Union-Freeness Revisited — Between Deterministic and Nondeterministic Union-Free Languages

2021 ◽  
pp. 1-23
Author(s):  
Benedek Nagy
Keyword(s):  
Normal Form ◽  
Regular Language ◽  
Regular Expression ◽  
Finite Automata ◽  
Finite Union ◽  
Regular Languages ◽  
Regular Expressions ◽  
Closure Properties ◽  
Language Class ◽  
Union Operation

Union-free expressions are regular expressions without using the union operation. Consequently, (nondeterministic) union-free languages are described by regular expressions using only concatenation and Kleene star. The language class is also characterised by a special class of finite automata: 1CFPAs have exactly one cycle-free accepting path from each of their states. Obviously such an automaton has exactly one accepting state. The deterministic counterpart of such class of automata defines the deterministic union-free (d-union-free, for short) languages. In this paper [Formula: see text]-free nondeterministic variants of 1CFPAs are used to define n-union-free languages. The defined language class is shown to be properly between the classes of (nondeterministic) union-free and d-union-free languages (in case of at least binary alphabet). In case of unary alphabet the class of n-union-free languages coincides with the class of union-free languages. Some properties of the new subregular class of languages are discussed, e.g., closure properties. On the other hand, a regular expression is in union normal form if it is a finite union of union-free expressions. It is well known that every regular expression can be written in union normal form, i.e., all regular languages can be described as finite unions of (nondeterministic) union-free languages. It is also known that the same fact does not hold for deterministic union-free languages, that is, there are regular languages that cannot be written as finite unions of d-union-free languages. As an important result here we show that every regular language can be defined by a finite union of n-union-free languages. This fact also allows to define n-union-complexity of regular languages.

COMPLEXITY IN UNION-FREE REGULAR LANGUAGES

2011 ◽  
Vol 22 (07) ◽  
pp. 1639-1653 ◽  
Author(s):  
GALINA JIRÁSKOVÁ ◽  
TOMÁŠ MASOPUST
Keyword(s):  
Free Path ◽  
Finite Automata ◽  
Regular Languages ◽  
Regular Expressions ◽  
Deterministic Finite Automata ◽  
Union Operation

We continue the investigation of union-free regular languages that are described by regular expressions without the union operation. We also define deterministic union-free languages as languages accepted by one-cycle-free-path deterministic finite automata, and show that they are properly included in the class of union-free languages. We prove that (deterministic) union-freeness of languages does not accelerate regular operations, except for the reversal in the nondeterministic case.

scholarly journals Software Toolchain for Large-Scale RE-NFA Construction on FPGA

2009 ◽  
Vol 2009 ◽  
pp. 1-10 ◽  
Author(s):  
Yi-Hua E. Yang ◽  
Viktor K. Prasanna
Keyword(s):  
High Performance ◽  
Large Scale ◽  
Regular Expression ◽  
Finite Automata ◽  
Fixed Number ◽  
Regular Expressions ◽  
Pattern Complexity ◽  
Regular Expression Matching ◽  
Area Increase ◽  
Prototype Software

We present a software toolchain for constructing large-scaleregular expression matching(REM) on FPGA. The software automates the conversion of regular expressions into compact and high-performance nondeterministic finite automata (RE-NFA). Each RE-NFA is described as an RTL regular expression matching engine (REME) in VHDL for FPGA implementation. Assuming a fixed number of fan-out transitions per state, ann-statem-bytes-per-cycle RE-NFA can be constructed inO(n×m)time andO(n×m)memory by our software. A large number of RE-NFAs are placed onto a two-dimensionalstaged pipeline, allowing scalability to thousands of RE-NFAs with linear area increase and little clock rate penalty due to scaling. On a PC with a 2 GHz Athlon64 processor and 2 GB memory, our prototype software constructs hundreds of RE-NFAs used by Snort in less than 10 seconds. We also designed a benchmark generator which can produce RE-NFAs with configurable pattern complexity parameters, including state count, state fan-in, loop-back and feed-forward distances. Several regular expressions with various complexities are used to test the performance of our RE-NFA construction software.

ORDINAL AUTOMATA AND CANTOR NORMAL FORM

2012 ◽  
Vol 23 (01) ◽  
pp. 87-98
Author(s):  
ZOLTÁN ÉSIK
Keyword(s):  
Normal Form ◽  
Polynomial Time ◽  
Regular Language ◽  
Finite Automata ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Order Type ◽  
Deterministic Finite Automaton ◽  
Lexicographic Ordering ◽  
Language L

It is known that an ordinal is the order type of the lexicographic ordering of a regular language if and only if it is less than ωω. We design a polynomial time algorithm that constructs, for each well-ordered regular language L with respect to the lexicographic ordering, given by a deterministic finite automaton, the Cantor Normal Form of its order type. It follows that there is a polynomial time algorithm to decide whether two deterministic finite automata accepting well-ordered regular languages accept isomorphic languages. We also give estimates on the state complexity of the smallest "ordinal automaton" representing an ordinal less than ωω, together with an algorithm that translates each such ordinal to an automaton.

UNWEIGHTED AND WEIGHTED HYPER-MINIMIZATION

2012 ◽  
Vol 23 (06) ◽  
pp. 1207-1225 ◽  
Author(s):  
ANDREAS MALETTI ◽  
DANIEL QUERNHEIM
Keyword(s):  
Finite Automata ◽  
Reduction Technique ◽  
Regular Languages ◽  
Minimization Algorithm ◽  
Compression Method ◽  
Closure Properties ◽  
State Reduction ◽  
Deterministic Finite Automata ◽  
Input Strings ◽  
Weighted Finite Automata

Hyper-minimization of deterministic finite automata (DFA) is a recently introduced state reduction technique that allows a finite change in the recognized language. A generalization of this lossy compression method to the weighted setting over semifields is presented, which allows the recognized weighted language to differ for finitely many input strings. First, the structure of hyper-minimal deterministic weighted finite automata is characterized in a similar way as in classical weighted minimization and unweighted hyper-minimization. Second, an efficient hyper-minimization algorithm, which runs in time [Formula: see text], is derived from this characterization. Third, the closure properties of canonical regular languages, which are languages recognized by hyper-minimal DFA, are investigated. Finally, some recent results in the area of hyper-minimization are recalled.

Roots and Powers in Regular Languages: Recognizing Nonregular Properties by Finite Automata

2020 ◽  
Vol 175 (1-4) ◽  
pp. 173-185
Author(s):  
Fabian Frei ◽  
Juraj Hromkovič ◽  
Juhani Karhumäki
Keyword(s):  
Lower Bound ◽  
Regular Language ◽  
Finite Automata ◽  
Regular Languages ◽  
Fractional Powers ◽  
Concrete Construction ◽  
Square Roots ◽  
Language L ◽  
Interesting Behavior ◽  
Fractional Length

It is well known that the set of powers of any given order, for example squares, in a regular language need not be regular. Nevertheless, finite automata can identify them via their roots. More precisely, we recall that, given a regular language L, the set of square roots of L is regular. The same holds true for the nth roots for any n and for the set of all nontrivial roots; we give a concrete construction for all of them. Using the above result, we obtain decision algorithms for many natural problems on powers. For example, it is decidable, given two regular languages, whether they contain the same number of squares at each length. Finally, we give an exponential lower bound on the size of automata identifying powers in regular languages. Moreover, we highlight interesting behavior differences between taking fractional powers of regular languages and taking prefixes of a fractional length. Indeed, fractional roots in a regular language can typically not be identified by finite automata.

Set Automata

2016 ◽  
Vol 27 (02) ◽  
pp. 187-214 ◽  
Author(s):  
Martin Kutrib ◽  
Andreas Malcher ◽  
Matthias Wendlandt
Keyword(s):  
Finite Automata ◽  
Point Of View ◽  
Storage Medium ◽  
Deterministic Finite Automaton ◽  
Closure Properties ◽  
Language Class ◽  
Trade Offs ◽  
Double Exponential ◽  
Order Of Magnitude ◽  
Context Free

We consider the model of deterministic set automata which are basically deterministic finite automata equipped with a set as an additional storage medium. The basic operations on the set are the insertion of elements, the removing of elements, and the test whether an element is in the set. We investigate the computational power of deterministic set automata and compare the language class accepted with the context-free languages and classes of languages accepted by queue automata. As result the incomparability to all classes considered is obtained. Furthermore, we examine the closure properties under several operations. Then we show that deterministic set automata may be an interesting model from a practical point of view by proving that their regularity problem as well as the problems of emptiness, finiteness, infiniteness, and universality are decidable. Finally, the descriptional complexity of deterministic and nondeterministic set automata is investigated. A conversion procedure that turns a deterministic set automaton accepting a regular language into a deterministic finite automaton is developed which leads to a double exponential upper bound. This bound is proved to be tight in the order of magnitude by presenting also a double exponential lower bound. In contrast to these recursive bounds we obtain non-recursive trade-offs when nondeterministic set automata are considered.

Finite Automata on Transfinite Sequences and Regular Expressions

1985 ◽  
Vol 8 (3-4) ◽  
pp. 379-396
Author(s):  
Jerzy Wojciechowski
Keyword(s):  
Regular Expression ◽  
Finite Automata ◽  
Characterization Theorem ◽  
Regular Expressions ◽  
Emptiness Problem

In this paper the notion of regular expression for finite automata on transfinite sequences /TF-automata/ is introduced. The characterization theorem for TF-automata is proved. From this theorem we conclude the decidability of the emptiness problem for TF-automata and the characterization theorem for finite automata on transfinite sequences of bounded lenght.

ON THE AVERAGE STATE COMPLEXITY OF PARTIAL DERIVATIVE AUTOMATA: AN ANALYTIC COMBINATORICS APPROACH

2011 ◽  
Vol 22 (07) ◽  
pp. 1593-1606 ◽  
Author(s):  
SABINE BRODA ◽  
ANTÓNIO MACHIAVELO ◽  
NELMA MOREIRA ◽  
ROGÉRIO REIS
Keyword(s):  
Lower Bound ◽  
Asymptotic Behaviour ◽  
Partial Derivative ◽  
Regular Expression ◽  
Finite Automata ◽  
Regular Expressions ◽  
Alphabet Size ◽  
State Complexity ◽  
Analytic Combinatorics ◽  
Average State

The partial derivative automaton ([Formula: see text]) is usually smaller than other nondeterministic finite automata constructed from a regular expression, and it can be seen as a quotient of the Glushkov automaton ([Formula: see text]). By estimating the number of regular expressions that have ε as a partial derivative, we compute a lower bound of the average number of mergings of states in [Formula: see text] and describe its asymptotic behaviour. This depends on the alphabet size, k, and for growing k's its limit approaches half the number of states in [Formula: see text]. The lower bound corresponds to consider the [Formula: see text] automaton for the marked version of the regular expression, i.e. where all its letters are made different. Experimental results suggest that the average number of states of this automaton, and of the [Formula: see text] automaton for the unmarked regular expression, are very close to each other.

scholarly journals Ambiguity in Finite Automata

1977 ◽  
Vol 6 (82) ◽  
Author(s):  
Erik Meineche Schmidt
Keyword(s):  
Regular Expression ◽  
Finite Automata ◽  
Regular Languages ◽  
Deterministic Finite Automata

<p>The gain in succinctness of descriptions of regular languages when nondeterministic (unambiguous) finite automata are used rather than unambiguous (deterministic) finite automata, is not bounded by any polynomium.</p><p>The problem of deciding whether an unambiguous regular expression does not generate all words over its terminal alphabet, is in NP.</p>

Roots and Powers in Regular Languages: Recognizing Nonregular Properties by Finite Automata

2020 ◽  
Author(s):  
Fabian Frei ◽  
Juraj Hromkovič ◽  
Juhani Karhumäki
Keyword(s):  
Lower Bound ◽  
Regular Language ◽  
Finite Automata ◽  
Regular Languages ◽  
Fractional Powers ◽  
Concrete Construction ◽  
Square Roots ◽  
Language L ◽  
Interesting Behavior ◽  
Fractional Length

It is well known that the set of powers of any given order, for example squares, in a regular language need not be regular. Nevertheless, finite automata can identify them via their roots. More precisely, we recall that, given a regular language L, the set of square roots of L is regular. The same holds true for the nth roots for any n and for the set of all nontrivial roots; we give a concrete construction for all of them. Using the above result, we obtain decision algorithms for many natural problems on powers. For example, it is decidable, given two regular languages, whether they contain the same number of squares at each length. Finally, we give an exponential lower bound on the size of automata identifying powers in regular languages. Moreover, we highlight interesting behavior differences between taking fractional powers of regular languages and taking prefixes of a fractional length. Indeed, fractional roots in a regular language can typically not be identified by finite automata.

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