Monoids are equivalent to automata
A language is called
-recognisable if there is some homomorphism of
-monoids
such that has finite universe, and which recognizes
in the sense that membership
is uniquely determined by
. The following theorem shows that in the special (but not all that special) case of languages that contain only
-words, this notion coincides with the notion of
-regularity.
Theorem. A language is
-recognisable if and only if it is
-regular.
Proof. For define
to be the finite nonempty words that are mapped by
to
, and define
to be the
-words that are mapped by
to
. It is easy to see that each language
is a regular language of finite words, because it is recognised by a finite monoid, namely the monoid obtained from
by ignoring products for infinite words. Furthermore, using the Ramsey theorem in the same way as in the previous lemma, we conclude that
is equal to the
-regular language
with the union ranging over elements which satisfy
. Every set of
-words recognised by
is a finite union of languages of the form
, and therefore the left-to-right implication follows by closure of
-regular languages under finite union.
For the right-to-left implication, suppose that is recongised by a nondeterministic Büchi automaton with states
. Define
defined as follows. The empty word gets mapped to . Infinite words are mapped to
, namely an infinite word is mapped to the set of those states
such that the word admits an accepting run that begins with
. Finite words are mapped to
in the same way as in the proof of complementation for Büchi automata. It is not difficult to see that this mapping is compositional, and therefore its image can be equipped with the structure of an
-monoid which makes it into a homomorphism.
Leave a Reply