Alternative Automata Characterization of Piecewise Testable Languages

Klíma, Ondřej; Polák, Libor

doi:10.1007/978-3-642-38771-5_26

Cited by 26 publications

(53 citation statements)

References 11 publications

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“…Recall that the depth of a complete DFA is the maximal length of an acyclic path from some initial to some reachable state. When L is regular, we write dp(L) for the depth of the canonical DFA for L. Since h(L) ≤ dp(L) holds for all PT languages [KP13], one could try to bound dp(L ¡ w) in terms of dp(L) and w. This does not seem very promising: First, for L fixed, dp(L ¡w) cannot be bounded in O(|w|). Furthermore, dp(L) can be much larger than h(L): if L is k-PT and |A| = n then the depth of the minimal DFA for L can be as large as k+n k − 1 [MT17, Thm.…”

Section: A General Upper Bound?mentioning

confidence: 99%

On shuffle products, acyclic automata and piecewise-testable languages

Halfon¹,

Schnoebelen²

2019

Information Processing Letters

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Section: A General Upper Bound?mentioning

confidence: 99%

On shuffle products, acyclic automata and piecewise-testable languages

Halfon¹,

Schnoebelen²

2019

Information Processing Letters

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“…In [11], this definition was studied in the context of acyclic (semi)automata, in which case several equivalent conditions were described. One of them can be rephrased in the following way.…”

Section: Acyclic Confluent Automatamentioning

confidence: 99%

“…⊓ ⊔ Using the condition from Lemma 8, one can prove that the class of all acyclic confluent semiautomata is a variety of semiautomata similarly as in Proposition 2. Finally, the main result from [11] can be formulated in the following way. It is mentioned in [11] that the defining condition is testable in a polynomial time.…”

Section: Acyclic Confluent Automatamentioning

confidence: 99%

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Language and Automata Theory and Applications

2019

Lecture Notes in Computer Science

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The Eilenberg correspondence relates varieties of regular languages to pseudovarieties of finite monoids. Various modifications of this correspondence have been found with more general classes of regular languages on one hand and classes of more complex algebraic structures on the other hand. It is also possible to consider classes of automata instead of algebraic structures as a natural counterpart of classes of languages.Here we deal with the correspondence relating positive C-varieties of languages to positive C-varieties of ordered automata and we present various specific instances of this correspondence. These bring certain well-known results from a new perspective and also some new observations. Moreover, complexity aspects of the membership problem are discussed both in the particular examples and in a general setting.

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“…This variety of J -trivial monoids is characterized by the identities x ω+1 = x ω and (xy) ω = (yx) ω , or, alternatively, by the identities y(xy) ω = (xy) ω = (xy) ω x. Simon's original proof is based on a very nice argument of combinatorics on words. Simon's theorem inspired a lot of subsequent research and a number of alternative proofs have been proposed [97,107,1,2,40,42,43,44]. Let me just mention two important consequences in semigroup theory.…”

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confidence: 99%

The Dot-Depth Hierarchy, 45 Years Later

2017

The Role of Theory in Computer Science

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In 1970, R. S. Cohen and Janusz A. Brzozowski introduced a hierarchy of star-free languages called the dot-depth hierarchy. This hierarchy and its generalisations, together with the problems attached to them, had a long-lasting influence on the development of automata theory. This survey article reports on the numerous results and conjectures attached to this hierarchy.This paper is a follow-up of the survey article Open problems about regular languages, 35 years later [57]. The dot-depth hierarchy, also known as Brzozowski hierarchy, is a hierarchy of star-free languages first introduced by Cohen and Brzozowski [25] in 1971. It immediately gave rise to many interesting questions and an account of the early results can be found in Brzozowski's survey [20] from 1976. Terminology, notation and backgroundMost of the terminology used in this paper was introduced in [57]. We just complete these definitions by giving the ordered versions of the notions of syntactic monoid and variety of finite monoids. Syntactic order and positive varietiesAn ordered monoid is a monoid equipped with an order compatible with the multiplication: x y implies zx zy and xz yz.The syntactic preorder 1 of a language L of A * is the relation L defined on A * by u L v if and only if, for every x, y ∈ A * , xuy ∈ L ⇒ xvy ∈ L. * The author was funded from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 670624).1 Unfortunately, the author used the opposite order in earlier papers (from 1995 to 2011). 1The syntactic congruence of L is the associated equivalence relation ∼ L , defined by u ∼ L v if and only if u L v and v L u.The syntactic monoid of L is the quotient M (L) of A * by ∼ L and the natural morphism η : A * → A * /∼ L is called the syntactic morphism of L. The syntactic preorder L induces an order on the quotient monoid M (L). The resulting ordered monoid is called the syntactic ordered monoid of L.For instance, the syntactic monoid of the language {a, aba} is the monoid M = {1, a, b, ab, ba, aba, 0} presented by the relations a 2 = b 2 = bab = 0. Its syntactic order is given by the relations 0 < ab < 1, 0 < ba < 1, 0 < aba < a, 0 < b.The syntactic ordered monoid of a language was first introduced by Schützenberger [86] in 1956, but thereafter, he apparently only used the syntactic monoid.A positive variety of languages is a class of languages closed under finite unions, finite intersections, left and right quotients and inverses of morphisms. A variety of languages is a positive variety closed under complementation.Similarly, a variety of finite ordered monoids is a class of finite ordered monoids closed under taking ordered submonoids, quotients and finite products. Varieties of finite (ordered) semigroups are defined analogously. If V is a variety of ordered monoids, let V d denote the dual variety, consisting of all ordered monoids (M, ) such that (M, ) ∈ V. We refer the reader to the books [2, 28, 62] for more details.Eilenberg's variety theor...

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Alternative Automata Characterization of Piecewise Testable Languages

Cited by 26 publications

References 11 publications

On shuffle products, acyclic automata and piecewise-testable languages

On shuffle products, acyclic automata and piecewise-testable languages

Language and Automata Theory and Applications

The Dot-Depth Hierarchy, 45 Years Later

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