Related papers: Constructive Separations and Their Consequences
A subset $B \subset Y$ is constructible if it is an element of the smallest family that contains all open sets and is stable under finite intersections and complements. A function $f : X \to Y$ is said to be piece-wise closed if $X$ can be…
As the use of machine learning models has increased, numerous studies have aimed to enhance fairness. However, research on the intersection of fairness and explainability remains insufficient, leading to potential issues in gaining the…
This chapter studies the problem of decomposing a tensor into a sum of constituent rank one tensors. While tensor decompositions are very useful in designing learning algorithms and data analysis, they are NP-hard in the worst-case. We will…
Large Language Models (LLMs) increasingly exhibit strong reasoning abilities, often attributed to their capacity to generate chain-of-thought-style intermediate reasoning. Recent work suggests that exposure to code can further enhance these…
The downward closure of a language is the set of all (not necessarily contiguous) subwords of its members. It is well-known that the downward closure of every language is regular. Moreover, recent results show that downward closures are…
Structural resolution (or S-resolution) is a newly proposed alternative to SLD-resolution that allows a systematic separation of derivations into term-matching and unification steps. Productive logic programs are those for which…
Bottom-up knowledge compilation is a paradigm for generating representations of functions by iteratively conjoining constraints using a so-called apply function. When the input is not efficiently compilable into a language - generally a…
Robust Principal Component Analysis (PCA) (Candes et al., 2011) and low-rank matrix completion (Recht et al., 2010) are extensions of PCA to allow for outliers and missing entries respectively. It is well-known that solving these problems…
A closed convex subset of a normed linear space is said to have the strong separation property if it can be strongly separated from every other disjoint closed and convex set by a closed hyperplane. In this paper we give some results on the…
Given a function based on the computation of an NP machine, can one in general eliminate some solutions? That is, can one in general decrease the ambiguity? This simple question remains, even after extensive study by many researchers over…
We study constructively the relations between the finite cases of Dickson's lemma. Although there are many constructive proofs of them, the novel aspect of our proofs is the extraction of a corresponding bound. We provide some new one-step…
We connect the study of pseudodeterministic algorithms to two major open problems about the structural complexity of $\mathsf{BPTIME}$: proving hierarchy theorems and showing the existence of complete problems. Our main contributions can be…
We consider some distinguished classes of elements of a multiplicative lattice endowed with coarse lower topologies, and call them lower spaces. The primary objective of this paper is to study the topological properties of these lower…
Many constructions in computability theory rely on "time tricks". In the higher setting, relativising to some oracles shows the necessity of these. We construct an oracle~$A$ and a set~$X$, higher Turing reducible to~$X$, but for which…
Humans do not just find mistakes after the fact -- we often catch them mid-stream because 'reflection' is tied to the goal and its constraints. Today's large language models produce reasoning tokens and 'reflective' text, but is it…
Construction Grammar (CxG) has recently been used as the basis for probing studies that have investigated the performance of large pretrained language models (PLMs) with respect to the structure and meaning of constructions. In this…
The aim of this thesis is to determine classes of NP relations for which random generation and approximate counting problems admit an efficient solution. Since efficient rank implies efficient random generation, we first investigate some…
Over recent years, devising classification algorithms that are robust to adversarial perturbations has emerged as a challenging problem. In particular, deep neural nets (DNNs) seem to be susceptible to small imperceptible changes over test…
A binary matrix has the Consecutive Ones Property (C1P) if its columns can be ordered in such a way that all 1's on each row are consecutive. A Minimal Conflicting Set is a set of rows that does not have the C1P, but every proper subset has…
Lateral predictive coding (LPC) is a simple theoretical framework to appreciate feature detection in biological neural circuits. Recent theoretical work [Huang et al., Phys.Rev.E 112, 034304 (2025)] has successfully constructed optimal LPC…