Mikołaj Bojańczyk


Monoids are equivalent to automata

A language L \subseteq \Sigma^\infty is called \infty-recognisable if there is some homomorphism of \infty-monoids

    \[h : \Sigma^\infty \to M\]

such that M has finite universe, and which recognizes L in the sense that membership w \in L is uniquely determined by h(w).  The following theorem shows that in the special (but not all that special) case of languages that contain only \omega-words, this notion coincides with the notion of \omega-regularity.

Theorem. A language L \subseteq \Sigma^\omega is \infty-recognisable if and only if it is \omega-regular.

Proof. For m \in M define L_{m,+} to be the finite nonempty words that are mapped by h to m, and define L_{m,\omega} to be the \omega-words that are mapped by h to m. It is easy to see that each language L_{m,+} is a regular language of finite words, because it is recognised by a finite monoid, namely the monoid obtained from M by ignoring products for infinite words. Furthermore, using the Ramsey theorem in the same way as in the previous lemma, we conclude that L_{m,\omega} is equal to the \omega-regular language

    \[\bigcup_{s,t} L_{s,+} \cdot (L_{t,+})^\omega\]

with the union ranging over elements s,t \in M which satisfy st^\omega = m. Every set of \omega-words recognised by h is a finite union of languages of the form L_{m,\omega}, and therefore the left-to-right implication follows by closure of \omega-regular languages under finite union.

For the right-to-left implication, suppose that L is recongised by a nondeterministic Büchi automaton with states Q. Define

    \[h : \Sigma^\infty \to P(Q) \cup P(Q \times \set{0,1} \times Q) \cup \set \epsilon\]

defined as follows. The empty word gets mapped to \epsilon. Infinite words are mapped to P(Q), namely an infinite word is mapped to the set of those states p such that the word admits an accepting run that begins with p. Finite words are mapped to P(Q \times \set{0,1} \times Q) 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 \infty-monoid which makes it into a homomorphism. \Box




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