Ben Yaacov Research Articles

Approximate isomorphism of metric structures

AbstractWe give a formalism for approximate isomorphism in continuous logic simultaneously generalizing those of two papers by Ben Yaacov [2] and by Ben Yaacov, Doucha, Nies, and Tsankov [6], which are largely incompatible. With this we explicitly exhibit Scott sentences for the perturbation systems of the former paper, such as the Banach‐Mazur distance and the Lipschitz distance between metric spaces. Our formalism is simultaneously characterized syntactically by a mild generalization of perturbation systems and semantically by certain elementary classes of two‐sorted structures that witness approximate isomorphism. As an application, we show that the theory of any ‐tree or ultrametric space of finite radius is stable, improving a result of Carlisle and Henson [8].

Metric spaces are universal for bi-interpretation with metric structures

In the context of metric structures introduced by Ben Yaacov, Berenstein, Henson, and Usvyatsov [3], we exhibit an explicit encoding of metric structures in countable signatures as pure metric spaces in the empty signature, showing that such structures are universal for bi-interpretation among metric structures with positive diameter. This is analogous to the classical encoding of arbitrary discrete structures in finite signatures as graphs, but is stronger in certain ways and weaker in others. There are also certain fine grained topological concerns with no analog in the discrete setting.

Complexity of distances: Theory of generalized analytic equivalence relations

We generalize the notion of analytic/Borel equivalence relations, orbit equivalence relations, and Borel reductions between them to their continuous and quantitative counterparts: analytic/Borel pseudometrics, orbit pseudometrics, and Borel reductions between them. We motivate these concepts on examples and we set some basic general theory. We illustrate the new notion of reduction by showing that the Gromov–Hausdorff distance maintains the same complexity if it is defined on the class of all Polish metric spaces, spaces bounded from below, from above, and from both below and above. Then we show that [Formula: see text] is not reducible to equivalences induced by orbit pseudometrics, generalizing the seminal result of Kechris and Louveau. We answer in negative a question of Ben Yaacov, Doucha, Nies, and Tsankov on whether balls in the Gromov–Hausdorff and Kadets distances are Borel. In appendix, we provide new methods using games showing that the distance-zero classes in certain pseudometrics are Borel, extending the results of Ben Yaacov, Doucha, Nies, and Tsankov. There is a complementary paper of the authors where reductions between the most common pseudometrics from functional analysis and metric geometry are provided.

Stability in a group

We develop local stable group theory directly from topological dynamics, and extend the main tools in this subject to the setting of stability “in a model.” Specifically, given a group G , we analyze the structure of sets A \subseteq G such that the bipartite relation xy\in A omits infinite half-graphs. Our proofs rely on the characterization of model-theoretic stability via Grothendieck's “double-limit” theorem (as shown by Ben Yaacov), and the work of Ellis and Nerurkar on weakly almost periodic G -flows.

FRAÏSSÉ LIMITS FOR RELATIONAL METRIC STRUCTURES

AbstractThe general theory developed by Ben Yaacov for metric structures provides Fraïssé limits which are approximately ultrahomogeneous. We show here that this result can be strengthened in the case of relational metric structures. We give an extra condition that guarantees exact ultrahomogenous limits. The condition is quite general. We apply it to stochastic processes, the class of diversities, and its subclass of $L_1$ diversities.

Analog reducibility

Abstract In this paper we introduce and characterize two ‘analog reducibility’ notions for $[0,1]$-valued oracles on $\omega $ obtained by applying the syntactic characterizations of Turing and enumeration reducibility in terms of (positive) relatively $\varSigma _1$ and $\varPi _1$ formulas to formulas in continuous logic (Ben Yaacov, Berenstein, Henson and Usvyatsov, 2008, Model Theory for Metric Structures, vol. 2 of London Mathematical Society Lecture Note Series, pp. 315–427. Cambridge University Press.). The resulting analog and analog enumeration degree structures, $\mathscr {D}_a$ and $\mathscr {D}_{ae}$, naturally extend $\mathscr {D}_T$ and $\mathscr {D}_e$ in a compatible way. To show that these extensions are proper we prove that a sufficiently generic total $[0,1]$-valued oracle does not ‘analog enumerate’ any non-c.e. discrete set and that a sufficiently generic positive $[0,1]$-valued oracle neither ‘analog enumerates’ a non-c.e. discrete set nor ‘analog computes’ a non-trivial total $[0,1]$-valued oracle. We also provide a characterization of the continuous degrees among $\mathscr {D}_{ae}$ as precisely $\mathscr {D}_e \cap \mathscr {D}_a$. Finally we characterize a generalization of r.i.c.e. relations to metric structures via $\varSigma _1$ formulas in the ‘hereditarily compact superstructure’, which was the original motivation for the concepts in this paper.

Maximally highly proximal flows

For $G$ a Polish group, we consider $G$-flows which either contain a comeager orbit or have all orbits meager. We single out a class of flows, the maximally highly proximal (MHP) flows, for which this analysis is particularly nice. In the former case, we provide a complete structure theorem for flows containing comeager orbits, generalizing theorems of Melleray, Nguyen Van Thé, and Tsankov and of Ben Yaacov, Melleray, and Tsankov. In the latter, we show that any minimal MHP flow with all orbits meager has a metrizable factor with all orbits meager, thus ‘reflecting’ complicated dynamical behavior to metrizable flows. We then apply this to obtain a structure theorem for Polish groups whose universal minimal flow is distal.

Bounds on continuous Scott rank

An analog of Nadel's effective bound for the continuous Scott rank of metric structures, developed by Ben Yaacov, Doucha, Nies, and Tsankov, will be established: Let $\mathscr{L}$ be a language of continuous logic with code $\hat{\mathscr{L}}$. Let $\Omega$ be a weak modulus of uniform continuity with code $\hat{\Omega}$. Let $\mathcal{D}$ be a countable $\mathscr{L}$-pre-structure. Let $\bar{\mathcal{D}}$ denote the completion structure of $\mathcal{D}$. Then $\mathrm{SR}_\Omega(\bar{D}) \leq \omega_1^{\hat{\mathscr{L}}\oplus\hat{\Omega}\oplus\mathcal{D}}$, the Church-Kleene ordinal relative to $\hat{\mathscr{L}}\oplus\hat{\Omega}\oplus\mathcal{D}$.

On Amalgamation in NTP2 Theories and Generically Simple Generics

We prove a couple of results on NTP2 theories. First, we prove an amalgamation statement and deduce from it that the Lascar distance over extension bases is bounded by 2. This improves previous work of Ben Yaacov and Chernikov. We propose a line of investigation of NTP2 theories based on S1 ideals with amalgamation and ask some questions. We then define and study a class of groups with generically simple generics, generalizing NIP groups with generically stable generics.

Decision procedures for the conditions true in certain metric structures

After establishing a completeness theorem for continuous logic, Ben Yaacov and Pedersen conclude that if T is a complete recursive L-theory in continuous logic, and v(φ) is the truth value of the L-sentence φ in models of T, then v(φ) is a recursive real uniformly recursive in φ. Some of the examples to which the latter result applies are theories of the following structures: atomless probability structures, the Urysohn space of diameter 1, Hilbert space, the lattice-ordered group or ring of real-valued continuous functions on the Cantor set, and the complex ⁎-algebra of continuous functions on the Cantor set. This paper will explain why these examples obey much stronger results, yielding (for example) decision procedures for the conditions true in these structures.

Model Theory Methods for Topological Groups

This thesis gathers different works approaching subjects of topological dynamics by means of logic and descriptive set theory, and conversely. The first part is devoted to the study of Roelcke precompact Polish groups, which are the same as the automorphism groups of $\aleph_0$-categorical structures. They form a rich family of examples of infinite-dimensional topological groups, including several interesting permutation groups, isometry groups and homeomorphism groups of distinguished mathematical objects. Building on previous work of Ben Yaacov and Tsankov, we develop a model-theoretic translation of several dynamical aspects of these groups. Then we use this translation to obtain a precise understanding, in this case, of the dynamical hierarchy studied by Glasner and Megrelishvili. Later, with I. Ben Yaacov and T. Tsankov, we provide a model-theoretic description of the Hilbert-compactification of oligomorphic groups, and we give a characterization of Eberlein oligomorphic groups. We also study automorphism groups of randomized structures, as well the separable models of the theory of beautiful pairs of randomizations. The second part, with J. Melleray, studies full groups of minimal homeomorphisms of the Cantor space and their invariant measures. We show that full groups of minimal homeomorphisms do not admit a Polish group topology, and are moreover non-Borel subsets of the homeomorphism group of the Cantor space. We then study the closures of full groups by means of Fraisse theory. Finally, we give a characterization of the sets of invariant measures of minimal homeomorphisms of the Cantor space.

Omitting types in logic of metric structures

This paper is about omitting types in logic of metric structures introduced by Ben Yaacov, Berenstein, Henson and Usvyatsov. While a complete type is omissible in some model of a countable complete theory if and only if it is not principal, this is not true for the incomplete types by a result of Ben Yaacov. We prove that there is no simple test for determining whether a type is omissible in a model of a theory [Formula: see text] in a countable language. More precisely, we find a theory in a countable language such that the set of types omissible in some of its models is a complete [Formula: see text] set and a complete theory in a countable language such that the set of types omissible in some of its models is a complete [Formula: see text] set. Two more unexpected examples are given: (i) a complete theory [Formula: see text] and a countable set of types such that each of its finite sets is jointly omissible in a model of [Formula: see text], but the whole set is not and (ii) a complete theory and two types that are separately omissible, but not jointly omissible, in its models.

A completeness theorem for continuous predicate modal logic

We study a modal extension of the Continuous First-Order Logic of Ben Yaacov and Pedersen (J Symb Logic 75(1):168–190, 2010). We provide a set of axioms for such an extension. Deduction rules are just Modus Ponens and Necessitation. We prove that our system is sound with respect to a Kripke semantics and, building on Ben Yaacov and Pedersen (2010), that it satisfies a number of properties similar to those of first-order predicate logic. Then, by means of a canonical model construction, we get that every consistent set of formulas is satisfiable. From the latter result we derive an Approximated Strong Completeness Theorem, in the vein of Continuous Logic, and a Compactness Theorem.

A direct solution to the Generic Point Problem

We provide a new proof of a recent theorem of Ben Yaacov, Melleray, and Tsankov. If G G is a Polish group and X X is a minimal, metrizable G G -flow with all orbits meager, then the universal minimal flow M ( G ) M(G) is nonmetrizable. In particular, we show that given X X as above, the universal highly proximal extension of X X is nonmetrizable.

I. Ben Yaacov, J. Melleray, and T. Tsankov, Metrizable universal minimal flows of Polish groups have a comeagre orbit.Geometric and Functional Analysis, vol. 27 (2017), no. 1, pp. 67–77. - J. Melleray, L. Nguyen Van Thé, and T. Tsankov, Polish groups with metrizable universal minimal flows. International Mathematics Research Notices, vol. 2016, no. 5, pp. 1285–1307.

I. Ben Yaacov, J. Melleray, and T. Tsankov, <i>Metrizable universal minimal flows of Polish groups have a comeagre orbit</i>.Geometric and Functional Analysis, vol. 27 (2017), no. 1, pp. 67–77. - J. Melleray, L. Nguyen Van Thé, and T. Tsankov, <i>Polish groups with metrizable universal minimal flows</i>. International Mathematics Research Notices, vol. 2016, no. 5, pp. 1285–1307.

RETRACTED ARTICLE: A completeness theorem for continuous predicate modal logic

We study a modal extension of the Continuous First-Order Logic of Ben Yaacov and Pedersen (J Symb Logic 75(1):168–190, 2010). We provide a set of axioms for such an extension. Deduction rules are just Modus Ponens and Necessitation. We prove that our system is sound with respect to a Kripke semantics and, building on Ben Yaacov and Pedersen (2010), that it satisfies a number of properties similar to those of first-order predicate logic. Then, by means of a canonical model construction, we get that every consistent set of formulas is satisfiable. From the latter result we derive an Approximated Strong Completeness Theorem, in the vein of Continuous Logic, and a Compactness Theorem.

AUTOMATIC CONTINUITY FOR ISOMETRY GROUPS

We present a general framework for automatic continuity results for groups of isometries of metric spaces. In particular, we prove automatic continuity property for the groups of isometries of the Urysohn space and the Urysohn sphere, i.e. that any homomorphism from either of these groups into a separable group is continuous. This answers a question of Ben Yaacov, Berenstein and Melleray. As a consequence, we get that the group of isometries of the Urysohn space has unique Polish group topology and the group of isometries of the Urysohn sphere has unique separable group topology. Moreover, as an application of our framework we obtain new proofs of the automatic continuity property for the group $\text{Aut}([0,1],\unicode[STIX]{x1D706})$, due to Ben Yaacov, Berenstein and Melleray and for the unitary group of the infinite-dimensional separable Hilbert space, due to Tsankov.

The dynamical hierarchy for Roelcke precompact Polish groups

We study several distinguished function algebras on a Polish group G, under the assumption that G is Roelcke precompact. We do this by means of the model-theoretic translation initiated by Ben Yaacov and Tsankov: we investigate the dynamics of No-categorical metric structures under the action of their automorphism group. We show that, in this context, every strongly uniformly continuous function (in particular, every Asplund function) is weakly almost periodic. We also point out the correspondence between tame functions and NIP formulas, deducing that the isometry group of the Urysohn sphere is Tame ∩ UC-trivial.

AXIOMS FOR THE THEORY OF RANDOM VARIABLE STRUCTURES: AN ELEMENTARY APPROACH

The theory of random variable structures was first studied by Ben Yaacov in [2]. Ben Yaacov's axiomatization of the theory of random variable structures used an early result on the completeness theorem for Lukasiewicz's [0, 1]-valued propositional logic. In this paper, we give an elementary approach to axiomatizing the theory of random variable structures. Only well-known results from probability theory are required here.

Omitting types for infinitary [formula omitted]-valued logic

We describe an infinitary logic for metric structures which is analogous to Lω1,ω. We show that this logic is capable of expressing several concepts from analysis that cannot be expressed in finitary continuous logic. Using topological methods, we prove an omitting types theorem for countable fragments of our infinitary logic. We use omitting types to prove a two-cardinal theorem, which yields a strengthening of a result of Ben Yaacov and Iovino concerning separable quotients of Banach spaces.