On Decidability of Intermediate Levels of Concatenation Hierarchies

Almeida, Jorge; Bartoňová, Jana; Klíma, Ondřej; Kunc, Michal

doi:10.1007/978-3-319-21500-6_4

Cited by 14 publications

(27 citation statements)

References 14 publications

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“…Since |W n,n | may not be a multiple of p(|x|) + 1, we add $ to fill up any remaining space after the last configuration. For a word w = a i 1 , δ 1 · · · a i ℓ , δ ℓ ∈ Π ℓ , we define w [1] = a i 1 · · · a i ℓ ∈ Σ ℓ n and w [2] = δ 1 · · · δ ℓ ∈ ∆ ℓ #$ . Conversely, for a word v ∈ ∆ * #$ , we write enc(v) to denote the set of all words w ∈ Π |v| with w [2] = v. Similarly, for v ∈ Σ * n , enc(v) denotes the words w ∈ Π |v| with w [1] = v. We extend this notation to sets of words.…”

Section: Deciding Universality Of Rponfasmentioning

confidence: 99%

“…We say that a word w encodes an accepting run of M on x if w [1] = W n,n and w [2] is of the form #w 1 # · · · #w m #$ j such that there is an i ∈ {1, . .…”

Section: Deciding Universality Of Rponfasmentioning

confidence: 99%

“…Condition (A) is not necessary, since we do not encode symbols in binary. Condition (B) can use the same expression as in (2), adjusted to our alphabet:…”

Section: Deciding Universality Of Rponfasmentioning

confidence: 99%

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Complexity of universality and related problems for partially ordered NFAs

Krötzsch¹,

Masopust²,

Thomazo³

2017

Information and Computation

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Partially ordered nondeterministic finite automata (poNFAs) are NFAs whose transition relation induces a partial order on states, that is, for which cycles occur only in the form of self-loops on a single state. A poNFA is universal if it accepts all words over its input alphabet. Deciding universality is PSpace-complete for poNFAs, and we show that this remains true even when restricting to a fixed alphabet. This is nontrivial since standard encodings of alphabet symbols in, e.g., binary can turn self-loops into longer cycles. A lower coNP-complete complexity bound can be obtained if we require that all self-loops in the poNFA are deterministic, in the sense that the symbol read in the loop cannot occur in any other transition from that state. We find that such restricted poNFAs (rpoNFAs) characterize the class of R-trivial languages, and we establish the complexity of deciding if the language of an NFA is R-trivial. Nevertheless, the limitation to fixed alphabets turns out to be essential even in the restricted case: deciding universality of rpoNFAs with unbounded alphabets is PSpace-complete. Based on a close relation between universality and the problems of inclusion and equivalence, we also obtain the complexity results for these two problems. Finally, we show that the languages of rpoNFAs are definable by deterministic (one-unambiguous) regular expressions, which makes them interesting in schema languages for XML data.

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Section: Deciding Universality Of Rponfasmentioning

confidence: 99%

“…We say that a word w encodes an accepting run of M on x if w [1] = W n,n and w [2] is of the form #w 1 # · · · #w m #$ j such that there is an i ∈ {1, . .…”

Section: Deciding Universality Of Rponfasmentioning

confidence: 99%

See 1 more Smart Citation

Complexity of universality and related problems for partially ordered NFAs

Krötzsch¹,

Masopust²,

Thomazo³

2017

Information and Computation

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“…Very recently, Almeida, Bartonova, Klíma and Kunc [5] improved the result of Weil and the author [72] to get the following decidability result.…”

Section: Level 3/2mentioning

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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“…Although significant progress has been recently achieved on the membership problem for the levels of the Straubing-Thérien hierarchy [25,26,5], it remains an open problem whether it is decidable at all levels. A key tool that has been used in such works is the following lifting problem: to determine when a pair of elements of a finite monoid can be realized as values in the monoid of the sides of a pseudoinequality which is valid in a given level of the corresponding hierarchy of pseudovarieties.…”

Section: Introductionmentioning

confidence: 99%