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%%% Primary Title: automaton
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%%% Filename: Automaton2.tex
%%% Version: 11
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\begin{document}
\begin{definition}
A {\em (classical) automaton, s-automaton} $\A$, or sequential machine, is defined as a quintuple of sets, $I$,$O$ and $S$, and set-theoretical mappings,
$$(I, O, S, \delta: I \times S \rightarrow S; \lambda: S \times S \rightarrow O),$$
where $I$ is the set of s-automaton inputs, $S$ is the set of states (or the state space of the s-automaton), $O$ is the set of s-automaton outputs, $\delta$ is the {\em transition function} that maps an s-automaton state $s_i$ onto its next state $s_{i+1}$ in response to a specific s-automaton input $i \in I$, and $\lambda$ is the \emph{output function} that maps couples of consecutive (or sequential) s-automaton states $(s_i, s_{i+1})$ onto s-automaton outputs $o_{i+1}$:
$$(s_i, s_{i+1}) \mapsto o_{i+1}$$
(hence the older name of sequential machine for an s-automaton).
\end{definition}
\begin{definition}
A categorical automaton can also be defined by a \htmladdnormallink{commutative square diagram}{http://planetphysics.us/encyclopedia/Commutativity.html} containing all of the above components.
\end{definition}
With the above automaton definition(s) one can now also define \htmladdnormallink{morphisms}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} between automata and their \htmladdnormallink{composition}{http://planetphysics.us/encyclopedia/Cod.html}.
\begin{definition} A \emph{\htmladdnormallink{homomorphism}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} of automata} or {\em automata homomorphism} 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 one now has sufficient data to define the \htmladdnormallink{category of automata}{http://planetphysics.us/encyclopedia/AAT.html} and automata homomorphisms.
\begin{definition}
A \emph{category of automata} is defined as a \htmladdnormallink{category}{http://planetphysics.us/encyclopedia/Cod.html} of quintuples
$(I, O, X, \delta: I \times X \rightarrow X; \lambda: X \times S \rightarrow O)$ 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 \emph{Automata homomorphisms} can be considered also as automata transformations
or as \htmladdnormallink{semigroup}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} homomorphisms, when the state space, $X$, of the automaton is defined
as a \emph{semigroup} $S$.
\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 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{category 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 that of a {\em stable automaton} when all its state transitions are {\em reversible}; then its state space can be seen to possess a \htmladdnormallink{groupoid}{http://planetphysics.us/encyclopedia/GroupoidHomomorphism2.html} (\htmladdnormallink{algebraic}{http://planetphysics.us/encyclopedia/CoIntersections.html}) structure. 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}
\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{category 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 that of a {\em stable automaton} when all its state transitions are {\em reversible}; then its state space can be seen to possess a groupoid (algebraic) structure. The {\em category of reversible automata} is then a 2-category, and also a subcategory of the 2-category of groupoids, or the groupoid category.
\end{example}
\end{document}
