Complexity of Right-Ideal, Prefix-Closed, and Prefix-Free Regular Languages

Brzozowski, Janusz; Sinnamon, Corwin

doi:10.14232/actacyb.23.1.2017.3

Cited by 9 publications

(19 citation statements)

References 22 publications

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“…The authors of [9] proved this result with t = (1 ′ , 2 ′ ), but we prove a slightly more general statement.…”

Section: Examplesmentioning

confidence: 55%

“…, n − 1}. The inductive proof given by the authors of [9] has a different structure from the type of argument captured by Theorem 1. To reach a state (q ′ , S), in Theorem 1 we start from some state (q ′ , B) and apply a word that fixes the first component q ′ .…”

Section: Examplesmentioning

confidence: 99%

“…Example 1 (Prefix-Closed Witness. Brzozowski and Sinnamon, 2017 [9]). Our technique does not seem to work well with the following witness languages.…”

Section: Examplesmentioning

confidence: 99%

“…Subclass Complexity Regular [1,9,15,19] (m − 1)2 n + 2 n−1 Prefix-closed [6,9] (m + 1)2 n−2 Unary [16,17,19] ∼mn (asymptotically) Prefix-free [9,13,14] m + n − 2 Finite unary [10,18] m + n − 2 Suffix-closed [6,8] mn − n + 1 Finite binary [10] (m − n + 3)2 n−2 − 1 Suffix-free [8,12] (m − 1)2 n−2 + 1 Star-free [7] (m − 1)2 n + 2 n−1 Right ideal [4,5,9] m + 2 n−2 Non-returning [3,11] (m − 1)2 n−1 + 1 Left ideal [4,5,8] m + n − 1 This suggests that our technique is widely applicable and should be considered as an viable alternative to the traditional induction argument when attempting reachability proofs in concatenation automata. The rest of the paper is structured as follows.…”

Section: Subclass Complexitymentioning

confidence: 99%

“…For example, if t is the constant transformation (Q A → 1 ′ ) then (p ′ , Q B ) and (q ′ , Q B ) are indistinguishable. However, in [9] the authors take t to be the transposition (1 ′ , 2 ′ ) and find that all reachable states are pairwise distinguishable.…”

Section: Examplesmentioning

confidence: 99%

See 4 more Smart Citations

A New Technique for Reachability of States in Concatenation Automata

Davies

2018

Descriptional Complexity of Formal Systems

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We present a new technique for demonstrating the reachability of states in deterministic finite automata representing the concatenation of two languages. Such demonstrations are a necessary step in establishing the state complexity of the concatenation of two languages, and thus in establishing the state complexity of concatenation as an operation. Typically, ad-hoc induction arguments are used to show particular states are reachable in concatenation automata. We prove some results that seem to capture the essence of many of these induction arguments. Using these results, reachability proofs in concatenation automata can often be done more simply and without using induction directly.

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“…The authors of [9] proved this result with t = (1 ′ , 2 ′ ), but we prove a slightly more general statement.…”

Section: Examplesmentioning

confidence: 55%

Section: Examplesmentioning

confidence: 99%

“…Example 1 (Prefix-Closed Witness. Brzozowski and Sinnamon, 2017 [9]). Our technique does not seem to work well with the following witness languages.…”

Section: Examplesmentioning

confidence: 99%

Section: Subclass Complexitymentioning

confidence: 99%

Section: Examplesmentioning

confidence: 99%

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A New Technique for Reachability of States in Concatenation Automata

Davies

2018

Descriptional Complexity of Formal Systems

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Complexity of Suffix-Free Regular Languages

Brzozowski

Szykuła

2015

Fundamentals of Computation Theory

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We study various complexity properties of suffix-free regular languages. The quotient complexity of a regular language L is the number of left quotients of L; this is the same as the state complexity of L, which is the number of states in a minimal deterministic finite automaton (DFA) accepting L. A regular language L ′ is a dialect of a regular language L if it differs only slightly from L (for example, the roles of the letters of L ′ are a permutation of the roles of the letters of L). The quotient complexity of an operation on regular languages is the maximal quotient complexity of the result of the operation expressed as a function of the quotient complexities of the operands. A sequence (L k , L k+1 , . . . ) of regular languages in some class C, where n is the quotient complexity of Ln, is called a stream. A stream is most complex in class C if its languages Ln meet the complexity upper bounds for all basic measures, namely, they meet the quotient complexity upper bounds for star and reversal; they have largest syntactic semigroups; they have the maximal numbers of atoms, each of which has maximal quotient complexity; and (possibly together with their dialects L ′ m ) they meet the quotient complexity upper bounds for boolean operations and product (concatenation). It is known that there exist such most complex streams in the class of regular languages, in the class of prefix-free languages, and also in the classes of right, left, and two-sided ideals. In contrast to this, we prove that there does not exist a most complex stream in the class of suffix-free regular languages. However, we do exhibit one ternary suffix-free stream that meets the bound for product and whose restrictions to binary alphabets meet the bounds for star and boolean operations. We also exhibit a quinary stream that meets the bounds for boolean operations, reversal, size of syntactic semigroup, and atom complexities. Moreover, we solve an open problem about the bound for the product of two languages of quotient complexities m and n in the binary case by showing that it can be met for infinitely many m and n. Two transition semigroups play an important role for suffix-free languages: semigroup T 5 (n) is a suffix-free semigroup that has maximal cardinality for 2 n 5, while semigroup T 6 (n) has maximal cardinality for n = 2, 3, and n 6. We prove that all witnesses meeting the bounds for the star and the second witness in a product must have transition semigroups in T 5 (n). On the other hand, witnesses meeting the bounds for reversal, size of syntactic semigroup, and the complexity of atoms must have semigroups in T 6 (n).

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Complexity of Proper Prefix-Convex Regular Languages

Brzozowski

Sinnamon

2017

Implementation and Application of Automata

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A language L over an alphabet Σ is prefix-convex if, for any words x, y, z ∈ Σ * , whenever x and xyz are in L, then so is xy. Prefix-convex languages include right-ideal, prefix-closed, and prefix-free languages, which were studied elsewhere. Here we concentrate on prefixconvex languages that do not belong to any one of these classes; we call such languages proper. We exhibit most complex proper prefix-convex languages, which meet the bounds for the size of the syntactic semigroup, reversal, complexity of atoms, star, product, and boolean operations.

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Complexity of Right-Ideal, Prefix-Closed, and Prefix-Free Regular Languages

Cited by 9 publications

References 22 publications

A New Technique for Reachability of States in Concatenation Automata

A New Technique for Reachability of States in Concatenation Automata

Complexity of Suffix-Free Regular Languages

Complexity of Proper Prefix-Convex Regular Languages

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