Formal argumentation gives a toolkit to analyze the problem of reasoning with conflicting information. I’ll introduce the basic definitions and ideas of formal argumentation theory. In particular, I’ll discuss how various "semantics" describe different reasoning types, and how, depending on our setting, we may desire our semantics to have certain properties. Among these desirable properties are existence (that our reasoning will come to some conclusion), directionality (any set S of arguments which are not contradicted from outside S can be reasoned on independently from the arguments outside S), and weak reinstatement (slightly more than saying that we should accept as true any argument that is not contradicted at all).
SCC-recursiveness, and in particular the semantics cf2 and stg2 were developed to satisfy these criteria among others. In the setting where we reason over finitely many arguments, this yields a desirable collection of properties for the semantics, suggesting that cf2 and stg2 represent forms of reasoning with conflicting information which are suitable to address many natural real-world (automated) reasoning problems. In this talk, I’ll explore what works and what doesn’t work when applying these same ideas in the setting where we reason over infinitely many arguments. Though the topic is slightly foreign to computability-theorists, the toolkit will be familiar, including transfinite recursion, compactness, and ill-founded trees.
This work is joint with Luca San Mauro.

Computable presentations of compact spaces can behave in surprising ways: while every computable presentation of a Stone space is homeomorphic to a computably compact one, the homeomorphism itself need not be computable. Because the inverse of a computable homeomorphism is not in general computable, attempts to classify presentations of computably compact Polish spaces up to computable homeomorphism naturally yield a reducibility: Given two presentations M and N of the same space up to homeomorphism then we define M is computably-topologically reducible to N – $M \leq_{ct} N$ – if there is a computable homeomorphism $f:N \to M$. In this talk, we'll investigate the ct-degrees of various Stone spaces. No background in computable topology will be assumed. This is joint work with Noam Greenberg, Sasha Melnikov, and Dan Turetsky.

True stages are a method for organizing complex constructions in computability theory. Over several lectures, I'll explain the method of true stages by working through some examples in computable structure theory. We'll start with some necessary computability background. Time permitting, I may discuss some of the applications of true stages to descriptive set theory.

True stages are a method for organizing complex constructions in computability theory. Over several lectures, I'll explain the method of true stages by working through some examples in computable structure theory. We'll start with some necessary computability background. Time permitting, I may discuss some of the applications of true stages to descriptive set theory.

Cardinal characteristics (CCs) are uncountable cardinals defined as the least size of a complete solution set for some topological or combinatorial problem on Baire space. Recent research shows that, if ZFC proves A ≤ B (where A and B are CCs) then the problem associated to A is computationally easier than that associated to B. This latter fact, in turn, corresponds to saying that NL(A) is included in NL(B), where NL(X) is the non-lowness class of X, i.e. the class of reals which compute "hard" instances of the problem associated to X. The converse is false, and indeed there are CCs A and B which are consistently different but have NL(A)=NL(B). Since the definition of non-lowness class can be easily relativized to any countable Turing ideal I, it is natural to ask which ideals give rise to the "correct" diagram of inclusions between NL classes. In my talk I will briefly explain the computational view of cardinal characteristics, and I will introduce two forcing notions which work well over ideals closed under hyperarithmetical reducibility, and can be used to separate three non-lowness classes relative to these ideals.

The descriptive and computational complexity of equivalence relations has been a topic of intense study for decades.

Recently, primitive recursive equivalence relations on ω under primitive recursive reduction have received attention. They form a countable distributive lattice with greatest but no least element.

I will present joint work with Vittorio Cipriani and Luca San Mauro exploring this structure. In particular, I will focus on how the analysis of its finite substructures leads to the decidability of its ∀∃-theory in the language of partial ordering.

The sets that encode themselves refer to regressive, retraceable, introenumerable, majorenumerable, introreducible, and majorreducible sets. Dekker showed that every regressive set has a retraceable subset, and Greenberg, Harrison-Trainor, Patey, and Turetsky showed that every uniformly introenumerable set has a uniformly introreducible subset. These facts are related to the well-known fact that every infinite c.e. set has an infinite computable subset. We focus on those sets’ properties and which sets always have a subset of the other. A new method of building an introenumerable set is introduced. This talk is about my first project under advising of Joseph Miller.

Assuming the axiom of choice, cardinal arithmetic is extremely well-behaved: any two sets are comparable in size, and there is no infinite strictly decreasing sequence of cardinals. Moreover, for any nonempty sets X and Y, X injects into Y if and only if Y surjects onto X—so the injective and surjective orderings coincide. Without choice, much of this structure breaks down: there may exist incomparable sets and infinite strictly decreasing sequences of cardinals. Although the Cantor-Schröder-Bernstein theorem ensures that if two sets inject into each other then they are in bijective correspondence, no analogous result need hold for surjections, so the injective and surjective orderings may also no longer agree. In this talk, we examine the surjective ordering on sets in the absence of choice, focusing on results that highlight just how bad the situation can be. We also discuss some results surrounding the surjective well-foundedness of cardinals. We draw on recent works of Shen and Zhou and on joint work of the speaker with Andreas Blass.

Scott analysis is a branch of computable structure theory and descriptive set theory that aims to calibrate the complexity of the isomorphism relation. Two of the most fundamental notions studied in Scott analysis are Scott rank and the ordinal-indexed back-and-forth relations. The Scott rank of a countable structure tells you how difficult it is to understand the isomorphism type of that structure. Meanwhile, the ordinal indexed back-and-forth relations measure how difficult it is to tell different structures apart from each other. We introduce the concept of a Scott filtration, which connects these two fundamental concepts in Scott analysis. Given a structure M, a Scott filtration uses simpler structures (i.e., structures of smaller Scott rank) to concretely understand the back-and-forth classes of M. We will discuss the existence of Scott filtrations up to the Vaught ordinal, along with some other related theorems. This is joint work with Dino Rossegger and Dan Turetsky.

The complexity of back-and-forth relations Two structures are n-back-and-forth equivalent if and only if they satisfy the same Pi_n sentences of the infinitary logic. These relations play a key role in computable structure theory. However, the relation of n-back-and-forth equivalence cannot be characterized in a Pi_n way, but rather in a Pi_2n way. I will talk about several contexts in which this shows up including as a key ingredient of the construction of the non-arithmetic degrees as a degree spectrum.

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