PlanetPhysics/Automaton 2

A (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 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.

A \htmladdnormallink{homomorphism {http://planetphysics.us/encyclopedia/TrivialGroupoid.html} of automata} or automata homomorphism is a morphism of automata quintuples that preserves commutativity of the set-theoretical mapping compositions of both the transition 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 $$h:{\A}_i \rightarrow {\A}_j$$, such that these homomorphisms commute with both the transition and the output functions of any automata $${\A}_i$$ and $${\A}_j$$.

Remarks:

or as semigroup homomorphisms, when the state space, $$X$$, of the automaton is defined as a semigroup $$S$$.
 * 1) Automata homomorphisms can be considered also as automata transformations

computers, supercomputers, always considered as discrete state space sequential machines.\\
 * 1) Abstract automata have numerous realizations in the real world as : machines, robots, devices,
 * 1) Fuzzy or analog devices are not included as standard automata.
 * 2) Similarly, variable (transition function) automata are not included, but Universal Turing machines are.

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 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.

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 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.