The Online Logic Seminar meets weekly on Thursdays at 1pm US Central Time (currently UTC-5) on Zoom. You can connect to the live seminar by clicking here, or by joining the Zoom meeting with Meeting ID 122 323 340.
In cases where I have slides, or a link to them, I have a link from the title of the talk.
The earliest examples of linear orders not Medvedev above some of their endpointed intervals (due to the speaker) were well-orders - roughly speaking, any two distinct "sufficiently $\omega_1$-like" well-orders are Medvedev-incomparable, and we can always view the smaller as an endpointed initial segment of the larger. Annoyingly, these examples are quite large and the nontrivial relationships between even relatively concrete well-orders (e.g. the $\omega_n^{CK}$s for finite $n$) are still open; later, Julia Knight and Alexandra Soskova gave a much simpler example of a linear order having an endpointed interval which it does not uniformly compute and is low in the arithmetical hierarchy.
I will present two additional examples, intended to flesh out this picture:
If time remains, I will present some separate work motivated by trying to find simpler Medvedev-incomparable well-orders.
The pursuit of improvements to the Pila--Wilkie Theorem remains an active area of research, in several different directions. These are motivated by potential diophantine applications, but are centred on the inherent nature of the definable sets involved. Time permitting, we will discuss various aspects of this pursuit, related to geometric smooth parameterization, effectivity, and the importance of certain key systems of differential equations.
I will then describe how we can leverage SAT solving to accomplish other automated-reasoning tasks. Sampling uniformly at random satisfying truth assignments of a given Boolean formula or counting the number of such assignments are both fundamental computational problems in computer science with applications in software testing, software synthesis, machine learning, personalized learning, and more. While the theory of these problems has been thoroughly investigated since the 1980s, approximation algorithms developed by theoreticians do not scale up to industrial-sized instances. Algorithms used by the industry offer better scalability, but give up certain correctness guarantees to achieve scalability. We describe a novel approach, based on universal hashing and Satisfiability Modulo Theory, that scales to formulas with hundreds of thousands of variables without giving up correctness guarantees.
In this talk we will study the class of pseudo real closed fields (PRC-fields) from a model theoretical point of view and we will explain some of the main results obtained. We know that the complete theory of a bounded PRC field (i.e., with finitely many algebraic extensions of degree m, for each m > 1) is NTP_2 and we have a good description of forking.
Also, in a joint work with Alf Onshuus and Pierre Simon we describe the definable groups in the case that they have f-generics types.
In the end of the talk we will explain some results obtained with Silvain Rideau. Where we generalize the notion of PRC fields to a more general class of fields. In particular, this class includes fields that have orders and valuations at the same time.
Their algebraic semantics are based on residuated lattices. The class of these ordered algebraic structures is quite big and hard to study, but it contains some proper subclasses that are well-known such as Boolean algebras, Heyting algebras, MV-algebras. In this talk we will see different constructions of new residuated lattices based on better-known algebras.
In previous work, I proposed a model-theoretic framework of "semantic constraints," where there is no strict distinction between logical and nonlogical vocabulary. The form of sentences in a formal language is determined rather by a set of constraints on models. In the talk, I will show how this framework can also be used in the process of formalization, where the semantic constraints are conceived of as commitments made with respect to the language. I will extend the framework to include "formalization constraints" on functions taking arguments from a source language to a target language, and I will consider various meta-constraints on both the process of formalization and its end result.
In spite of the great diversity of those features, there are some eerie similarities between them. These were observed and made more precise in the case of graph homomorphisms by Brightwell and Winkler, who showed that dismantlability of the target graph, connectedness of the set of homomorphisms, and good mixing properties of the corresponding spin system are all equivalent. In this talk we go a step further and demonstrate similar connections for arbitrary CSPs. This requires a much deeper understanding of dismantling and the structure of the solution space in the case of relational structures, and also new refined concepts of mixing. In addition, we develop properties related to the study of valid extensions of a given partially defined homomorphism, an approach that turns out to be novel even in the graph case. We also add to the mix the combinatorial property of finite duality and its logic counterpart, FO-definability, studied by Larose, Loten, and Tardif. This is joint work with Andrei Bulatov, Víctor Dalmau, and Benoît Larose.
I will explain how to extend these results to the broader class of henselian valued fields of equicharacteristic zero, residue field algebraically closed and poly- regular value group. This includes many interesting mathematical structures such as the Laurent Series over the Complex numbers, but more importantly extends the results to valued fields with finitely many definable convex subgroups.
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