Abstract

Semi-flower languages are those of the form $$L^*$$ for some finite maximal prefix code L, or equivalently, those recognizable by a so-called semi-flower automaton, in which all the cycles have a common state $$q_0$$ , which happens to be the initial state and the only accepting state. We show that the syntactic complexity of these languages is exactly $$n^n-n!+n$$ (where n stands for the state complexity as usual) and that this bound is reachable with an alphabet of size n.

Highlights

  • The state complexity of a regular language is the number of states of its minimal automaton, or equivalently, the number of classes of its syntactic rightcongruence

  • When C is a class of regular languages, its syntactic complexity is a function over the single integer variable n, namely it is the maximum possible syntactic complexity of a language belonging to C and having state complexity at most n

  • We showed that the syntactic complexity of semi-flower languages is exactly nn − n! + n and that in order to reach this complexity, an alphabet of linear size suffices

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Summary

Introduction

The state complexity of a regular language is the number of states of its minimal automaton, or equivalently, the number of classes of its syntactic rightcongruence. In [12] they showed that if a circular semi-flower automaton over a binary alphabet has a single ,,branching point in”, or bpi (a state q is called a bpi if there are at least two tuples (p, a) ∈ Q × Σ with pa = q), it has a linear syntactic complexity, and if it possesses exactly two bpis, 2n(n + 1) is a sharp bound on its syntactic complexity This is a serious restriction: it essentially restricts the elementary transformations to a circular permutation and some semi-flower transformation of rank at most 2 (for the definitions, see the Notation section). In this paper we determine that the syntactic complexity of languages recognizable by semi-flower automata (without placing any restriction on the number of their “branch points going in”) is nn − n! + n and show that this bound is reachable by an alphabet of size n

Notation and some facts
Conclusion

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