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27
State Complexity of Basic Operations on Nondeterministic Finite Automata
 In Implementation and Application of Automata (CIAA ’02), LNCS 2608
, 2001
"... The state complexities of basic operations on nondeterministic finite automata (NFA) are investigated. In particular, we consider Boolean operations, catenation operations  concatenation, iteration, free iteration  and the reversal on NFAs that accept finite and infinite languages over arbitrar ..."
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Cited by 28 (3 self)
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The state complexities of basic operations on nondeterministic finite automata (NFA) are investigated. In particular, we consider Boolean operations, catenation operations  concatenation, iteration, free iteration  and the reversal on NFAs that accept finite and infinite languages over arbitrary alphabets. Most of the shown bounds are tight in the exact number of states, i.e. the number is sufficient and necessary in the worst case. For the intersection of finite languages and the complementation tight bounds in the order of magnitude are proved.
Nondeterministic Descriptional Complexity of Regular Languages
 International Journal of Foundations of Computer Science
, 2002
"... We investigate the descriptional complexity of operations on finite and infinite regular languages over unary and arbitrary alphabets. The languages are represented by nondeterministic finite automata (NFA). In particular, we consider Boolean operations, catenation operations  concatenation, itera ..."
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Cited by 12 (2 self)
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We investigate the descriptional complexity of operations on finite and infinite regular languages over unary and arbitrary alphabets. The languages are represented by nondeterministic finite automata (NFA). In particular, we consider Boolean operations, catenation operations  concatenation, iteration, free iteration  and the reversal. Most of the shown bounds are tight in the exact number of states, i.e. the number is sucient and necessary in the worst case. Otherwise tight bounds in the order of magnitude are shown.
Descriptional Complexity of Machines with Limited Resources
 J. UNIVERSAL COMPUTER SCI
, 2002
"... Over the last 30 years or so many results have appeared on the descriptional complexity of machines with limited resources. Since these results have appeared in a variety of different contexts, o rgo4 here is to pro vide a survey o these results. Particular emphasis is put o limiting reso rces (e.g. ..."
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Cited by 9 (3 self)
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Over the last 30 years or so many results have appeared on the descriptional complexity of machines with limited resources. Since these results have appeared in a variety of different contexts, o rgo4 here is to pro vide a survey o these results. Particular emphasis is put o limiting reso rces (e.g., no ndeterminism, ambiguity,lo o ahead, etc.) fo vario s types o finite state machines, pushdo wn auto mata, parsers and cellular auto mata ando n the e#ect it haso n their descriptio nal co mplexity. We also address the questio no f how descriptional complexity might help in the future to solve practical issues, such as software reliability.
Quotient complexity of regular languages
 J. Autom. Lang. Comb
, 2010
"... The past research on the state complexity of operations on regular languages is examined, and a new approach based on an old method (derivatives of regular expressions) is presented. Since state complexity is a property of a language, it is appropriate to define it in formallanguage terms as the nu ..."
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Cited by 9 (3 self)
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The past research on the state complexity of operations on regular languages is examined, and a new approach based on an old method (derivatives of regular expressions) is presented. Since state complexity is a property of a language, it is appropriate to define it in formallanguage terms as the number of distinct quotients of the language, and to call it “quotient complexity”. The problem of finding the quotient complexity of a language f(K,L) is considered, where K and L are regular languages and f is a regular operation, for example, union or concatenation. Since quotients can be represented by derivatives, one can find a formula for the typical quotient of f(K,L) in terms of the quotients of K and L. To obtain an upper bound on the number of quotients of f(K,L) all one has to do is count how many such quotients are possible, and this makes automaton constructions unnecessary. The advantages of this point of view are illustrated by many examples. Moreover, new general observations are presented to help in the estimation of the upper bounds on quotient complexity of regular operations. 1
Unary Language Operations and their Nondeterministic State Complexity
 DLT 2002. LNCS
, 2001
"... We investigate the costs, in terms of states, of operations on infinite and finite unary regular languages where the languages are represented by nondeterministic finite automata. In particular, we consider Boolean operations, concatenation, iteration, and free iteration. Most of the bounds are tig ..."
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Cited by 8 (1 self)
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We investigate the costs, in terms of states, of operations on infinite and finite unary regular languages where the languages are represented by nondeterministic finite automata. In particular, we consider Boolean operations, concatenation, iteration, and free iteration. Most of the bounds are tight in the exact number of states, i.e. the number is sufficient and necessary in the worst case. For the complementation of infinite languages a tight bound in the order of magnitude is shown.
State complexity of basic operations on suffixfree regular languages
, 2007
"... We investigate the state complexity of basic operations for suffixfree regular languages. The state complexity of an operation for regular languages is the number of states that are necessary and sufficient in the worstcase for the minimal deterministic finitestate automaton that accepts the lan ..."
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Cited by 7 (2 self)
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We investigate the state complexity of basic operations for suffixfree regular languages. The state complexity of an operation for regular languages is the number of states that are necessary and sufficient in the worstcase for the minimal deterministic finitestate automaton that accepts the language obtained from the operation. We establish the precise state complexity of catenation, Kleene star, reversal and the Boolean operations for suffixfree regular languages.
Salomaa: State complexity of orthogonal catenation
 in: Proceedings of DCFS 08
, 2008
"... A language L is the orthogonal catenation of languages L1 and L2 if every word of L can be written in a unique way as a catenation of a word in L1 and a word in L2. We establish a tight bound for the state complexity of orthogonal catenation of regular languages. The bound is smaller than the bound ..."
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Cited by 7 (0 self)
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A language L is the orthogonal catenation of languages L1 and L2 if every word of L can be written in a unique way as a catenation of a word in L1 and a word in L2. We establish a tight bound for the state complexity of orthogonal catenation of regular languages. The bound is smaller than the bound for arbitrary catenation. 1
State complexity of shuffle on trajectories
 In Descriptional Complexity of Formal Systems (DCFS) (2002
, 2004
"... Abstract It is easy to get an upper bound for the state complexity of shuffle on trajectories that generalizes the bound for unrestricted shuffle. We establish improved bounds for slender trajectories. For trajectories with USL index 1 (or 1thin trajectories) we obtain an asymptotically tight lower ..."
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Cited by 6 (4 self)
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Abstract It is easy to get an upper bound for the state complexity of shuffle on trajectories that generalizes the bound for unrestricted shuffle. We establish improved bounds for slender trajectories. For trajectories with USL index 1 (or 1thin trajectories) we obtain an asymptotically tight lower bound when the state complexity of the trajectory grows with respect to the state complexity of the component languages. Some estimations are improved by considering nondeterministic state complexity. 1 Introduction The notion of shuffle on trajectories was introduced as an extension of the existing notions of shuffle by Mateescu et al. [5] to provide an abstraction of parallel composition of words, an important operation in parallel computation.
State complexity of union and intersection of finite languages
 In Proceedings of DLT’07, Lecture Notes in Computer Science 4588
, 2007
"... 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 finitestate automata (DFAs) for fin ..."
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Cited by 4 (2 self)
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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 finitestate 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
Yu: State complexity of combined operations with two basic operations
 Theoretical Computer Science
"... This paper studies the state complexity of (L1L2) R, L R 1 L2, L ∗ 1L2, (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, MartinVide, Salomaa, Yu, 2008] for the state complexity of (L1L2) R coincides with the l ..."
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Cited by 3 (1 self)
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This paper studies the state complexity of (L1L2) R, L R 1 L2, L ∗ 1L2, (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, MartinVide, Salomaa, Yu, 2008] 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 L R 1 L2, L ∗ 1L2, 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. Keywords: automata state complexity, combined operations, regular languages, finite 1.