Semiautomaton Explained

In mathematics and theoretical computer science, a semiautomaton is a deterministic finite automaton having inputs but no output. It consists of a set Q of states, a set Σ called the input alphabet, and a function T: Q × Σ → Q called the transition function.

Associated with any semiautomaton is a monoid called the characteristic monoid, input monoid, transition monoid or transition system of the semiautomaton, which acts on the set of states Q. This may be viewed either as an action of the free monoid of strings in the input alphabet Σ, or as the induced transformation semigroup of Q.

In older books like Clifford and Preston (1967) semigroup actions are called "operands".

In category theory, semiautomata essentially are functors.

Transformation semigroups and monoid acts

See main article: semigroup action.

A transformation semigroup or transformation monoid is a pair

(M,Q)

consisting of a set Q (often called the "set of states") and a semigroup or monoid M of functions, or "transformations", mapping Q to itself. They are functions in the sense that every element m of M is a map

m\colonQ\toQ

. If s and t are two functions of the transformation semigroup, their semigroup product is defined as their function composition

(st)(q)=(s\circt)(q)=s(t(q))

.

Some authors regard "semigroup" and "monoid" as synonyms. Here a semigroup need not have an identity element; a monoid is a semigroup with an identity element (also called "unit"). Since the notion of functions acting on a set always includes the notion of an identity function, which when applied to the set does nothing, a transformation semigroup can be made into a monoid by adding the identity function.

M-acts

Let M be a monoid and Q be a non-empty set. If there exists a multiplicative operation

\mu\colonQ x M\toQ

(q,m)\mapstoqm=\mu(q,m)

which satisfies the properties

q1=q

for 1 the unit of the monoid, and

q(st)=(qs)t

for all

q\inQ

and

s,t\inM

, then the triple

(Q,M,\mu)

is called a right M-act or simply a right act. In long-hand,

\mu

is the right multiplication of elements of Q by elements of M. The right act is often written as

QM

.

A left act is defined similarly, with

\mu\colonM x Q\toQ

(m,q)\mapstomq=\mu(m,q)

and is often denoted as

MQ

.

An M-act is closely related to a transformation monoid. However the elements of M need not be functions per se, they are just elements of some monoid. Therefore, one must demand that the action of

\mu

be consistent with multiplication in the monoid (i.e.

\mu(q,st)=\mu(\mu(q,s),t)

), as, in general, this might not hold for some arbitrary

\mu

, in the way that it does for function composition.

Once one makes this demand, it is completely safe to drop all parenthesis, as the monoid product and the action of the monoid on the set are completely associative. In particular, this allows elements of the monoid to be represented as strings of letters, in the computer-science sense of the word "string". This abstraction then allows one to talk about string operations in general, and eventually leads to the concept of formal languages as being composed of strings of letters.

Another difference between an M-act and a transformation monoid is that for an M-act Q, two distinct elements of the monoid may determine the same transformation of Q. If we demand that this does not happen, then an M-act is essentially the same as a transformation monoid.

M-homomorphism

For two M-acts

QM

and

BM

sharing the same monoid

M

, an M-homomorphism

f\colonQM\toBM

is a map

f\colonQ\toB

such that

f(qm)=f(q)m

for all

q\inQM

and

m\inM

. The set of all M-homomorphisms is commonly written as

Hom(QM,BM)

or

HomM(Q,B)

.

The M-acts and M-homomorphisms together form a category called M-Act.[1]

Semiautomata

A semiautomaton is a triple

(Q,\Sigma,T)

where

\Sigma

is a non-empty set, called the input alphabet, Q is a non-empty set, called the set of states, and T is the transition function

T\colonQ x \Sigma\toQ.

(Q,\Sigma,T,q0,A)

, but without the initial state

q0

or set of accept states A. Alternately, it is a finite state machine that has no output, and only an input.

Any semiautomaton induces an act of a monoid in the following way.

Let

\Sigma*

be the free monoid generated by the alphabet

\Sigma

(so that the superscript * is understood to be the Kleene star); it is the set of all finite-length strings composed of the letters in

\Sigma

.

For every word w in

\Sigma*

, let

Tw\colonQ\toQ

be the function, defined recursively, as follows, for all q in Q:

w=\varepsilon

, then

T\varepsilon(q)=q

, so that the empty word

\varepsilon

does not change the state.

w=\sigma

is a letter in

\Sigma

, then

T\sigma(q)=T(q,\sigma)

.

w=\sigmav

for

\sigma\in\Sigma

and

v\in\Sigma*

, then

Tw(q)=Tv(T\sigma(q))

.

Let

M(Q,\Sigma,T)

be the set

M(Q,\Sigma,T)=\{Tw\vertw\in\Sigma*\}.

The set

M(Q,\Sigma,T)

is closed under function composition; that is, for all

v,w\in\Sigma*

, one has

Tw\circTv=Tvw

. It also contains

T\varepsilon

, which is the identity function on Q. Since function composition is associative, the set

M(Q,\Sigma,T)

is a monoid: it is called the input monoid, characteristic monoid, characteristic semigroup or transition monoid of the semiautomaton

Properties

If the set of states Q is finite, then the transition functions are commonly represented as state transition tables. The structure of all possible transitions driven by strings in the free monoid has a graphical depiction as a de Bruijn graph.

CPn

, and individual states are referred to as n-state qubits. State transitions are given by unitary n×n matrices. The input alphabet

\Sigma

remains finite, and other typical concerns of automata theory remain in play. Thus, the quantum semiautomaton may be simply defined as the triple
n,\Sigma,\{U
(CP
\sigma1
,U
\sigma2
,...c,U
\sigmap

\})

when the alphabet

\Sigma

has p letters, so that there is one unitary matrix

U\sigma

for each letter

\sigma\in\Sigma

. Stated in this way, the quantum semiautomaton has many geometrical generalizations. Thus, for example, one may take a Riemannian symmetric space in place of

CPn

, and selections from its group of isometries as transition functions.

The syntactic monoid of a regular language is isomorphic to the transition monoid of the minimal automaton accepting the language.

Literature

Notes and References

  1. Moghbeli-Damaneh . Halimeh . July 2020 . The symmetric monoidal closed category of cpo M-sets . Categories and General Algebraic Structures with Applications . 13 . 1.