State Complexity of Boundary of Prefix-Free Regular Languages

2015 ◽  
Vol 26 (06) ◽  
pp. 697-707 ◽  
Author(s):  
Hae-Sung Eom ◽  
Yo-Sub Han
Keyword(s):  
Structural Properties ◽  
The State ◽  
Regular Languages ◽  
Tight Bound ◽  
State Complexity ◽  
Complexity Function ◽  
Prefix Codes

Recently, researchers studied the state complexity of boundary — [Formula: see text] — of regular languages L motivated from the famous Kuratowski’s 14-theorem. Prefix codes — a set of languages — play an important role in several applications. We consider prefix-free regular languages and investigate the state complexity of two operations, [Formula: see text] and [Formula: see text] for prefix-free regular languages. Based on the unique structural properties of a prefix-free minimal DFA, we compute the precise state complexity of [Formula: see text] and [Formula: see text]. We then present the tight bound over a quaternary alphabet for [Formula: see text] and [Formula: see text]. Our results are smaller than the composition of the state complexity function for individual operations of prefix-free regular languages.

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 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.

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 State Complexity of Overlap Assembly

2020 ◽  
pp. 1-20
Author(s):  
Janusz A. Brzozowski ◽  
Lila Kari ◽  
Bai Li ◽  
Marek Szykuła
Keyword(s):  
Self Assembly ◽  
Binary Operation ◽  
The State ◽  
Regular Languages ◽  
Deterministic Finite Automaton ◽  
State Complexity ◽  
Dna Strands ◽  
Unary Languages ◽  
Number Formula ◽  
Dna Strand

The state complexity of a regular language [Formula: see text] is the number [Formula: see text] of states in a minimal deterministic finite automaton (DFA) accepting [Formula: see text]. The state complexity of a regularity-preserving binary operation on regular languages is defined as the maximal state complexity of the result of the operation where the two operands range over all languages of state complexities [Formula: see text] and [Formula: see text], respectively. We determine, for [Formula: see text], [Formula: see text], the exact value of the state complexity of the binary operation overlap assembly on regular languages. This operation was introduced by Csuhaj-Varjú, Petre, and Vaszil to model the process of self-assembly of two linear DNA strands into a longer DNA strand, provided that their ends “overlap”. We prove that the state complexity of the overlap assembly of languages [Formula: see text] and [Formula: see text], where [Formula: see text] and [Formula: see text], is at most [Formula: see text]. Moreover, for [Formula: see text] and [Formula: see text] there exist languages [Formula: see text] and [Formula: see text] over an alphabet of size [Formula: see text] whose overlap assembly meets the upper bound and this bound cannot be met with smaller alphabets. Finally, we prove that [Formula: see text] is the state complexity of the overlap assembly in the case of unary languages and that there are binary languages whose overlap assembly has exponential state complexity at least [Formula: see text].

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.

scholarly journals On the State Complexity of the Shuffle of Regular Languages

2016 ◽  
pp. 73-86 ◽  
Author(s):  
Janusz Brzozowski ◽  
Galina Jirásková ◽  
Bo Liu ◽  
Aayush Rajasekaran ◽  
Marek Szykuła
Keyword(s):  
The State ◽  
Regular Languages ◽  
State Complexity

STATE COMPLEXITY OF UNION AND INTERSECTION OF FINITE LANGUAGES

2008 ◽  
Vol 19 (03) ◽  
pp. 581-595 ◽  
Author(s):  
YO-SUB HAN ◽  
KAI SALOMAA
Keyword(s):  
Structural Properties ◽  
Upper Bounds ◽  
The State ◽  
Finite State Automata ◽  
State Complexity ◽  
Finite State

We investigate the state complexity of union and intersection for finite languages. Note that the problem of obtaining the tight bounds for both operations was open. First we compute upper bounds using structural properties of minimal deterministic finite-state automata for finite languages. Then, we show that the upper bounds are tight if we have a variable sized alphabet that can depend on the size of input deterministic finite-state automata. In addition, we prove that the upper bounds are unreachable for any fixed sized alphabet.

Sign in / Sign up

Export Citation Format

Share Document