# PlanetPhysics/Automaton 2

A (classical) automaton, s-automaton Failed to parse (unknown function "\A"): {\displaystyle \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 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 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).
A categorical automaton can also be defined by a commutative square diagram containing all of the above components.
With the above automaton definition(s) one can now also define morphisms between automata and their composition.
function $\delta$ and the output function $\lambda$.
With the above two definitions one now has sufficient data to define the category of automata and automata homomorphisms.
A category of automata is defined as a category of quintuples $(I,O,X,\delta :I\times X\rightarrow X;\lambda :X\times S\rightarrow O)$ and automata homomorphisms Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikiversity.org/v1/":): {\displaystyle h:{\A}_i \rightarrow {\A}_j} , such that these homomorphisms commute with both the transition and the output functions of any automata Failed to parse (unknown function "\A"): {\displaystyle {\A}_i} and Failed to parse (unknown function "\A"): {\displaystyle {\A}_j} .
Remarks:
or as semigroup homomorphisms, when the state space, $X$, of the automaton is defined as a semigroup $S$.
computers, supercomputers, always considered as discrete state space sequential machines.\\
* Fuzzy or analog devices are not included as standard automata.
* Similarly, variable (transition function) automata are not included, but Universal Turing machines are.
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 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 category of abstract automata can be defined by specifying automata homomorphisms in terms of the morphisms between five-tuples representing such abstract automata.
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 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 category of abstract automata can be defined by specifying automata homomorphisms in terms of the morphisms between five-tuples representing such abstract automata.
A special case of automaton is that of a stable automaton when all its state transitions are reversible ; then its state space can be seen to possess a groupoid (algebraic) structure. The category of reversible automata is then a 2-category, and also a subcategory of the 2-category of groupoids, or the groupoid category.