Decidability of Definability

Bodirsky, Manuel; Pinsker, Michael; Tsankov, Todor

doi:10.1109/lics.2011.11

Cited by 49 publications

(85 citation statements)

References 20 publications

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“…, P n ) is homogeneous and a Ramsey structure, since its automorphism group is, as a topological group, isomorphic to Aut(Q; <) n , and since being a Ramsey structure is a property of the automorphism group (as a topological group) [KPT05]. Thus, by [BP11,BPT13], we have the following analogous statement to Proposition 12 for this structure. In the statement, we may drop the mention of the auxiliary relations P 1 , .…”

Section: Homogeneous Equivalence Relationsmentioning

confidence: 76%

“…Achieving canonicity in Ramsey structures. The next proposition, which is an instance of more general statements from [BP11,BPT13], provides us with the main combinatorial tool for analyzing functions on Henson graphs. Equip H n with a total order ≺ in such a way that (H n , E, ≺) is homogeneous; up to isomorphism, there is only one such structure (H n , E, ≺), called the random ordered K n -free graph.…”

Section: Homogeneous Equivalence Relationsmentioning

confidence: 99%

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Constraint Satisfaction Problems for Reducts of Homogeneous Graphs

Bodirsky¹,

Martin²,

Pinsker³

et al. 2019

SIAM J. Comput.

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For n ≥ 3, let (Hn, E) denote the n-th Henson graph, i.e., the unique countable homogeneous graph with exactly those finite graphs as induced subgraphs that do not embed the complete graph on n vertices. We show that for all structures Γ with domain Hn whose relations are first-order definable in (Hn, E) the constraint satisfaction problem for Γ is either in P or is NP-complete.We moreover show a similar complexity dichotomy for all structures whose relations are first-order definable in a homogeneous graph whose reflexive closure is an equivalence relation.Together with earlier results, in particular for the random graph, this completes the complexity classification of constraint satisfaction problems of structures first-order definable in countably infinite homogeneous graphs: all such problems are either in P or NP-complete.An extended abstract of this paper has appeared at the 43rd International Colloquium on Automata, Languages and Programming (ICALP) Track B, 2016. and sinks [HN90, BKN09]). Various methods, combinatorial (graph-theoretic), logical, and universal-algebraic were brought to bear on this classification project, with many remarkable consequences. A conjectured delineation for the dichotomy was given in the algebraic language in [BKJ05], and finally the conjecture, and in particular this delineation, has recently been proven to be accurate [Bul17,Zhu17].When the domain is infinite, the complexity of the CSP can be outside NP, and even undecidable [BN06]. But for natural classes of such CSPs there is often the potential for structured classifications, and this has proved to be the case for structures first-order definable over the order (Q, <) of the rationals [BK09] or over the integers with successor [BMM17]. Another classification of this type has been obtained for CSPs where the constraint language is first-order definable over the random (Rado) graph [BP15a], making use of structural Ramsey theory. This paper was titled 'Schaefer's theorem for graphs' and it can be seen as lifting the famous classification of Schaefer [Sch78] from Boolean logic to logic over finite graphs, since the random graph is universal for the class of finite graphs.1.2. Homogeneous graphs and their reducts. The notion of homogeneity from model theory plays an important role when applying techniques from finite-domain constraint satisfaction to constraint satisfaction over infinite domains. A relational structure is homogeneous if every isomorphism between finite induced substructures can be extended to an automorphism of the entire structure. Homogeneous structures are uniquely (up to isomorphism) given by the class of finite structures that embed into them. The structure (Q, <) and the random graph are among the most prominent examples of homogeneous structures. The class of structures that are first-order definable over a homogeneous structure with finite relational signature is a very large generalization of the class of all finite structures, and CSPs for those structures have been studied independently in many dif...

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Section: Homogeneous Equivalence Relationsmentioning

confidence: 76%

Section: Homogeneous Equivalence Relationsmentioning

confidence: 99%

Constraint Satisfaction Problems for Reducts of Homogeneous Graphs

Bodirsky¹,

Martin²,

Pinsker³

et al. 2019

SIAM J. Comput.

Self Cite

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show abstract

“…Recall that a topological group

G

is extremely amenable if whenever

X

is a

G

‐flow , that is, a non‐empty, compact Hausdorff

G

‐space on which

G

acts continuously, then there is a

G

‐fixed point in

X

. The starting point for this paper is the following question, raised in [; , Question 1.1]. Question Suppose

G

is a closed, oligomorphic permutation group on a countable set.…”

Section: Introductionmentioning

confidence: 99%

Automorphism groups and Ramsey properties of sparse graphs

Evans

Hubička

Nešetřil

2019

Proc. London Math. Soc.

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We study automorphism groups of sparse graphs from the viewpoint of topological dynamics and the Kechris, Pestov, Todorčević correspondence. We investigate amenable and extremely amenable subgroups of these groups using the space of orientations of the graph and results from structural Ramsey theory. Resolving one of the open questions in the area, we show that Hrushovski's example of an ω‐categorical sparse graph has no ω‐categorical expansion with extremely amenable automorphism group.

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“…The Henson-graphs have no nontrivial reducts [17]. Later in [3] and [4] a general technique was introduced to investigate first order definable reducts of homogeneous structures on a finite language. Although the strategy works only under very special conditions, the first few reducts were possible to determine in several cases.…”

Section: Introductionmentioning

confidence: 99%