category of automata
Definition 0.2
A
categorical automaton

or
discrete and finite/countable, categorical dynamic system is defined by a
commutative square diagram
containing all of the above components and assuming that

is either a countable or finite set of discrete states:
With the above definition one can now define morphisms
between automata and their composition. If the automata are defined by square diagrams
such as
the one shown above, and diagrams
are defined by their associated functors, then automata homomorphisms are in fact defined as natural transformations
between diagram functors. One also has a consistent, simpler definition
as follows.
Definition 0.3
A
homomorphism
of automata 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 now we have sufficient data to define the category of automata and automaton homomorphisms.
Definition 0.4
The
category of automata is a category of automata 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 (UT) 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 when all its transitions are
reversible; then its state space is a
groupoid. The
category of reversible automata is then a
2-category, and also a subcategory of the 2-category of groupoids, or the
groupoid category.
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 category of quantum automata
is not a subcategory of the automata category defined here.
Contributors to this entry (in most recent order):
As of this snapshot date, this entry was owned by bci1.