MAGIC NUMBERS FOR SYMMETRIC DIFFERENCE NFAS

Iwama et al. showed that there exists an n-state binary nondeterministic finite automaton such that its equivalent minimal deterministic finite automaton has exactly 2n - α states, for all n ≥ 7 and 5 ≤ α ≤ 2n-2, subject to certain coprimality conditions. We investigate the same question for both unary and binary symmetric difference nondeterministic finite automata. In the binary case, we show that for any n ≥ 4, there is an n-state symmetric difference nondeterministic finite automaton for which the equivalent minimal deterministic finite automaton has 2n - 1 + 2k - 1 - 1 states, for 2 < k ≤ n - 1. In the unary case, we consider a large practical subclass of unary symmetric difference nondeterministic finite automata: for all n ≥ 2, we argue that there are many values of α such that there is no n-state unary symmetric difference nondeterministic finite automaton with an equivalent minimal deterministic finite automaton with 2n - α states, where 0 < α < 2n - 1. For each n ≥ 2, we quantify such values of α precisely.

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DETERMINISTIC BLOW-UPS OF MINIMAL NONDETERMINISTIC FINITE AUTOMATA OVER A FIXED ALPHABET

International Journal of Foundations of Computer Science ◽

10.1142/s0129054108005851 ◽

2008 ◽

Vol 19 (03) ◽

pp. 617-631 ◽

Cited By ~ 6

Author(s):

JOZEF JIRÁSEK ◽

GALINA JIRÁSKOVÁ ◽

ALEXANDER SZABARI

Keyword(s):

Finite Automaton ◽

Finite Automata ◽

Magic Numbers ◽

Deterministic Finite Automaton ◽

Nondeterministic Finite Automata ◽

Letter Alphabet ◽

Input Alphabet ◽

Nondeterministic Finite Automaton

We show that for all integers n and α such that n ⩽ α ⩽ 2n, there exists a minimal nondeterministic finite automaton of n states with a four-letter input alphabet whose equivalent minimal deterministic finite automaton has exactly α states. It follows that in the case of a four-letter alphabet, there are no "magic numbers", i.e., the holes in the hierarchy. This improves a similar result obtained by Geffert for a growing alphabet of size n + 2.

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Classes of Transfinite Sequences Accepted by Nondeterministic Finite Automata

Fundamenta Informaticae ◽

10.3233/fi-1984-7203 ◽

1984 ◽

Vol 7 (2) ◽

pp. 191-223

Author(s):

Jerzy Wojciechowski

Keyword(s):

Finite Automaton ◽

Finite Automata ◽

Basic Properties ◽

Nondeterministic Finite Automata ◽

The Family ◽

Finite Sequences ◽

Emptiness Problem ◽

Nondeterministic Finite Automaton ◽

Defining Sets

In this paper the notion of a nondeterministic finite automaton acting on arbitrary transfinite sequences is introduced. It is a generalization of the finite automaton on finite sequences and the finite automaton on ω-sequences. The basic properties of the behaviour of such automata are proved. The methods are shown how to construct automata accepting classes A ⋃ B, A ⋂ B, A ∘ B, A*, Aω, A# if we have automata accepting classes A and B. We prove that if a TF-automaton having k states accepts anything then it accepts an α-sequence for a certain, α ∈ { ∑ i = 0 m ω i · a i : ∑ i = 1 m i · a i + a 0 ⩽ k }. Using the foregoing fact, we show that the family of classes definable by TF-automata is not closed with respect to the complement operation, that nondeterministic automata are not equivalent to the deterministic ones and that the emptiness problem for TP-automata is decidable. In the last section we show the construction of TP-automata defining sets {∗α} for α < ω ω and having as few states as possible.

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The Degree of Irreversibility in Deterministic Finite Automata

International Journal of Foundations of Computer Science ◽

10.1142/s0129054117400044 ◽

2017 ◽

Vol 28 (05) ◽

pp. 503-522

Author(s):

Holger Bock Axelsen ◽

Markus Holzer ◽

Martin Kutrib

Keyword(s):

Finite Automaton ◽

Regular Language ◽

Finite Automata ◽

Deterministic Finite Automaton ◽

Deterministic Finite Automata ◽

Infinite Hierarchy ◽

Complete Problem ◽

Nondeterministic Finite Automata ◽

Forbidden Patterns

Recently, a method to decide the NL-complete problem of whether the language accepted by a given deterministic finite automaton (DFA) can also be accepted by some reversible deterministic finite automaton (REV-DFA) has been derived. Here, we show that the corresponding problem for nondeterministic finite automata (NFA) is PSPACE-complete. The recent DFA method essentially works by minimizing the DFA and inspecting it for a forbidden pattern. We here study the degree of irreversibility for a regular language, the minimal number of such forbidden patterns necessary in any DFA accepting the language, and show that the degree induces a strict infinite hierarchy of language families. We examine how the degree of irreversibility behaves under the usual language operations union, intersection, complement, concatenation, and Kleene star, showing tight bounds (some asymptotically) on the degree.

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ON THE REGULARITY OF SETS OF MULTI-ACCEPTED STRINGS OF A NON-DETERMINISTIC FINITE AUTOMATON

Asian-European Journal of Mathematics ◽

10.1142/s1793557109000595 ◽

2009 ◽

Vol 02 (04) ◽

pp. 717-726

Author(s):

L. K. Waters ◽

J. K. Grieshop

Keyword(s):

Finite Automaton ◽

Finite Automata ◽

Equivalence Classes ◽

Deterministic Finite Automaton ◽

Finite Sets ◽

Nondeterministic Finite Automata ◽

Labeling Approach ◽

Acceptance Levels ◽

Labeling Method

In this paper we establish the regularity of various sets of multi-accepted strings of nondeterministic finite automata. Regularity follows from the existence of accepting automata constructed by introducing a vector labeling method which generalizes the subset labeling approach. In each set the acceptance levels of the strings correspond to finite sets of additive equivalence classes of non-negative integers.

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MAGIC NUMBERS AND TERNARY ALPHABET

International Journal of Foundations of Computer Science ◽

10.1142/s0129054111008076 ◽

2011 ◽

Vol 22 (02) ◽

pp. 331-344 ◽

Cited By ~ 10

Author(s):

GALINA JIRÁSKOVÁ

Keyword(s):

Finite Automaton ◽

Magic Numbers ◽

Deterministic Finite Automaton ◽

Alphabet Size ◽

Input Alphabet ◽

Nondeterministic Finite Automaton

A number α, in the range from n to 2n, is magic for n with respect to a given alphabet size s, if there is no minimal nondeterministic finite automaton of n states and s input symbols whose equivalent minimal deterministic finite automaton has α states. We show that in the case of a ternary alphabet, there are no magic numbers. For all n and α satisfying n ⩽ α ⩽ 2n, we define an n-state nondeterministic finite automaton with a three-letter input alphabet that requires exactly α deterministic states.

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THE MAGIC NUMBER PROBLEM FOR SUBREGULAR LANGUAGE FAMILIES

International Journal of Foundations of Computer Science ◽

10.1142/s0129054112400084 ◽

2012 ◽

Vol 23 (01) ◽

pp. 115-131 ◽

Cited By ~ 2

Author(s):

MARKUS HOLZER ◽

SEBASTIAN JAKOBI ◽

MARTIN KUTRIB

Keyword(s):

Finite Automaton ◽

Magic Number ◽

Regular Languages ◽

Magic Numbers ◽

Deterministic Finite Automaton ◽

Number Problem ◽

Nondeterministic Finite Automaton

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 α states, for all n and α satisfying n ≤ α ≤ 2n. A number α not satisfying this condition is called a magic number (for n). It was shown that no magic numbers exist for general regular languages, whereas trivial and non-trivial magic numbers for unary regular languages were identified. We obtain similar results for automata accepting subregular languages like, for example, star-free languages, 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.

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Finite Automata Implementations Considering CPU Cache

Acta Polytechnica ◽

10.14311/1008 ◽

2007 ◽

Vol 47 (6) ◽

Author(s):

J. Holub

Keyword(s):

Mathematical Models ◽

Finite Automaton ◽

Finite Automata ◽

The Other ◽

Practical Implementation ◽

Deterministic Finite Automaton ◽

Input Text ◽

Finite State ◽

Nondeterministic Finite Automaton

The finite automata are mathematical models for finite state systems. More general finite automaton is the nondeterministic finite automaton (NFA) that cannot be directly used. It is usually transformed to the deterministic finite automaton (DFA) that then runs in time O(n), where n is the size of the input text. We present two main approaches to practical implementation of DFA considering CPU cache. The first approach (represented by Table Driven and Hard Coded implementations) is suitable forautomata being run very frequently, typically having cycles. The other approach is suitable for a collection of automata from which various automata are retrieved and then run. This second kind of automata are expected to be cycle-free.

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SIMULATIONS OF UNARY ONE-WAY MULTI-HEAD FINITE AUTOMATA

International Journal of Foundations of Computer Science ◽

10.1142/s0129054114400139 ◽

2014 ◽

Vol 25 (07) ◽

pp. 877-896 ◽

Cited By ~ 3

Author(s):

MARTIN KUTRIB ◽

ANDREAS MALCHER ◽

MATTHIAS WENDLANDT

Keyword(s):

Lower Bound ◽

Lower Bounds ◽

Finite Automaton ◽

Finite Automata ◽

Upper And Lower Bounds ◽

Unary Languages ◽

Order Of Magnitude ◽

Simulation Results ◽

Special Case ◽

Nondeterministic Finite Automaton

We investigate the descriptional complexity of deterministic one-way multi-head finite automata accepting unary languages. It is known that in this case the languages accepted are regular. Thus, we study the increase of the number of states when an n-state k-head finite automaton is simulated by a classical (one-head) deterministic or nondeterministic finite automaton. In the former case upper and lower bounds that are tight in the order of magnitude are shown. For the latter case we obtain an upper bound of O(n2k) and a lower bound of Ω(nk) states. We investigate also the costs for the conversion of one-head nondeterministic finite automata to deterministic k-head finite automata, that is, we trade nondeterminism for heads. In addition, we study how the conversion costs vary in the special case of finite and, in particular, of singleton unary lanuages. Finally, as an application of the simulation results, we show that decidability problems for unary deterministic k-head finite automata such as emptiness or equivalence are LOGSPACE-complete.

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Square on Deterministic, Alternating, and Boolean Finite Automata

International Journal of Foundations of Computer Science ◽

10.1142/s0129054119400318 ◽

2019 ◽

Vol 30 (06n07) ◽

pp. 1117-1134

Author(s):

Galina Jirásková ◽

Ivana Krajňáková

Keyword(s):

Upper Bound ◽

Finite Automaton ◽

Finite Automata ◽

Upper Bounds ◽

The State ◽

Final States ◽

Deterministic Finite Automaton ◽

State Complexity

We investigate the state complexity of the square operation on languages represented by deterministic, alternating, and Boolean automata. For each [Formula: see text] such that [Formula: see text], we describe a binary language accepted by an [Formula: see text]-state deterministic finite automaton with [Formula: see text] final states meeting the upper bound [Formula: see text] on the state complexity of its square. We show that in the case of [Formula: see text], the corresponding upper bound cannot be met. Using the binary deterministic witness for square with [Formula: see text] states where half of them are final, we get the tight upper bounds [Formula: see text] and [Formula: see text] on the complexity of the square operation on alternating and Boolean automata, respectively.

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Parallel Algorithms for Minimal Nondeterministic Finite Automata Inference

Fundamenta Informaticae ◽

10.3233/fi-2021-2004 ◽

2021 ◽

Vol 178 (3) ◽

pp. 203-227

Author(s):

Tomasz Jastrzab ◽

Zbigniew J. Czech ◽

Wojciech Wieczorek

Keyword(s):

Parallel Algorithms ◽

Constraint Satisfaction ◽

Finite Automaton ◽

Regular Language ◽

Finite Automata ◽

Constraint Satisfaction Problem ◽

Computational Experiments ◽

Learning Sample ◽

Automata Inference ◽

Nondeterministic Finite Automaton

The goal of this paper is to develop the parallel algorithms that, on input of a learning sample, identify a regular language by means of a nondeterministic finite automaton (NFA). A sample is a pair of finite sets containing positive and negative examples. Given a sample, a minimal NFA that represents the target regular language is sought. We define the task of finding an NFA, which accepts all positive examples and rejects all negative ones, as a constraint satisfaction problem, and then propose the parallel algorithms to solve the problem. The results of comprehensive computational experiments on the variety of inference tasks are reported. The question of minimizing an NFA consistent with a learning sample is computationally hard.

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