Semiautomaton

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 : Q → Q . If s and t are two functions of the transformation semigroup, their semigroup product is defined as their function composition ( s t ) ( q ) = ( s ∘ t ) ( 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

which satisfies the properties

for 1 the unit of the monoid, and

for all q ∈ Q and s , t ∈ M , then the triple ( Q , M , μ ) is called a right M-act or simply a right act. In long-hand, μ is the right multiplication of elements of Q by elements of M. The right act is often written as Q M .

The left act is defined similarly, with

and is often denoted as M Q .

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 μ be consistent with multiplication in the monoid (i.e. μ ( q , s t ) = μ ( μ ( q , s ) , t ) ), as, in general, this might not hold for some arbitrary μ , 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 Q M and B M sharing the same monoid M , an M-homomorphism f : Q M → B M is a map f : Q → B such that

for all q ∈ Q and m ∈ M . The set of all M-homomorphisms is commonly written as H o m ( Q M , B M ) or H o m M ( Q , B ) .

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

Semiautomata

A semiautomaton is a triple ( Q , Σ , T ) where Σ 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

When the set of states Q is a finite set (it need not be!), a semiautomaton may be thought of as a deterministic finite automaton ( Q , Σ , T , q 0 , A ) , but without the initial state q 0 or set of accept states A. Alternately, it is a finite state machine which has no output, and only an input.

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

Let Σ ∗ be the free monoid generated by the alphabet Σ (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 Σ .

For every word w in Σ ∗ , let T w : Q → Q be the function, defined recursively, as follows, for all q in Q:

Let M ( Q , Σ , T ) be the set

The set M ( Q , Σ , T ) is closed under function composition; that is, for all v , w ∈ Σ ∗ , one has T w ∘ T v = T v w . It also contains T ε , which is the identity function on Q. Since function composition is associative, the set M ( Q , Σ , T ) is a monoid: it is called the input monoid, characteristic monoid, characteristic semigroup or transition monoid of the semiautomaton ( Q , Σ , T ) .

Properties

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

The set of states Q need not be finite, or even countable. As an example, semiautomata underpin the concept of quantum finite automata. There, the set of states Q are given by the complex projective space C P n , and individual states are referred to as n-state qubits. State transitions are given by unitary n×n matrices. The input alphabet Σ remains finite, and other typical concerns of automata theory remain in play. Thus, the quantum semiautomaton may be simply defined as the triple ( C P n , Σ , { U σ 1 , U σ 2 , … , U σ p , } ) when the alphabet Σ has p letters, so that there is one unitary matrix U σ for each letter σ ∈ Σ . Stated in this way, it is obvious that the quantum semiautomaton has many geometrical generalizations. Thus, for example, one may take a Riemannian symmetric space in place of C P n , and selections from its group of isometries as transition functions.

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