Well-quasi-orders and regular ω-languages

Ogawa, Mizuhito

doi:10.1016/j.tcs.2004.03.052

Cited by 7 publications

(12 citation statements)

References 3 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…Future directions include leveraging well-quasiorders for infinite words [21] to shed new light on the inclusion problem between ω languages. Our results could also be extended to inclusion of tree languages by relying on the extensions of Myhill-Nerode theorems for tree languages [20].…”

Section: Discussionmentioning

confidence: 99%

Static Analysis

2019

Lecture Notes in Computer Science

View full text Add to dashboard Cite

We study the language inclusion problem L1 ⊆ L2 where L1 is regular. Our approach relies on abstract interpretation and checks whether an overapproximating abstraction of L1, obtained by successively overapproximating the Kleene iterates of its least fixpoint characterization, is included in L2. We show that a language inclusion problem is decidable whenever this overapproximating abstraction satisfies a completeness condition (i.e. its loss of precision causes no false alarm) and prevents infinite ascending chains (i.e. it guarantees termination of least fixpoint computations). Such overapproximating abstraction function on languages can be defined using quasiorder relations on words where the abstraction gives the language of all words "greater than or equal to" a given input word for that quasiorder. We put forward a range of quasiorders that allow us to systematically design decision procedures for different language inclusion problems such as regular languages into regular languages or into trace sets of one-counter nets. In the case of inclusion between regular languages, some of the induced inclusion checking procedures correspond to well-known state-of-the-art algorithms like the so-called antichain algorithms. Finally, we provide an equivalent greatest fixpoint language inclusion check which relies on quotients of languages and, to the best of our knowledge, was not previously known.

show abstract

Section: Discussionmentioning

confidence: 99%

Static Analysis

2019

Lecture Notes in Computer Science

View full text Add to dashboard Cite

show abstract

“…We believe we have only scratched the surface of the use of well-quasiorders on words for solving language inclusion problems. Future directions include leveraging wellquasiorders for infinite words [3], [26] to shed new light on the inclusion problem between ω-regular languages. Our results could also be extended to inclusion of tree languages by relying on the extensions of Myhill-Nerode theorems for tree languages [25].…”

Section: Discussionmentioning

confidence: 99%

Complete Abstractions for Checking Language Inclusion

Ganty¹,

Ranzato²,

Valero³

2019

Preprint

View full text Add to dashboard Cite

We study the language inclusion problem L1 ⊆ L2 where L1 is regular or context-free. Our approach relies on abstract interpretation and checks whether an overapproximating abstraction of L1, obtained by successively overapproximating the Kleene iterates of its least fixpoint characterization, is included in L2. We show that a language inclusion problem is decidable whenever this overapproximating abstraction satisfies a completeness condition (i.e. its loss of precision causes no false alarm) and prevents infinite ascending chains (i.e. it guarantees termination of least fixpoint computations). Such overapproximating abstraction function on languages can be defined using quasiorder relations on words where the abstraction gives the language of all words "greater than or equal to" a given input word for that quasiorder. We put forward a range of quasiorders that allow us to systematically design decision procedures for different language inclusion problems such as context-free languages into regular languages and regular languages into trace sets of one-counter nets. We also provide quasiorders for which the induced inclusion checking procedure corresponds to well-known state-of-theart algorithms like the so-called antichain algorithms. Finally, we provide an equivalent greatest fixpoint language inclusion check which relies on quotients of languages and, to the best of our knowledge, was not previously known.

show abstract

“…Another development of Theorem 8.1 was initiated in [85] where some analogues of this theorem for infinite words were found. A QO on A ω is a periodic extension of a QO ≤ on A * if ∀i < ω(u i ≤ v i ) implies u 0 u 1 · · · v 0 v 1 · · · and ∀p ∈ A ω ∃u, v ∈ A * (p uv ω ∧ uv ω p).…”

Section: Well Quasiorders and Regular Languagesmentioning

confidence: 99%

“…For instance, the subword relation on infinite words is a periodic extension of the subword relation on finite words and is therefore WQO. A basic fact in [85] is the following characterization of regular ω-languages. Relate to any monotone WQO ≤ on A * the class L ω ≤ of upward closed sets in (A ω ; ), for some periodic extension of ≤.…”

Section: Well Quasiorders and Regular Languagesmentioning

confidence: 99%

Well-Quasi Orders and Hierarchy Theory

Selivanov

2020

Trends in Logic

View full text Add to dashboard Cite

We discuss some applications of WQOs to several fields were hierarchies and reducibilities are the principal classification tools, notably to Descriptive Set Theory, Computability theory and Automata Theory. While the classical hierarchies of sets usually degenerate to structures very close to ordinals, the extension of them to functions requires more complicated WQOs, and the same applies to reducibilities. We survey some results obtained so far and discuss open problems and possible research directions.

show abstract

Well-quasi-orders and regular ω-languages

Cited by 7 publications

References 3 publications

Static Analysis

Static Analysis

Complete Abstractions for Checking Language Inclusion

Well-Quasi Orders and Hierarchy Theory

Contact Info

Product

Resources

About