Definable and invariant types in enrichments of NIP theories

Let T be an NIP L-theory and T' be an enrichment. We give a sufficient condition on T' for the underlying L-type of any definable (respectively invariant) type over a model of T' to be definable (respectively invariant) as an L-type. Besides, we generalise work of Simon and Starchenko on the density of definable types among non forking types to this relative setting. These results are then applied to Scanlon's model completion of valued differential fields.Comment: 9 pages. An error was pointed out in section 2 of the previous version so that section was removed. So was Proposition 3.8 that depended on i

Imaginaries in separably closed valued fields

We show that separably closed valued fields of finite imperfection degree (either with lambda-functions or commuting Hasse derivations) eliminate imaginaries in the geometric language. We then use this classification of interpretable sets to study stably dominated types in those structures. We show that separably closed valued fields of finite imperfection degree are metastable and that the space of stably dominated types is strict pro-definable

Beautiful pairs

We introduce an abstract framework to study certain classes of stably embedded pairs of models of a complete $\mathcal{L}$-theory $T$, called beautiful pairs, which comprises Poizat's belles paires of stable structures and van den Dries-Lewenberg's tame pairs of o-minimal structures. Using an amalgamation construction, we relate several properties of beautiful pairs with classical Fra\"{i}ss\'{e} properties. After characterizing beautiful pairs of various theories of ordered abelian groups and valued fields, including the theories of algebraically, $p$-adically and real closed valued fields, we show an Ax-Kochen-Ershov type result for beautiful pairs of henselian valued fields. As an application, we derive strict pro-definability of particular classes of definable types. When $T$ is one of the theories of valued fields mentioned above, the corresponding classes of types are related to classical geometric spaces such as Berkovich and Huber's analytifications. In particular, we recover a result of Hrushovski-Loeser on the strict pro-definability of stably dominated types in algebraically closed valued fields.Comment: 40 page

Invariant types in model theory

We study how the product of global invariant types interacts with the preorder of domination, i.e. semi-isolation by a small type, and the induced equivalence relation, domination-equivalence. We provide sufficient conditions for the latter to be a congruence with respect to the product, and show that this holds in various classes of theories. In this case, we develop a general theory of the quotient semigroup, the domination monoid, and carry out its computation in several cases of interest. Notably, we reduce its study in o-minimal theories to proving generation by 1-types, and completely characterise it in the case of Real Closed Fields. We also provide a full characterisation for the theory of dense meet-trees, and moreover show that the domination monoid is well-defined in certain expansions of it by binary relations. We give an example of a theory where the domination monoid is not commutative, and of one where it is not well-defined, correcting some overly general claims in the literature. We show that definability, finite satisfiability, generic stability, and weak orthogonality to a fixed type are all preserved downwards by domination, hence are domination-equivalence invariants. We study the dependence on the choice of monster model of the quotient of the space of global invariant types by domination-equivalence, and show that if the latter does not depend on the former then the theory under examination is NIP

PSA 2018

These preprints were automatically compiled into a PDF from the collection of papers deposited in PhilSci-Archive in conjunction with the PSA 2018

Foundations of Software Science and Computation Structures

This open access book constitutes the proceedings of the 24th International Conference on Foundations of Software Science and Computational Structures, FOSSACS 2021, which was held during March 27 until April 1, 2021, as part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2021. The conference was planned to take place in Luxembourg and changed to an online format due to the COVID-19 pandemic. The 28 regular papers presented in this volume were carefully reviewed and selected from 88 submissions. They deal with research on theories and methods to support the analysis, integration, synthesis, transformation, and verification of programs and software systems