On deterministic finite automata and syntactic monoid size

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On deterministic finite automata and syntactic monoid size (2003)

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by Markus Holzer , Barbara König

Venue:	PROCEEDINGS OF THE 7TH INTERNATIONAL CONFERENCE ON DEVELOPMENTS IN LANGUAGE THEORY, NUMBER 2710 IN LNCS
Citations:	18 - 1 self

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@INPROCEEDINGS{Holzer03ondeterministic,
    author = {Markus Holzer and Barbara König},
    title = {On deterministic finite automata and syntactic monoid size},
    booktitle = {PROCEEDINGS OF THE 7TH INTERNATIONAL CONFERENCE ON DEVELOPMENTS IN LANGUAGE THEORY, NUMBER 2710 IN LNCS},
    year = {2003},
    pages = {349--360},
    publisher = {Springer}
}

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Abstract

We investigate the relationship between regular languages and syntactic monoid size. In particular, we consider the transformation monoids of n-state (minimal) deterministic finite automata. We show tight upper bounds on the syntactic monoid size, proving that an nstate deterministic finite automaton with singleton input alphabet (input alphabet with at least three letters, respectively) induces a linear (n n, respectively) size syntactic monoid. In the case of two letter input alphabet, we can show a lower bound of n n − ` ´ ` ´ n k n k ℓ ℓ!n − k ℓ, for some ℓ ℓ, for the size of the syntactic monoid of natural numbers k and ℓ close to n 2 a language accepted by an n-state deterministic finite automaton. This induces a family of deterministic finite automata such that the fraction of the size of the induced syntactic monoid and n n tends to 1 as n goes to infinity.

Keyphrases

deterministic finite automaton syntactic monoid size letter input tight upper bound transformation monoids natural number nstate deterministic finite automaton induced syntactic monoid singleton input alphabet syntactic monoid size syntactic monoid n-state deterministic finite automaton regular language

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