Definability in the Homomorphic Quasiorder of Finite Labeled Forests

Kudinov, Oleg V.; Selivanov, Victor L.

doi:10.1007/978-3-540-73001-9_45

Cited by 9 publications

(13 citation statements)

References 4 publications

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“…This result is a non-trivial corollary of some results in [Her93,Se04,Se07a,KS07]. In the rest of this section we recall some relevant information from those papers and deduce from them the main result.…”

Section: Proof 1 It Suffices To Show Thatsupporting

confidence: 55%

“…In [Her93,Se07a] the quotient-structures of ((BC(Σ 0 1 )) k ; ≤) and ((∆ 0 2 ) k ; ≤) over N were characterized in terms of a natural preorder ≤ h on the finite and countable well-founded k-labeled forests, respectively. These characterizations clarified the corresponding structures considerably and led to deep definability theories for both structures in [KS07,KS09,KSZ09]. These results show that, similar to the structure of Wadge degrees, the structures of degrees of k-partitions may serve as tools to measure the topological complexity of natural problems.…”

Section: Representations and Reducibilitiesmentioning

confidence: 71%

“…By induction, p(x ′ ) = h(x ′ ) for some p ∈ Aut(k). By Lemma 5 in [KS07], the function a → a ′ on the join-irreducible elements a with |M (a)| ≥ 3 is injective, hence y ′ = h(x ′ ) and p(x) = y.…”

Section: Proof 1 It Suffices To Show Thatmentioning

confidence: 97%

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Total Representations

Selivanov¹

2013

Logical Methods in Computer Science

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Abstract. Almost all representations considered in computable analysis are partial. We provide arguments in favor of total representations (by elements of the Baire space). Total representations make the well known analogy between numberings and representations closer, unify some terminology, simplify some technical details, suggest interesting open questions and new invariants of topological spaces relevant to computable analysis.

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Section: Proof 1 It Suffices To Show Thatsupporting

confidence: 55%

Section: Representations and Reducibilitiesmentioning

confidence: 71%

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Total Representations

Selivanov¹

2013

Logical Methods in Computer Science

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“…It is also worth observing that (i)-(iii) yield a canonical representation for t . PROPOSITION 2.5 [3,9]. Let t = p i ( U) for some i < k and for (at most) countable…”

Section: Proposition 24mentioning

confidence: 99%

“…The structure (F k ; ≤) of finite k-labeled forests (k ≥ 2) with the h-quasiorder ≤ was studied in [1][2][3][4]. The structure ( F k ; ≤) was introduced and explored in [5].…”

Section: Preliminariesmentioning

confidence: 99%

Definability of closure operations in the h-quasiorder of labeled forests

2010

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We prove that natural closure operations on quotient structures of the h-quasiorder of finite and (at most) countable k-labeled forests (k ≥ 3) are definable provided that minimal nonsmallest elements are allowed as parameters. This strengthens our previous result which holds that each element of the h-quasiorder of finite k-labeled forests is definable in the first-order language, and each element of the h-quasiorder of (at most) countable k-labeled forests is definable in the language L ω 1 ω ; in both cases k ≥ 3 and minimal nonsmallest elements are allowed as parameters. Similar results hold true for two other relevant structures: the h-quasiorder of finite (resp. countable) k-labeled trees and k-labeled trees with a fixed label on the root element.

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Well-Quasi Orders and Hierarchy Theory

Selivanov

2020

Trends in Logic

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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.

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Definability in the Homomorphic Quasiorder of Finite Labeled Forests

Cited by 9 publications

References 4 publications

Total Representations

Total Representations

Definability of closure operations in the h-quasiorder of labeled forests

Well-Quasi Orders and Hierarchy Theory

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