A Constructive Proof of the Topological Kruskal Theorem

Goubault-Larrecq, Jean

doi:10.1007/978-3-642-40313-2_3

Cited by 4 publications

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References 43 publications

(64 reference statements)

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“…Noetherian spaces generalize well-quasi orders: The Alexandrov topology on a quasi-order is Noetherian iff the quasi-order is a well-quasi order. As shown by Goubault-Larrecq [21], results on the preservation of well-quasi orders under various constructions (such as Higman's Lemma or Kruskal's Tree Theorem [20]) extend to Noetherian spaces; furthermore, Noetherian spaces exhibit some additional closure properties, e.g. the Hoare space of a Noetherian space is Noetherian again [18].…”

Section: Introductionmentioning

confidence: 99%

Quasi-Polish spaces

Brecht

2013

Annals of Pure and Applied Logic

124

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In the presence of suitable power spaces, compactness of X can be characterized as the singleton {X} being open in the space O(X) of open subsets of X. Equivalently, this means that universal quantification over a compact space preserves open predicates.Using the language of represented spaces, one can make sense of notions such as a Σ 0 2subset of the space of Σ 0 2 -subsets of a given space. This suggests higher-order analogues to compactness: We can, e.g. , investigate the spaces X where {X} is a ∆ 0 2 -subset of the space of ∆ 0 2 -subsets of X. Call this notion ∇-compactness. As ∆ 0 2 is self-dual, we find that both universal and existential quantifier over ∇-compact spaces preserve ∆ 0 2 predicates. Recall that a space is called Noetherian iff every subset is compact. Within the setting of Quasi-Polish spaces, we can fully characterize the ∇-compact spaces: A Quasi-Polish space is Noetherian iff it is ∇-compact. Note that the restriction to Quasi-Polish spaces is sufficiently general to include plenty of examples.

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Section: Introductionmentioning

confidence: 99%

Quasi-Polish spaces

Brecht

2013

Annals of Pure and Applied Logic

124

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“…; alternatively, this is an easy exercise from the characterization of [non-strict] inclusion in [ACABJ04].) It follows that if P is strictly below P ′ in Idl(Σ * ), then µ(P ) is strictly below µ(Q) in the multiset extension of ⊏, where, for P = e 1 e 2 • • • e m , µ(P ) is the multiset {e 1 , e 2 , • • • , e m }, a fact already used in [Gou13].…”

Section: Because the Empty Word Belongs To The Removed Prefix (Amentioning

confidence: 99%

Forward Analysis for WSTS, Part III: Karp-Miller Trees

Blondin,

Finkel,

Goubault-Larrecq

2017

Preprint

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This paper is a sequel of "Forward Analysis for WSTS, Part I: Completions" [STACS 2009, LZI Intl. Proc. in Informatics 3, 433-444] and "Forward Analysis for WSTS, Part II: Complete WSTS" [Logical Methods in Computer Science 8(3), 2012]. In these two papers, we provided a framework to conduct forward reachability analyses of WSTS, using finite representations of downwards-closed sets. We further develop this framework to obtain a generic Karp-Miller algorithm for the new class of very-WSTS. This allows us to show that coverability sets of very-WSTS can be computed as their finite ideal decompositions. Under natural effectiveness assumptions, we also show that LTL model checking for very-WSTS is decidable. The termination of our procedure rests on a new notion of acceleration levels, which we study. We characterize those domains that allow for only finitely many accelerations, based on ordinal ranks.

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