automaton
Definition 0.1
A
(classical) automaton, s-automaton

, or sequential machine, is defined as a quintuple of sets,

,

and

, and set-theoretical mappings,
where
is the set of s-automaton inputs,
is the set of states (or the state space of the s-automaton),
is the set of s-automaton outputs,
is the transition function that maps an s-automaton state
onto its next state
in response to a specific s-automaton input
, and
is the output function that maps couples of consecutive (or sequential) s-automaton states
onto s-automaton outputs
:
(hence the older name of sequential machine for an s-automaton).
Definition 0.2
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.
Definition 0.3
A
homomorphism 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

and the output function

.
With the above two definitions one now has sufficient data to define the category of automata
and automata homomorphisms.
Definition 0.4
A
category of automata is defined as a
category
of quintuples

and
automata homomorphisms

,
such that these homomorphisms commute with both the transition and the output functions of any automata

and

.
Remarks:
- Automata homomorphisms can be considered also as automata transformations
or as semigroup homomorphisms, when the state space,
, of the automaton is defined
as a semigroup
.
- Abstract automata have numerous realizations in the real world as : machines, robots, devices,
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.
Definition 0.5
An alternative definition of an automaton is also in use:
as a five-tuple

, where

is a non-empty set of symbols

such that one can define a
configuration of the automaton as a couple

of a state

and a symbol

. Then

defines a “next-state relation, or a transition relation” which associates to each configuration

a subset

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.
Example 0.1
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.
Definition 0.6
An alternative definition of an automaton is also in use:
as a five-tuple

, where

is a non-empty set of symbols

such that one can define a
configuration of the automaton as a couple

of a state

and a symbol

. Then

defines a “next-state relation, or a transition relation” which associates to each configuration

a subset

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.
Example 0.2
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.
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As of this snapshot date, this entry was owned by bci1.