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12
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 formal-language terms as the nu ..."
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Cited by 36 (23 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 formal-language 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
Quotient complexity of ideal languages
- In: LATIN 2010. LNCS 6034, Springer-Verlag
, 2010
"... Abstract. We study the state complexity of regular operations in the class of ideal languages. A language L ⊆ Σ ∗ is a right (left) ideal if it satisfies L = LΣ ∗ (L = Σ ∗ L). It is a two-sided ideal if L = Σ ∗ LΣ ∗ , and an all-sided ideal if L = Σ ∗ L, the shuffle of Σ ∗ with L. We prefer the term ..."
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Cited by 22 (12 self)
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Abstract. We study the state complexity of regular operations in the class of ideal languages. A language L ⊆ Σ ∗ is a right (left) ideal if it satisfies L = LΣ ∗ (L = Σ ∗ L). It is a two-sided ideal if L = Σ ∗ LΣ ∗ , and an all-sided ideal if L = Σ ∗ L, the shuffle of Σ ∗ with L. We prefer the term “quotient complexity ” instead of “state complexity”, and we use derivatives to calculate upper bounds on quotient complexity, whenever it is convenient. We find tight upper bounds on the quotient complexity of each type of ideal language in terms of the complexity of an arbitrary generator and of its minimal generator, the complexity of the minimal generator, and also on the operations union, intersection, set difference, symmetric difference, concatenation, star and reversal of ideal languages.
Syntactic complexity of ideal and closed languages
, 2011
"... The state complexity of a regular language is the number of states in the minimal deterministic automaton accepting the language. The syntactic complexity of a regular language is the cardinality of its syntactic semigroup. The syntactic complexity of a subclass of regular languages is the worst-c ..."
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Cited by 20 (15 self)
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The state complexity of a regular language is the number of states in the minimal deterministic automaton accepting the language. The syntactic complexity of a regular language is the cardinality of its syntactic semigroup. The syntactic complexity of a subclass of regular languages is the worst-case syntactic complexity taken as a function of the state complexity n of languages in that class. We prove that nn−1 is a tight upper bound on the complexity of right ideals and prefix-closed languages, and that there exist left ideals and suffix-closed languages of syntactic complexity nn−1+n−1, and two-sided ideals and factor-closed languages of syntactic complexity nn−2 + (n − 2)2n−2 + 1.
Quotient complexity of bifix-, factor-, and subword-free regular languages
, 2011
"... A language L is prefix-free if, whenever words u and v are in L and u is a prefix of v, then u = v. Suffix-, factor-, and subword-free languages are defined similarly, where “subword ” means “subsequence”. A language is bifix-free if it is both prefix- and suffix-free. We study the quotient com-plex ..."
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Cited by 11 (7 self)
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A language L is prefix-free if, whenever words u and v are in L and u is a prefix of v, then u = v. Suffix-, factor-, and subword-free languages are defined similarly, where “subword ” means “subsequence”. A language is bifix-free if it is both prefix- and suffix-free. We study the quotient com-plexity, more commonly known as state complexity, of operations in the classes of bifix-, factor-, and subword-free regular languages. We find tight upper bounds on the quotient complexity of intersection, union, difference, symmetric difference, concatenation, star, and reversal in these three classes of languages.
Quotient complexity of star-free languages
, 2010
"... The quotient complexity, also known as state complexity, of a regular language is the number of distinct left quotients of the language. The quo-tient complexity of an operation is the maximal quotient complexity of the language resulting from the operation, as a function of the quotient com-plexiti ..."
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Cited by 7 (3 self)
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The quotient complexity, also known as state complexity, of a regular language is the number of distinct left quotients of the language. The quo-tient complexity of an operation is the maximal quotient complexity of the language resulting from the operation, as a function of the quotient com-plexities 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 complex-ities 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 the bound for reversal is 2n − 1.
Complexity in convex languages
- Proceedings of the 4th International Conference on Language and Automata Theory (LATA). Volume 6031 of LNCS
, 2010
"... Abstract. A language L is prefix-convex if, whenever words u and w are in L with u a prefix of w, then every word v which has u as a prefix and is a prefix of w is also in L. Similarly, we define suffix-, factor-, and subword-convex languages, where by subword we mean subsequence. To-gether, these l ..."
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Cited by 4 (1 self)
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Abstract. A language L is prefix-convex if, whenever words u and w are in L with u a prefix of w, then every word v which has u as a prefix and is a prefix of w is also in L. Similarly, we define suffix-, factor-, and subword-convex languages, where by subword we mean subsequence. To-gether, these languages constitute the class of convex languages which contains interesting subclasses, such as ideals, closed languages (includ-ing factorial languages) and free languages (including prefix-, suffix-, and infix-codes, and hypercodes). There are several advantages of studying the class of convex languages and its subclasses together. These classes are all definable by binary relations, in fact, by partial orders. Closure properties of convex languages have been studied in this general frame-work of binary relations. The problems of deciding whether a language is convex of a particular type have been analyzed together, and have been solved by similar methods. The state complexities of regular operations in subclasses of convex languages have been examined together, with considerable economies of effort. This paper surveys the recent results on convex languages with an emphasis on complexity issues.
Complexity in prefix-free regular languages
- University of Saskatchewan
, 2010
"... We examine deterministic and nondeterministic state complexities of regular operations on prefix-free languages. We strengthen several results by providing witness languages over smaller alphabets, usually as small as possible. We next provide the tight bounds on state complexity of symmetric differ ..."
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We examine deterministic and nondeterministic state complexities of regular operations on prefix-free languages. We strengthen several results by providing witness languages over smaller alphabets, usually as small as possible. We next provide the tight bounds on state complexity of symmetric difference, and deterministic and nondeterministic state complexity of difference and cyclic shift of prefix-free languages. 1
The Size of One-Way Cellular Automata
"... We investigate the descriptional complexity of basic operations on real-time one-way cellular automata with an unbounded as well well as a fixed number of cells. The size of the automata is measured by their number of states. Most of the bounds shown are tight in the order of magnitude, that is, the ..."
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We investigate the descriptional complexity of basic operations on real-time one-way cellular automata with an unbounded as well well as a fixed number of cells. The size of the automata is measured by their number of states. Most of the bounds shown are tight in the order of magnitude, that is, the sizes resulting from the effective constructions given are optimal with respect to worst case complexity. Conversely, these bounds also show the maximal savings of size that can be achieved when a given minimal real-time OCA is decomposed into smaller ones with respect to a given operation. From this point of view the natural problem of whether a decomposition can algorithmically be solved is studied. It turns out that all decomposition problems considered are algorithmically unsolvable. Therefore, a very restricted cellular model is studied in the second part of the paper, namely, real-time one-way cellular automata with a fixed number of cells. These devices are known to capture the regular languages and, thus, all the problems being undecidable for general one-way cellular automata become decidable. It is shown that these decision problems are NLOGSPACE-complete and thus share the attractive computational complexity of deterministic finite automata. Furthermore, the state complexity of basic operations for these devices is studied and upper and lower bounds are given.
Departamento de Ciência de Computadores Faculdade de Ciências da Universidade do Porto
, 2011
"... Abstract. The state complexity of a regular language is the number of states of its minimal determinitisc finite automaton. The complexity of a language operation is the complexity of the resulting language seen as a function of the complexities of the operation arguments. In this report we review s ..."
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Abstract. The state complexity of a regular language is the number of states of its minimal determinitisc finite automaton. The complexity of a language operation is the complexity of the resulting language seen as a function of the complexities of the operation arguments. In this report we review some of the results of state complexity of individual operations for regular and some subregular languages. 1 State Complexity and Nondeterministic State Complexity The state complexity of a regular language L, sc(L), is the number of states of its minimal DFA. The nondeterministic state complexity of a regular language L, nsc(L), is the number of states of a minimal NFA that accepts L. Since a DFA is in particular an NFA, for any regular language L one has sc(L) ≤ nsc(L). It is well knownthatanym-stateNFAcanbeconverted,viathesubset construction,inanequivalentDFAwith at most 2 m states [114] (we call this conversion determination). Thus, sc(L) ≤ 2 nsc(L). To show that (i) (ii) (iii)
