Add Determinism, reversibility, stochasticity.

bib/dealb.bib

@@ -397,3 +397,50 @@ keywords = {Boolean P systems, Boolean networks, Reachability, Complexity},
   biburl       = {https://dblp.org/rec/phd/hal/Riva22.bib},
   bibsource    = {dblp computer science bibliography, https://dblp.org}
 }
+
+@article{DotyKLOSW2023,
+  author       = {David Doty and
+                  Niels Kornerup and
+                  Austin Luchsinger and
+                  Leo Orshansky and
+                  David Soloveichik and
+                  Damien Woods},
+  title        = {Harvesting Brownian Motion: Zero Energy Computational Sampling},
+  journal      = {CoRR},
+  volume       = {abs/2309.06957},
+  year         = {2023},
+  url          = {https://doi.org/10.48550/arXiv.2309.06957},
+  doi          = {10.48550/ARXIV.2309.06957},
+  eprinttype    = {arXiv},
+  eprint       = {2309.06957},
+  timestamp    = {Mon, 05 Feb 2024 20:19:04 +0100},
+  biburl       = {https://dblp.org/rec/journals/corr/abs-2309-06957.bib},
+  bibsource    = {dblp computer science bibliography, https://dblp.org}
+}
+
+@article{Bennett1973,
+  author =	 {Charles H. Bennett},
+  title =	 {Logical reversibility of computation},
+  journal =	 {IBM Journal of Research and Development},
+  volume =	 {17(6)},
+  pages =	 {525--532},
+  year =	 {1973}
+}
+
+@misc{CarrollArrowFAQ2007,
+  author =	 {Sean Carroll},
+  title =	 {Arrow of Time {FAQ}},
+  howpublished =
+                  {\url{https://www.preposterousuniverse.com/blog/2007/12/03/arrow-of-time-faq/}},
+  year =	 {2007}
+}
+
+@misc{wikiStochastic,
+  author =	 "{Wikipedia contributors}",
+  title =	 "Stochastic process --- {Wikipedia}{,} The Free
+                  Encyclopedia",
+  year =	 "2024",
+  howpublished =
+                  "\url{https://en.wikipedia.org/w/index.php?title=Stochastic_process&oldid=1194369849}",
+  note =	 "[Online; accessed 9-April-2024]"
+}

deal.tex

@@ -564,6 +564,106 @@ reasons:
 Last but not least, this choice is motivated by my own expertise as
 a computer scientist lying in the field of discrete dynamical systems.
 
+\subsection{Determinism, reversibility, stochasticity}
+\label{sec:det-rever-stoch}
+
+The type of the function $\Phi$ as shown in the previous section
+imposes \emph{deterministic} behavior: for any time interval $t\in T$,
+$\Phi : X \times T \to X$ assigns to every state $s \in X$ exactly one
+state $\Phi(s, t)$.  Furthermore, when $T = \mathbb{Z}$ or
+$T = \mathbb{R}$, the system is \emph{reversible}: to any state $s$
+and any time interval $t > 0$, $\Phi$ associates exactly one state
+$\Phi(s, -t)$.  In the case $T = \mathbb{Z}$, this implies that every
+state $s \in X$ has exactly one preimage under the function $F$:
+$|F^{-1}(s)| = 1$.
+
+While it is true that any non-deterministic Turing machine
+$\mathcal{N}$ can be simulated by a deterministic Turing machine which
+explores all non-deterministic branches of the computations of
+$\mathcal{N}$, and Bennett showed in~\cite{Bennett1973} how any Turing
+machine can be made reversible, determinism and reversibility are
+quite strong restrictions.  In particular, for discrete-time dynamical
+systems, the state graph generated by $F$ is a set of
+chains~\cite{DotyKLOSW2023}, i.e., it can be represented in the plane
+as a set of linear parallel non-intersecting oriented paths.
+Despite the fact that non-determinism and reversibility do not
+increase the computational power of Turing machines, these
+restrictions may have a significant impact in weaker models of
+computation.  On the other hand, non-determinism and irreversibility
+are useful ingredients to have for practical reasons when building
+models of reality: indeed, our intuitive perception is that many
+phenomena in the world around us are non-deterministic, and most of
+them appear irreversible\footnote{Here, I will only point the reader
+  to~\cite{CarrollArrowFAQ2007} for an entry point into the
+  fascinating discussion of the irreversibility observed in the
+  macroscopic world as opposed to the reversibility of the microscopic
+  world.}.
+
+A general way to capture the fact that a phenomenon may go different
+ways at some point are stochastic processes, which can be defined as
+sequences of random variables, each of which describes the possible
+outcomes at a given moment of time~\cite{wikiStochastic}.  In general,
+stochastic processes can also describe irreversible processes, e.g.,
+because the total number of possible outcomes may decrease over time.
+In the case of discrete-time dynamical systems with $T = \mathbb{N}$,
+non-determinism can be represented in an arguably simpler way by
+modifying the type of $F$ to be $F : X \to 2^X$, meaning that for
+a state $s$, $F(s) \subseteq X$ represents the set of possible next
+states.  The notion of the state graph can be naturally extended by
+defining it as graph whose set of nodes is $X$ and which contains an
+edge from $s_1 \in X$ to $s_2 \in X$ if $s_2 \in F(s_1)$.
+Modifying the type of $F$ in this way enriches the state graph in two
+ways:
+\begin{enumerate}
+\item a state $s \in X$ may have no successor states, i.e.,
+  $F(s) = \emptyset$,
+\item they may be multiple edges originating at the same state
+  $s \in X$ if $|F(s)| > 1$.
+\end{enumerate}
+It is further possible to enrich the structure of $F$ by annotating
+the next states in $F(s)$ with probabilities. More concretely, the
+type of $F$ can be further extended to
+$F : X \to 2^{X \times \mathbb{R}}$, thus making $F$ produce sets of
+pairs (state, probability), with the normalization condition that for
+every $s$ for which $F(s) \neq \emptyset$ the probabilities of the
+next states amount to 1: $\sum_{(s', p) \in F(s)} p = 1$.
+
+In the rest of the manuscript I use the following terminology for
+designating different types of discrete-time dynamical systems with
+$T = \mathbb{N}$, depending on the type of $F$:
+\begin{itemize}
+\item $F : X \to X$: \emph{deterministic} discrete-time dynamical
+  system, or deterministic DDS;
+\item $F : X \to 2^X$: \emph{non-deterministic} discrete-time
+  dynamical system, or non-deterministic DDS;
+\item $F : X \to 2^{X \times \mathbb{R}}$ with the normalization
+  condition from the previous paragraph: \emph{stochastic}
+  discrete-time dynamical system, or stochastic DDS.
+\end{itemize}
+
+Note how understanding discrete-time dynamical systems as a function
+assigning states to states (i.e., of one of the types above) allows
+separating non-determinism from stochasticity.  Indeed, according to
+the preceding discussion, non-determinism describes the possibility of
+a state to have more than one successor state, while stochasticity
+means that, in addition, a probability distribution is defined on the
+set of successor states.  Transposing this separation between
+non-determinism and stochasticity to general dynamical systems defined
+as $\Phi : X \times T \to X$ seems at least syntactically cumbersome,
+and the traditional way of capturing non-determinism via stochastic
+processes as sequences of random variables includes probability
+distributions from the start.
+
+The definition of a trajectory in the case of non-deterministic and
+irreversible dynamical systems depends on the choice of formalism, but
+any definition should respect the intuition that a trajectory is a set
+of states the dynamical system may traverse. In the case of
+non-deterministic DDS defined as above, a trajectory is a sequence of
+states $(s_i)_{i \in T}$ such that $s_{i+1} \in F(s_i)$.  Similarly,
+in the case of stochastic DDS, a trajectory is a sequence of state
+$(s_i)_{i \in T}$ such that there exists a probability $p$ yielding
+the pair $(s_{i+1}, p) \in F(s_i)$.
+
 \printbibliography[heading=subbibliography]
 
 \end{refsection}