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\begin{document}
\begin{definition}
A {\em \htmladdnormallink{classical automaton}{http://planetphysics.us/encyclopedia/StableAutomaton.html}} $\A$, (or simply {\em automaton}, or
{\em sequential machine}, is defined as a quintuple of sets, $I,O$ and $S$, and set-theoretical mappings,
$$(I, O, S, \delta: I \times S \rightarrow S; \lambda: S \times S \rightarrow O),$$
with $\delta$ being called the {\em \htmladdnormallink{transition function}{http://planetphysics.us/encyclopedia/StableAutomaton.html}} and $\lambda$ being called the {\em \htmladdnormallink{output function}{http://planetphysics.us/encyclopedia/StableAutomaton.html}}.
\end{definition}
\begin{definition} A {\em categorical automaton} $\A_C$ or {\em discrete and finite/countable, categorical \htmladdnormallink{dynamic system}{http://planetphysics.us/encyclopedia/GenericityInOpenSystems.html}} is defined by a \htmladdnormallink{commutative square diagram}{http://planetphysics.us/encyclopedia/Commutativity.html} containing all of the above components and assuming that $S_A$ is either a countable or finite set of discrete states:
$$\begin{xy}
*!C\xybox{
\xymatrix{
{I \times S }\ar[r]^{\delta}\ar[d]_{t}&{S}\ar[d]^{o}\\
{S \times S}\ar[r]_{\lambda}&{O}
} }\end{xy}$$
\end{definition}
With the above definition one can now define \htmladdnormallink{morphisms}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} between automata and their \htmladdnormallink{composition}{http://planetphysics.us/encyclopedia/Cod.html}. If the automata are defined by \htmladdnormallink{square diagrams}{http://planetphysics.us/encyclopedia/Commutativity.html} such as
the one shown above, and \htmladdnormallink{diagrams}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} are defined by their associated \htmladdnormallink{functors}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html}, then automata homomorphisms are in fact defined as \htmladdnormallink{natural transformations}{http://planetphysics.us/encyclopedia/VariableCategory2.html}
between diagram functors. One also has a consistent, simpler definition
as follows.
\begin{definition} A \emph{\htmladdnormallink{homomorphism}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} of automata} is a morphism of automata quintuples that preserves \htmladdnormallink{commutativity}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} of the set-theoretical mapping compositions of both the transition function $\delta$ and the output function $\lambda$.
\end{definition}
With the above two definitions now we have sufficient data to define the category of automata and automaton homomorphisms.
\begin{definition}
The \emph{category of automata} is a category of automata quintuples
$(I_X, O_X, X, {\delta}_X: I_X \times X \rightarrow X; {\lambda}_X: X \times X \rightarrow O)$ and automata homomorphisms $h:{\A}_i \rightarrow {\A}_j$,
such that these homomorphisms \htmladdnormallink{commute}{http://planetphysics.us/encyclopedia/Commutator.html} with both the transition and the output functions of any automata ${\A}_i$ and ${\A}_j$.
\end{definition}
\textbf{Remarks:}
\begin{enumerate}
\item Automata homomorphisms can be considered also as automata {\em transformations} or as \htmladdnormallink{semigroup}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} homomorphisms, when the \htmladdnormallink{state space}{http://planetphysics.us/encyclopedia/StableAutomaton.html}, $X$, of the automaton is defined as a \emph{semigroup} $\mathcal{S}$.
\item Abstract automata have numerous realizations in the real world as : machines, \htmladdnormallink{robots}{http://planetphysics.us/encyclopedia/Program3.html}, devices, computers, \htmladdnormallink{supercomputers}{http://planetphysics.us/encyclopedia/SupercomputerArchitercture.html}, always considered as \emph{discrete} state space sequential machines.
\item Fuzzy or analog devices are not included as standard automata.
\item Similarly, \emph{variable (transition function)} automata are not included, but Universal Turing (UT) machines are.
\end{enumerate}
\begin{definition} An alternative definition of an automaton is also in use:
as a five-tuple $(S, \Sigma, \delta, I, F)$, where $\Sigma$ is a non-empty set of symbols
$\alpha$ such that one can define a {\em configuration} of the automaton as a couple
$(s,\alpha)$ of a state $s \in S $ and a symbol $\alpha \in \Sigma $. Then $\delta$
defines a ``next-state relation, or a transition relation'' which associates to each configuration
$(s, \alpha)$ a subset $\delta (s,\alpha)$ of S- the state space of the automaton.
With this formal automaton definition, the \emph{\htmladdnormallink{category}{http://planetphysics.us/encyclopedia/Cod.html} of abstract automata} can be defined by specifying automata homomorphisms in terms of the morphisms between five-tuples representing such abstract automata.
\end{definition}
\begin{example}
A special case of automaton is when all its transitions are {\em reversible}; then its state space is a \htmladdnormallink{groupoid}{http://planetphysics.us/encyclopedia/QuantumOperatorAlgebra5.html}. The {\em category of reversible automata} is then a \htmladdnormallink{2-category}{http://planetphysics.us/encyclopedia/2Category.html}, and also a subcategory of the 2-category of groupoids, or the \htmladdnormallink{groupoid category}{http://planetphysics.us/encyclopedia/GroupoidCategory3.html}.
\end{example}
\subsection{Remarks:}
Other definitions of automata, sequential machines, semigroup automata or cellular automata lead to subcategories of the category of automata defined above. On the other hand, the \htmladdnormallink{category of quantum automata}{http://planetphysics.us/encyclopedia/CategoryOfQuantumAutomata.html} is not a subcategory of the automata category defined here.
\end{document}
