Varieties Of Formal Languages
Revised Papers from the Second International Workshop on Implementing Automata
WIA '97 Revised Papers from the Second International Workshop on Implementing Automata
On deterministic finite automata and syntactic monoid size, continued
DLT'03 Proceedings of the 7th international conference on Developments in language theory
State complexity and the monoid of transformations of a finite set
CIAA'04 Proceedings of the 9th international conference on Implementation and Application of Automata
Three learnable models for the description of language
LATA'10 Proceedings of the 4th international conference on Language and Automata Theory and Applications
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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 n- state deterministic finite automaton with singleton input alphabet (input alphabet with at least three letters, respectively) induces a linear (nn, respectively) size syntactic monoid. In the case of two letter input alphabet, we can show a lower bound of nn - (ln) l!nk - (ln)kk ll, for some natural numbers k and l close to n/2, for the size of the syntactic monoid of 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 nn tends to 1 as n goes to infinity.