Date/Time : 02 Mar 2022 16:00
Venue : via Zoom
Speaker : Lavinia Picollo, National University of Singapore

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Abstract: Truth predicates are widely believed to be capable of serving a certain logical or quasi-logical function. There is little consensus, however, on the exact nature of this function. We offer a series of formal results in support of the thesis that disquotational truth is a device to simulate higher-order resources in a first-order setting. More specifically, we show that any theory formulated in a higher-order language can be naturally and conservatively interpreted in a first-order theory with a disquotational truth or truth-of predicate. In the first part of the talk we focus on the relation between truth and full impredicative sentential quantification. The second part is devoted to the relation between truth-of and full impredicative predicate quantification.

This is joint work with Thomas Schindler

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Date/Time : 16 Feb 2022 16:30
Venue : via Zoom
Speaker : Rupert Hoelzl, Universitaet der Bundeswehr, Muenchen

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Abstract: A Martin-Loef test U is universal if it captures all sequences which are not Martin-Loef random. It is optimal if for every Martin-Loef test V there is a constant c such that, for all n, V_{n+c} subseteq U_n. We study the computational differences between universal and optimal Martin-Loef tests as well as the effects of these differences on both the notions of layerwise computability and the Weihrauch degree of LAY, the function that produces a bound on the randomness deficiency of a given Martin-Loef random sequence. We prove several robustness and idempotence results concerning the Weihrauch degree of LAY and we show that layerwise computability is more restrictive than Weihrauch reducibility to LAY. Along similar lines, we also study the principle RD, a variant of LAY outputting the precise randomness deficiency of sequences instead of only an upper bound as LAY.

This is joint work with Paul Shafer. The paper is available as in the Annals of Pure and Applied Logic, https://doi.org/10.1016/j.apal.2015.05.006

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Date/Time : 09 Feb 2022 16:00
Venue : via Zoom
Speaker : Xie Ruofei,

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Meeting ID: 830 4925 8042
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Abstract: Consider the sum of the series (-1)^{x_n}a_n over n, where x_n is a binary sequence and a_n is a sequence of real numbers.
We say the sequence x_n has convergence property if the sum above converges for every computable sequence of real numbers whose sum \sum_n a_n^2 converges computably.

Downey, Greenberg, and Tanggara showed in their unpublished paper that every Schnorr random series x_n has convergence property.
In this talk, we will focus on convergence property and show its similarities and differences with randomness.

This is joint work with Noam Greenberg.

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Date/Time : 03 Feb 2022 16:00
Venue : via Zoom
Speaker : Andre Nies, The University of Auckland

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Abstract: The K-trivial sets are antirandom in the sense that the initial segment complexity in terms of prefix-free Kolmogorov complexity K grows as slowly as possible. Chaitin introduced this notion in about 1975, and showed that each K-trivial is Turing below the halting set. Shortly after, Solovay proved that a K-trivial set can be noncomputable.

In the past two decades, many alternative characterisations of this class have been found: properties such as being low for K, low for Martin-Loef (ML) randomness, and a basis for ML randomness, which state in one way or the other that the set is close to computable.

Initially, the class of noncomputable K-trivials appeared to be amorphous. More recently, an internal structure has been found. Most of these results can be phrased in the language of a reducibility on the K-trivials which is weaker than Turing’s:

A is ML-below B if each ML-random computing B also computes A.

Bienvenu, Greenberg, Kucera, Nies and Turetsky (JEMS 2016) showed that there an ML complete K-trivial set. Greenberg, Miller and Nies (JML, 2019) established a dense hierarchy of subclasses of the K-trivials based on fragments of Omega computing the set, and each such subclass is an initial segment for ML. More recent results (see arxiv.org/abs/1707.00258, updated and submitted Dec 2020) generalise these approaches using cost functions. They also show that each K-trivial set is ML-equivalent to a c.e. K-trivial.

The extreme lowness of K-trivials, far from being an obstacle, allows for methods which don’t work in a wider setting. The talk provides an overview and discusses open questions. For instance, is ML-completeness an arithmetical property of K-trivials?

This is joint work with Noam Greenberg, Joseph Miller and Dan Turetsky.

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Date/Time : 26 Jan 2022 16:00
Venue : via Zoom
Speaker : Zhang Jing, National University of Singapore

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Abstract:  Structures like trees, linear orders, partial orders, graphs have been widely studied in different areas of mathematics. The Ramsey-type theorems we will discuss usually take the following form: for any given coloring of the structure, there exists a “large sub-structure” that intersects “very few” color classes. One example is a theorem of Laver that states for any finite coloring of Q x Q (ordered pairs of rationals), there exists X, Y order isomorphic to Q, such that X x Y intersects at most 2 color classes. We will discuss the uncountable analogues of these statements and their consistency. In particular, a diagonal version of the Halpern-Lauchli theorem plays a key role. The differences between countable combinatorics and uncountable combinatorics will be highlighted.

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Date/Time : 19 Jan 2022 16:00
Venue : via Zoom
Speaker : Ye Jinhe, Institut de Mathematiques de Jussieu - Paris Rive Gauche

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Abstract: Consider the class of fields with Char(K)=0 and x^4+y^4=1 has only 4 solutions in K, we show that this class has a model companion, which we denote by curve-excluding fields. Curve-excluding fields provides (counter)examples to various questions. Model theoretically, they are model complete. Field theoretically, they are not large and unbounded. Time permitting, we will discuss other aspects such as decidability of such fields.

Joint work with Will Johnson and Erik Walsberg.

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Date/Time : 10 Nov 2021 16:00
Venue : via Zoom
Speaker : Manat Mustafa, Nazarbayev University

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Abstract: The talk employs the punctuality paradigm in the studies of numberings. We consider the punctual numberings, i.e. uniform computations for families of primitive recursive functions. The punctual reducibility between numberings is induced by primitive recursive functions. This approach gives rise to upper semilattices of degrees, which are called Rogers pr-semilattices. The main focus of the talk will be the structural properties of Rogers pr-semilattices. We will show several examples, which highlight further contrasts between the punctual framework and the classical theory of computable numberings.

All results are obtained in joint work with Nikolay Bazhenov and Sergei Ospichev.

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Date/Time : 03 Nov 2021 16:00
Venue : via Zoom
Speaker : Wu Guohua, Nanyang Technological University

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Abstract:   Domain theory was contrived by Dana Scott in 1969 as a theory of approximating computations. This topic was also raised by Ershov independently. In this talk, I will give a brief introduction of domain theory first, hopefully, from the beginning, and then move to the comparison of various substitutions of Hausdorff property, including sobriety, well-filteredness and monotone convergence.

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Date/Time : 27 Oct 2021 17:00
Venue : via Zoom
Speaker : Mars Yamaleev, Kazan Federal University

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Abstract: In our talk we consider relative enumerability of 2-c.e. Turing degrees in c.e. degrees below them. We focus on an old question which arises from the well-known work of Soare and Stob in 1982. The Soare-Stob theorem says that for any noncomputable low c.e. Turing degree a there exists a non-c.e. Turing degree d which is above a and relative enumerable in a. The question is whether the degree d can always be chosen as 2-c.e. We answer this question by showing that for some the degree d must be beyond 2-c.e., and discuss the ideas of proof.  Also we consider possible generalizations of this result.

All results are obtained in a joint work with Arslanov M.M. and Batyrshin I.I.

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