## State complexity of union and intersection of finite languages (2007)

Venue: | In Proceedings of DLT’07, Lecture Notes in Computer Science 4588 |

Citations: | 5 - 3 self |

### BibTeX

@INPROCEEDINGS{Han07statecomplexity,

author = {Yo-sub Han and Kai Salomaa},

title = {State complexity of union and intersection of finite languages},

booktitle = {In Proceedings of DLT’07, Lecture Notes in Computer Science 4588},

year = {2007},

pages = {2007}

}

### OpenURL

### Abstract

Abstract. 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. We compute the upper bounds based on the structural properties of minimal deterministic finite-state automata (DFAs) 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 DFAs. In addition, we prove that the upper bounds are unreachable for any fixed sized alphabet. 1

### Citations

4144 |
Introduction to Automata Theory, Languages, and Computation
- Hopcroft, Ullman
- 1979
(Show Context)
Citation Context ...ach is again based on the structural properties of minimal DFAs of finite languages. We start from the Cartesian product of states for the intersection of two DFAs. Proposition 3 (Hopcroft and Ullman =-=[7]-=-). Given two DFAs A =(Q1,Σ,δ1, s1,F1) and B =(Q2,Σ,δ2,s2,F2), letM∩ =(Q1 × Q2,Σ,δ,(s1,s2),F1 × F2), where for all p ∈ Q1 and q ∈ Q2 and a ∈ Σ, Then, L(M∩)=L(A) ∩ L(B). δ((p,q),a)=(δ1(p,a),δ2(q,a)). Le... |

511 | Formal Languages
- Salomaa
- 1973
(Show Context)
Citation Context ...inaries Here we recall some definitions needed in the later sections. For all unexplained notions related to formal languages and finite automata we refer the reader to Hopcroft and Ullman [10] or Yu =-=[17]-=-. In the following E always denotes a finite alphabet of characters and E* is the set of all strings over E. The number of characters in E is denoted by |E|. A language over E is any subset of E*. The... |

58 |
The state complexities of some basic operations on regular languages, Theoret
- Salomaa, Yu, et al.
- 1994
(Show Context)
Citation Context ...or L(A)r\L(B). Yu et al. [f 9] gave a systematic study of state complexity of regular language operations. Campeanu et al. [1] investigated the state complexity of finite languages. The known results =-=[1, 19]-=- are summarized in Table 1. In the figure, TO, n > 1 denote the state complexity of L\ and L2, respectively. Table 1. The state complexity of basic operations on finite languages and regular languages... |

45 |
Regular expressions: new results and open problems
- Ellul, Krawetz, et al.
(Show Context)
Citation Context ...mplexity of L and studied the state complexity of basic operations on regular languages and finite languages. Holzer and Kutrib [5,6] investigated the state complexity of NFAs. Recently, Ellul et al. =-=[3]-=- examined the size of the shortest regular expression for a given regular language. There are many other results on state complexity with different viewpoints [4,8,9,10,11]. We focus on the measure of... |

35 | State complexity of regular languages
- Yu
(Show Context)
Citation Context ... a regular language L is accepted by a deterministic finite-state automaton (DFA) or a nondeterministic finitestate automaton (NFA). L is also described by a regular expression. Yu and his co-authors =-=[1,13,14]-=- regarded the number of states in the minimal DFA for L as the complexity of L and studied the state complexity of basic operations on regular languages and finite languages. Holzer and Kutrib [5,6] i... |

30 | State complexity of basic operations on finite languages
- Câmpeanu, Salomaa, et al.
- 2001
(Show Context)
Citation Context ... a regular language L is accepted by a deterministic finite-state automaton (DFA) or a nondeterministic finitestate automaton (NFA). L is also described by a regular expression. Yu and his co-authors =-=[1,13,14]-=- regarded the number of states in the minimal DFA for L as the complexity of L and studied the state complexity of basic operations on regular languages and finite languages. Holzer and Kutrib [5,6] i... |

25 |
Unary language operations, state complexity and Jacobsthal’s function
- Pighizzini, Shallit
(Show Context)
Citation Context ...exity of NFAs. Recently, Ellul et al. [3] examined the size of the shortest regular expression for a given regular language. There are many other results on state complexity with different viewpoints =-=[4,8,9,10,11]-=-. We focus on the measure of Yu [13]: The state complexity of a regular language L is the number of states of the minimal DFA for L. The state complexity of an operation ⋆ Han was supported by the KIS... |

23 | Average State Complexity of Operations on Unary Automata
- Nicaud
- 1999
(Show Context)
Citation Context ...exity of NFAs. Recently, Ellul et al. [3] examined the size of the shortest regular expression for a given regular language. There are many other results on state complexity with different viewpoints =-=[4,8,9,10,11]-=-. We focus on the measure of Yu [13]: The state complexity of a regular language L is the number of states of the minimal DFA for L. The state complexity of an operation ⋆ Han was supported by the KIS... |

22 | Grail: A C++ library for automata and expressions
- Raymond, Wood
- 1995
(Show Context)
Citation Context ...tions such as vi, emacs and Perl. Furthermore, researchers developed a number of software libraries for manipulating formal language objects with the emphasis on regular languages; examples are Grail =-=[12]-=- and Vaucanson [2]. The applications and implementations of regular languages motivate the study of the descriptional complexity of regular languages. The descriptional complexity of regular languages... |

20 |
State complexity of proportional removals
- Domaratzki
(Show Context)
Citation Context ...Therefore, if rn = n = 10s is the size of the minimal DFAs for the finite languages L\ and L2, then from equations (1) and (2) we know that the minimal DFA for L = L\ U L2 needs at least 1 -n states. =-=(3)-=- Too' Notice that we have used an alphabet S with four characters. If we encode the languages L\ and L2 over a binary alphabet, then we get a similar lower bounds590 Y.-S. Han & K. Salomaa for the sta... |

18 |
State complexity of concatenation and complementation
- Jirásek, Jirásková, et al.
(Show Context)
Citation Context ...exity of NFAs. Recently, Ellul et al. [3] examined the size of the shortest regular expression for a given regular language. There are many other results on state complexity with different viewpoints =-=[4,8,9,10,11]-=-. We focus on the measure of Yu [13]: The state complexity of a regular language L is the number of states of the minimal DFA for L. The state complexity of an operation ⋆ Han was supported by the KIS... |

14 | Nondeterministic descriptional complexity of regular languages
- Holzer, Kutrib
(Show Context)
Citation Context ...,13,14] regarded the number of states in the minimal DFA for L as the complexity of L and studied the state complexity of basic operations on regular languages and finite languages. Holzer and Kutrib =-=[5,6]-=- investigated the state complexity of NFAs. Recently, Ellul et al. [3] examined the size of the shortest regular expression for a given regular language. There are many other results on state complexi... |

12 |
Estimates of the number of states of finite automata
- Maslov
- 1970
(Show Context)
Citation Context ...died the state complexity of basic operations on regular languages and finite languages. Similar results where state complexity is defined using incomplete DFAs already appeared in the work by Maslov =-=[13]-=-. Holzer and Kutrib [8, 9] investigated the state complexity of NFAs. Recently, Ellul et al. [5] examined the size of the shortest regular expression for a given regular language. There are many other... |

8 | Unary language operations and their nondeterministicstate complexity
- Holzer, Kutrib
- 2002
(Show Context)
Citation Context ...,13,14] regarded the number of states in the minimal DFA for L as the complexity of L and studied the state complexity of basic operations on regular languages and finite languages. Holzer and Kutrib =-=[5,6]-=- investigated the state complexity of NFAs. Recently, Ellul et al. [3] examined the size of the shortest regular expression for a given regular language. There are many other results on state complexi... |

8 | State complexity of basic operations on suffix-free regular languages - Han, Salomaa |

7 | State complexity of prefix-free regular languages
- Han, Salomaa, et al.
- 2006
(Show Context)
Citation Context |

7 | State complexity of shuffle on trajectories
- DOMARATZKI, SALOMAA
- 2004
(Show Context)
Citation Context ... of L has at least 2 2r > s 2 classes. (Recall that 2 r > s.) Thus, if m = n = 14s is the size of the minimal DFAs for L3 and L4, then the minimal DFA for L = L3 n L4 needs at least 1 2 n states. 196 =-=(4)-=-s594 Y.-S. Han & K. Salomaa This construction uses an alphabet of size four. The same construction works if we encode the characters over a binary alphabet and the modification only changes the consta... |

3 |
A.: Union and intersection of regular languages and descriptional complexity
- Hricko, Jirásková, et al.
- 2005
(Show Context)
Citation Context |

2 | Inside Vaucanson
- Claveirole, Lombardy, et al.
- 2006
(Show Context)
Citation Context ... Perl. Furthermore, researchers have developed a number of software libraries for manipulating formal language objects with an emphasis on regular languages; examples include Grail [16] and Vaucanson =-=[2]-=-. *A preliminary version of this paper appeared in Proceedings of 11th International Conference Developments in Language Theory, DLT 2007, Lect. Notes Comput. Sci. 4588, Springer-Verlag, 2007, pp. 217... |

1 |
Grail: A CH—h library for automata and expressions
- Raymond, Wood
- 1994
(Show Context)
Citation Context ...ch as vi, emacs and Perl. Furthermore, researchers have developed a number of software libraries for manipulating formal language objects with an emphasis on regular languages; examples include Grail =-=[16]-=- and Vaucanson [2]. *A preliminary version of this paper appeared in Proceedings of 11th International Conference Developments in Language Theory, DLT 2007, Lect. Notes Comput. Sci. 4588, Springer-Ver... |