hdr/deal.tex

\chapter{A Deal with Life}
\begin{refsection}[bib/sivanov-dblp-mod.bib,bib/sivanov-extra.bib,bib/dealb.bib]

Life is one of the most beautiful things in the universe.  Arguably,
it is because we humans belong to the kingdom of Life that it
fascinates us so.  Beyond its intrinsic beauty to which our sensory
organs are attuned, it also deeply attracts us because of the
self-referentiality of its contemplation: when thinking about Life, we
often think about our interactions with it, and ultimately
about ourselves.

Self-referentiality is also a hurdle: it is intrinsically difficult to
conceive of oneself.  Even though theoretical computer science is no
substitute for philosophy, I enjoy taking Gödel's incompleteness
theorems\footnote{\url{https://en.wikipedia.org/wiki/Gödel's_incompleteness_theorems}}
and especially Hilbert's
\emph{Entscheidungsproblem}\footnote{\url{https://en.wikipedia.org/wiki/Entscheidungsproblem}}
and the halting
problem\footnote{\url{https://en.wikipedia.org/wiki/Halting_problem}}
as vivid examples: Turing's famous proof states that a Turing machine
cannot generally decide whether another Turing machine will ever halt.
Since abstract computing devices can be seen as distant
mathematizations of the human brain, this formal result hints that
entirely conceiving of our mind---and by extension of Life itself---is
borderline intractable.

The difficulty of self-referiantiality is also deeply disturbing,
especially because understanding how our bodies function within their
environments has so many essential implications: dealing with the
climate crisis, tackling diseases, improving the quality of life, to
only cite the foremost ones.  To avoid the worry of looking into the
mirror for too long, one can brutally build a wall between oneself and
``the rest'' of Life, and adopt what may be called the Engineer's
position: a living organism is a machine constituted out of mechanical
pieces, whilst the human disassembles, adjusts, and reassembles them
again, improved.

Modern biology, medicine, biotechnology illustrate the high
performance of the Engineer's approach, and this text is not
a criticism of mechanicism per se.  Nevertheless, its efficiency does
not entail total truthfulness, nor even exclusivity about truth.
In other words, mechanistic views allowing for impressive technical
achievements does not mean that these views fully reflect reality, nor
that mechanicism is the final stop on our journey to understanding
Life.  In my research, I aim for exploring different approaches to
Life and tools supporting such approaches.  I take particular
enthusiasm in thinking about striking \emph{a deal with Life}:
establishing \emph{mutually beneficial} interactions with living
systems.

Concluding deals as opposed to taking the Engineer's position resets
the power balance in our relationship with Life: instead of seeking to
control, hack, or otherwise dominate living organisms, the goal is to
further take into account their well-being.  I believe that
approaching Life from this viewpoint is essential if we are after true
solutions to fundamental problems such as the climate crisis or
complex diseases.  On a more philosophical note, the framework of
mutually beneficial interactions should remind us that our
intelligence in no way warrants an extraction of the human being into
an exceptional superior stance---we are part of Life, and we ought to
think and act accordingly.

\newpage

\section{Mechanicism: Where engineering meets biology}
\label{sec:mechanicism}

In the 20th century, biology was dramatically affected by physics and
engineering, and this has brought revolutionary advances in
understanding Life and interaction with
it~\cite{CornishBowdenCLSA2007,Glade22,Nicholson2019,Woese2004}.
Grounding the function of biological structures in the physical
reality allowed for convergence of worldview between physics and
biology, thereby conferring to the latter the gravitas of a ``real''
science.  A remarkable tool physics and engineering brought to biology
is reductionism---to understand a system, decompose it into parts,
understand each of the parts, and understand the interactions between
the parts to get back to the big picture.  Reductionism in turn
fostered the emergence of mechanicism, the modern proponents of which
``conceive of the cell as an intricate piece of machinery whose
organization reflects a pre-existing design, whose structure is wholly
intelligible in reductionistic terms, and whose operation is governed
by deterministic laws, rendering its behaviour predictable and
controllable—at least in principle.''\cite{Nicholson2019}

With all due recognition of the major advances yielded by reductionism
and mechanicism, it appears hard to believe that this is the final
stop on the way to understanding Life.  I recall first of all the
discussion in~\cite[page~2]{Woese2004} of reductionism as an
operational tool allowing to tackle complexity (empirical
reductionism), as opposed to the belief that it actually corresponds
to the organization of the living matter (fundamental reductionism).
Fundamental reductionism makes therefore an additional strong
assumption, which impacts the ``sense of what is important'':
molecular biology established the molecular level as fundamental, and
demoted the status of larger structures---e.g. organisms, ecosystems,
etc.  These are deemed emergent, and therefore less important,
secondary, directly derivable from more fundamental matters.

While the notion of emergence in natural sciences is fraught, and its
objective qualities can be debated (e.g.~\cite{RonaldSC99}), it has
the merit of putting in focus the hierarchy of scales.  It is
a hierarchy in the sense that, while physics teaches us that the whole
is always necessarily the sum of its parts (plus the interactions), it
is often irrelevant to put the whole away, and only peer at the
components.  It is therefore important to not always fall through to
the underlying levels, and specifically to avoid Laplace's daemon
abuse: the Laplace's daemon\footnote{Laplace's daemon is a thought
  experiment introducing an imaginary creature which knows exactly the
  positions and momenta of every atom in the universe.  The original
  conclusion conceived by of Pierre-Simon Laplace in 1814 is that this
  absolute knowledge should entail full knowledge of past and future
  positions of these particles~\cite{wikiLaplace}.  In modern days,
  Laplace's daemon is often used as a metaphor for absolute knowledge
  of the minutae of a complex system, down to its elementary
  particles.} cannot practically exist, but should it exist, it would
in no way have any influence on the fact that we as humans find it
extremely useful to operate with concepts situated at higher
scales\footnote{An informal inspiration for these observations comes
  from~\cite{Carroll}.}.  It is physics again, and statistical
mechanics in particular, that recalls this saliently by deeply relying
upon thinking about systems such as gasses in terms of macrostates
(volume, pressure, temperature) and microstates (positions and momenta
of all particles)~\cite{SusskindCourse,wikiEntropy}.  In other words,
while one might argue that microstates are more ``fundamental'' in
some way, it is of little practical importance, and addressing
multiple scales is still pertinent.

Fundamental reductionism as a belief is strongly related to
engineering, and specifically the practice of constructing complex
structures and mechanisms out of simpler building blocks.
The multiple ways in which engineering has been durably changing our
lives and our surroundings naturally fuels extending its reach beyond
human creation, onto living matter.  A spectacular manifestation is
the Machine Conception of the Cell (MCC) as introduced
in~\cite{Nicholson2019}: the cell is seen as an intricate machine,
somewhat similar to a computer, which makes it appropriate to use
engineering terms to designate the cellular components visible by
microscopy: molecular motors, Golgi apparatus, genetic program, pumps,
locks, keys, gates, circuitry, etc.  The choice of terms is in
principle contingent, and it is natural to use words evoking familiar
structures, but in practice this reinforces the belief in the
truthfulness of the engineering approach.  Indeed, scientific papers
ubiquitously summarize knowledge in the form of circuits or maps.
As stated in~\cite[page~6]{Mayer2009}, ``the typical ‘cartoons’ of
signaling pathways, with their reassuring arrows and limited number of
states [...] could be the real villain of the piece.'' The Wikipedia
page on molecular motors literally starts with the sentence
``Molecular motors are [...] molecular
\emph{machines}''\cite{wikiMotors} (the emphasis is mine), and
features several animations which would look appropriate in a book on
the construction of mechanical toys.  The last illustration---and
probably the most verbose---of the relationship between reductionism
and the Engineer's work I bring here is the very term
``biological engineering''.

In fact, widely admitted considerations easily uncover some flaws in
the belief in the fundamental nature of the MCC~\cite{Nicholson2019}.
To cite two of the most salient ones, the cell is a milieu which is
better described as liquid, rather than solid.  It is densely packed
with various molecules, which do not always strictly respect a certain
conformation, but rather continuously evolve across a spectrum of
shapes.  It being impossible for a human to observe the cellular
processes with the naked eye, the researcher is tempted to follow the
mindset suggested by the available technology conceived for conceiving
of and observing microscopic machines~\cite{Glade22}, a mindset which
also happens to be mainstream.  Unsurprisingly, if one looks for
machines, one finds machines, as the animation ``The Inner Life of the
Cell'' conveniently illustrates~\cite{lifeOfTheCell}.

Avoiding conceptual frameworks other than fundamental reductionism and
mechanicism not only forces our thinking into a certain box which
partially corresponds to reality, but also biases our methodology of
interactions with Life.  When one imagines the cell as a machine, one
expects mechanistic explanations, building upon strong causality.
When the computer screen shows a picture or a car modifies its
trajectory, it is always possible to indicate a satisfactory set of
causes.  This is because the engineers who built the device had
a specific intention in mind, which can be relatively easily unpacked.
Biological systems originated from spontaneous evolution, without
anyone human baking in specific goals, implying that causality is much
harder to establish convincingly.  Yet, reductionism and mechanicism
tempt the researches to only look for correlations which may be
interpreted as causal: ``It is much easier to write and publish
a paper suggesting Protein X is necessary for transmitting a signal
from A to B, than one showing that Protein X is one of many potential
components of a heterogeneous ensemble of signaling complexes that
together couple A to B.''~\cite{Mayer2009}.

While the Machine Conception of the Cell and similar mechanistic
points of view are not oblivious to the intrinsic noise of the
respective biological systems, seeing them as machines invites to
treating noise as a nuisance which the biological systems manage to
successfully combat in every moment of their existence.  However,
multiple indications exist that noise plays an essential role, as
a matter of fact making some processes possible.  We cite as an
example the Brownian ratchet model of intracellular transport, which
has been gaining considerable traction recently~\cite{Nicholson2019},
and which essentially consists in hypothesising that molecular motors
feature two distinct conformations of the energy landscape---a flat
one and a saw-toothed one.  By periodically switching between the two,
the motor buffeted by thermal fluctuations will tend to advance along
the cytoskeletal track it is attached to
(Figure~\ref{fig:ratchet-motor}).

\begin{figure}
  \centering

  \tikzstyle axis=[->]
  \tikzstyle movement=[-{Latex[width=1.2mm]},semithick]
  \tikzstyle landscape=[very thick,cap=round]
  \tikzstyle motor=[draw,circle,thick,minimum size=3.5mm]
  \tikzstyle motorFlip=[motor]
  \tikzstyle motorFlop=[motor,fill=black!40]
  \tikzstyle motorGhost=[motor,densely dotted]
  \newcommand{\landscapeXOff}{.2mm}
  \newcommand{\landscapeYOff}{1mm}
  \newcommand{\xLength}{56mm}
  \newcommand{\yLength}{11mm}
  \newcommand{\graphSkip}{\vspace{-3mm}}
  \newcommand{\stepLabOff}{-7mm}
  \begin{tikzpicture}
    \draw[axis] (0,0) --
    node[midway,xshift=\stepLabOff,minimum width=7mm] {\small (1)}
    (0,\yLength)
    node[xshift=3mm] {$U$};
    \draw[axis] (0,0) -- (\xLength, 0) node[yshift=-2mm,xshift=-1mm] {$x$};
    \draw[landscape] (\landscapeXOff,\landscapeYOff) -- +(52mm,0);

    \node[motorFlip] (motor) at (11mm,3mm) {};
    \node[motorGhost] at ($(motor)-(3.5mm,0)$) {};
    \node[motorGhost] at ($(motor)-(6mm,0)$) {};
    \node[motorGhost] at ($(motor)+(3.5mm,0)$) {};
    \node[motorGhost] at ($(motor)+(6mm,0)$) {};

    \coordinate[above=2mm of motor] (arrowAnchor);
    \draw[movement] ($(arrowAnchor)-(2mm,0)$) -- +(-6mm,0);
    \draw[movement] ($(arrowAnchor)+(2mm,0)$) -- +(6mm,0);
  \end{tikzpicture}

  \graphSkip

  \begin{tikzpicture}
    \draw[axis] (0,0) --
    node[midway,xshift=\stepLabOff,minimum width=7mm] {\small (2)}
    (0,\yLength)
    node[xshift=3mm] {$U$};
    \draw[axis] (0,0) -- (\xLength, 0) node[yshift=-2mm,xshift=-1mm] {$x$};
    \draw[landscape] (\landscapeXOff,\landscapeYOff)
    -- ++(2mm,5mm) -- ++(11mm,-5mm)
    -- ++(2mm,5mm) -- ++(11mm,-5mm)
    -- ++(2mm,5mm) -- ++(11mm,-5mm)
    -- ++(2mm,5mm) -- ++(11mm,-5mm);

    \node[motorFlop] (motor) at (25.2mm,3.7mm) {};
    \coordinate[above=2mm of motor] (arrowAnchor);
    \draw[movement] ($(arrowAnchor)-(2mm,0)$) -- +(-4.5mm,0);
    \draw[movement] ($(arrowAnchor)+(2mm,0)$) -- +(9mm,0);
  \end{tikzpicture}

  \graphSkip

  \begin{tikzpicture}
    \draw[axis] (0,0) --
    node[midway,xshift=\stepLabOff,minimum width=7mm] {\small (3)}
    (0,\yLength)
    node[xshift=3mm] {$U$};
    \draw[axis] (0,0) -- (\xLength, 0) node[yshift=-2mm,xshift=-1mm] {$x$};
    \draw[landscape] (\landscapeXOff,\landscapeYOff) -- +(52mm,0);

    \node[motorFlip] (motor) at (25.2mm,3mm) {};
    \node[motorGhost] at ($(motor)-(3.5mm,0)$) {};
    \node[motorGhost] at ($(motor)-(6mm,0)$) {};
    \node[motorGhost] at ($(motor)+(3.5mm,0)$) {};
    \node[motorGhost] at ($(motor)+(6mm,0)$) {};

    \coordinate[above=2mm of motor] (arrowAnchor);
    \draw[movement] ($(arrowAnchor)-(2mm,0)$) -- +(-6mm,0);
    \draw[movement] ($(arrowAnchor)+(2mm,0)$) -- +(6mm,0);
  \end{tikzpicture}

  \graphSkip

  \begin{tikzpicture}
    \draw[axis] (0,0) --
    node[midway,xshift=\stepLabOff,minimum width=7mm] {\small (4)}
    (0,\yLength)
    node[xshift=3mm] {$U$};
    \draw[axis] (0,0) -- (\xLength, 0) node[yshift=-2mm,xshift=-1mm] {$x$};
    \draw[landscape] (\landscapeXOff,\landscapeYOff)
    -- ++(2mm,5mm) -- ++(11mm,-5mm)
    -- ++(2mm,5mm) -- ++(11mm,-5mm)
    -- ++(2mm,5mm) -- ++(11mm,-5mm)
    -- ++(2mm,5mm) -- ++(11mm,-5mm);

    \node[motorFlop] (motor) at (38.2mm,3.7mm) {};
    \coordinate[above=2mm of motor] (arrowAnchor);
    \draw[movement] ($(arrowAnchor)-(2mm,0)$) -- +(-4.5mm,0);
    \draw[movement] ($(arrowAnchor)+(2mm,0)$) -- +(9mm,0);
  \end{tikzpicture}

  \caption{A schematic illustration of the Brownian ratchet model of
    molecular motors.  A motor is shown as a circle
    (\protect\tikz[baseline,yshift=1.2mm]\protect\node[motorFlip,minimum
    size=2.5mm]{}; or
    \protect\tikz[baseline,yshift=1.2mm]\protect\node[motorFlop,minimum
    size=2.5mm]{};), and its energy landscape is shown as a thick line
    \protect\tikz[baseline,yshift=.2em]\protect\draw[landscape]
    (0,0) -- (2ex,0);.  The horizontal axis $x$ represents the motor's
    position on the cytoskeletal track, while the vertical axis $U$
    illustrates the motor's free energy.  The motor is hypothesized to
    feature two distinct potential energy landscapes, depending on its
    conformational state.  In the flip conformation
    \protect\tikz[baseline,yshift=1.2mm]\protect\node[motorFlip,minimum
    size=2.5mm]{};, the energy landscape is flat so the protein may
    slide freely in one of the two directions, with equal probability
    for both directions.  In the flop conformation
    \protect\tikz[baseline,yshift=1.2mm]\protect\node[motorFlop,minimum
    size=2.5mm]{};, the saw-tooth shape of the landscape favors the
    motor moving to the right, illustrated by a longer arrow pointing
    to the right.  When cycles of ATP hydrolysis make the motor
    periodically switch between the two conformations, thermal
    fluctuations will tend to push it to the right.  (The original
    figure is~\cite[Figure~4]{Nicholson2019}, itself a reproduction
    from~\cite{Kurakin2006}.)}
  \label{fig:ratchet-motor}
\end{figure}

\section{The Deal: Mutually beneficial interactions}
\label{sec:deals}

Seeing Life as an ensemble of machines biases how we expect to collect
profit from acting on it.  Machine means control: we are constantly
looking for knobs which we could turn this or that way, and which
could modify the behavior of the system to fit our needs and
expectations.  This can be seen both at the very practical level,
where bioengineers seek to modify bacteria to produce chemicals,
e.g.~\cite{berkleyBio}, and also at the theoretical level, where
researchers develop methodologies to support looking for the coveted
knobs, e.g.~\cite{PardoID21,Vogel2008,Zanudo2015}.  If we admit that
the reductionistic and mechanistic approach is not globally true, we
must accept that these knobs may not necessarily have a definitive
shape, but rather be a complex assemblage of factors, affecting the
trajectory of the system in multiple non-trivial ways, and possibly
shifting in time.  Finally, this control mindset introduces an
asymmetric relationship between the controller and the controlled,
which is unnatural biological context because both the controller and
the controlled are made out of the same kind of matter, and are
ultimately embedded in the same environment.

This chapter outlines a conceptual framework putting symmetry back
into the picture, \emph{the Deal with Life}: instead of
surreptitiously lifting the human above and out of the living matter
by self-designating ourselves as superior Engineers, I propose to
account for the fact that we act \emph{within} Life and its complex
feedback loops by looking to organize \emph{mutually beneficial
  interactions} with the living systems, as opposed to trying to
control, hack, or engineer them.  Since we are talking about the
general mindset, the choice of words in not contingent: controlling,
hacking, and engineering impose a vertical power relationship, while
thinking in terms of mutual benefit admits that our target system has
a trajectory of its own, which we would like to preserve it to some
degree.  Playing with words and summarizing the control-hack-engineer
mindset as ``We control, Life obeys'' makes the power imbalance even
more striking.

Today, the most obvious inspiration for considering mutual benefit
comes from the climate crisis: for centuries, we have acted on the
environment expecting it to behave like a heat bath, i.e., to absorb
whatever we throw at it without essentially changing its state.
Besides brandishing a certain naïveté, this point of view is so
difficult to abandon that is has become the epitome of science denial
according to certain studies,
e.g.~\cite{BjornbergKGH2017,ONeillB2010,wikiClimate}, as well
as~\cite[page~155]{DryzekNS2011}.  As these references and multiple
others show, refusing to admit human cause as central to the climate
crisis has been invariably and strongly supported by the fossil fuel
industry. I suspect nevertheless that one of the reasons for the
resilience of the denialist mindset is the deeply anchored feeling
that we are engineers and the environment a mere tool.  My own
inspiration for the Deal with Life comes from theoretical biology
discussions with Nicolas \textsc{Glade} at the TIMC lab in
Grenoble\footnote{\url{https://www.timc.fr/}}, and specifically from
the remarks outlined in Section~\ref{sec:mechanicism} above concerning
the dominance of the engineering mindset in modern biology, especially
in molecular biology, and the ruts it forces our thinking in.

Thinking about mutual benefit in dealing with Life unpacks multiple
different levels of caring about the destiny of the system of
interest:
\begin{itemize}
\item \emph{Level 0}: This is the Engineer's mindset: fundamental
  reductionism and mechanicism---we control, Life obeys.  At this
  level, we do not conceive of any kind of benefit to the
  target system.
\item \emph{Level 1}: We aim to preserve the destiny of the target
  system to a certain degree.  If it is a yeast population, we may
  want to not allow its size below a certain threshold, or if it is
  a farm animal, we may want to ensure a certain quality of life
  according to a set of measures.
\item \emph{Level 2}: We aim to benefit the target system to a certain
  degree, while also extracting our own profit from the interaction.
  In the case of a farm animal, we may want to ensure that its the
  state of well-being be \emph{improved} in the context of its
  interaction with respect to a life without any human intervention.
\end{itemize}

All three levels of this hierarchy of mutual benefit are in fact
already present in our interactions with living organisms.
Respecting Level 1 is almost ubiquitously needed, since otherwise we
may kill the system of interest before it is capable of producing the
deliverable we are after.  Level~2 manifests itself to different
degrees in interactions with domesticated animals, especially in the
context of increased awareness of the conditions to which livestock
are typically treated in modern agriculture.  Levels 1 and 2 are also
progressively making their way to prominence in dealing with
ecosystems: cutting down forests brings about various kinds of
catastrophes, so it is now laudable to curb deforestation, and even to
conduct reforestation campaigns.

It would seem on the other hand that biomedical research is stubbornly
fond of ignoring Levels~1 and 2, and instead focuses on proudly
brandishing the Engineer's Level 0, claiming that if something does
not work out today, it will certainly work out tomorrow, provided that
tomorrow brings around more energy, more computing power, more
workforce, more data.  Yet again, in no way do I aim to deny or
minimize the benefits of mechanicism and reductionism in
biology---which has been instrumental in multiple groundbreaking
achievements over the 20th century and beyond.  I insist nevertheless
that exclusively sticking to Level 0 of the hierarchy of mutual
benefit is a fundamental limitation of thought.  Lifting this
limitation will undoubtedly open up a multitude of new approaches and
solutions, as this chapter attempts to outline.

A final argument for taking into consideration the destiny of the
system of interest which has been lurking around the corner the whole
time is that we as humans do not often have a choice on this matter:
the living system serving as a target is often required to survive our
intervention, and sometimes to maintain the majority of the functions
it had before the intervention.  That biomedical research is reluctant
to go from accepting this obvious constraint to taking more holistic
approaches including mutual benefit is possibly due to the complexity
that awaits us on the very threshold of the comfortable Engineer's
mindset.  In other words, it is much easier to see the disease as
separate from the carrying organism, and imagine curative strategies
tightly focused on a well defined set of diseased structures than to
admit that the onset of the disease is a consequence of a complex
interplay of multiple factors.  Indeed, conceiving of diseases from
this more holistic viewpoint is often prohibitively complex with the
currently available data and knowledge, all while the reductionist
approach gives at least some solutions.  I claim however that this is
no way should hinder our motivation to tackle the complexity of the
more holistic approach.

Finally, the way I employ the terms ``deal'' and ``mutual benefit''
corroborates no particular social ideology.  The Deal with Life simply
calls for including the potential benefit of the target system into
the picture by establishing a measure of it.  It is up to the
protagonists of the concrete context, problem, or practical
application to decide whether, how much, and in which way to
prioritize this benefit over the profit we humans are expecting
to extract.

\section{The Deal: A conceptual maquette}
\label{sec:maquette}

This section aims to provide an overview of the Deal with Life
framework for thinking about mutually beneficial interactions with
living systems, without focusing on the formal details.
Properly formalizing the ideas exposed in this section is
a significant part of my future research.

A formalization of the Deal with Life is structured into two stages:
\begin{enumerate}
\item abstract general framework,
\item concrete implementations.
\end{enumerate}
The abstract general framework rigorously defines the main concepts:
the interacting systems, their benefits, etc., while concrete
implementations instantiate these definitions for concrete systems,
supplying additional concepts and tools applicable to the concrete
case.  Concrete implementations in their own turn occur on two levels:
concrete formal dynamical systems---e.g., membrane systems, Boolean
networks, string rewriting systems\footnote{The subsequent chapters of
  this manuscript describe these formal dynamical systems in more
  detail.}---and concrete biological systems---e.g., a human,
population of yeasts, a plant, an ecosystem, etc.  Note that while the
introductory discourse heavily focuses on how we as humans interact
with Life, the Deal with Life framework can be adapted to
characterizing and evaluating the interactions between other living
systems, not necessarily involving the human.

\subsection{Dynamical systems}
\label{sec:ds}

Following the spirit of~\cite{Thom1974,Zeeman1976}, I propose to use
the language of abstract dynamical systems for the general framework
of the Deal.  In this subsection I quickly recall the main notions,
and I refer to sources like~\cite{Brown2018,wikiDS} for more general
and more detailed definitions.

An \emph{abstract dynamical system} over the state space $X$ and time
$T$ is a (partial) function $\Phi : X \times T \to X$ assigning to
a state $s \in X$ the new state $\Phi(s, t)$ in which the system will
be after time $t \in T$.  The function $\Phi$ satisfies the following
two natural properties:
\begin{enumerate}
\item $\Phi(s, 0) = s$: the system does not change its state in
  0 time,
\item $\Phi(\Phi(s, t_1), t_2) = \Phi(s, t_1 + t_2)$: evolving the
  system for time $t_1$, then for time $t_2$ leads to the same state
  as evolving the systems for time $t_1 + t_2$.
\end{enumerate}

A common shape for a state is an $n$-vector of real parameters of
a system---in which case $X \subseteq \mathbb{R}^n$---but no
particular restrictions are imposed: $X$ may be a more a general
algebraic object.  In the most general setting, $T$ is a monoid, but
it is rather customary for the time domain to be a contiguous infinite
subset of either $\mathbb{Z}$ or $\mathbb{R}$.  In the first case,
$\Phi$ is usually called a discrete-time dynamical system, and in the
second case $\Phi$ is called a continuous-time dynamical system.
If additionally the state space $X$ of a discrete-time dynamical
system $\Phi : X \times T \to T$ with $T \subseteq \mathbb{Z}$ is
discrete (countable), then $\Phi$ is called a discrete dynamical
system.  Finally, if $X$ is a finite set, then $\Phi$ is called
a finite dynamical system.

In the case of discrete-time dynamical systems, the time variable can
be interpreted as the number of evolution steps.  More concretely, for
a $t > 0$, $\Phi(s, t)$ can be seen as the state of the system after
$t$ steps, while $\Phi(s, -t)$ can be seen as the state the system
\emph{was in} $t$ steps ago.  When $T = \mathbb{N}$, it is customary
to see the discrete-time dynamical systems as the function
$F : X \to X$, giving the state to which the dynamical system
transitions from state $s \in X$: $F(s) = \Phi(s, 1)$
(e.g,~\cite[Chapter~1]{Riva22}).

Given a dynamical system $\Phi : X \times T \to X$ and a state
$s \in X$, it is possible to construct the restricted function
$\Phi_s : T \to X$, $\Phi_s(t) = \Phi(s, t)$, determining the
\emph{trajectory} of $\Phi$ through $s$.
$\Ima \Phi_s = \{\Phi(s, t) \mid t\in T\}$.  If $T = \mathbb{Z}$ or
$T = \mathbb{R}$, then the trajectory $\Ima \Phi_s$ can be interpreted
as the set of all states through which $\Phi$ goes before and after
reaching $s$.  If $T = \mathbb{N}$, then the trajectory through $s$ is
the iteration of $F$: $\{F^k(s) \mid k \in \mathbb{N}\}$, where
$F^0(s) = s$, $F^1(s) = F(s)$, $F^2(s) = F(F(s))$, etc.

\printbibliography[heading=subbibliography]

\end{refsection}

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