STATE COMPLEXITY OF TWO COMBINED OPERATIONS: CATENATION-UNION AND CATENATION-INTERSECTION

2011 ◽  
Vol 22 (08) ◽  
pp. 1797-1812 ◽  
Author(s):  
BO CUI ◽  
YUAN GAO ◽  
LILA KARI ◽  
SHENG YU
Keyword(s):  
The State ◽  
State Complexity ◽  
Combined Operations ◽  
Combined Operation

In this paper, we study the state complexities of two particular combinations of operations: catenation combined with union and catenation combined with intersection. We show that the state complexity of the former combined operation is considerably less than the mathematical composition of the state complexities of catenation and union, while the state complexity of the latter one is equal to the mathematical composition of the state complexities of catenation and intersection.

ON THE STATE COMPLEXITY OF COMBINED OPERATIONS AND THEIR ESTIMATION

2007 ◽  
Vol 18 (04) ◽  
pp. 683-698 ◽  
Author(s):  
KAI SALOMAA ◽  
SHENG YU
Keyword(s):  
Large Class ◽  
The State ◽  
Estimation Methods ◽  
State Complexity ◽  
Combined Operations ◽  
Combined Operation ◽  
Nondeterministic State ◽  
General Estimation

We consider the state complexity of several combined operations. Those results show that the state complexity of a combined operation is in general very different from the composition of the state complexities of the participating individual operations. We also consider general estimation methods for the state complexity of combined operations. In particular, estimation through nondeterministic state complexity is studied. It is shown that the method is very promising for a large class of combined operations.

STATE COMPLEXITY AND APPROXIMATION

2012 ◽  
Vol 23 (05) ◽  
pp. 1085-1098 ◽  
Author(s):  
YUAN GAO ◽  
SHENG YU
Keyword(s):  
The State ◽  
General Algorithm ◽  
Future Research ◽  
State Complexity ◽  
Combined Operations ◽  
Essential Questions

We discuss a number of essential questions concerning the state complexity research. The questions include why many basic problems were not studied earlier, whether there is a general algorithm for state complexity of combined operations, and whether there is a new and effective approach in this area of research. The concept of state complexity approximation is also discussed. We show that state complexity approximation can be used to obtain good results when the exact state complexities are difficult to find and when the exact state complexities are too complex to comprehend. We also list a number of questions for future research in this area.

STATE COMPLEXITY OF TWO COMBINED OPERATIONS: CATENATION-STAR AND CATENATION-REVERSAL

2012 ◽  
Vol 23 (01) ◽  
pp. 51-66 ◽  
Author(s):  
BO CUI ◽  
YUAN GAO ◽  
LILA KARI ◽  
SHENG YU
Keyword(s):  
Research Work ◽  
The State ◽  
State Complexity ◽  
Combined Operations

This paper is a continuation of our research work on state complexity of combined operations. Motivated by applications, we study the state complexities of two particular combined operations: catenation combined with star and catenation combined with reversal. We show that the state complexities of both of these combined operations are considerably less than the compositions of the state complexities of their individual participating operations.

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 Neighbourhoods and Approximate Pattern Matching

2018 ◽  
Vol 29 (02) ◽  
pp. 315-329 ◽  
Author(s):  
Timothy Ng ◽  
David Rappaport ◽  
Kai Salomaa
Keyword(s):  
Lower Bound ◽  
Pattern Matching ◽  
Finite Automaton ◽  
The State ◽  
Worst Case ◽  
State Complexity ◽  
Approximate Pattern Matching ◽  
The Given ◽  
Nondeterministic Finite Automaton ◽  
Distance Formula

The neighbourhood of a language [Formula: see text] with respect to an additive distance consists of all strings that have distance at most the given radius from some string of [Formula: see text]. We show that the worst case deterministic state complexity of a radius [Formula: see text] neighbourhood of a language recognized by an [Formula: see text] state nondeterministic finite automaton [Formula: see text] is [Formula: see text]. In the case where [Formula: see text] is deterministic we get the same lower bound for the state complexity of the neighbourhood if we use an additive quasi-distance. The lower bound constructions use an alphabet of size linear in [Formula: see text]. We show that the worst case state complexity of the set of strings that contain a substring within distance [Formula: see text] from a string recognized by [Formula: see text] is [Formula: see text].

scholarly journals Most Complex Non-Returning Regular Languages

2019 ◽  
Vol 30 (06n07) ◽  
pp. 921-957
Author(s):  
Janusz A. Brzozowski ◽  
Sylvie Davies
Keyword(s):  
Star Product ◽  
Upper Bounds ◽  
The State ◽  
Boolean Operations ◽  
Deterministic Finite Automaton ◽  
Initial State ◽  
Complexity Measures ◽  
State Complexity ◽  
Complex Sequence ◽  
Binary Operations

A regular language [Formula: see text] is non-returning if in the minimal deterministic finite automaton accepting it there are no transitions into the initial state. Eom, Han and Jirásková derived upper bounds on the state complexity of boolean operations and Kleene star, and proved that these bounds are tight using two different binary witnesses. They derived tight upper bounds for concatenation and reversal using three different ternary witnesses. These five witnesses use a total of six different transformations. We show that for each [Formula: see text], there exists a ternary witness of state complexity [Formula: see text] that meets the bound for reversal, and restrictions of this witness to binary alphabets meet the bounds for star, product, and boolean operations. Hence all of these operations can be handled simultaneously with a single witness, using only three different transformations. We also derive tight upper bounds on the state complexity of binary operations that take arguments with different alphabets. We prove that the maximal syntactic semigroup of a non-returning language has [Formula: see text] elements and requires at least [Formula: see text] generators. We find the maximal state complexities of atoms of non-returning languages. We show that there exists a most complex sequence of non-returning languages that meet the bounds for all of these complexity measures. Furthermore, we prove there is a most complex sequence that meets all the bounds using alphabets of minimal size.

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