Characterizations of monadic NIP

Braunfeld, Samuel; Laskowski, M.

doi:10.1090/btran/94

Cited by 13 publications

(14 citation statements)

References 17 publications

(33 reference statements)

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“…It is defined in terms of a variant of forking independence -a central concept in stability theory, generalizing independence in vector spaces or algebraic independence. The equivalence (1) ↔ ( 2) is due to Baldwin and Shelah [4], the implications (2) → (3) → ( 4) are due to Shelah [46] and the implication (4) → ( 2) is a very recent result of Braunfeld and Laskowski [14]. Our contribution is the implication (4) → ( 5), and the overall equivalence in the case of ordered binary structures.…”

Section: Second Main Resultsmentioning

confidence: 77%

“…Intuitively, if C defines large grids then the product of two sets 𝐴 × 𝐵 can be represented by a set of single elements 𝐶 in some structure S ∈ C. Hence an arbitrary relation 𝑅 ⊆ 𝐴 × 𝐵 can be represented by some subset of 𝐶, so C is monadically independent. This is stated in the following lemma, due to Shelah [46] (see [14]). Lemma 20.…”

Section: Model Theorymentioning

confidence: 96%

“…There are other known grid theorems, including the Marcus-Tardos theorem itself. The recent result of Braunfeld and Laskowski [14] characterizes monadically NIP classes as exactly those that do not define large grids, in the sense of Theorem 8.…”

Section: Consequences and Related Workmentioning

confidence: 99%

“…Theorem 8 generalizes the implications (ii)→ (iii)→ (v) from Theorem 9 to arbitrary classes of structures -finite or infinite, ordered or unordered, and over an arbitrary signature. It also involves a notion which we call 1-dimensionality, which is a model-theoretic notion originating from Shelah (it is called finite satisfiability dichotomy in [14]), and is defined below (see Def. 25).…”

Section: Model Theorymentioning

confidence: 99%

See 3 more Smart Citations

Twin-width IV: ordered graphs and matrices

Bonnet

Giocanti

Mendez

et al. 2022

Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing

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We establish a list of characterizations of bounded twin-width for hereditary classes of totally ordered graphs: as classes of at most exponential growth studied in enumerative combinatorics, as monadically NIP classes studied in model theory, as classes that do not transduce the class of all graphs studied in finite model theory, and as classes for which model checking first-order logic is fixedparameter tractable studied in algorithmic graph theory.This has several consequences. First, it allows us to show that every hereditary class of ordered graphs either has at most exponential growth, or has at least factorial growth. This settles a question first asked by Balogh, Bollobás, and Morris [Eur. J. Comb. '06] on the growth of hereditary classes of ordered graphs, generalizing the Stanley-Wilf conjecture/Marcus-Tardos theorem. Second, it gives a fixed-parameter approximation algorithm for twin-width on ordered graphs. Third, it yields a full classification of fixed-parameter tractable first-order model checking on hereditary classes of ordered binary structures. Fourth, it provides a model-theoretic characterization of classes with bounded twin-width. Finally, it settles our small conjecture [SODA '21] in the case of ordered graphs. CCS CONCEPTS• Theory of computation → Finite Model Theory; • Mathematics of computing → Graph algorithms.

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Section: Second Main Resultsmentioning

confidence: 77%

Section: Model Theorymentioning

confidence: 96%

Section: Consequences and Related Workmentioning

confidence: 99%

Section: Model Theorymentioning

confidence: 99%

See 2 more Smart Citations

Twin-width IV: ordered graphs and matrices

Bonnet

Giocanti

Mendez

et al. 2022

Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing

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“…On the other hand, Braunfeld and Laskowski [BL22] very recently proved that for hereditary classes of structures C that are not monadically stable or monadically dependent, the required obstructions (total orders or arbitrary graphs) can be exhibited by an existential formula ϕ( x, ȳ) in the signature of C , without any additional unary predicates. Among other things, this shows that for hereditary classes of structures, the notions of monadic stability coincides with the more well-known notion of stability, and similarly, monadic dependence coincides with dependence (NIP).…”

Section: Introductionmentioning

confidence: 99%

Flipper games for monadically stable graph classes

Gajarský¹,

Nikolas²,

McCarty³

et al. 2023

Preprint

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A class of graphs C is monadically stable if for any unary expansion C of C , one cannot interpret, in first-order logic, arbitrarily long linear orders in graphs from C . It is known that nowhere dense graph classes are monadically stable; these encompass most of the studied concepts of sparsity in graphs, including classes of graphs that exclude a fixed topological minor. On the other hand, monadic stability is a property expressed in purely model-theoretic terms and hence it is also suited for capturing structure in dense graphs.For several years, it has been suspected that one can construct a structure theory for monadically stable graph classes that mirrors the theory of nowhere dense graph classes in the dense setting. In this work we provide a next step in this direction by giving a characterization of monadic stability through the Flipper game: a game on a graph played by Flipper, who in each round can complement the edge relation between any pair of vertex subsets, and Connector, who in each round is forced to localize the game to a ball of bounded radius. This is an analog of the Splitter game, which characterizes nowhere dense classes of graphs (Grohe, Kreutzer, and Siebertz, J. ACM '17).We give two different proofs of our main result. The first proof is based on tools borrowed from model theory, and it exposes an additional property of monadically stable graph classes that is close in spirit to definability of types. Also, as a byproduct, we give an alternative proof of the recent result of Braunfeld and Laskowski (arXiv 2209.05120) that monadic stability for graph classes coincides with existential monadic stability. The second proof relies on the recently introduced notion of flip-wideness (Dreier, Mählmann, Siebertz, and Toru ńczyk, arXiv 2206.13765) and provides an efficient algorithm to compute Flipper's moves in a winning strategy.

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Vector spaces with a union of independent subspaces

Berarducci,

Mamino,

Mennuni

2024

Arch. Math. Logic

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We study the theory of K-vector spaces with a predicate for the union X of an infinite family of independent subspaces. We show that if K is infinite then the theory is complete and admits quantifier elimination in the language of K-vector spaces with predicates for the n-fold sums of X with itself. If K is finite this is no longer true, but we still have that a natural completion is near-model-complete.

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Characterizations of monadic NIP

Cited by 13 publications

References 17 publications

Twin-width IV: ordered graphs and matrices

Twin-width IV: ordered graphs and matrices

Flipper games for monadically stable graph classes

Vector spaces with a union of independent subspaces

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