Recognizable Set

In computer science, more precisely in automata theory, a recognizable set of a monoid is a subset that can be distinguished by some homomorphism to a finite monoid. Recognizable sets are useful in automata theory, formal languages and algebra.

This notion is different from the notion of recognizable language. Indeed, the term "recognizable" has a different meaning in computability theory.

Definition

Let $N$ be a monoid, a subset $S\subseteq N$ is recognized by a monoid $M$ if there exists a homomorphism $\phi$ from $N$ to $M$ such that $S=\phi^(\phi(S))$, and recognizable if it is recognized by some finite monoid. This means that there exists a subset $T$ of $M$ (not necessarily a submonoid of $M$) such that the image of $S$ is in $T$ and the image of $N \setminus S$ is in $M \setminus T$.

Example

Let $A$ be an alphabet: the set $A^*$ of words over $A$ is a monoid, the free monoid on $A$. The recognizable subsets of $A^*$ are precisely the regular languages. Indeed, such a language is recognized by the transition monoid of any automaton that recognizes the language.

The recognizable subsets of $\mathbb N$ are the ultimately periodic sets of integers.

Properties

A subset of $N$ is recognizable if and only if its syntactic monoid is finite.

The set $\mathrm(N)$ of recognizable subsets of $N$ is closed under:
* union
* intersection
* complement
* right and left quotient

Mezei's theorem states that if $M$ is the product of the monoids $M_1, \dots, M_n$, then a subset of $M$ is recognizable if and only if it is a finite union of subsets of the form $R_1 \times \cdots \times R_n$, where each $R_i$ is a recognizable subset of $M_i$. For instance, the subset $\$ of $\mathbb N$ is rational and hence recognizable, since $\mathbb N$ is a free monoid. It follows that the subset $S=\$ of $\mathbb N^2$ is recognizable.

McKnight's theorem states that if $N$ is finitely generated then its recognizable subsets are rational subsets.
This is not true in general, since the whole $N$ is always recognizable but it is not rational if $N$ is infinitely generated.

Conversely, a rational subset may not be recognizable, even if $N$ is finitely generated.
In fact, even a finite subset of $N$ is not necessarily recognizable. For instance, the set $\$ is not a recognizable subset of $(\mathbb Z, +)$. Indeed, if a homomorphism $\phi$ from $\mathbb Z$ to $M$ satisfies $\=\phi^(\phi(\))$, then $\phi$ is an injective function; hence $M$ is infinite.

Also, in general, $\mathrm(N)$ is not closed under Kleene star. For instance, the set $S=\$ is a recognizable subset of $\mathbb N^2$, but $S^*=\$ is not recognizable. Indeed, its syntactic monoid is infinite.

The intersection of a rational subset and of a recognizable subset is rational.

Recognizable sets are closed under inverse of homomorphisms. I.e. if $N$ and $M$ are monoids and $\phi:N\rightarrow M$ is a homomorphism then if $S\in\mathrm(M)$ then $\phi^(S)=\\in\mathrm(N)$.

For finite groups the following result of Anissimov and Seifert is well known: a subgroup ''H'' of a finitely generated group ''G'' is recognizable if and only if ''H'' has finite index in ''G''. In contrast, ''H'' is rational if and only if ''H'' is finitely generated.preprint
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See also

* Rational set
* Rational monoid

References

*

*Jean-Éric Pin
Mathematical Foundations of Automata Theory
Chapter IV: Recognisable and rational sets

Further reading

* {{cite book , last=Sakarovitch , first=Jacques , title=Elements of automata theory , others=Translated from the French by Reuben Thomas , location=Cambridge , publisher=Cambridge University Press , year=2009 , isbn=978-0-521-84425-3 , zbl=1188.68177 , at = Part II: The power of algebra

Automata (computation)