State Complexity of k-Union and k-Intersection for Prefix-Free Regular Languages

2015 ◽  
Vol 26 (02) ◽  
pp. 211-227 ◽  
Author(s):  
Hae-Sung Eom ◽  
Yo-Sub Han ◽  
Kai Salomaa
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Finite Automata ◽  
Constant Factor ◽  
The State ◽  
Regular Languages ◽  
Deterministic Finite Automata ◽  
State Complexity ◽  
Binary Alphabet ◽  
Multiple Intersections

We investigate the state complexity of multiple unions and of multiple intersections for prefix-free regular languages. Prefix-free deterministic finite automata have their own unique structural properties that are crucial for obtaining state complexity upper bounds that are improved from those for general regular languages. We present a tight lower bound construction for k-union using an alphabet of size k + 1 and for k-intersection using a binary alphabet. We prove that the state complexity upper bound for k-union cannot be reached by languages over an alphabet with less than k symbols. We also give a lower bound construction for k-union using a binary alphabet that is within a constant factor of the upper bound.

The complexity of concatenation on deterministic and alternating finite automata

2018 ◽  
Vol 52 (2-3-4) ◽  
pp. 153-168
Author(s):  
Michal Hospodár ◽  
Galina Jirásková
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Open Problem ◽  
Finite Automata ◽  
The State ◽  
Regular Languages ◽  
Final States ◽  
State Complexity ◽  
Deterministic Automata

We study the state complexity of the concatenation operation on regular languages represented by deterministic and alternating finite automata. For deterministic automata, we show that the upper bound m2n − k2n−1 on the state complexity of concatenation can be met by ternary languages, the first of which is accepted by an m-state DFA with k final states, and the second one by an n-state DFA with ℓ final states for arbitrary integers m, n, k, ℓ with 1 ≤ k ≤ m − 1 and 1 ≤ ℓ ≤ n − 1. In the case of k ≤ m − 2, we are able to provide appropriate binary witnesses. In the case of k = m − 1 and ℓ ≥ 2, we provide a lower bound which is smaller than the upper bound just by one. We use our binary witnesses for concatenation on deterministic automata to describe binary languages meeting the upper bound 2m + n + 1 for the concatenation on alternating finite automata. This solves an open problem stated by Fellah et al. [Int. J. Comput. Math. 35 (1990) 117–132].

STATE COMPLEXITY OF ADDITIVE WEIGHTED FINITE AUTOMATA

2007 ◽  
Vol 18 (06) ◽  
pp. 1407-1416 ◽  
Author(s):  
KAI SALOMAA ◽  
PAUL SCHOFIELD
Keyword(s):  
Upper Bound ◽  
Finite Automaton ◽  
Regular Language ◽  
Finite Automata ◽  
The State ◽  
Deterministic Finite Automata ◽  
State Complexity ◽  
Weighted Finite Automata

It is known that the neighborhood of a regular language with respect to an additive distance is regular. We introduce an additive weighted finite automaton model that provides a conceptually simple way to reprove this result. We consider the state complexity of converting additive weighted finite automata to deterministic finite automata. As our main result we establish a tight upper bound for the state complexity of the conversion.

State Complexity of Insertion

2016 ◽  
Vol 27 (07) ◽  
pp. 863-878 ◽  
Author(s):  
Yo-Sub Han ◽  
Sang-Ki Ko ◽  
Timothy Ng ◽  
Kai Salomaa
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Regular Language ◽  
Finite Automata ◽  
The State ◽  
Tight Bound ◽  
State Complexity ◽  
Nondeterministic State

It is well known that the resulting language obtained by inserting a regular language to a regular language is regular. We study the nondeterministic and deterministic state complexity of the insertion operation. Given two incomplete DFAs of sizes m and n, we give an upper bound (m+2)·2mn−m−1·3m and find a lower bound for an asymp-totically tight bound. We also present the tight nondeterministic state complexity by a fooling set technique. The deterministic state complexity of insertion is 2Θ(mn) and the nondeterministic state complexity of insertion is precisely mn+2m, where m and n are the size of input finite automata. We also consider the state complexity of insertion in the case where the inserted language is bifix-free or non-returning.

scholarly journals Regular Expressions with Lookahead

2021 ◽  
Vol 27 (4) ◽  
pp. 324-340
Author(s):  
Martin Berglund ◽  
Brink van der Merwe ◽  
Steyn van Litsenborgh
Keyword(s):  
Finite Automata ◽  
The State ◽  
Regular Expressions ◽  
Deterministic Finite Automata ◽  
State Complexity

This paper investigates regular expressions which in addition to the standard operators of union, concatenation, and Kleene star, have lookaheads. We show how to translate regular expressions with lookaheads (REwLA) to equivalent Boolean automata having at most 3 states more than the length of the REwLA. We also investigate the state complexity when translating REwLA to equivalent deterministic finite automata (DFA).

Square on Deterministic, Alternating, and Boolean Finite Automata

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.

scholarly journals Two Extensions of Cover Automata

Axioms ◽  
2021 ◽  
Vol 10 (4) ◽  
pp. 338
Author(s):  
Cezar Câmpeanu
Keyword(s):  
Computational Complexity ◽  
Finite Automata ◽  
The State ◽  
Finite Cover ◽  
Deterministic Finite Automata ◽  
State Complexity ◽  
Compact Representations ◽  
Minimization Algorithms ◽  
Deterministic Automata ◽  
Representation Transformation

Deterministic Finite Cover Automata (DFCA) are compact representations of finite languages. Deterministic Finite Automata with “do not care” symbols and Multiple Entry Deterministic Finite Automata are both compact representations of regular languages. This paper studies the benefits of combining these representations to get even more compact representations of finite languages. DFCAs are extended by accepting either “do not care” symbols or considering multiple entry DFCAs. We study for each of the two models the existence of the minimization or simplification algorithms and their computational complexity, the state complexity of these representations compared with other representations of the same language, and the bounds for state complexity in case we perform a representation transformation. Minimization for both models proves to be NP-hard. A method is presented to transform minimization algorithms for deterministic automata into simplification algorithms applicable to these extended models. DFCAs with “do not care” symbols prove to have comparable state complexity as Nondeterministic Finite Cover Automata. Furthermore, for multiple entry DFCAs, we can have a tight estimate of the state complexity of the transformation into equivalent DFCA.

THE COMPLEXITY OF DECIDING CODE AND MONOID PROPERTIES FOR REGULAR SETS

1992 ◽  
Vol 02 (01) ◽  
pp. 39-55
Author(s):  
DUNG T. HUYNH
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Finite Automata ◽  
Free Monoid ◽  
Deterministic Finite Automaton ◽  
Deterministic Finite Automata ◽  
Nondeterministic Finite Automata ◽  
Maximal Code ◽  
Regular Sets ◽  
Np Hardness

In this paper, we study the complexity of deciding code and monoid properties for regular sets specified by deterministic or nondeterministic finite automata. The results are as follows. The code problem for regular sets specified by deterministic or nondeterministic finite automata is NL-complete under NC(1) reducibilities. The problems of determining whether a regular set given by a deterministic finite automaton is a monoid or a free monoid or a finitely generated monoid are all NL-complete under NC(1) reducibilities. These monoid problems become PSPACE-complete if the regular sets are specified by nondeterministic finite automata instead. The maximal code problem for deterministic finite automata is shown to be in DET and NL-hard, while a PSPACE upper bound and NP-hardness lower bound hold for the case of nondeterministic finite automata.

STATE COMPLEXITY OF CONCATENATION AND COMPLEMENTATION

2005 ◽  
Vol 16 (03) ◽  
pp. 511-529 ◽  
Author(s):  
JOZEF JIRÁSEK ◽  
GALINA JIRÁSKOVÁ ◽  
ALEXANDER SZABARI
Keyword(s):  
Upper Bound ◽  
Entire Range ◽  
Upper Bounds ◽  
The State ◽  
Regular Languages ◽  
State Complexity ◽  
Nondeterministic State

We investigate the state complexity of concatenation and the nondeterministic state complexity of complementation of regular languages. We show that the upper bounds on the state complexity of concatenation are also tight in the case that the first automaton has more than one accepting state. In the case of nondeterministic state complexity of complementation, we show that the entire range of complexities, up to the known upper bound can be produced.

On the state complexity of intersection of regular languages

1991 ◽  
Vol 22 (3) ◽  
pp. 52-54 ◽  
Author(s):  
Sheng Yu ◽  
Qingyu Zhuang
Keyword(s):  
The State ◽  
Regular Languages ◽  
State Complexity

scholarly journals Lower bounds for the state complexity of probabilistic languages and the language of prime numbers

2020 ◽  
Vol 30 (1) ◽  
pp. 175-192
Author(s):  
NathanaËl Fijalkow
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Prime Numbers ◽  
Regular Languages ◽  
Automata Theory ◽  
Seminal Paper ◽  
Probabilistic Automata ◽  
State Complexity ◽  
Probabilistic Languages ◽  
Alternating Automata

Abstract This paper studies the complexity of languages of finite words using automata theory. To go beyond the class of regular languages, we consider infinite automata and the notion of state complexity defined by Karp. Motivated by the seminal paper of Rabin from 1963 introducing probabilistic automata, we study the (deterministic) state complexity of probabilistic languages and prove that probabilistic languages can have arbitrarily high deterministic state complexity. We then look at alternating automata as introduced by Chandra, Kozen and Stockmeyer: such machines run independent computations on the word and gather their answers through boolean combinations. We devise a lower bound technique relying on boundedly generated lattices of languages, and give two applications of this technique. The first is a hierarchy theorem, stating that there are languages of arbitrarily high polynomial alternating state complexity, and the second is a linear lower bound on the alternating state complexity of the prime numbers written in binary. This second result strengthens a result of Hartmanis and Shank from 1968, which implies an exponentially worse lower bound for the same model.

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