The Online Logic Seminar meets weekly on Thursdays at 1pm US Central Daylight Time 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.
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.
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.
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