18,268 research outputs found
The Magic Number Problem for Subregular Language Families
We investigate the magic number problem, that is, the question whether there
exists a minimal n-state nondeterministic finite automaton (NFA) whose
equivalent minimal deterministic finite automaton (DFA) has alpha states, for
all n and alpha satisfying n less or equal to alpha less or equal to exp(2,n).
A number alpha not satisfying this condition is called a magic number (for n).
It was shown in [11] that no magic numbers exist for general regular languages,
while in [5] trivial and non-trivial magic numbers for unary regular languages
were identified. We obtain similar results for automata accepting subregular
languages like, for example, combinational languages, star-free, prefix-,
suffix-, and infix-closed languages, and prefix-, suffix-, and infix-free
languages, showing that there are only trivial magic numbers, when they exist.
For finite languages we obtain some partial results showing that certain
numbers are non-magic.Comment: In Proceedings DCFS 2010, arXiv:1008.127
Automata Minimization: a Functorial Approach
In this paper we regard languages and their acceptors - such as deterministic
or weighted automata, transducers, or monoids - as functors from input
categories that specify the type of the languages and of the machines to
categories that specify the type of outputs. Our results are as follows:
A) We provide sufficient conditions on the output category so that
minimization of the corresponding automata is guaranteed.
B) We show how to lift adjunctions between the categories for output values
to adjunctions between categories of automata.
C) We show how this framework can be instantiated to unify several phenomena
in automata theory, starting with determinization, minimization and syntactic
algebras. We provide explanations of Choffrut's minimization algorithm for
subsequential transducers and of Brzozowski's minimization algorithm in this
setting.Comment: journal version of the CALCO 2017 paper arXiv:1711.0306
Theory of Atomata
We show that every regular language defines a unique nondeterministic finite
automaton (NFA), which we call "\'atomaton", whose states are the "atoms" of
the language, that is, non-empty intersections of complemented or
uncomplemented left quotients of the language. We describe methods of
constructing the \'atomaton, and prove that it is isomorphic to the reverse
automaton of the minimal deterministic finite automaton (DFA) of the reverse
language. We study "atomic" NFAs in which the right language of every state is
a union of atoms. We generalize Brzozowski's double-reversal method for
minimizing a deterministic finite automaton (DFA), showing that the result of
applying the subset construction to an NFA is a minimal DFA if and only if the
reverse of the NFA is atomic. We prove that Sengoku's claim that his method
always finds a minimal NFA is false.Comment: 29 pages, 2 figures, 28 table
Index problems for game automata
For a given regular language of infinite trees, one can ask about the minimal
number of priorities needed to recognize this language with a
non-deterministic, alternating, or weak alternating parity automaton. These
questions are known as, respectively, the non-deterministic, alternating, and
weak Rabin-Mostowski index problems. Whether they can be answered effectively
is a long-standing open problem, solved so far only for languages recognizable
by deterministic automata (the alternating variant trivializes).
We investigate a wider class of regular languages, recognizable by so-called
game automata, which can be seen as the closure of deterministic ones under
complementation and composition. Game automata are known to recognize languages
arbitrarily high in the alternating Rabin-Mostowski index hierarchy; that is,
the alternating index problem does not trivialize any more.
Our main contribution is that all three index problems are decidable for
languages recognizable by game automata. Additionally, we show that it is
decidable whether a given regular language can be recognized by a game
automaton
Testing the Equivalence of Regular Languages
The minimal deterministic finite automaton is generally used to determine
regular languages equality. Antimirov and Mosses proposed a rewrite system for
deciding regular expressions equivalence of which Almeida et al. presented an
improved variant. Hopcroft and Karp proposed an almost linear algorithm for
testing the equivalence of two deterministic finite automata that avoids
minimisation. In this paper we improve the best-case running time, present an
extension of this algorithm to non-deterministic finite automata, and establish
a relationship between this algorithm and the one proposed in Almeida et al. We
also present some experimental comparative results. All these algorithms are
closely related with the recent coalgebraic approach to automata proposed by
Rutten
From Finite Automata to Regular Expressions and Back--A Summary on Descriptional Complexity
The equivalence of finite automata and regular expressions dates back to the
seminal paper of Kleene on events in nerve nets and finite automata from 1956.
In the present paper we tour a fragment of the literature and summarize results
on upper and lower bounds on the conversion of finite automata to regular
expressions and vice versa. We also briefly recall the known bounds for the
removal of spontaneous transitions (epsilon-transitions) on non-epsilon-free
nondeterministic devices. Moreover, we report on recent results on the average
case descriptional complexity bounds for the conversion of regular expressions
to finite automata and brand new developments on the state elimination
algorithm that converts finite automata to regular expressions.Comment: In Proceedings AFL 2014, arXiv:1405.527
Small NFAs from Regular Expressions: Some Experimental Results
Regular expressions (res), because of their succinctness and clear syntax,
are the common choice to represent regular languages. However, efficient
pattern matching or word recognition depend on the size of the equivalent
nondeterministic finite automata (NFA). We present the implementation of
several algorithms for constructing small epsilon-free NFAss from res within
the FAdo system, and a comparison of regular expression measures and NFA sizes
based on experimental results obtained from uniform random generated res. For
this analysis, nonredundant res and reduced res in star normal form were
considered.Comment: Proceedings of 6th Conference on Computability in Europe (CIE 2010),
pages 194-203, Ponta Delgada, Azores, Portugal, June/July 201
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