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.

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.

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.

Descriptional Complexity of the Forever Operator

2019 ◽  
Vol 30 (01) ◽  
pp. 115-134 ◽  
Author(s):  
Michal Hospodár ◽  
Galina Jirásková ◽  
Peter Mlynárčik
Keyword(s):  
Regular Language ◽  
Finite Automata ◽  
Deterministic Finite Automata ◽  
State Complexity ◽  
Descriptional Complexity ◽  
Trade Offs ◽  
Number Formula ◽  
Deterministic Automata ◽  
Initial States ◽  
Nondeterministic State

We examine the descriptional complexity of the forever operator, which assigns the language [Formula: see text] to a regular language [Formula: see text], and we investigate the trade-offs between various models of finite automata. We consider complete and partial deterministic finite automata, nondeterministic finite automata with single or multiple initial states, alternating, and Boolean finite automata. We assume that the argument and the result of this operation are accepted by automata belonging to one of these six models. We investigate all possible trade-offs and provide a tight upper bound for 32 of 36 of them. The most interesting result is the trade-off from nondeterministic to deterministic automata given by the Dedekind number [Formula: see text]. We also prove that the nondeterministic state complexity of [Formula: see text] is [Formula: see text] which solves an open problem stated by Birget [The state complexity of [Formula: see text] and its connection with temporal logic, Inform. Process. Lett. 58 (1996) 185–188].

scholarly journals State Complexity of Suffix Distance

2019 ◽  
Vol 30 (06n07) ◽  
pp. 1197-1216
Author(s):  
Timothy Ng ◽  
David Rappaport ◽  
Kai Salomaa
Keyword(s):  
Lower Bound ◽  
Finite Automaton ◽  
Regular Language ◽  
Upper Bounds ◽  
The State ◽  
Deterministic Finite Automaton ◽  
Tight Bound ◽  
State Complexity

The neighbourhood of a regular language with respect to the prefix, suffix and subword distance is always regular and a tight bound for the state complexity of prefix distance neighbourhoods is known. We give upper bounds for the state complexity of the neighbourhood of radius [Formula: see text] of an [Formula: see text]-state deterministic finite automaton language with respect to the suffix distance and the subword distance, respectively. For restricted values of [Formula: see text] and [Formula: see text] we give a matching lower bound for the state complexity of suffix distance neighbourhoods.

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).

The Degree of Irreversibility in Deterministic Finite Automata

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.

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.

Operations on Unambiguous Finite Automata

2018 ◽  
Vol 29 (05) ◽  
pp. 861-876 ◽  
Author(s):  
Jozef Jirásek ◽  
Galina Jirásková ◽  
Juraj Šebej
Keyword(s):  
Upper Bound ◽  
Finite Automaton ◽  
Finite Automata ◽  
Upper Bounds ◽  
Input String ◽  
State Complexity ◽  
Letter Alphabet ◽  
Left And Right ◽  
Nondeterministic Finite Automaton ◽  
Intersection Formula

A nondeterministic finite automaton is unambiguous if it has at most one accepting computation on every input string. We investigate the state complexity of basic regular operations on languages represented by unambiguous finite automata. We get tight upper bounds for reversal ([Formula: see text]), intersection ([Formula: see text]), left and right quotients ([Formula: see text]), positive closure ([Formula: see text]), star ([Formula: see text]), shuffle ([Formula: see text]), and concatenation ([Formula: see text]). To prove tightness, we use a binary alphabet for intersection and left and right quotients, a ternary alphabet for star and positive closure, a five-letter alphabet for shuffle, and a seven-letter alphabet for concatenation. For complementation, we reduce the trivial upper bound [Formula: see text] to [Formula: see text]. We also get some partial results for union and square.

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].

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