Automorphism groups of randomized structures

We study automorphism groups of randomizations of separable structures, with focus on the $\aleph_0$-categorical case. We give a description of the automorphism group of the Borel randomization in terms of the group of the original structure. In the $\aleph_0$-categorical context, this provides a new source of Roelcke precompact Polish groups, and we describe the associated Roelcke compactifications. This allows us also to recover and generalize preservation results of stable and NIP formulas previously established in the literature, via a Banach-theoretic translation. Finally, we study and classify the separable models of the theory of beautiful pairs of randomizations, showing in particular that this theory is never $\aleph_0$-categorical (except in basic cases).Comment: 28 page

On theories of random variables

We study theories of spaces of random variables: first, we consider random variables with values in the interval $[0,1]$, then with values in an arbitrary metric structure, generalising Keisler's randomisation of classical structures. We prove preservation and non-preservation results for model theoretic properties under this construction: i) The randomisation of a stable structure is stable. ii) The randomisation of a simple unstable structure is not simple. We also prove that in the randomised structure, every type is a Lascar type

Generically Stable Measures and Distal Regularity in Continuous Logic

We develop a theory of generically stable and smooth Keisler measures in NIP metric theories, generalizing the case of classical logic. Using smooth extensions, we verify that fundamental properties of (Borel)-definable measures and the Morley product hold in the NIP metric setting. With these results, we prove that as in discrete logic, generic stability can be defined equivalently through definability properties, statistical properties, or behavior under the Morley product. We also examine weakly orthogonal Keisler measures, characterizing weak orthogonality in terms of various analytic regularity properties. We then examine Keisler measures in distal metric theories, proving that as in discrete logic, distality is characterized by all generically stable measures being smooth, or by all pairs of generically stable measures being weakly orthogonal. We then use this, together with our results on weak orthogonality and a cutting lemma, to find analytic versions of distal regularity and the strong Erd\H{o}s-Hajnal property

Approximate isomorphism of randomization pairs

We study approximate $\aleph_0$-categoricity of theories of beautiful pairs of randomizations, in the sense of continuous logic. This leads us to disprove a conjecture of Ben Yaacov, Berenstein and Henson, by exhibiting $\aleph_0$-categorical, $\aleph_0$-stable metric theories $Q$ for which the corresponding theory $Q_P$ of beautiful pairs is not approximately $\aleph_0$-categorical, i.e., has separable models that are not isomorphic even up to small perturbations of the smaller model of the pair. The theory $Q$ of randomized infinite vector spaces over a finite field is such an example. On the positive side, we show that the theory of beautiful pairs of randomized infinite sets is approximately $\aleph_0$-categorical. We also prove that a related stronger property, which holds in that case, is stable under various natural constructions, and formulate our guesswork for the general case.Comment: 15 page

Definable groups, NIP theories, and the Ellis Group Conjecture

DEFINABLE GROUPS, NIP THEORIES, AND THE ELLIS GROUP CONJECTURE A \emph{definable group} $G$ is a group which is definable in a first-order structure. Despite the name, it is not a single group, but a family of groups given by interpreting the defining formulas in \emph{elementary extensions} of the structure defining the group. For instance, algebraic groups are definable in the complex field using first-order formulas. These include matrix groups and abelian varieties such as elliptic curves. Among groups which are definable with first-order formulas in the real field there are $\operatorname{GL}(n, \mathbb R)$, $\operatorname{SO}(n, \mathbb R)$, and other Lie groups. The two families of examples above are, in a sense, orthogonal. The field $\mathbb C$ falls into the class of \emph{stable} structures, which are, in a nutshell, the ones that do not define an order relation on an infinite set. Stable theories have been a central and fruitful topic in the model theory of the past decades (e.g. [1,11]), and there is a huge literature on stable groups (for instance [3,13]). Unfortunately, since stability is destroyed by the presence of a total infinite order, the field structure on $\mathbb R$ lives outside this realm, and more generally \emph{o-minimal} structures, another important class in which it is possible to provide a framework for \emph{tame geometry} (see [14]), are not stable. Model theorists have therefore tried to generalize methods from stability theory to broader contexts. One robust, simultaneous generalization of both stability and o-minimality is found in the class of \emph{dependent}, or \textsc{nip} theories. \textsc{Nip} structures can be roughly described as the ones that do not code a membership relation on an infinite set; this viewpoint is intimately connected to \textsc{vc}-dimension, a fundamental tool of statistical learning theory. This thesis explores a problem, which we are now going to outline, concerning the relation between two groups that can be attached to any group definable in a \textsc{nip} structure. In \textsc{nip} theories, to every definable group is associated a concrete compact Hausdorff topological group called $G/G^{00}$. As an example, it can be proven that if $G$ is a definably compact group definable over $\emptyset$ in a real closed field, for instance $\operatorname{SO}(3, M)$ for $M\succ \mathbb R$ an hyperreal field, then $G/G^{00}$ is exactly $G(\mathbb R)$, and the projection to $G/G^{00}$ behaves like a ``standard part'' map. If $G$ is not compact then this may not be true, as in the case of $\operatorname{SL}(n, M)$ where $G/G^{00}$ is trivial. In general (see [2]), for a group which is definable in an o-minimal structure, $G/G^{00}$ is a real Lie group. As a stable example, if $G$ is the additive group in the structure of the integers with sum (but without product), then $G/G^{00}$ is isomorphic to $\hat{\mathbb Z}=\varprojlim \mathbb Z/n\mathbb Z$. All these isomorphisms preserve the topology, i.e. are isomorphisms of topological groups. This canonical quotient is the first protagonist of the problem studied in the thesis. In order to introduce the second one, some preliminary explanations are needed. An important concept in the study of stable groups is the one of a \emph{generic type}. Trying to find a well-behaved analogue in the unstable context, Newelski noticed that a certain notion, namely that of a \emph{weak generic type}, is well understood when bringing topological dynamics into the picture\footnot{Briefly, in the dynamical context ``generic'' becomes ``syndetic'', and ``weak generic'' corresponds to ``piecewise syndetic''.}. In topological dynamics one is often interested in $G$-flows, actions of a group $G$ on compact Hausdorff spaces by homeomorphisms; soon one turns the attention to the ones that have a dense orbit ($G$-ambits) and to the ones in which all orbits are dense (minimal flows). A very special $G$-flow is the universal $G$-ambit $\beta G$ of ultrafilters on $G$: every $G$-ambit can be seen as a quotient of $\beta G$, and its minimal subflows enjoy a similar universal property. A ``tame'' counterpart of $\beta G$ is the space $S_G(M)$ of types over a model $M$ concentrating on $G$, i.e. the ultrafilters on definable subsets of $G(M)$, and one could develop a theory of \emph{tame topological dynamics} ([7,12]) and hope for $S_G(M)$ to be universal with respect to \emph{definable} $G(M)$-flows. Now, one important tool in the study of a $G$-flow $X$ is its \emph{enveloping semigroup} $E(X)$; it turns out that $\beta G\cong E(\beta G)$ and this equips the former with a semigroup structure. Once some technical obstacles are overcome, this construction can be carried out for $S_G(M)$ too, or at least for a certain bigger type space called $S_G^{\textnormal{ext}}(M)$. Applying the theory of enveloping semigroups to $E(\beta G)\cong\beta G$ produces a certain family of sub-semigroups that are indeed groups, and furthermore all in the same isomorphism class: this is the \emph{ideal group}, or \emph{Ellis group} associated to the flow. Modulo the complications mentioned above, an Ellis group can also be associated to $S_G(M)$. Even if this may depend on $M$, a comparison with $G/G^{00}$ can be made, and indeed the latter is always a quotient of the former, the projection $\pi$ being the restriction of a certain natural map $S_G(M)\to G/G^{00}$. Since in stable groups a similar situation arises replacing the Ellis group with the subspace of generic types of $S_G(M)$, and in that case the relevant map is injective, the next question is: is this $\pi$ an isomorphism? Even in tame context, this need not be the case: it was shown in [8] that the Ellis group of $\operatorname{SL}(2,\mathbb R)$ is the group with two elements, but its $G/G^{00}$ is trivial. A property that is \emph{not} satisfied by $\operatorname{SL}(2, \mathbb R)$ is amenability: there is no finitely additive, left-translation-invariant probability measure defined on $\mathscr P(\operatorname{SL}(2, \mathbb R))$. Another group lacking amenability is $\operatorname{SO}(3, \mathbb R)$; this is essentially the Banach-Tarski paradox. The reasons behind the non-amenability of these two groups are, however, different. If one searches for a left-translation-invariant (finitely additive) measure defined not on the whole power-set, but only on the Boolean algebra of \emph{definable} subsets of $\operatorname{SO}(3, \mathbb R)$, then such a measure \emph{does} exist, and we say that $\operatorname{SO}(3, \mathbb R)$ is \emph{definably amenable}. A similar thing happens with free groups on at least two generators. This is due to the fact that the non-measurable sets arising from the Banach-Tarski paradox are very complicated, and certainly not definable in the first-order structure of $\mathbb R$, and so this kind of obstructions to amenability disappear when we only want a measure on an algebra of ``simple'' sets. On the contrary, $\operatorname{SL}(2,\mathbb R)$ is not even definably amenable, thus being more inherently pathological under this point of view. In [4] Pillay then proposed the \emph{Ellis Group Conjecture}. Several special cases were proven in the same paper and, thereafter, the conjecture was proven true in [5] by Chernikov and Simon, hence we state it as a Theorem. \begin{theorem}[5, Theorem 5.6] If $G$ is a definably amenable \textsc{nip} group, the restriction of the natural map $S_G^{\textnormal{ext}}(M)\to G/G^{00}$ to any ideal group of $G$ is an isomorphism. \end{theorem} Remarkably, the model-theoretic techniques involved in stating, approaching, and proving the conjecture are anything but peculiar to this particular problem, and the main focus of this thesis is on the development and understanding of said techniques. This is reflected in the fact that we will deal with Ellis semigroups only in the first chapter and in the closing section. We start in Chapter 1 by studying enveloping semigroups, first in the classical context ([6]) and then in the definable one, without any kind of tameness assumption ([9, 10]). In Chapter 2 we introduce some techniques, still without assuming anything on the underlying theory beyond being first-order complete. In Chapter 3 we introduce dependent theories, see how the previously introduced tools behave in this context, and explore some constructions that heavily exploit the \textsc{nip} hypothesis. In Chapter 4 we bring in the last ingredient, i.e. definable amenability, see that under our hypotheses it is preserved when passing to Shelah's expansion, characterize it in terms of f-generic types, and conclude by studying the proof of the Ellis Group Conjecture. [1] J. T. Baldwin. Fundamentals of Stability Theory, volume 12. Springer-Verlag, 1988. [2] A. Berarducci, M. Otero, Y. Peterzil, and A. Pillay. A descending chain condition for groups definable in o-minimal structures. Annals of Pure and Applied Logic, 134:303–313, 2005. [3] A. Borovik and A. Nesin. Groups of Finite Morley Rank, volume 26 of Oxford Logic Guides. Oxford University Press, 1994. [4] A. Chernikov, A. Pillay, and P. Simon. External definability and groups in NIP theories. Journal of the London Mathematical Society, 2014. [5] A. Chernikov and P. Simon. Definably amenable NIP groups. http://arxiv.org/abs/1502.04365, submitted. [6] R. Ellis. Lectures on Topological Dynamics. Mathematics Lecture Note Series. W.A. Benjamin, 1969. [7] J. Gismatullin, D. Penazzi, and A. Pillay. On compactifications and the topological dynamics of definable groups. Annals of Pure and Applied Logic, 165:552–562, 2014. [8] J. Gismatullin, D. Penazzi, and A. Pillay. Some model theory of SL(2, R). Fundamenta Mathematicae, 229:117–128, 2015. [9] L. Newelski. Topological dynamics of definable group actions. Journal of Symbolic Logic, 74:50–72, 2009. [10] L. Newelski. Model theoretic aspects of the ellis semigroup. Israel Journal of Mathematics, 190:477–507, 2012. [11] A. Pillay. Geometric Stability Theory, volume 32 of Oxford Logic Guides. Oxford University Press, 1996. [12] A. Pillay. Topological dynamics and definable groups. The Journal of Symbolic Logic, 78:657–666, 2013. [13] B. Poizat. Stable Groups, volume 87 of Mathematical Surveys and Monographs. American Mathematical Society, 2001. translated from the 1987 original. [14] L. van den Dries. Tame Topology and O-minimal Structures, volume 248. Cambridge University Press, 1998

GNU epsilon - an extensible programming language

Reductionism is a viable strategy for designing and implementing practical programming languages, leading to solutions which are easier to extend, experiment with and formally analyze. We formally specify and implement an extensible programming language, based on a minimalistic first-order imperative core language plus strong abstraction mechanisms, reflection and self-modification features. The language can be extended to very high levels: by using Lisp-style macros and code-to-code transforms which automatically rewrite high-level expressions into core forms, we define closures and first-class continuations on top of the core. Non-self-modifying programs can be analyzed and formally reasoned upon, thanks to the language simple semantics. We formally develop a static analysis and prove a soundness property with respect to the dynamic semantics. We develop a parallel garbage collector suitable to multi-core machines to permit efficient execution of parallel programs.Comment: 172 pages, PhD thesi