\mathbb{N}(f)}} (\langle \mathbf{x} \rangle)$.

**5 Conclusion**

It had been known that the power of Boolean network can be enhanced by MP semantics for improved coverage and attractor detection, while completely preserving the discrete nature of the model and accepting only a mild increase in the size of the data structure. Recent results have now shown that MP semantics can be given a compagnon representation in the form of continuous Petri nets, where the MP dynamics is faithfully reflected. While the admission of *infinite* state spaces and *transfinite* transition sequences may seem an inacceptable price to pay, the consistent use of adequate abstraction and purely structural methods for Petri nets actually allows to obtain discrete behavioural representations of bounded size and at a reasonable computational cost. We hope these arguments motivate our colleagues from the AUTOMATA community to join our journey - *to infinity and beyond*.
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