scholarly journals State complexity of combined operations involving catenation and binary Boolean operations: Beyond the Brzozowski conjectures

2019 ◽  
Vol 800 ◽  
pp. 15-30
Author(s):  
Pascal Caron ◽  
Jean-Gabriel Luque ◽  
Bruno Patrou
Keyword(s):  
Boolean Operations ◽  
State Complexity ◽  
Combined Operations

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.

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.

scholarly journals QUOTIENT COMPLEXITY OF STAR-FREE LANGUAGES

2012 ◽  
Vol 23 (06) ◽  
pp. 1261-1276 ◽  
Author(s):  
JANUSZ BRZOZOWSKI ◽  
BO LIU
Keyword(s):  
Lower Bound ◽  
Regular Language ◽  
Regular Languages ◽  
Symmetric Difference ◽  
Boolean Operations ◽  
State Complexity

The quotient complexity, also known as state complexity, of a regular language is the number of distinct left quotients of the language. The quotient complexity of an operation is the maximal quotient complexity of the language resulting from the operation, as a function of the quotient complexities of the operands. The class of star-free languages is the smallest class containing the finite languages and closed under boolean operations and concatenation. We prove that the tight bounds on the quotient complexities of union, intersection, difference, symmetric difference, concatenation and star for star-free languages are the same as those for regular languages, with some small exceptions, whereas 2n-1 is a lower bound for reversal.

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.

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