In this talk, I will discuss a wide range of results, obtained with various co-authors, 

that either aim to classify individual continuous functions with certain

algorithmic properties or address various algorithmic problems for the space

$C[0,1]$, and, more generally, $C(K)$, where $K$ is (locally) compact.

Among other results, I will discuss a characterisation of all (Büchi) automatic functions

$f \colon [0,1] \to \mathbb{R}$, the recognition problem for $C([0,1]; \mathbb{R})$ among all Banach spaces, the algorithmic universality of $C([0,1]; \mathbb{R})$,

and the existence of a computably presented space $C(K; \mathbb{R})$, where the compact set $K \subseteq [0,1]$ is pathological (not algorithmically presented).

Useful links: 

https://homepages.ecs.vuw.ac.nz/~melnikal/C_zero_one.pdf

https://homepages.ecs.vuw.ac.nz/~melnikal/Effective_Gelfand-2.pdf

https://arxiv.org/pdf/2508.06187

https://homepages.ecs.vuw.ac.nz/~melnikal/main_1.pdf

I will provide a concise introduction to team-based logics and discuss recent developments in probabilistic team-based logics, which operate on finite probability distributions and have interesting connections to metafinite model theory, Blum–Shub–Smale computations, and the existential theory of the reals.

Oligomorphic groups  are the automorphism groups of omega-categorical structures. The continuous isomorphisms between two groups correspond to bi-interpretations between the underlying structures (Coquand). Work with Paolini and Schlicht from 2024 onwards explores this connection to understand continuous automorphisms of an oligomorphic group G. We show that the group of outer automorphisms of G, corresponding to the self bi-interpretations up to interdefinability of the underlying structure M, is locally compact. It is unknown whether this group is profinite. The talk will review so me examples such as G= Aut(Q,<) where this group has in fact two elements. Recent work  seeks similar results when replacing G by a different topological structure, the polymorphism clone of M.

Reference: Paolini and Nies, Oligomorphic groups, their automorphism groups, and the complexity of their isomorphism, arxiv.org/pdf/2410.02248.pdf 

We recall key examples of non-monotonic logics, together with their syntax and semantics, and present a generic approach to their proof theory. Rather than working logic by logic, our framework yields uniform constructions and results across a wide class of non-monotonic systems. We show how to systematically derive sequent calculi and establish cut elimination, and how resolution-style proof systems can in turn be obtained from these calculi.

Algebraic machine learning offers a novel alternative to conventional neural network approaches that is "without parameters, nor relying on fitting, regression, backtracking, constraint satisfiability, logical rules, production rules or error minimization” (from the company Algebraic AI).  Instead, the approach is built on universal algebra and model theory.   

We provide a gentle overview to this fascinating new application of algebra and logic, providing a duality-theoretic exposition of the underlying framework.

Answer Set Programming (ASP) has become a popular paradigm for declarative problem solving and is about to

find its way into industry.  This is due to its expressive yet easy knowledge representation language powered

by highly performant (Boolean) solving technology.  As with many other such paradigms before, the transition

from academia to industry calls for more versatility. Hence, many real-world applications are not tackled by

pure ASP but rather hybrid ASP.  The corresponding ASP systems are usually augmented by foreign language

constructs from which additional inferences can be drawn.  Examples include linear equations or temporal

formulas.  For the design of “sound” systems, however, it is indispensable to provide semantic underpinnings

right from the start.  To this end, we will discuss the vital role of ASP’s logical foundations, the logic of

Here-and-There and its non-monotonic extension, Equilibrium Logic, in designing hybrid ASP systems and highlight some of the resulting industrial applications.

Modal logics have been extensively studied, as they can express non-trivial problems ranging from domains in Mathematics and Philosophy to the representation of computational systems. The implementation of reasoning engines for those logics is, therefore, highly desirable. However, the satisfiability problem for even the most basic of the multimodal logics, the modal logic Kn, is not tractable: local and global reasoning in the multiagent set are PSPACE-complete and EXPTime-complete, respectively. We are, thus, interested in calculi that can be effectively employed for reasoning within those logics in an efficient way. In this talk, we will discuss two different resolution-based calculi for the multimodal Kn with particular focus on the characteristics that have an impact on automatic theorem-proving. We will report on experimental results and the influence of proof strategies and processing techniques for both calculi, and how different techniques impact the efficiency of theorem-proving in practice.