State Complexity Of Combined Operations Research Articles

Descriptional Complexity of Finite Automata – Selected Highlights

The state complexity, respectively, nondeterministic state complexity of a regular language L is the number of states of the minimal deterministic, respectively, of a minimal nondeterministic finite automaton for L. Some of the most studied state complexity questions deal with size comparisons of nondeterministic finite automata of differing degree of ambiguity. More generally, if for a regular language we compare the size of description by a finite automaton and by a more powerful language definition mechanism, such as a context-free grammar, we encounter non-recursive trade-offs. Operational state complexity studies the state complexity of the language resulting from a regularity preserving operation as a function of the complexity of the argument languages. Determining the state complexity of combined operations is generally challenging and for general combinations of operations that include intersection and marked concatenation it is uncomputable.

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

We exhaustively investigate possible combinations of a Boolean operation together with a catenation. In many cases we prove and improve some conjectures by Brzozowski. For each family of operations, we endeavour to provide a common witness with a small size alphabet.

State complexity of combined operations for suffix-free regular languages

We investigate the state complexity of combined operations for suffix-free regular languages. Suffix-free deterministic finite-state automata have a unique structural property that is crucial for obtaining the precise state complexity of basic operations. Based on the same property, we establish the state complexity of four combined operations: star-of-union, star-of-intersection, star-of-reversal and star-of-catenation. In the case of star-of-intersection, we only have an upper bound and the lower bound is open.

STATE COMPLEXITY AND APPROXIMATION

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 union and intersection of square and reversal on [formula omitted] regular languages

In this paper, we continue our study on the state complexity of combined operations on regular languages. We study the state complexities of ⋃i=1kLi2, ⋂i=1kLi2, ⋃i=1kLiR, and ⋂i=1kLiR, for regular languages Li, 1≤i≤k, k≥2. We obtain the exact bounds for these combined operations and show that the state complexities of ⋃i=1kLi2 and ⋂i=1kLi2 are the same as the mathematical compositions of the state complexities of their component individual operations, while, on the other hand, the state complexities of ⋃i=1kLiR and ⋂i=1kLiR are lower than the corresponding mathematical compositions.

State complexity of combined operations with two basic operations

This paper studies the state complexity of (L1L2)R, L1RL2, L1∗L2, (L1∪L2)L3, (L1∩L2)L3, L1L2∩L3, and L1L2∪L3 for regular languages L1, L2, and L3. We first show that the upper bound proposed by Liu et al. (2008) [18] for the state complexity of (L1L2)R coincides with the lower bound and is thus the state complexity of this combined operation by providing some witness DFAs. Also, we show that, unlike most other cases, due to the structural properties of the result of the first operation of the combinations L1RL2, L1∗L2, and (L1∪L2)L3, the state complexity of each of these combined operations is close to the mathematical composition of the state complexities of the component operations. Moreover, we show that the state complexities of (L1∩L2)L3, L1L2∩L3, and L1L2∪L3 are exactly equal to the mathematical compositions of the state complexities of their component operations in the general cases. We also include a brief survey that summarizes all state complexity results for combined operations with two basic operations.

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

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.

State complexity of union and intersection of star on [formula omitted] regular languages

In this paper, we continue our study on state complexity of combined operations. We study the state complexities of L1∗∪L2∗, ⋃i=1kLi∗, L1∗∩L2∗, and ⋂i=1kLi∗ for regular languages Li, 1≤i≤k. We obtain the exact bounds for these combined operations and show that the bounds are different from the mathematical compositions of the state complexities of their component individual operations.

On the State Complexity of Star of Union and Star of Intersection

The state complexity of the star of union of an m-state DFA language and an n-state DFA language is proved to be 2 m+n−1−2 m−1−2 n−1+1 for every alphabet of at least two letters. The state complexity of the star of intersection is established as 3/4 2 mn for every alphabet of six or more letters. This improves the recent results of A. Salomaa, K. Salomaa and Yu (“State complexity of combined operations”, Theoret. Comput. Sci., 383 (2007) 140-152).

State Complexity of Catenation Combined with Star and Reversal

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.

Estimation of state complexity of combined operations

It appears that the state complexity of each combined operation has its own special features. Thus, it is important and practical to obtain good estimates for some commonly used general cases. In this paper, we consider the state complexity of combined Boolean operations and give an exact bound for all of them in the case when the alphabet is not fixed. Moreover, we show that for any fixed alphabet, this bound can be reached in infinitely many cases. We also consider the state complexity of multiple catenations. The state complexities are obtained in the cases of the catenations of three and four languages. An estimate for the catenation of an arbitrary number of languages is given, which is very close to the state complexities in the three and four languages cases.

State complexity of unique rational operations

For each basic language operation we define its “unique” counterpart as being the operation that results in a language whose words can be obtained uniquely through the given operation. These unique operations can arguably be viewed as combined basic operations, placing this work in the popular area of state complexity of combined operations on regular languages. We study the state complexity of unique rational operations and we provide upper bounds and empirical results meant to cast light into this matter. Equally important, we hope to have provided a generic methodology for estimating their state complexity.

State complexity of basic language operations combined with reversal

We study the state complexity of combined operations on regular languages. Each of the combined operations is a basic operation combined with reversal. We show that their state complexities are all very different from the compositions of state complexities of individual operations.

ON THE STATE COMPLEXITY OF COMBINED OPERATIONS AND THEIR 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 of combined operations

We study the state complexity of combined operations. Two particular combined operations are studied: star of union and star of intersection. It is shown that the state complexity of a combined operation is not necessarily similar to the combination of the individual state complexities of the participating operations.