On Threshold Problems for Orbits of Semigroup Actions by Eike

Today Eike Neumann will give a talk on Threshold Problems for Orbits of Semigroup Actions as a part of our theory seminar series.

Abstract:

Consider the following computational problem: Given a real function g on a space X, a compactly generated semi-group S acting on X, and a point x in X, is g positive on every point of the orbit of x under S?

This generalises a large number of widely studied problems, such as safety and liveness verification for discrete-time dynamical systems (corresponding to semi-groups with a single generator), threshold problems for stochastic (corresponding stochastic matrices acting on probability distributions) or quantum automata (corresponding to unitary operators acting on Hilbert spaces) and more.

When the objects above are presented via rational or algebraic data, the associated problems quickly become undecidable or very sensitive to the problem formulation. For example, threshold problems for stochastic automata are undecidable in general, and threshold problems for quantum automata are decidable if and only if they are formulated using strict inequality.

I will consider the above problem in its general form from the computable analysis perspective, replacing decidability with maximal partial decidability. I will give a sound algorithm that partially decides the problem over effectively locally compact spaces. I will show that the algorithm is complete when the space is zero-dimensional or locally contractible, and give some examples of spaces where the algorithm is not complete but the problem is maximally partially decidable and spaces where the problem is not maximally partially decidable at all.

Giorgio Genovesi on Characterizing Regular Countable Second Countable Spaces in Second Order Arithmetic

Today's seminar talk is by Giorgio Genovesi from Leeds, who will be talking about countable second countable topological spaces in the context of reverse mathematics. 

Title: Characterizing Regular Countable Second Countable Spaces in Second Order Arithmetic

Abstract: Regular countable second countable (CSC) spaces admit rather nice characterizations and can easily be formalized in second order arithmetic. It is natural to ask what set existence axioms are needed to ensure regular CSC spaces remain nice. It turns out many theorems which characterize regular CSC are equivalent to one of the big five subsystems of second order arithmetic.

Hideki Tsuiki visiting Swansea

Professor Hideki Tsuiki from Kyoto University is visiting Swansea University from 22nd to 24th of January. Yesterday, he gave a talk on Coinductive View of Shadows of 3D Fractals. 

The talk did not only give insight into a fascinating area of research, but was also entertaining. You may want to check out some of Prof. Tsuiki's videos, e.g. https://www.youtube.com/watch?v=VsFD37f-2ck  

Elvira Mayordomo visiting Swansea

Elvira Mayordomo is visiting us this week. Today she gave a talk as a part of our seminar series.

Title: On information theory in geometric measure theory

Abstract

Effective and resource-bounded dimensions were defined by Lutz in 2003 and have proven to be useful and meaningful for quantitative analysis in the contexts of algorithmic randomness, computational complexity and fractal geometry.

The point-to-set principle (PSP) of J. Lutz and N. Lutz (2018) fully characterizes Hausdorff and packing dimensions in terms of effective dimensions in the Euclidean space, enabling effective dimensions to be used to answer open questions about fractal geometry, with already an interesting list of geometric measure theory results.

In this talk I will review the point-to-set principles focusing on recent applications and extensions and presenting open questions as well as further application opportunities.

Theory Away Day

Our Theory Research Group kicked off the year with an inspiring away day, providing the perfect opportunity to reconnect as a team after the festive break. It was a chance to reflect on past successes, align our goals, and spark fresh ideas to drive our work forward in 2025. Here's to an exciting year of collaboration and innovation!

 

Oliver Kullmann's talk

Today Oliver Kullmann will give a talk on Automated search for special Latin squares as a part of our Seminar series.

Abstract: 

Latin squares have been studied since the days of Euler. After some overview on the history and background, an effort for a complete enumeration of special types of Latin squares of order 13, by as completely automated means as possible (which is currently actually not possible), will be presented and evaluated. The main method here is Cube-and-Conquer, a kind of 2-stage SAT-solving (as invented by the presenter). Quite some fine-tuning of representation and choice of solver was needed, and will be discussed (at some high level)

Matteo Acclavio visiting Swansea

Next week's theory seminar will be given by Matteo Acclavio from the University of Sussex, who is visiting us for a few days. The topic will be a new logical framework for concurrent programs, abstract below.

Title: A new logical framework for concurrent programs

Abstract:
Designing logical frameworks to reason about the properties of concurrent programs while accurately capturing the essence of concurrency is a challenging task.
The main difficulties can be traced back to the syntactic constraints of the languages used for this purpose.

In this talk, I will present my ongoing line of research, which aims to provide a new computation-as-deduction paradigm for the study of concurrent programs.
In particular, I will show you a non-commutative logic where we can interpret proofs as computation trees for the pi-calculus, and use proof nets to provide canonical representations of these trees modulo interleaving concurrency.

This work is based on joint works with Giulia Manara and Fabrizio Montesi