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Add Determinism, reversibility, stochasticity.

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Sergiu Ivanov 2024-04-24 14:44:23 +02:00
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47
bib/dealb.bib
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@ -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]"
}

100
deal.tex
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@ -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}