A crucial question in analyzing a concurrent system is to determine its long-run behaviour, and in particular, whether there are irreversible choices in its evolution, leading into parts of the reachability space from which there is no return to other parts. Casting this problem in the unifying framework of safe Petri nets, our previous work [3] has provided techniques for identifying attractors, i.e. terminal strongly connected components of the reachability space, whose attraction basins we wish to determine. Here, we provide a solution for the case of safe Petri nets.
Our algorithm uses net unfoldings and provides a map of all of the system's configurations  (concurrent executions)  that act as cliff-edges, i.e. any maximal extension for those configurations lies in some basin that is considered fatal. The computation turns out to require only a relatively small prefix of the unfolding, just twice the depth of Esparza's complete prefix.

§ INTRODUCTION

Unfoldings of Petri nets [6], which are essentially event structures in the sense of Winskel et al. [18] with additional information about states,

are an acyclic representation of the possible sequences of transitions, akin to Mazurkiewicz traces

but enriched with branching information.

Many reachability-related verification problems for concurrent systems have been successfully addressed by Petri-net unfolding methods over the past decades, see [15, 7, 6].

However, questions of long-term behaviour and stabilization have received relatively little attention.  With the growing interest in formal methods for biology, the key feature of  multistability of systems [30, 24, 20, 23] comes into focus. It has been studied in other
qualitative models such as Boolean and multivalued networks [29, 28, 26].
Multistability characterizes many fundamental biological processes, such as cellular differentiation, cellular reprogramming, and cell-fate decision; in fact, stabilization of a cell regulatory network corresponds to reaching one of the - possibly many - phenotypes of the cell, thus explaining the important role of multistability in cell biology. However, multistability emerges also in  many other branches of the life sciences; our own motivation is  the qualitative analysis of the fate of ecosystems, see [25].

Multistability can be succinctly described as the presence of several attractors in the system under study.
Attractors characterize the stable behaviours, given as the smallest subsets of states from which the system cannot escape;  in other words, they are terminal strongly connected components of the associated transition system. In the long run, the system will enter one of its attractors and remain inside; multi-stability arises when there is more than one such attractor.
The basin of attractor $\attn$ consists of  the states from  which the system inevitably reaches $\attn$.

The  basin includes the attractor itself, and possibly one or several transient states [14].

We aim at finding the tipping points in which the system switches from an undetermined or free state into some basin; while interesting beyond that domain, this is a recurrent question in the analysis of
signalling and gene regulatory networks [5, 16].
In [8], the authors provide a method for identifying, in a boolean network model, the states in which one transition
leads to losing the reachability of a given attractor (called bifurcation transitions there; we prefer to speak of tipping points instead).
However, enumerating the states in which the identified transitions make the system branch away from the attractor can be highly combinatorial and hinders a fine understanding of the branching.
Thus, the challenge resides in identifying the specific contexts and sequences of transitions leading to a strong basin.

Using a bounded unfolding prefix, all reachable  attractors [3] can be extracted. Also, we have exhibited ([11]) the particular shape of basins that are visible in a concurrent model.

In the present paper, we build on this previous analyses;
the point of view taken here is that all attractors correspond to the end of the system's free behaviour, in other words to its doom. We will give characterizations of  basin boundaries (called cliff-edges below), and  of those behaviours that remain free, in terms of properties of the unfolding, reporting also on practical experiments with an implementation of the algorithms derived.  We finally introduce a novel type of quantitative measure, called protectedness,
to indicate how far away (or close) a system is from doom, in a state that is still free per se. General discussions and outlook will conclude this paper.

§ PETRI NETS AND UNFOLDINGS

We begin now by recalling the basic definitions needed below.
A Petri net is a bipartite directed graph whose nodes are either places or transitions, and places may carry tokens.
In this paper, we consider only safe Petri nets where  a place carries either one or no token in any reachable marking.
The set of currently active places form the state, or marking, of the net.

Note. Some remarks are in order concerning our use of Petri nets versus that of  boolean networks, which are more widely used in systems biology.
Safe (or 1-bounded) Petri nets [17] are close to Boolean and multivalued networks [4], yet  enable a more fine-grained specification of the conditions for triggering value changes. Focussing on safe PNs entails no limitation of generality of the model, as two-way behaviour-preserving translations between Boolean and multivalued models exist (see [4] and the appendix of [3] for discussion). We are thus entitled to move between these models without loss of expressiveness; however, Petri nets provide more convenient ways to develop and present the theory and the algorithms here.

Formally, a net is a tuple $\net=\tup{\place,\trans,\flow}$,  where  $\trans$ is a finite set of  transitions, $\place$ a finite set of places,

and  $\flow\subseteq(\place\times \trans)\cup(\trans\times \place)$ is a flow relation whose elements are called arcs.
In figures,  places are represented by circles and the transitions by boxes (each one with a label identifying it).

For any node $x\in \place\cup \trans$, we call pre-set of $x$ the set $\preset{x}=\{y\in \place\cup \trans\mid \tup{y,x}\in \flow\}$ and post-set of $x$ the set $\postset{x}=\{y\in \place\cup \trans\mid \tup{x,y}\in \flow\}$.
A subset $\markn\subseteq \place$ of the places is called a marking. A Petri net is a tuple $\petrinet=\tup{\place,\trans,\flow,\markn_0}$, with $\markn_0\subseteq\place$ an initial marking.
Markings are represented by dots (or tokens) in the marked places.
A transition $\transn\in \trans$ is enabled at a marking $\markn$, denoted $\markn\move{\transn}$, if and only if $\preset{\transn}\subseteq \markn$. An enabled transition $\transn$ can fire, leading to the new marking $\markn'=(\markn\setminus\preset{\transn})\cup\postset{\transn}$;[This definition does not correspond to the standard semantics of Petri nets, but is equivalent for safe Petri nets, and we prefer it for the sake of simplicity.] in that case we write $\markn\move{\transn}\markn'$.
A firing sequence from a marking $M'_0$ is a (finite or infinite) sequence $w=t_1t_2t_3\dots$ over $\trans$ such that there exist markings $\markn'_1,\markn'_2,\dots$ with $\markn'_0\move{t_1}\markn'_1\move{t_2}\markn'_2\move{t_3}\dots$. If $w$ is finite and of length $n$, we write $\markn'_0\move{w}\markn'_n$, and we say that $\markn'_n$ is reachable from $\markn'_0$, also simply written $\markn'_0\rightarrow\markn'_n$. We denote the set of markings reachable from some marking $\markn$ in a net $\net$ by $\reach_\net(\markn)$.
A Petri net $\tup{\net,\markn_0}$ is considered safe if every marking in $\markn\in\reach_\net(\markn_0)$ and every transition $t$ enabled in $M$ satisfy $(\markn\cap\postset{\transn})\subseteq\preset{\transn}$. In this paper, we assume that all our Petri nets are safe.

From an initial marking of the net, one can recursively derive all possible
transitions and reachable markings, resulting in a marking graph
(Def. <ref>).

Let $\net=\tup{\place,\trans,\flow}$ be a net and $\states$ a set of markings. The marking graph induced by $\states$ is
a directed graph $\tup{\states,\edges}$ such that $\edges\subseteq \states\times \states$ contains $\tup{\markn,\markn'}$ iff $\markn\move{t}\markn'$ for some $\transn\in \trans$;
the arc $\tup{\markn,\markn'}$ is then labeled by $\transn$.

The reachability graph of a Petri net $\tup{\net,M_0}$ is the graph induced by $\reach_\net(\markn_0)$.

The reachability graph is always finite for safe Petri nets.

Petri net example from [11] in fig:basinexample, and its reachability graph in fig:basinstategr.

Figure <ref> shows the reachability graph for our running example <ref>.

Roughly speaking, the unfolding of
a Petri net $\petrinet$ is an acyclic Petri net (with particular structural properties), $\unfolding$, that reproduces exactly the
same behaviours as $\petrinet$.

We now give some technical definitions to introduce unfoldings formally. A  more extensive treatment  can be found, e.g., in [7, 6].

Let $\net=\tup{P,T,F}$ be a net and $x,y\in P\cup T$ two nodes
of $\net$.
We say that
$x$ is a causal predecessor of $y$, noted $x<y$, if there exists a
non-empty path of arcs from $x$ to $y$. We note $x\leq y$ if $x<y$ or $x=y$.
If $x\leq y$ or $y\leq x$, then $x$ and $y$ are said to be
causally related.
Transitions $u$ and $v$ are in direct conflict, noted $u\dircf v$, iff $\preset{u}\cap\preset{v}\ne
\emptyset$; nodes
$x$ and $y$ are in conflict, noted $x\conflict y$, if there exist
$u,v\in T$ such that $u\ne v$, $u\le x$, $v\le y$, and $u\dircf v$. We call
$x$ and $y$ concurrent, noted $x\concurrent y$, if they are neither
causally related nor in conflict.
A set of concurrent places is called a co-set.

Let $\onet=\tup{\oplace,\otrans,\oflow,\cutn_0}$ be a Petri net. We say that $\onet$
is an occurrence net if it satisfies the following properties:

* The causality relation $<$ is acyclic;

* $|\preset{\oplacen}|\le 1$ for all places $\oplacen\in\oplace$,
and $\oplacen\in \cutn_0$ iff $|\preset{\oplacen}|=0$;

* For every transition $\otransn\in\otrans$, $\otransn\conflict\otransn$ does not hold, and \(\{x \mid x \leq \otransn\}\) is finite.

Following the  convention in the unfolding literature, we refer to the
places of an occurrence net as conditions and to its transitions
as events. Due to the structural constraints, the firing
sequences of occurrence nets have special properties: if some condition $\oplacen$ is
marked during a run, then the token on $\oplacen$ was either present initially
or produced by one particular event (the single event in $\preset{\oplacen}$);
moreover, once the token on $\oplacen$ is consumed, it can never be replaced by
another token, due to acyclicity of $<$.

Let $\onet=\tup{\oplace,\otrans,\oflow,\cutn_0}$ be an occurrence net.
A set $\confign\subseteq \otrans$ is called a
configuration of $\onet$ if (i) $\confign$ is causally
closed, i.e. $e'<e$ and $e\in \confign$ imply $e'\in \confign$; and (ii) $\confign$ is conflict-free, i.e. if $e,e'\in\confign$,
then $\neg(e\conflict e')$.
In particular, for any $e\in E$, $\cone{e}\define\{e'\in E:~e'\myleq e\}$ and $\trunk{e}\define\{e'\in E:~e'< e\}$ are configurations, called the cone and stump of $\otransn$, respectively; any $\confign$ such that $\exists~e\in E:~\confign=\cone{e}$ is called a prime configuration.
Denote the set of all configurations of $\onet$ as $\configs(\onet)$, and its subset containing all finite configurations as $\finfigs(\onet)$, where we drop the reference to $\onet$ if no confusion can arise. The cut of a finite $\confign$, denoted $\cut(\confign)$, is the set of
conditions $(\cutn_0\cup\postset{\confign})\setminus\preset{\confign}$.
A run is a maximal element of $\configs(\onet)$ w.r.t. set inclusion; denote the set of $\onet$'s runs as $\Omega=\Omega(\onet)$, and its elements generically by $\omega$.
If $\confign\in\finfigs$, let the crest of $\confign$ be the set
$\crest{\confign}\define \max_<(\confign)$ of its maximal events.
We say that configuration $\confign$ enables event $\otransn$, written $ \confign\omove{\otransn}$, iff i) $\otransn\not\in\confign$ and ii) $\confign\cup\{\otransn\}$ is a configuration.

Configurations $\confign_1,\confign_2$ are in conflict, written $\confign_1\conflict\confign_2$, iff $(\confign_1\cup\confign_2)\not\in\configs$ or, equivalently, iff there exist $\otransn_1\in\confign_1$ and $\otransn_2\in\confign_2$ such that $\otransn_1\conflict\otransn_2$.[The use of the same symbol $\conflict$ is motivated by the fact that $\confign_1=\cone{\otransn_1}$ and $\confign_2=\cone{\otransn_2}$ implies $\confign_1\conflict\confign_2\Leftrightarrow \otransn_1\conflict\otransn_2$.]

Intuitively, a configuration is a set of events that can fire during
a firing sequence of $\petrinet$, and its cut is the set of conditions
marked after that firing sequence. Note that $\emptyset$ is a configuration, that $\crest{\emptyset}=\emptyset$,
and that $\cutn_0$ is the cut of the configuration $\emptyset$. The crest of a prime configuration $\cone{e}$ is $\{e\}$.

A prefix of the unfolding for the Petri net of Figure <ref>.

In Figure <ref>, the initial cut is $\cutn_0=\{b_1^1,b_2^1\}$;
we have prime configurations, e.g., $\{\alpha_1\}$, $\{\beta_1\}$, $\{\xi_1\}$, $\{\zeta_1\}$ etc,
and non-prime configurations $\{\alpha_1,\gamma_1\}$, $\{\alpha_1,\delta_1\}$ etc.

Definition of Unfoldings. Let $\petrinet=\tup{\place,\trans,\flow,\markn_0}$
be a safe Petri net. The unfolding $\unfolding=\tup{\oplace,\otrans,\oflow,\cutn_0}$ of
$\petrinet$ is an
occurrence net (equipped with a mapping $\fold$)
such that
the firing sequences and reachable markings of $\unfolding$ are exactly
the firing sequences and reachable markings of $\petrinet$ (modulo $\fold$), see below.
$\unfolding$ may be infinite; it can be inductively constructed as follows:

* The condition set $\oplace$ is a subset of $(\otrans\cup\{\bot\})\times \place$.
For a condition $\oplacen=\tup{\otransn,\placen}$, we will have $\otransn=\bot$ iff $\oplacen\in \cutn_0$;
otherwise $\otransn$ is the singleton event in $\preset{\oplacen}$. Moreover, $\fold(\oplacen)=\placen$.
The initial cut $\cutn_0$ contains exactly one condition $\tup{\bot,\placen}$
for each initially marked place $\placen\in\markn_0$ of $\petrinet$.

* The events of $\otrans$ are a subset of $2^\oplace\times \trans$.
More precisely, for every co-set $\oplace'\subseteq\oplace$ such that
%\{\,\fold(\oplacen)\mid \oplacen\in \oplace'\,\}=
\preset{\transn}$,

we have an event $\otransn=\tup{\oplace',t}$.
In this case, we add edges $\tup{\oplacen,\otransn}$
for each $\oplacen\in \oplace'$ (i.e. $\preset{\otransn}=\oplace'$), we set $\fold(\otransn)=\transn$, and for each
$\placen\in\postset{\transn}$, we add to $\oplace$ a condition $\oplacen=\tup{\otransn,\placen}$ connected by
an edge $\tup{\otransn,\oplacen}$.

Intuitively, a condition $\tup{\otransn,\placen}$ represents the possibility of putting
a token onto place $\placen$ through a particular set of events, while an event
$\tup{\oplace',\otransn}$ represents a possibility of firing transition $\otransn$ in a particular

Configurations and Markings.
The following fact from the literature  will be used below:

Fix a safe Petri net  $\petrinet=\tup{\place,\trans,\flow,\markn_0}$  and its unfolding $\unfolding=\tup{\oplace,\otrans,\oflow,\cutn_0, \fold}$.
Then for any two conditions (events) $\oplacen,\oplacen'$ ($\otransn,\otransn'$) such that $\oplacen\concurrent\oplacen'$ ($\otransn \concurrent \otransn'$), one has $\fold(\oplacen)\ne\fold(\oplacen')$ ($\fold(\otransn)\ne\fold(\otransn')$).
Moreover, every
finite configuration $\confign$ of $\unfolding$ represents a possible firing
sequence whose resulting marking corresponds, due to the construction of
$\unfolding$, to a reachable marking of $\petrinet$. This marking is defined as
$\marking{\confign}\define\{\,\fold(\oplacen)\mid \oplacen\in\cut(\confign)\,\}$.

This means, informally speaking, that any configuration of the system can be split into consecutive parts in such a way that each part is itself a configuration obtained by unfolding the Petri net `renewed' with the marking reached by the previous configuration. The following definition formalizes this.

Let $\onet=\tup{\oplace,\otrans,\oflow,\cutn_0}$ be an occurrence net.

For any finite configuration $\confign\in\finfigs(\onet)$, denote by $\onet_\confign\define\unfolding(\tup{\net,\marking{\confign}})$ the shift of $\onet$ by $\confign$.
$\confign$ is the concatenation of $\confign_1$ and $\confign_2$, written $\confign=\confign_1\concat\confign_2$,  iff one has

* $\confign_1\in\finfigs(\onet)$ and

* $\confign_2\in\finfigs(\onet_{\confign_1})$ and

Clearly, the empty configuration $\emptyset$ satisfies $\confign\concat\emptyset=\emptyset\concat\confign=\confign$. If $\confign =\confign_1\concat\confign_2$, write $\confign_1=\confign\leftchop\confign_2$
and $\confign_2=\confign\rightchop\confign_1$. Moreover, write

\begin{eqnarray*}
\confign=\konkat_{i=1}^n\confign_i&\text{iff}&
\confign=\confign_1\concat\ldots\concat\confign_n.
\end{eqnarray*}

In figure <ref>, setting $\confign_1\define\{\beta_1,\gamma_1\}$ , $\confign_2\define\{\zeta_1,\kappa_1\}$ and $\confign_3\define\{\beta_1,\gamma_1,\zeta_1,\kappa_1\}$, one has $\confign_3=\confign_1\concat\confign_2$ and consequently $\confign_1=\confign_3\leftchop\confign_2$ and $\confign_2=\confign_3\rightchop\confign_1$.

Complete Prefix.
In general, $\unfolding$
is an infinite net, but if $\petrinet$ is safe, then it is possible
to compute a finite prefix $\prefn$
of $\unfolding$ that is “complete”
in the sense that every reachable marking of $\petrinet$ has a reachable
counterpart in $\prefn$,  and vice

Let $\petrinet=\tup{\net,\markn_0}$ be a safe Petri net and
its unfolding. A finite occurrence net ${\prefn}=\tup{\oplace',\otrans',\oflow',\cutn_0}$
is said to be a prefix of $\unfolding$ if $\otrans'\subseteq \otrans$ is
causally closed, $\oplace'=\cutn_0\cup\postset{\otrans'}$, and $\oflow'$ is the restriction
of $\oflow$ to $\oplace'$ and $\otrans'$. A prefix $\prefn$ is said to be complete
if for every reachable marking $\markn$ of $\petrinet$ there exists a
configuration $\confign$ of $\prefn$ such that (i) $\marking{\confign}=\markn$,
and (ii) for each transition $\transn\in \trans$ enabled in $\markn$, there is an event
$\tup{\oplace'',\transn}\in \otrans'$ enabled in $\cut(\confign)$.

We shall write $\prefixn_0=\prefixn_0(\petrinet)$ to denote an arbitrary complete prefix of the unfolding of $\netn$.
It is known ([15, 7]) that the construction of such a complete prefix
is indeed possible, and efficient tools such as Mole ([27]) exist for this purpose.
While the precise details of this construction are out of scope for this paper;  some ingredients of it will play a role below, so we sketch them here.

Complete prefix scheme. The unfolding is stopped on each branch when some cutoff event is added. The criterion for classifying an event $\otransn$ as cutoff is given by  Marking equivalence: the marking $\marking{\cone{\otransn}}$ that $\otransn$ `discovers' has already been discovered by a smaller configuration. Now, the ordering relation $\adequate$ to compare two configurations must be an adequate order, i.e. $\confign_1\subseteq\confign_2$ must imply $\confign_1\adequate\confign_2$, to ensure the completeness of the prefix obtained. As shown in [7], for some  choices of $\prec$, the obtained prefix may be bigger than the reachability graph for some safe nets; however, if $\prec$ is a total order, the number of non-cutoff events of the prefix $\prefixn_0$ thus obtained never exceeds the size of the reachability graph.

We will assume throughout this paper that complete prefixes are computed according to some adequate total order, as is done in particular in the Mole tool [27]. Below, we will propose a new such order relation that underlies a novel concept of distance between markings.

The nested family $(\prefixn_n)_{n\geq 0}$ of finite prefixes. Denote the  complete prefix for $\petrinet$ obtained according to definition <ref> as $\prefixn_0$; we extend $\prefixn_0$ to increasing prefixes $\prefixn_1,\prefixn_1, \ldots$ as follows.
Starting at $n=0$,

* let $\configs^n\define\max(\configs(\prefixn_n))$,

* set $
\markis_n\define \{\markn\in 2^\place:~\exists~\confign\in\configs^n:~\markn=\marking{\confign}\}$,

* for all $\markn\in\markis_n$, compute a complete prefix $\prefix^\markn$ of $\tup{\net,\markn}$;

* obtain $\prefixn_{n+1}$ by appending, to every $\confign\in\configs^n$, a copy of $\prefix^{\marking{\confign}}$ to every $\confign\in\configs^n$.

§ DOOMED CONFIGURATIONS, AND HOW TO AVOID THEM

§.§ Bad, Free and Doomed Configurations and Markings.

In this section, we present an algorithm that identifies precisely those configurations of a Petri net unfolding from which one can no longer avoid reaching a certain long-term behaviour, its theoretical foundations, and some experimental results.
The formal setting here contains and extends the one established in [10], specialized to the 1-safe case. We assume that we are given a set of bad markings $\badstates\subseteq 2^{\places}$. Since we are interested in long-term behaviours, we assume that $\badstates$ is reachability-closed, i.e. $\markn\in\badstates$ and $\markn\move{}\markn'$ imply $\markn'\in\badstates$.

Define $\badconfs\define\{\confign\in\finfigs:\marking{\confign}\in\badstates\}$ as the set of bad configurations, and let $\badconfs_0$ be the set of configurations in $\badconfs$ that are contained in $\prefixn_0$.
$\badconfs\subseteq\configs$  is absorbing or upward closed, that is,
for all $\confign_1\in\badconfs$ and $\confign_2\in\finfigs$ such that $\confign_1\subseteq\confign_2$,  one must have $\confign_2\in\badconfs$.

For any  $\confign\in\configs$, let $\runs_\confign\define \left\{ \runn \in\runs:~\confign\subseteq \runn\right\}$ denote the maximal runs into which $\confign$ can evolve.
We are interested in
those finite configurations all of whose maximal extensions are `bad', where we consider infinite configurations as bad if they contain a bad finite configuration. We will call such configurations doomed, since from them, the system cannot avoid entering a bad marking sooner or later (and from then on, all reachable markings are bad).
Configuration $\confign\in\finfigs$ is doomed iff

\begin{eqnarray}
\label{eq:latbad} \forall~\runn\in\runs_\confign:~\exists~\confign'\in\finfigs:
\left\{\begin{array}{l}
\confign\subseteq\confign'\subseteq\runn\\
\land~\marking{\confign'}\in\badstates
\end{array}
\right.
\end{eqnarray}

The set of doomed configurations is denoted $\doofigs$; denote the set of minimal elements in $\doofigs$ by $\mindofigs$.
If $\confign$ is not doomed, it has at least one maximal extension that never reaches bad markings. We  call configurations that are not doomed free, and denote the set of free configurations by $\viables$.

All reachable markings are represented by at least one configuration.
Moreover, since the future evolution of $\netn$ depends only on the current marking,
$\marking{\confign_1}=\marking{\confign_2}$ for two configurations $\confign_1$ and $\confign_2$ implies that either both $\confign_1$ and $\confign_2$ are free, or both are doomed.
Therefore, by extension, we call $\marking{\confign}$ free or doomed whenever $\confign$ is.

Running Example. In the context of Figures <ref> and <ref>, we consider $\badstates$ the singleton set containing the marking $\markn_8=\{\place_8\}$. Clearly, $\confign_1\define\{\alpha_1,\gamma_1,\xi_1\}$ and $\confign_2=\{\beta_1,\delta_1,\eta_1\}$ satisfy $\marking{\confign_1}=\marking{\confign_2}=\markn_8$ and therefore $\confign_1,\confign_2\in\badconfs$. But note that $\confign_1'\define\{\alpha_1,\gamma_1\}$ and $\confign_2'=\{\beta_1,\delta_1\}$ produce markings outside  $\badstates$, but they are doomed since any extension of these configurations leads into $\badstates$. Therefore, $\confign_1',\confign_2'\in\badconfs$. On the other hand,
$\emptyset$ is free, as well as $\{\beta_1,\gamma_1\}$, $\{\alpha_1,\delta_1\}$, etc.
We note in passing that the Petri net in Fig <ref> allows to refine the understanding of the `tipping point' by showing that doom is not brought about by a single transition but rather the combined effect of two independent choices; this fact is obscured, or at least far from obvious, in the state graph shown in Figure <ref>.

Identifying free and doomed configurations belongs to the core objectives of this paper. In a first step towards that, Theorem <ref>  below uses a similar proof idea as Lemma 8 in [12] in the context of fault  diagnosis.
Let us first recall the notion of spoilers, introduced in   [12]:

A spoiler of transition $\transn$ (or  event $\otransn$) is any $\transn'\in\trans$ ($\otransn'\in\otrans$) such that $\preset{\transn'}\cap\preset{\transn}\ne\emptyset$ ($\preset{\otransn}\cap\preset{\otransn'}\ne\emptyset$).
We write $\spoil{\transn}$  ($\spoil{\otransn}$) for the set of $\transn'$s ($\otransn$'s) spoilers.

Note that $\transn\in\spoil\transn$ for all $\transn\in\trans$. The spoilers of $\transn$ are characterized by the fact that their firing cancels any enabling of $\transn$; that is, by being either in conflict with $\transn$, or identical with $\transn$.
A configuration $\confign\in\finfigs$ is free iff either a) there exists a finite maximal configuration $\confign'$ such that $\confign\subseteq\confign'\not\in\badconfs$, or b) there exist  configurations $\confign_1,\confign_2\in\finfigs$ such that

*  $\confign\subseteq\confign_1\subseteq \confign_2\notin\badconfs$;

* $\marking{\confign_1}=\marking{\confign_2}$;

*  for all events $\otransn\in\otrans$ such that $\confign_1\omove{\otransn}$, one has $\spoil{\otransn}\cap \confign_2\ne\emptyset$.

Some comments are in order before giving the proof of Theorem <ref>.
First of all, the requirement to check whether $\confign_2\not\in\badconfs$ can be met by checking whether $\marking{\confign_2}\in\clbadstates$.
Second, the spoiling condition (3) ensures that the process that takes $\confign_1$ to $\confign_2$ forms a loop whose iteration yields a run.

In the following, let $M:=\marking{C_1}$. We first prove the right-to-left implication. Case a) is obvious, so assume that b) holds.
Let $\confign_2=\confign_1\concat\hat{\confign}$; then by 2., we can append $\hat{\confign}$ to $\confign_2$, yielding a strictly increasing sequence of configurations $(\confign_n)_{n\in\NN}$
such that $\confign_{n+1}=\confign_n\concat \hat{\confign}$, and $\marking{\confign_n}=\markn$ for all $n$. By property 3, we know that $\hat{\confign}$ contains spoilers for all its initially enabled events , hence
no transition remains enabled forever, and $\omega\define\bigcup_{n\in\NN}\confign_n$ is a maximal configuration. It remains to show that $\omega$ contains no bad configuration:
Suppose that there is $C'\subseteq\omega$ with $C'\in\badconfs$. Since $C'$ is finite, we have $C'\subseteq C_n$ for some $n$. But then, $M=\marking{C_n}$ is reachable from $\marking{C'}\in\badstates$,
contradicting our assumptions. Thus $\omega$ never enters a bad state, and $\confign$ is free.

For the forward implication, assume that $\confign$ is free. Then there exists $\runn\in\runs_\confign$ such that $C'\notin\badconfs$ for all finite $C'\subseteq\runn$. If this $\runn$ can be chosen finite, then a) holds and we are done; so assume henceforth that $\runn$ must be chosen infinite. Clearly, there must exist a reachable marking $\markn$ that is visited an infinite number of times  by a family of  nested finite configurations $(\confign^n)_{n\in\NN}$ such that $\bigcup_{n\in\NN}\confign^n=\runn$. Let $\confign_1\define\confign^1$ and  $E':=\{\otransn:~\confign_1\omove{\otransn}\}$. Let $K$ be the smallest index such that for all $\otransn\in E'$, one has  $\spoil{\otransn}\cap\confign^K\ne\emptyset$; such a $K$ must exist since $\omega$ is maximal. Then $\confign_1$ and $\confign_2\define\confign^K$ have the required properties.

Notice that the proof could be restructured by observing that case a) of Theorem <ref> is indeed a special instance of case b). In fact, taking $\confign_1\define\confign_2\define\confign'$ with  $\confign'$ according to  case a), conditions 1 and 2 of part b) are obviously satisfied, and condition 3 holds vacuously since no event is enabled in $\confign_1=\confign'$.  We note in passeing that this observation is helpful in simplifying the implementation used for the experiments  below.

The interest of Theorem <ref> lies in the following fact:

For $\confign\in\finfigs$, checking whether $\confign$ is free can be done using finite prefix $\prefixn_1$ of $\unfolding(\net,\marking{\confign})$.

If $\confign$ is free, let $C_1$ and $C_2$ be the configurations witnessing this fact from Theorem <ref>, and let $M:=\marking{C_1}=\marking{C_2}$. If such configurations exist, then $C_1$ can be chosen from the complete prefix $\prefixn_0$, and $C_2$ can be chosen from $\prefixn_1$, notably
in the copy of  $\prefixn_{\marking{\confign_1}}$  appended after $\confign_1$.

Checking whether the  configuration $\confign_2$ thus found is in $\badconfs$ is immediate, since it suffices to check whether its marking is in $\badstates$, using the fact that  $\badstates$ is reachability-closed.
To check the spoiler condition (3) of Theorem <ref>, it suffices to check whether the conditions of the cut of $\confign_1$ that are not consumed by $\confign_2$ enable some event.

§.§ Finding Minimally Doomed Configurations: Algorithm Mindoo

Shaving and Rubbing. Let us start by observing that $\badconfs$, an upward closed set by construction, also has some downward closure properties, meaning one can restrict control to act on `small' configurations.

An event $\otransn$ is unchallenged iff there is no $\otransn'$ such that $\otransn\dircf\otransn'$, i.e. $\postset{(\preset{\otransn})}=\{\otransn\}$.

Let $\confign\in\finfigs$ and $\otransn\in\crest{\confign}$ unchallenged; set $\confign'\define\confign\backslash\{\otransn\}$. Then $\confign'\in\finfigs$, and $\runs_\confign=\runs_{\confign'}$.

$\confign'\in\finfigs$ holds by construction. Also, $\runs_{\confign}\subseteq\runs_{\confign'}$ follows from $\confign'\subseteq\confign$; it remains to show the reverse inclusion. Assume there exists $\runn\in\runs_{\confign'}\backslash\runs_\confign$; then $\confign\backslash\runn=\{\otransn\}$, and $\trunk{\otransn}\subseteq\runn$. By maximality, $\runn$ must contain some $\otransn'$ such that $\otransn\conflict\otransn'$. Then by definition, there are events $u\ne v$, $u\le e$, $v\le e'$, and $u\dircf v$.
In particular, $u\conflict e'$, and since $\{e'\}\cup\trunk{\otransn}\subseteq\runn$,
this implies $u=e$. But $e$ is unchallenged, so $v$ cannot exist, and neither can $\runn$.

[>=stealth,shorten >=1pt,node distance=cm,auto]
[cond] (b1) at (0, 0)  [label=left:$b_1$] ;
[cond] (b2) at (-1*,̱ -2*)̧  [label=left:$b_2$] ;
[cond] (b3) at (1*,̱ -2*)̧  [label=right:$b_3$] ;
[cond] (b4) at (-1*,̱ -4*)̧  [label=right:$b_4$] ;
[cond] (b5) at (1*,̱ -4*)̧  [label=left:$b_5$] ;
[cond] (b6) at (-3*,̱ -6*)̧  [label=left:$b_6$] ;
[cond] (b7) at (-1*,̱ -6*)̧  [label=left:$b_7$] ;
[cond] (b8) at (1*,̱ -6*)̧  [label=right:$b_8$] ;
[cond] (b9) at (3*,̱ -6*)̧  [label=right:$b_9$] ;
[cond] (b10) at (0*,̱ -8*)̧  [label=left:$b_{10}$] ;

[event] (x) at (0*,̱ -)̧ $x$;
[event] (y) at (-1*,̱ -3*)̧ $y$;
[event] (z) at (1*,̱ -3*)̧ $z$;
[event] (a) at (-3*,̱ -5*)̧ $\alpha$;
[event] (b) at (-1*,̱ -5*)̧ $\beta$;
[event] (c) at (1*,̱ -5*)̧ $\gamma$;
[event] (d) at (3*,̱ -5*)̧ $\delta$;
[event] (u) at (0*,̱ -7*)̧ $u$;

[->] (b1)  edge (x)
(x)  edge (b2)
(x)  edge (b3)
(b2) edge (y)
(b3) edge (z)
(y) edge (b4)
(z) edge (b5)
(b4) edge (a)
(b4) edge (b)
(b5) edge (c)
(b5) edge (d)
(a) edge (b6)
(b) edge (b7)
(c) edge (b8)
(d) edge (b9)
(b7) edge (u)
(b8) edge (u)
(u) edge (b10)

An occurrence net. With $\confign\define\{x,y,z,\beta,\gamma\}$ and $\confign'\define\confign\cup\{u\}$, suppose $\badstates=\{\marking{\confign'}\}=\fold(\{b_{10}\})$. Then  $\shaved{\confign'}=
\confign$, and $\confign$ is doomed. Moreover, $\confign\in\mindofigs$ since both
$ \confign_4\define\confign\backslash\{\beta\}$ and $ \confign_5\define\confign\backslash\{\gamma\}$ are free.

A configuration $\confign\in\finfigs$ such that $\crest{\confign}$ contains no unchallenged event is called shaved.

Clearly, every $\confign\in\finfigs$ contains a unique maximal shaved configuration, which we call $\shaved{\confign}$; it can be obtained from $\confign$ by recursively `shaving away' any unchallenged $\otransn\in\crest{\confign}$, and then continuing with the new crest, until no unchallenged events remain.

Example. In the context of Figure <ref>, for $\confign_1=\{x,y,z\}$ and $\confign_2=\confign_1\cup\{\beta,\gamma,u\}$, one has $\shaved{\confign_1}=\emptyset$ since $x$, $y$, and $z$ are unchallenged, and $\shaved{\confign_2}=\confign_1\cup\{\beta,\gamma\}$ since $u$ is unchallenged but neither $\beta$ nor $\gamma$ are.
Note that in the unfolding of the running example shown in Figure <ref>, the $\kappa$-labeled events are the only unchallenged ones.

As a consequence of Lemma <ref>, any $\confign\in\finfigs$ is in $\badconfs$ iff $\shaved{\confign}$ is. Still,  it may be possible that such a $\shaved{\confign}$ can still be reduced further by removing some of its crest events. This would be the case, e.g., if two conflicting events both lead to a bad state. Thus, given a crest event $e$, we test whether $\confign\backslash\{\otransn\}$ is free (e.g. because some event in conflict with $\otransn$ may allow to move away from doom) or still doomed. If the latter is the case, then $\confign$ was not minimally doomed, and analysis continues with $\confign\backslash\{\otransn\}$ (we say that we `rub away' $\otransn$). If $\confign\backslash\{\otransn\}$ is free, we leave $\otransn$ in place and test the remaining events from $\crest{\confign}$. A configuration that is shaved and from which no event can be rubbed away is minimally doomed.

Algorithm <ref> uses a `worklist' set $\worklist$ of doomed, shaved configurations to be explored; $\worklist$ is modified when a configuration is replaced by a set of rubbed (and again, shaved) versions of itself, or when a configuration $\confign$ is identified as minimally doomed, in which case it is removed from $\worklist$
and added to $\mindooout$.
Algorithm MinDoo
Safe Petri Net $\petrinet=\tup{\place,\trans,\flow,\markn_0}$ and $\badstates\subseteq 2^{\place}$
The set $\mindooout$ of $\petrinet's$ $\subseteq$-minimal doomed configurations
$\worklist \gets \emptyset$

Pick $\confign\in\worklist$;
$\addit\gets \true$
$(\confign\backslash\crest{\confign})$ is doomed
$\addit\gets \false$
$\worklist \gets \worklist\cup\{\confign'\}$

$(\confign\backslash\{\otransn\})$ is doomed
$\addit\gets \false$
$\worklist \gets \worklist\cup\{\confign'\}$

Every branch stops when a minimally doomed configuration is reached, i.e., a doomed configuration $\confign$ such by rubbing off any crest event $e$ from $\confign$ makes it free, i.e. $\confign\backslash\{e\}$ is free for all $e\in\crest{\confign}$. When the worklist is empty,  all minimally doomed configurations have been collected in $\mindooout$.
Note that if $\emptyset\in\worklist$ at any stage during the execution of Algorithm Mindoo, then
$\emptyset$ will be added to $\mindooout $, since Mindoo will not enter the second foreach-loop in that case.
In fact, if this situation arises, every configuration is doomed, and thus  $\emptyset$ is the unique minimally doomed configuration.

The configurations produced in the course of the search strictly decrease w.r.t both size and inclusion. Moreover, an upper bound on the prefixes explored at each step is given by $\badconfs$, itself strictly contained in the complete finite prefix used to find all bad markings.
According to [7], this prefix can be chosen of size equal or smaller (typically: considerably smaller) than the reachability graph of $\petrinet$.

For any safe Petri net $\petrinet=\tup{\net,\markn_0}$ and bad states set $\badstates\subseteq \reach_\net{(\markn_0)}$, Algorithm MinDoo terminates, with output set $\mindooout$ containing exactly all minimal doomed configurations, i.e. $\mindooout=\mindofigs$.

Termination follows from the finiteness of $\min_\subseteq(\badconfs_0)$,
since in  each round of MinDoo there is one configuration $\confign$ that is either replaced by a set of strict prefixes or removed from $\worklist$. Therefore, after a finite number of steps $\worklist$ is

According to Lemma <ref>, the status (doomed or free) of a given finite configuration can effectively be checked on a fixed finite prefix of $\unfolding$.
Assume that after termination of MinDoo, one has $\confign\in\mindooout$; we need to show $\confign\in\mindofigs$. Clearly, when $\confign$ was added to $\mindooout$, it had been detected as doomed; it remains to show that $\confign$ is also minimal with this property. Assume that there is $\confign'\subsetneq\confign$ that as doomed as well. But in that case  there exists $\otransn\in\crest{\confign}$  such that $\confign'\subseteq(\confign\backslash\{\otransn\})\subsetneq\confign$, which implies that this  $(\confign\backslash\{\otransn\})$ is doomed as well. But then $\addit$ has been set to $\false$ in the second foreach-loop, before $\confign$ could have been added to $\mindooout$.

Conversely, let $\confign\in\mindofigs$. Then $(\confign\backslash\{\otransn\})$ is free for all $\otransn\in\crest{\confign}$; the variable $\addit$ remains thus at the value $\true$ because no round of the second foreach-loop can flip it. Thus $\confign$ is added to $\mindooout$, from which MinDoo never removes any configuration.

§.§ Implementation and Experiments.

A prototype implementation of Mindoo is available at [21].
It takes as input a safe Petri net in the PEP format and relies on Mole [27] for computing the
initial finite prefix $\prefixn_0$ and its extensions.
Algorithm <ref> is implemented in Python, where the identification of maximal
configurations, bad configurations, as well as the verification of doomed status of a configuration
is performed in Answer-Set Programming (ASP) employing the Clingo solver [9], a logic programming technology close to SAT solving.

We illustrate in Table <ref> the behavior of the implementation on different
instances of Petri nets modeling biological processes.

Statistics of Algorithm <ref> on Petri net models of biological systems.
The size of $\prefixn_0$ and $\prefixn_1$ is the number of their events;
“# min doomed cfg” is the number of minimally doomed configurations;
“# doom checks” is the number of SAT checks for doom status of a configuration.
“time” is the total computation time on a 1.8Ghz CPU

Model     size $\prefixn_0$     size $\prefixn_1$     # min doomed cfg     # doom checks     time
Lambda switch     126     1,060     10     29     1s

Cell death receptor     791     19,262     57     407     37s

Budding yeast cell cycle     1,413     184,363      114     837     8m3s

In each case, we report the size (number of events) of prefixes $\prefixn_0$ and $\prefixn_1$
(including cut-off events), the number of minimally doomed configurations, and the number of
configurations which have been tested for being doomed.
The purpose of the conducted experiments was to study the tractability of our approach on literature models of biological systems for which the study of doomed configuration was relevant.
As exhibited in [3], one of the first potential bottleneck is the tractability of the computation of the finite complete prefix $\prefixn_0$ and the enumeration of maximal configurations, which is required for computing $\prefixn_1$. Then, our experiments have focused on assessing how evolved the number of minimally doomed configurations, the number of candidate configurations screened by Algorithm <ref>, and the overall computation time, with different sizes of prefixes $\prefixn_1$.

We selected 3 models published as Boolean networks, which can be translated as safe Petri nets using the encoding described in [3] implemented in the tool Pint [22].
The “Lambda switch” model [28] comprises 11 places and 41 transitions, and possesses two limit
behaviors, one being a deadlock, marked as a bad marking.
The “Cell death receptor” model [2] comprises 22 places and 33 transitions, and reproduces a bifurcation
process into different cell fates, one of which has been declared as bad (apoptosis).
In these two cases, the minimally doomed configurations identify configurations in which a decisive event has just taken place, committing the system to the attractor marked as bad.
The “Budding yeast cell cycle” model [19] comprises 18 places and 32 transitions, and represents the
oscillation of gene activity during the cell cycle. In this model, the cycle can exit and eventually
reach a marking corresponding to all genes being inactive, which is our bad marking.
In this later case, the minimally doomed configurations identify precisely when the system exits its oscillatory behavior.

It appears that the computation time for identifying minimally doomed configurations seems mostly affected by the size
of $\prefixn_1$ for the verification of the doom property of a configuration by ASP solving,
implementing the conditions of Theorem <ref>.
In each case, the number of minimally doomed configurations is a fraction of the size of the finite complete prefix $\prefixn_0$. Future work may explore compact representations of the set of minimally doomed configurations, as they typically share a large amount of events, and may ease biological interpretations.

§ PROTECTEDNESS

§.§ Cliff-Edges and Ridges.
From the minimal doomed configurations, we derive the critical `points' at which a run becomes doomed:

An event set $\watershed\subseteq\otrans$ is called a cliff-edge iff there exists a minimally doomed configuration $\confign\in\mindofigs$ such that $\watershed=\crest{\confign}$. The set of cliff-edges is denoted $\tips$. The folding $\ridge\define\fold(\watershed)\subseteq\trans$ of a cliff-edge $\watershed$ is called a ridge.

To complete the map of the evolutional landscape for $\netn$, it is important to find, in a bounded prefix of the unfolding, all ridges that determine the viability of a trajectory. Notice that the completeness of prefix $\prefixn_0$ only guarantees that all reachable markings of $\netn$ are represented by at least one configuration of $\prefixn_0$; this does not extend to a guarantee that all concurrent steps that lead into a doomed marking can be found in $\prefixn_0$ as well. Fortunately, one has:

For every ridge $\ridge$ of $\netn$ there is a witness in $\prefixn_0$, i.e. there exists a minimally doomed configuration $\confign$ in $\prefixn_1$ such that $\fold(\crest{\confign})=\ridge$.

Fix $\ridge$, and let $\confign_\ridge$ be any configuration such that $\fold(\crest{\confign_\ridge})=\ridge$; set $\markn^\confign\define\marking{\confign_\ridge}$, and let $\markn_\ridge^\confign$ the unique reachable marking such that $\markn_\ridge^\confign\move{\ridge}\markn^\confign$. Then any such  $\markn_\ridge^\confign$ is represented by some $\confign^\ridge$ in $\prefixn_0$. By construction, there exists a cliff-edge $\watershed$ such that $\confign^\ridge\omove{\watershed}$ and $\fold(\watershed)=\ridge$. Then $\confign\define\confign^\ridge\cup\watershed$ is a minimally doomed configuration that lies within $\prefixn_1$.

§.§ Measuring the Distance from Doom

With the above, we have the tools to draw a map of the `landscape' in which the system evolves, with doomed zones and cliff-edges highlighted. What we wish to add now is to assist navigation in this landscape: we intend to give a meaningful measure of how well, or badly, a current system state is protected against falling from a cliff-edge. We chose to measure this distance not in terms of the length of paths, or similar notions, but rather in terms of the choices that are made by the system in following  a particular path.

Consider a  configuration $\confign$ and the nonsequential process that it represents. Some of the events in $\confign$ can be seen as representing a decision, in the sense that their occurrence took place in  conflict with some event that was enabled by some prefix of  $\confign$. The number of such events gives a measure of the information contained in $\confign$, in terms of the decisions necessary to obtain $\confign$:

Let $\confign\in\finfigs$, and define

\begin{eqnarray*}\label{eq:decision}
\dheight{\confign}&\define&\left|\left\{\otransn\in\confign:~\exists~\otransn'\in\otrans:~
\otransn\strcf^\confign\otransn'
\right\}\right|,
\end{eqnarray*}

where $\strcf$ is the strict $\confign$-conflict relation defined, for all $\otransn\in\confign$, by

\begin{eqnarray*}
\otransn\strcf^\confign\otransn'&\setgdw&\otransn\dircf\otransn'~\land~\trunk{\otransn'}\subseteq\confign.
\end{eqnarray*}

$\dheight{\confign}$ is  called the decisional height of $\confign$.

In Figure <ref>, the configuration $\confign_1=\{\xi_1,\alpha_1,\gamma_1\}$  satisfies $\dheight{\confign_1}=2$, whereas for $\confign_0=\{\beta_1\}$, one has $\dheight{\confign_0}=1$.

Note that $\strcf^\confign$ is more restrictive than direct conflict $\dircf$; it is also more restrictive than  the
immediate conflict in the literature (e.g. [1]). It is closely dependent on the configuration $\confign$ under study, and describes precisely those events against which the process had to decide in performing $\confign$.

Illustration of direct conflict.

Now, for any free marking $\markn$ (or, equivalently, any free configuration $\confign$ such that $\marking{\confign}=\markn$),
we wish to measure the threat represented by doomed markings reachable from $\markn$: how far away from doom is the system when it is in $\markn$ ? Using the decisional height introduced above, we can define a height difference in terms of the conflicts that lead from one marking to another:

For $\confign\in\finfigs$, let

\begin{eqnarray}
\mindofigs_\confign&\define & \left\{ \begin{array}{lcr}
\{\confign'\in\mindofigs:~\confign\subseteq\confign'\}&: &
\confign\in\viables\\
\{\confign\} &:& \confign\in\badconfs
\end{array}\right.
\end{eqnarray}

The protectedness of $\confign$ is then

\begin{eqnarray}
\schutz{\confign}&\define& \min_{\confign'\in\mindofigs_\confign}\left\{\dheight{\confign'\rightchop\confign}\right\}
\end{eqnarray}

In Figure <ref>, with the definitions introduced there, $\schutz{\confign}=\schutz{\confign'}=0$. Setting $\confign_1\define\{x\}$, $\confign_2\define\{x,y\}$, $\confign_3\define\{x,z\}$, $\confign_4\define\confign_2\cup\confign_3$, $\confign_5\define\confign_4\cup\{\beta\}$,
and $\confign_6\define\confign_4\cup\{\gamma\}$,
one further has

\begin{eqnarray*}
\schutz{\confign_1}=\schutz{\confign_2}=\schutz{\confign_3}&=&2\\
\schutz{\confign_4}=\schutz{\confign_5}&=&1.
\end{eqnarray*}

Returning to Figure <ref>, suppose that $\confign'=\{x,y,\beta\}$ is the only minimally doomed configuration. Then for $\confign=\{x,z,\alpha\}$ as above, we have $\schutz{\confign}=1$, because the only direct conflict here is the one between $z$ and $y$.

Note that the definition of protectedness is parametrized by the choice of conflict relation in computing $\dheight{\bullet}$. Using direct conflict instead of strict conflict would increase $\dheight{\bullet}$ and lead to an overevaluation of protectedness.

To see the point, consider the occurrence net in Figure <ref>. Let $\confign_\alpha=\{x,z,\alpha\}$, $\confign_\beta=\{x,y,\beta\}$  and $\confign_\gamma=\{x,y,\alpha,\gamma\}$. We have
$\dheight{\confign_\alpha}=1$, $\dheight{\confign_\beta}=3$ and $\dheight{\confign_\gamma}=2$. Were $\strcf$ replaced by $\dircf$ in the computation of $\dheight{\bullet}$, these values would not change except for $\confign_\alpha$ where it would change to $2$. As a result, if $\confign\in\mindofigs$, the protectedness of the empty configuration would be evaluated as $2$, whereas by our definition $\schutz{\emptyset}=1$. Indeed, $\emptyset$ is just one wrong decision away from doom, and this is what protectness is meant to express.

§.§ Computing Protectedness is Feasible

Computation of $\schutz{\bullet}$ does not require any larger data structure than those already required for  computing $\mindofigs$ according to Lemma <ref>:

There is a complete prefix scheme  producing a complete prefix $\prefixn_0$  whose size is bounded by the number of reachable markings, and such that for every finite configuration $\confign,$ $\schutz{\confign}$ can be computed on $\prefixn_0(\marking{\confign})$.

If $\mindofigs\cap\configs(\prefixn_0)=\emptyset$, then all extensions of $\confign$ are free, and we are done. Otherwise,
the crucial step is to find an adequate total order $\adequate$ on finite configurations, that ensures that $\prefixn_0$ contains at least one  minimally doomed configuration that  minimizes $\dheight{\bullet}$ over all minimally doomed configurations in  $\unfolding(\marking{\confign})$.
The following order $\prec$ is obtained by modifying the total order $\prec_F$ introduced in [7], Def. 6.2.:
For $\confign_1,\confign_2\in\finfigs$, write $\confign_1\adequate\confign_2$ iff either

* $\dheight{\confign_1}<\dheight{\confign_2}$, or

* $\dheight{\confign_1}=\dheight{\confign_2}$ and $\confign_1\ll\confign_2$, or

* $\dheight{\confign_1}=\dheight{\confign_2}$ and $\confign_1\equiv\confign_2$, and $FC(\confign_1)\ll FC(\confign_2)$,

where $\ll$ ($\equiv$) denote lexicographic ordering (lexicographic equivalence) wrt some total ordering of the transition set $\trans$, and $FC$ denotes Cartier-Foata normal form. The proof of Theorem 6.4. of [7] extends immediately, proving that $\prec$ is an adequate total order; therefore, Lemma 5.3. of [7] applies, hence any complete prefix $\prefixn_0^\prec$ obtained via the scheme using $\prec$ is bounded in size by the reachability graph. Now, let $\configs^*$ be the set of configurations from $\mindofigs(\marking{\confign})$ that minimize $\dheight{\bullet}$; by construction of $\prec$, one has $\configs^*\cap\configs(\prefixn_0^\prec)\ne\emptyset$.

§ DISCUSSION

The results presented here give a toolkit for the analysis of tipping situations in a safe Petri net, i.e. when and how a basin boundary is crossed; an algorithmic method for finding minimally doomed configuration has been developed, implemented and tested.

Moreover, we have introduced a measure of protectedness that indicates the number of decisions that separate a free state from doom. It uses an intrinsic notion of decisional height that allows to warn about impending dangerous scenarios; at the same time, this height is also 'natural' for unfoldings, in the sense that it induces an adequate linear order that allows to compute complete prefixes of bounded size.

On a more general level, the results here are part of a broader effort to provide a discrete, Petri-net based framework for dynamical systems analysis in the life sciences. The applications that we target lie in systems biology and ecology.

Future work will investigate
possibilities for  Doom Avoidance Control, i.e. devising strategies that allow to steer away from doom; we expect to complement the existing approaches via structural methods of e.g. Antsaklis et al [13], and also the unfolding construction of Giua and Xie [10].  A crucial question is the knowledge that any control player can be assumed to have, as a basis for chosing control actions. We believe the protectedness measure is a valid candidate for coding this information, so that a controller may take action when the system is too close to doom (wrt some thresholds to be calibrated) but there still remain decisns that can be taken to avoid it.
Evaluating this option, along with other approaches, must, however, be left to future work.

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# A new Representation of $\alpha$-Bernstein Operators

Jamshid Saeidian<EMAIL_ADDRESS>Bahareh Nouri  Faculty of Mathematical
Sciences and Computer, Kharazmi University, No. 50, Taleghani ave., Tehran,
Iran.

###### Abstract

The $\alpha$-Bernstein operators were initially introduced in the paper by
Chen, X., Tan, J., Liu, Z., Xie, J. (2017) titled ”Approximation of Functions
by a New Family of Generalized Bernstein Operators” (Journal of Mathematical
Analysis and Applications, 450(1), 244-261). Since their introduction, these
operators have served as a source of inspiration for numerous research
endeavors. In this study, we propose a novel technique, founded on a recursive
relation, for constructing Bernstein-like bases. A special case of this new
representation yields a novel portrayal of Chen’s operators. This innovative
representation enables the discovery of additional properties of
$\alpha$-Bernstein operators and facilitates alternative and more
straightforward proofs for certain theorems.

###### keywords:

Bernstein-like operator , Generalized operators , Positive linear operator ,
Shape-preserving approximation , Voronovskaja-type theorem

###### MSC:

[2020] 41A25 , 26A15 , 47A58 , 41A10

##  1 Introduction

The Bernstein polynomials, rooted in the fertile ground of approximation
theory, stand as an enduring pillar of mathematical research and practice.
These polynomials, introduced by S.N. Bernstein in 1912, not only define a
fundamental class of algebraic polynomials but also wield a profound influence
in mathematical analysis. It was through the lens of Bernstein polynomials
that S.N. Bernstein provided the first constructive proof of Weierstrass’
approximation theorem, a seminal contribution that continues to reverberate
through the annals of mathematical history [1].

Defined as:

$B_{n,k}(z)={n\choose
k}z^{k}(1-z)^{n-k},~{}~{}k=0,...,n~{}~{}\&~{}~{}n\in\mathbb{N},$

these polynomials are instrumental in approximating functions over the
interval $[0,1]$. The core of this approximation is encapsulated in Bernstein
operators with the formula:

$\mathcal{B}_{n}(f;z)=\sum_{k=0}^{n}B_{n,k}(z)f(\frac{k}{n}),~{}~{}~{}z\in[0,1],$

where a weighted sum of function values at equidistant points on the interval
offers a versatile tool for approximation.

In 2017, Chen et al. [2] proposed an innovative modification of Bernstein
operators, centered around the concept of $\alpha$-Bernstein polynomials as
follow:

######  Definition 1.

[2] The $\alpha$-Bernstein polynomials of degree $n$, $B_{n,k}^{(\alpha)}(z)$,
are defined by $B_{1,0}^{(\alpha)}(z)=1-z$, $B_{1,1}^{(\alpha)}(z)=z$ and

$B_{n,k}^{(\alpha)}(z)=\left[\binom{n-2}{k}(1-\alpha)z+\binom{n-2}{k-2}(1-\alpha)\left(1-z\right)+\binom{n}{k}\alpha
z(1-z)\right]z^{k-1}(1-z)^{n-k-1}$ (1)

where $n\geq 2$, $z\in[0,1]$ and $\binom{m}{l}$ denotes the binomial
coefficients.

Then, the corresponding $\alpha$-Bernstein operators are defined to read

$\mathcal{B}_{n}^{\alpha}(f;z)=\sum_{k=0}^{n}B_{n,k}^{\alpha}(z)f(\frac{k}{n}),~{}~{}~{}z\in[0,1],$

for any $f\in C[0,1]$, i.e. continuous functions.

In recent years, the field of approximation theory has experienced noteworthy
advancements, propelled in part by the pioneering work of Chen and his
colleagues. Their contributions to the theory of Bernstein-like operators have
not only broadened our comprehension but also acted as a catalyst for
subsequent exploration and refinement. This seminal work has sparked
significant interest and inspired further investigations into various families
of Bernstein-like operators [3, 4, 5, 6].

The $\alpha$-Bernstein operators are commonly identified as generalized
Bernstein operators in the literature, and researchers have further developed
the underlying concept of this modification to generalize other prominent
operators in the field. Among these extensions, we can mention operators
linked to Schurer polynomials, Kantorovich polynomials, Stancu polynomials,
$q$-Bernstein polynomials, Durrmeyer polynomials, as well as other operators
like Favard-Szász-Mirakjan and Baskakov, these extensions, as well as other
related works could be traced in [3, 6, 7, 8, 9, 10].

Inspired by Chen’s research, this paper presents a novel framework for
generating Bernstein-like basis functions. This innovative structure can be
viewed as an extension of Bernstein polynomials, aligning with these classical
polynomials and their $\alpha$-Bernstein modifications [2] in specific
scenarios. This framework enables us to introduce a recursive representation
of the $\alpha$-Bernstein operators, a representation that promises to enhance
our comprehension of its underlying principles and applications. By revisiting
the operator’s structure through a fresh lens, we aim to offer a unique
perspective on its mathematical properties and computational implications.

Our primary focus lies in presenting this new representation and demonstrating
its utility through alternative proofs for selected theorems associated with
$\alpha$-Bernstein operators. Moreover, it provides us with the opportunity to
uncover some additional properties of $\alpha$-Bernstein operators.

The structure of the rest of this paper is as follows: Section 2 delineates
the novel representation and validates its equivalence with Chen’s formula.
Section 3 examines the properties of this new presentation within the context
of the Bernstein-like bases. In Section 4, on one hand, two novel properties
of the $\alpha$-Bernstein operators are derived, and subsequently, we provide
alternative proofs for key theorems, illustrating the practical implications
of our proposed representation. Finally, Section 5 presents concluding remarks
and proposes potential avenues for future research.

##  2 A general structure for constructing Bernstein-like bases

The idea of recursively constructing Bernstein-like basis functions has
already been employed by researchers, Yan and Liang [11] utilized this
approach to construct a Bezier-like model and Li [12] extended the idea to
define a parameter-based Bernstein-like basis, suitable for constructing
Bezier-type curves. Additionally, Bibi et al. [13] introduced the so-called
generalized hybrid trigonometric Bernstein basis, and Ameer and his colleagues
[14] developed generalized Bernstein-like basis functions. Both of these
studies further contribute to the recursive construction of Bernstein-like
bases.

The idea is straightforward: identify an appropriate set of initial bases,
such as $\\{F_{2,0},F_{2,1},F_{2,2}\\}$, and generate higher-order bases
through the established recursive relation of Bernstein polynomials [15]:

$F_{n,i}(z)=(1-z)F_{n-1,i}(z)+zF_{n-1,i-1}(z)\quad\text{for }z\in[0,1],\quad
i=0,1,2,\ldots,n.$ (2)

The count of initial bases, hereafter referred to as the starting basis, can
be any natural number. Yan and Liang [11] and Ameer et al. [14] both employed
three polynomials of degree 4, Li [12] selected four polynomials of degree 3
as the starting basis, and Bibi et al. [13] utilized three hybrid
trigonometric functions. It is essential for the starting basis functions to
exhibit as many fundamental properties of Bernstein bases as possible.

We stick to this technique and propose a general structure to define a family
of Bernstein-like bases.

######  Definition 2.

The starting Bernstein-like basis functions of order two are defined for
$z\in[0,1]$ as

$\displaystyle g_{2,0}(z)$ $\displaystyle=$ $\displaystyle az+b+\varphi(z),$
$\displaystyle g_{2,1}(z)$ $\displaystyle=$ $\displaystyle
1-2b-a-2\varphi(z),$ (3) $\displaystyle g_{2,2}(z)$ $\displaystyle=$
$\displaystyle-az+b+a+\varphi(z),$

where $a,b\in\mathbb{R}$ and the real valued function $\varphi$ must satisfy
some conditions. The higher order Bernstein-like bases are generated by
recursive relation (2).

######  Lemma 1.

The Bernstein-like basis functions of order $n\geq 2$, as defined in
Definition 2, satisfy the partition of unity property, expressed as
$\sum_{i=0}^{n}g_{n,i}(z)=1$. Additionally, if the function $\varphi$ is
constrained to satisfy $\varphi(1-z)=\varphi(z)$, then a symmetry feature
emerges: $g_{n,i}(z)=g_{n,n-i}(1-z)$ for $i=0,1,2,\ldots,n$.

One can impose further restrictions on the function $\varphi$ and the real
values $a$ and $b$ in order to satisfy more fundamental properties of
classical Bernstein polynomials. In fact, the non-negativity and end-point
coincidence with Bernstein bases are two unavoidable features.

######  Example 1.

When we set $a=-1$, $b=0$ and $\varphi(z)=z^{2}-z+1$, the polynomials in
equation (3) reduce to the Bernstein polynomials of degree 2, thus Definition
2 constructs the classical Bernstein polynomials.

######  Example 2.

Authors in [16] presented a special case of (3) where they used the function
$\varphi(z)=\sqrt{\left(1-\nu\right)\left(z^{2}-z\right)+\frac{1}{4}}$, along
with $a=-1$ and $b=\frac{1}{2}$. It is verified that the so-called $sq$-basis
functions also satisfy the non-negativity and end-point interpolation
properties. Furthermore, the corresponding operators exhibit the most common
features of the classical Bernstein operators [17].

##  3 New representation of the $\alpha$-Bernstein polynomials

Setting $a=-1$, $b=1$ and $\varphi(z)=\alpha(z^{2}-z)$, in equation (3), leads
us to a set of starting basis functions as follows:

$\displaystyle F_{2,0}(z)$ $\displaystyle=$ $\displaystyle
1-z+\alpha\left(z^{2}-z\right),$ $\displaystyle F_{2,1}(z)$ $\displaystyle=$
$\displaystyle-2\alpha\left(z^{2}-z\right),$ (4) $\displaystyle F_{2,2}(z)$
$\displaystyle=$ $\displaystyle z+\alpha\left(z^{2}-z\right).$

Now employing the recurrence relation (2), one constructs a set of basis
functions which its equivalence with $\alpha$-Bernstein polynomials of Chen
[2] are easily verified, thus we have:

######  Theorem 1.

The procedure presented in Definition 2, along with starting basis functions
(4), generates the $\alpha$-Bernstein polynomials.

The recursive definition of the $\alpha$-Bernstein polynomials provides an
opportunity to derive practical computational formulas, one of which is
articulated in the following lemma.

######  Lemma 2.

The $\alpha$-Bernstein polynomials, defined by equations (2) and (4), can be
derived through the following relation:

$\displaystyle
F_{n,i}(z)=\sum_{j=0}^{n-2}B_{n-2,j}(z)F_{2,i-j}(z),~{}~{}i=0,1,\cdots,n,$ (5)

where $B_{n-2,j}$ are the classical Bernstein polynomials of order $n-2$.

###### Proof.

We utilize the method of induction to establish the validity of the result,
initiating with the base case $n=3$:

$\displaystyle F_{n,0}(z)$ $\displaystyle=$
$\displaystyle\sum_{j=0}^{1}B_{1,j}(z)F_{2,-j}(z)=B_{1,0}(z)F_{2,0}(z)=(1-z)F_{2,0}(z),$
$\displaystyle F_{n,1}(z)$ $\displaystyle=$
$\displaystyle\sum_{j=0}^{1}B_{1,j}(z)F_{2,1-j}(z)=B_{1,0}(z)F_{2,1}(z)+B_{1,1}(z)F_{2,0}(z)=(1-z)F_{2,1}(z)+zF_{2,0}(z),$
$\displaystyle F_{n,2}(z)$ $\displaystyle=$
$\displaystyle\sum_{j=0}^{1}B_{1,j}(z)F_{2,2-j}(z)=B_{1,0}(z)F_{2,2}(z)+B_{1,1}(z)F_{2,1}(z)=(1-z)F_{2,2}(z)+zF_{2,1}(z),$
$\displaystyle F_{n,3}(z)$ $\displaystyle=$
$\displaystyle\sum_{j=0}^{1}B_{1,j}(z)F_{2,3-j}(z)=B_{1,1}(z)F_{2,2}(z)=zF_{2,2}(z),$

the desired outcome is achieved.

Now, assuming the validity of the relation for $n-1$, we obtain:

$\displaystyle
F_{n-1,i}(z)=\sum_{j=0}^{n-3}B_{n-3,j}(z)F_{2,i-j}(z),~{}~{}i=0,1,\cdots,n-1,$

In the next step, we aim to establish the validity of the relation for $n$.
Utilizing the recursive relation (2), for $i=0,1,\cdots,n$, we have:

$\displaystyle F_{n,i}(z)$ $\displaystyle=$
$\displaystyle(1-z)F_{n-1,i}(z)+zF_{n-1,i-1}(z)$ $\displaystyle=$
$\displaystyle(1-z)\sum_{j=0}^{n-3}B_{n-3,j}(z)F_{2,i-j}(z)+z\sum_{j=0}^{n-3}B_{n-3,j}(z)F_{2,i-1-j}(z)$
$\displaystyle=$
$\displaystyle(1-z)B_{n-3,0}(z)F_{2,i-0}(z)+\sum_{j=1}^{n-3}(1-z)B_{n-3,j}(z)F_{2,i-j}(z)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}+\sum_{j=1}^{n-3}zB_{n-3,j-1}(z)F_{2,i-j}(z)+zB_{n-3,n-3}(z)F_{2,i-(n-2)}(z)$
$\displaystyle=$
$\displaystyle(1-z)B_{n-3,0}(z)F_{2,i-0}(z)+\sum_{j=1}^{n-3}[(1-z)B_{n-3,j}(z)+zB_{n-3,j-1}(z)]F_{2,i-j}(z)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}+zB_{n-3,n-3}(z)F_{2,i-(n-2)}(z)$
$\displaystyle=$ $\displaystyle
B_{n-2,0}(z)F_{2,i-0}(z)+\sum_{j=1}^{n-3}B_{n-2,j}(z)F_{2,i-j}(z)+B_{n-2,n-2}(z)F_{2,i-(n-2)}(z)$
$\displaystyle=$ $\displaystyle\sum_{j=0}^{n-2}B_{n-2,j}(z)F_{2,i-j}(z).$

∎

######  Remark 1.

For any given value of $n$, the right-hand side of (5) has at most three non-
zero terms, providing a computational advantage. Moreover, relation (5) can be
represented in a matrix form as follows:

$\displaystyle\begin{bmatrix}F_{n,0}(z)\\\ F_{n,1}(z)\\\ \vdots\\\
F_{n,n}(z)\end{bmatrix}=\begin{bmatrix}B_{n-2,0}(z)&0&0\\\
B_{n-2,1}(z)&B_{n-2,0}(z)&0\\\ B_{n-2,2}(z)&B_{n-2,1}(z)&B_{n-2,0}(z)\\\
B_{n-2,3}(z)&B_{n-2,2}(z)&B_{n-2,1}(z)\\\ \vdots&\vdots&\vdots\\\
B_{n-2,n-3}(z)&B_{n-2,n-4}(z)&B_{n-2,n-5}(z)\\\
B_{n-2,n-2}(z)&B_{n-2,n-3}(z)&B_{n-2,n-3}(z)\\\
0&B_{n-2,n-2}(z)&B_{n-2,n-3}(z)\\\
0&0&B_{n-2,n-2}(z)\end{bmatrix}\begin{bmatrix}F_{2,0}(z)\\\ F_{2,1}(z)\\\
F_{2,2}(z)\end{bmatrix}.$

From the proof of Lemma 2, the following fact is straightforward:

######  Lemma 3.

For any set of basis functions $\\{F_{n,i}\\}_{i=0}^{n}$, which are
constructed using the recursive relation (2), any basis of order $n$ could be
represented in terms of the basis elements of order $m<n$, i.e.

$\displaystyle
F_{n,i}(z)=\sum_{j=0}^{n-m}B_{n-m,j}(z)F_{m,i-j}(z),~{}~{}~{}i=0,\cdots,n.$
(6)

######  Remark 2.

Lemma 3 provides a universal formula for expressing the basis functions in
terms of the starting basis functions, irrespective of the number of starting
bases employed.

The forthcoming result demonstrates that the $\alpha$-Bernstein polynomials
possess many of the properties of classical Bernstein polynomials, making them
well-suited for function approximation.

######  Theorem 2.

The $\alpha$-Bernstein polynomials derived from equations (2) and (4) exhibit
the following properties:

  * (a)

Nonnegativity: $F_{n,i}(z)\geq 0$ for $i=0,1,2,\ldots,n$.

  * (b)

Partition of unity: $\sum_{i=0}^{n}F_{n,i}(z)=1$.

  * (c)

Symmetry: $F_{n,i}(z)=F_{n,n-i}(1-z)$ for $i=0,1,2,\ldots,n$.

  * (d)

End-point values:

$\displaystyle F_{n,i}(0)=\begin{cases}1,&i=0,\\\ 0,&i\neq 0,\end{cases}\hskip
19.91684ptF_{n,i}(1)=\begin{cases}1,&i=n,\\\ 0,&i\neq n.\end{cases}$ (7)

###### Proof.

The proof of this theorem relies on the application of the mathematical
induction.

  * (a)

Non-negativity is obvious by (2) and (4).

  * (b)

It is already stated in Lemma 1.

  * (c)

The starting basis functions exhibit symmetry. Let’s assume that the
$\alpha$-Bernstein polynomials of order $k$ hold symmetry. Now, by considering
this inductive hypothesis along with the recurrence relation (2), we can
conclude that

$\displaystyle F_{k+1,i}(1-z)$ $\displaystyle=$
$\displaystyle(1-(1-z))F_{k,i}(1-z)+(1-z)F_{k,i-1}(1-z)$ $\displaystyle=$
$\displaystyle zF_{k,k-i}(z)+(1-z)F_{k,k-i+1}(z)=F_{k+1,k-i+1}(z).$

  * (d)

Based on a straightforward deduction from (4), it can be concluded that the
results in (7) apply to the case when $n=2$.
Let’s assume that the properties at the endpoints are satisfied by the
$\alpha$-Bernstein polynomials of order $k$. Consequently, by utilizing the
inductive hypothesis and equation (2), we can conclude that:

$\displaystyle F_{k+1,i}(0)$ $\displaystyle=$ $\displaystyle
F_{k,i}(0)=\begin{cases}1&i=0,\\\ 0,&i\neq 0,\end{cases}$ $\displaystyle
F_{k+1,i}(1)$ $\displaystyle=$ $\displaystyle
F_{k,i-1}(1)=\begin{cases}1&i=k+1,\\\ 0,&i\neq k+1.\end{cases}$

∎

In Figure 1, an illustration of the $\alpha$-Bernstein polynomials is
presented. These functions are generated for varying values of $\alpha$
(including $0,0.2,0.4,0.6,0.8$ and $1$), while the values of $n$ are set to
$2,3,$ and $4$.

(a)

(b)  (c)

Figure 1: The $\alpha$-Bernstein polynomials by adjusting $\alpha$ from
$0,0.2,\ldots,1$.

##  4 New results and alternative proofs

This section is dedicated to showcasing the benefits of the new representation
of $\alpha$-Bernstein operators. We introduce a new formula for computing the
moments and present an analogue of Vornovskaja’s theorem. Subsequently, we
verify the shape-preserving properties of these operators using entirely
different strategies.

######  Theorem 3.

For all $j\in\mathbb{N}\cup\left\\{0\right\\},n\in\mathbb{N}$,
$\alpha\in[0,1]$ and $z\in\left[0,1\right]$, we have the recurrence formula:

$\displaystyle\mathcal{B}_{n}^{\alpha}\left(z^{j+1};z\right)=\left(1-\frac{1}{n}\right)^{j+1}\left[\left(1-z\right)\mathcal{B}_{n-1}^{\alpha}\left(z^{j+1};z\right)+z\sum_{k=0}^{j+1}\binom{j+1}{k}\dfrac{1}{\left(n-1\right)^{k}}\mathcal{B}_{n-1}^{\alpha}\left(z^{j+1-k};z\right)\right].$

###### Proof.

Employing equation (2) we derive:

$\displaystyle\mathcal{B}_{n}^{\alpha}\left(z^{j+1};z\right)$ $\displaystyle=$
$\displaystyle\sum_{i=0}^{n}\left(\frac{i}{n}\right)^{j+1}F_{n,i}(z)$
$\displaystyle=$
$\displaystyle\sum_{i=0}^{n}\left(\frac{i}{n}\right)^{j+1}\left[\left(1-z\right)F_{n-1,i}(z)+zF_{n-1,i-1}(z)\right]$
$\displaystyle=$
$\displaystyle\frac{\left(n-1\right)^{j+1}}{n^{j+1}}\left[\left(1-z\right)\sum_{i=0}^{n}\left(\frac{i}{n-1}\right)^{j+1}F_{n-1,i}(z)+z\sum_{i=0}^{n}\left(\frac{i}{n-1}\right)^{j+1}F_{n-1,i-1}(z)\right]$
$\displaystyle=$
$\displaystyle\left(1-\frac{1}{n}\right)^{j+1}\left[\left(1-z\right)\mathcal{B}_{n-1}^{\alpha}\left(z^{j+1};z\right)+z\sum_{i=0}^{n}\left(\frac{i-1+1}{n-1}\right)^{j+1}F_{n-1,i-1}(z)\right]$
$\displaystyle=$
$\displaystyle\left(1-\frac{1}{n}\right)^{j+1}\left[\left(1-z\right)\mathcal{B}_{n-1}^{\alpha}\left(z^{j+1};z\right)+z\sum_{k=0}^{j+1}\binom{j+1}{k}\dfrac{1}{\left(n-1\right)^{k}}\sum_{i=0}^{n-1}\left(\frac{i}{n-1}\right)^{j+1-k}F_{n-1,i}(z)\right]$
$\displaystyle=$
$\displaystyle\left(1-\frac{1}{n}\right)^{j+1}\left[\left(1-z\right)\mathcal{B}_{n-1}^{\alpha}\left(z^{j+1};z\right)+z\sum_{k=0}^{j+1}\binom{j+1}{k}\dfrac{1}{\left(n-1\right)^{k}}\mathcal{B}_{n-1}^{\alpha}\left(z^{j+1-k};z\right)\right].$

We utilized the outcome of the following relation in the above calculations.

$\displaystyle\sum_{i=0}^{n}\left(\frac{i-1+1}{n-1}\right)^{j+1}F_{n-1,i-1}(z)$
$\displaystyle=$
$\displaystyle\sum_{i=0}^{n}\dfrac{1}{\left(n-1\right)^{j+1}}\left[\sum_{k=0}^{j+1}\binom{j+1}{k}\left(i-1\right)^{j+1-k}\right]F_{n-1,i-1}(z)$
$\displaystyle=$
$\displaystyle\sum_{i=0}^{n}\left[\sum_{k=0}^{j+1}\binom{j+1}{k}\dfrac{1}{\left(n-1\right)^{k}}\left(\dfrac{i-1}{n-1}\right)^{j+1-k}\right]F_{n-1,i-1}(z)$
$\displaystyle=$
$\displaystyle\sum_{k=0}^{j+1}\binom{j+1}{k}\dfrac{1}{\left(n-1\right)^{k}}\sum_{i=0}^{n-1}\left(\frac{i}{n-1}\right)^{j+1-k}F_{n-1,i}(z).$

∎

Next, we present a Grüss-Voronovskaja-type theorem for the $\alpha$-Bernstein
operators, utilizing the methodology outlined in [18].

######  Theorem 4.

Let $f,h\in C^{2}[0,1]$, then for any $z\in[0,1]$, we have

$\underset{n\rightarrow\infty}{lim}~{}~{}n\left[\mathcal{B}_{n}^{\alpha}(fh;z)-\mathcal{B}_{n}^{\alpha}(f;z)\mathcal{B}_{n}^{\alpha}(h;z)\right]=z(1-z){f}^{\prime}(z){h}^{\prime}(z),$
(8)

where $\alpha\in[0,1]$.

###### Proof.

It is straightforward to express

$\displaystyle\mathcal{B}_{n}^{\alpha}(fh;z)-\mathcal{B}_{n}^{\alpha}(f;z)\mathcal{B}_{n}^{\alpha}(h;z)$
$\displaystyle=$
$\displaystyle\mathcal{B}_{n}^{\alpha}(fh;z)-f\left(z\right)h\left(z\right)+f\left(z\right)h\left(z\right)-\mathcal{B}_{n}^{\alpha}(f;z)\mathcal{B}_{n}^{\alpha}(h;z)+\mathcal{B}_{n}^{\alpha}(f;z)h\left(z\right)-\mathcal{B}_{n}^{\alpha}(f;z)h\left(z\right)$
$\displaystyle=$
$\displaystyle\left[\mathcal{B}_{n}^{\alpha}(fh;z)-f\left(z\right)h\left(z\right)\right]+\left[h\left(z\right)\Big{(}f(z)-\mathcal{B}_{n}^{\alpha}(f;z)\Big{)}\right]+\left[\mathcal{B}_{n}^{\alpha}(f;z)\Big{(}h(z)-\mathcal{B}_{n}^{\alpha}(h;z)\Big{)}\right].$

Considering corresponding Korovkin [19] and Vornovskaja [20] theorems from the
original paper [2], we derive the following

$\displaystyle\underset{n\rightarrow\infty}{lim}n\left[\mathcal{B}_{n}^{\alpha}(fh;z)-\mathcal{B}_{n}^{\alpha}(f;z)\mathcal{B}_{n}^{\alpha}(h;z)\right]$
$\displaystyle=$
$\displaystyle\underset{n\rightarrow\infty}{lim}n\left[\mathcal{B}_{n}^{\alpha}(fh;z)-f\left(z\right)h\left(z\right)\right]+\left[h\left(z\right)\Big{(}\underset{n\rightarrow\infty}{lim}n[f(z)-\mathcal{B}_{n}^{\alpha}(f;z)]\Big{)}\right]$
$\displaystyle~{}~{}~{}~{}+\left[\underset{n\rightarrow\infty}{lim}\mathcal{B}_{n}^{\alpha}(f;z)\Big{(}\underset{n\rightarrow\infty}{lim}n[h(z)-\mathcal{B}_{n}^{\alpha}(h;z)]\Big{)}\right]$
$\displaystyle=$
$\displaystyle\dfrac{1}{2}z(1-z)(fh)^{\prime\prime}(z)-\dfrac{1}{2}z(1-z){f}^{\prime\prime}(z)h\left(z\right)-\dfrac{1}{2}z(1-z){h}^{\prime\prime}(z)f\left(z\right)$
$\displaystyle=$
$\displaystyle\dfrac{1}{2}z(1-z)\left[(fh)^{\prime\prime}(z)-{f}^{\prime\prime}(z)h\left(z\right)-{h}^{\prime\prime}(z)f\left(z\right)\right]=z(1-z){f}^{\prime}(z){h}^{\prime}(z).$

∎

###  4.1 Monotonicity preservation

######  Definition 3.

A system of functions $\\{h_{0},\cdots,h_{n}\\}$ is monotonicity preserving if
for any $\nu_{0}\leq\nu_{1}\leq\cdots\leq\nu_{n}$ in $\mathbb{R}$, the
function $\sum_{i=0}^{n}\nu_{i}h_{i}$ is increasing.

The Proposition 2.3 of [21] outlines the characterization of systems that
preserve monotonicity, as presented in the following result.

######  Proposition 1.

Let $\\{h_{0},\cdots,h_{n}\\}$ be a system of functions defined on an interval
$[a,b]$. Let $k_{i}:=\sum_{j=i}^{n}h_{j}$ for
$i\in\left\\{0,1,\cdots,n\right\\}$. Then $\\{h_{0},\cdots,h_{n}\\}$ is
monotonicity preserving if and only if $k_{0}$ is a constant function and the
functions $k_{i}$ are increasing for $i=1,\cdots,n$.

The subsequent result is derived from the preceding proposition.

######  Proposition 2.

The $\alpha$-Bernstein polynomials $\\{F_{n,0},F_{n,1},\cdots,F_{n,n}\\}$,
defined by equations (4) and (2), are monotonicity preserving.

###### Proof.

We use the mathematical induction to verify the result. Consider the
$\alpha$-Bernstein polynomials defined by equation (2) for $n=2$. According to
Proposition 1, the set $\\{F_{2,0},F_{2,1},F_{2,2}\\}$ preserves monotonicity
under the conditions that $k_{0}$ is a constant function, and both $k_{1}$ and
$k_{2}$ are increasing functions.

Utilizing the partition of unity of $\alpha$-Bernstein polynomials, we obtain
$k_{0}=\sum_{j=0}^{2}F_{2,j}(z)=1$. On the other hand

$\sum_{j=1}^{2}F_{2,j}(z)=z-\alpha(z^{2}-z)\Rightarrow\dfrac{d}{dz}\Big{(}\sum_{j=1}^{2}F_{2,j}(z)\Big{)}=1-\alpha(2z-1)\geq
1-\alpha,$

where $1-\alpha$ takes a non-negative value, therefore
$\sum_{j=1}^{2}F_{2,j}(z)$ is an increasing function. Similarly, we show the
monotonicity of the last function by taking derivative

$\sum_{j=2}^{2}F_{2,j}(z)=z+\alpha(z^{2}-z)\Rightarrow\dfrac{d}{dz}\Big{(}F_{2,2}(z)\Big{)}=1+\alpha(2z-1),$

where obviously $1+\alpha(2z-1)\geq 0$, so we have the desired result.

Now, we assume that the $\alpha$-Bernstein polynomials
$\\{F_{n-1,0},F_{n-1,1},\cdots,F_{n-1,n-1}\\}$ are monotonicity preserving,
and it remains to conclude the same result for any $\alpha$-Bernstein
polynomials of order $n$, $\\{F_{n,0},F_{n,1},\cdots,F_{n,n}\\}$.

To do so, we first observe that $k_{0}=\sum_{j=0}^{n}F_{n,j}(z)=1$, i.e. is a
constant function due to partition of unity. For
$k_{i}(z)=\sum_{j=i}^{n}F_{n,j}(z)$, we show the monotonicity by verifying the
non-negativity of its derivative,
$\dfrac{d}{dz}k_{i}(z)=\sum_{j=i}^{n}\dfrac{d}{dz}F_{n,j}(z)$.

Employing relation (2), we have

$\displaystyle\dfrac{d}{dz}k_{i}(z)$ $\displaystyle=$
$\displaystyle-\sum_{j=i}^{n}F_{n-1,j}(z)+(1-z)\sum_{j=i}^{n}\dfrac{d}{dz}F_{n-1,j}(z)+\sum_{j=i}^{n}F_{n-1,j-1}(z)+z\sum_{j=i}^{n}\dfrac{d}{dz}F_{n-1,j-1}(z)$
(9) $\displaystyle=$ $\displaystyle
F_{n-1,n}(z)+\left(-\sum_{j=i}^{n-1}F_{n-1,j}(z)+\sum_{j=i}^{n-1}F_{n-1,j}(z)\right)+F_{n-1,i-1}(z)$
$\displaystyle+(1-z)\sum_{j=i}^{n}\dfrac{d}{dz}F_{n-1,j}(z)+z\sum_{j=i}^{n}\dfrac{d}{dz}F_{n-1,j-1}(z)$
$\displaystyle=$ $\displaystyle
F_{n-1,i-1}(z)+(1-z)\sum_{j=i}^{n-1}\dfrac{d}{dz}F_{n-1,j}(z)+z\sum_{j=i-1}^{n-1}\dfrac{d}{dz}F_{n-1,j}(z).$

Now, we use the induction hypothesis to see that

$\sum_{j=i}^{n-1}\dfrac{d}{dz}F_{n-1,j}(z)\geq 0,\hskip
28.45274pt\sum_{j=i-1}^{n-1}\dfrac{d}{dz}F_{n-1,j}(z)\geq 0.$ (10)

and then it becomes evident that $\dfrac{d}{dz}k_{i}(z)\geq 0$. ∎

The monotonicity preservation of the $\alpha$-Bernstein operators follows
directly from the preceding proposition.

######  Theorem 5.

For a function $q$ that is continuous and monotonically increasing on the
interval $[0,1]$, the corresponding $\alpha$-Bernstein operators are also
increasing.

###### Proof.

Considering the monotonically increasing function $q$, we observe the
inequality $q\left(\frac{0}{n}\right)\leq
q\left(\frac{1}{n}\right)\leq\cdots\leq q\left(\frac{n}{n}\right)$.
Consequently, the result is directly derived from Proposition 2.

∎

###  4.2 Convexity preservation

The preservation of convexity by the $\alpha$-Bernstein operators is verified
by the subsequent theorem.

######  Theorem 6.

If $q\in C[0,1]$ is a convex function, then its $\alpha$-Bernstein operators
are all convex.

###### Proof.

We employ the method of induction to establish the result’s validity, starting
with the base case $n=2$. Consider a set of real values
$\lambda_{0},\lambda_{1},\lambda_{2}$ forming a convex data set, ensuring
$\lambda_{2}-2\lambda_{1}+\lambda_{0}\geq 0$. Our objective is to demonstrate
the convexity of the function
$G(z,\alpha)=\sum_{i=0}^{2}\lambda_{i}F_{2,i}(z)$ over the interval $[0,1]$.
Evaluating the second derivative, we get

$\dfrac{d^{2}}{dz^{2}}G(z,\alpha)=\sum_{i=0}^{2}\lambda_{i}\dfrac{d^{2}}{dz^{2}}F_{2,i}(z)=2\alpha\left(\lambda_{2}-2\lambda_{1}+\lambda_{0}\right)$

Given this expression and $\lambda_{2}-2\lambda_{1}+\lambda_{0}\geq 0$, we can
affirm that $\dfrac{d^{2}}{dz^{2}}G(z,\alpha)\geq 0$.

Presuming the convexity of a set of real values $\\{\beta_{i}\\}_{i=0}^{n}$,
we take as our induction hypothesis that the expression
$\sum_{i=0}^{n}\beta_{i}F_{n,i}(z)$ is convex.

Now, let’s introduce a new set of convex values $\\{\eta_{i}\\}_{i=0}^{n+1}$.
The goal is to establish the convexity of the function
$\sum_{i=0}^{n+1}\eta_{i}F_{n+1,i}(z)$. To accomplish this, we consider

$G(z,\alpha)=\sum_{i=0}^{n+1}\eta_{i}F_{n+1,i}(z)=\sum_{i=0}^{n+1}\eta_{i}\left[(1-z)F_{n,i}(z)+zF_{n,i-1}(z)\right],$

and establish that the function $\dfrac{d}{dz}G(z,\alpha)$ is increasing.

$\displaystyle\dfrac{d}{dz}G(z,\alpha)$ $\displaystyle=$
$\displaystyle\sum_{i=0}^{n+1}\eta_{i}\dfrac{d}{dz}\left[(1-z)F_{n,i}(z)+zF_{n,i-1}(z)\right]$
$\displaystyle=$
$\displaystyle\sum_{i=0}^{n+1}\eta_{i}\left[-F_{n,i}(z)+(1-z)\dfrac{d}{dz}F_{n,i}(z)+F_{n,i-1}(z)+z\dfrac{d}{dz}F_{n,i-1}(z)\right]$
$\displaystyle=$
$\displaystyle\sum_{i=0}^{n}(\eta_{i+1}-\eta_{i})F_{n,i}(z)+(1-z)\sum_{i=0}^{n}\eta_{i}\dfrac{d}{dz}F_{n,i}(z)+z\sum_{i=0}^{n}\eta_{i+1}\dfrac{d}{dz}F_{n,i}(z)$

Given that $\left\\{\eta_{i}\right\\}_{i=0}^{n+1}$ represents convex data, we
observe that $\eta_{i+1}-2\eta_{i}+\eta_{i-1}\geq 0,i=1,\cdots,n-1$. This
inequality implies
$\eta_{i+1}-\eta_{i}\geq\eta_{i}-\eta_{i-1},i=1,\cdots,n-1$. In simpler terms,
the differences $\eta_{i}-\eta_{i-1},i=1,\cdots,n-1$ form an increasing
sequence. By applying Proposition 2, we can affirm that the function
$\sum_{i=0}^{n}(\eta_{i+1}-\eta_{i})F_{n,i}(z)$ is an increasing function.

Assuming the induction hypothesis and given that
$\left\\{\eta_{i}\right\\}_{i=0}^{n+1}$ represents convex data, it follows
that both $\sum_{i=0}^{n}\eta_{i}F_{n,i}(z)$ and
$\sum_{i=0}^{n}\eta_{i+1}F_{n,i}(z)$ are convex functions. This implies that
their respective derivatives, $\sum_{i=0}^{n}\eta_{i}\dfrac{d}{dz}F_{n,i}(z)$
and $\sum_{i=0}^{n}\eta_{i+1}\dfrac{d}{dz}F_{n,i}(z)$, are increasing
functions.

By the derived results and considering $z\in[0,1]$, it follows that
$\dfrac{d}{dz}G(z,\alpha)$ is an increasing function. Consequently, this
implies $\dfrac{d^{2}}{dz^{2}}G(z,\alpha)\geq 0$, signifying that
$G(z,\alpha)$ is a convex function.

We observe that if $q$ is a convex function, then the values
$q\left(\frac{0}{n}\right),q\left(\frac{1}{n}\right),\ldots,q\left(\frac{n}{n}\right)$
form a set of convex data. Consequently, this directly implies that
$R_{n,\alpha}(q;z)$ is also a convex function. ∎

##  5 Conclusion

The current study makes two significant contributions to the field of
approximation theory, particularly concerning Bernstein-like operators.

Firstly, a somewhat generalized formula is introduced for generating
Bernstein-like functions. The study illustrates three specific families and
opens avenues for constructing new and innovative families.

Secondly, it is demonstrated that the newly presented structure offers a
recursive formula for generating $\alpha$-Bernstein polynomials. Utilizing
this representation, several properties are derived, and alternative proofs
are provided for established results.

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