On the State Complexity of Closures and Interiors of Regular Languages with Subwords

2014 ◽  
pp. 234-245
Author(s):  
Prateek Karandikar ◽  
Philippe Schnoebelen
Keyword(s):  
The State ◽  
Regular Languages ◽  
State Complexity

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 On the state complexity of closures and interiors of regular languages with subwords and superwords

2016 ◽  
Vol 610 ◽  
pp. 91-107 ◽  
Author(s):  
P. Karandikar ◽  
M. Niewerth ◽  
Ph. Schnoebelen
Keyword(s):  
The State ◽  
Regular Languages ◽  
State Complexity

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 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.

scholarly journals SYNTACTIC COMPLEXITY OF ℛ- AND 𝒥-TRIVIAL REGULAR LANGUAGES

2014 ◽  
Vol 25 (07) ◽  
pp. 807-821 ◽  
Author(s):  
JANUSZ BRZOZOWSKI ◽  
BAIYU LI
Keyword(s):  
Upper Bounds ◽  
The State ◽  
Regular Languages ◽  
Syntactic Complexity ◽  
State Complexity

The syntactic complexity of a subclass of the class of regular languages is the maximal cardinality of syntactic semigroups of languages in that class, taken as a function of the state complexity n of these languages. We prove that n! and ⌊e(n − 1)⌋. are tight upper bounds for the syntactic complexity of ℛ- and 𝒥-trivial regular languages, respectively.

scholarly journals IN SEARCH OF MOST COMPLEX REGULAR LANGUAGES

2013 ◽  
Vol 24 (06) ◽  
pp. 691-708 ◽  
Author(s):  
JANUSZ BRZOZOWSKI
Keyword(s):  
Positive Integer ◽  
Finite Automaton ◽  
Cyclic Permutation ◽  
The State ◽  
Regular Languages ◽  
Final States ◽  
Deterministic Finite Automaton ◽  
Small Positive Integer ◽  
Initial State ◽  
State Complexity

Sequences (Ln| n ≥ k), called streams, of regular languages Lnare considered, where k is some small positive integer, n is the state complexity of Ln, and the languages in a stream differ only in the parameter n, but otherwise, have the same properties. The following measures of complexity are proposed for any stream: (1) the state complexity n of Ln, that is, the number of left quotients of Ln(used as a reference); (2) the state complexities of the left quotients of Ln; (3) the number of atoms of Ln; (4) the state complexities of the atoms of Ln; (5) the size of the syntactic semigroup of Ln; and the state complexities of the following operations: (6) the reverse of Ln; (7) the star of Ln; (8) union, intersection, difference and symmetric difference of Lmand Ln; and (9) the concatenation of Lmand Ln. A stream that has the highest possible complexity with respect to these measures is then viewed as a most complex stream. The language stream (Un(a, b, c) | n ≥ 3) is defined by the deterministic finite automaton with state set {0, 1, … , n−1}, initial state 0, set {n−1} of final states, and input alphabet {a, b, c}, where a performs a cyclic permutation of the n states, b transposes states 0 and 1, and c maps state n − 1 to state 0. This stream achieves the highest possible complexities with the exception of boolean operations where m = n. In the latter case, one can use Un(a, b, c) and Un(b, a, c), where the roles of a and b are interchanged in the second language. In this sense, Un(a, b, c) is a universal witness. This witness and its extensions also apply to a large number of combined regular operations.

scholarly journals On a structural property in the state complexity of projected regular languages

2012 ◽  
Vol 449 ◽  
pp. 93-105 ◽  
Author(s):  
Galina Jirásková ◽  
Tomáš Masopust
Keyword(s):  
Structural Property ◽  
The State ◽  
Regular Languages ◽  
State Complexity
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