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Covering is a common type of data structure and covering-based rough set theory is an efficient tool to process this data. Lattice is an important algebraic structure and used extensively in investigating some types of generalized rough…
ABC algorithms involve a large number of simulations from the model of interest, which can be very computationally costly. This paper summarises the lazy ABC algorithm of Prangle (2015), which reduces the computational demand by abandoning…
The notion of associativity (which differs from the straightforward generalization of the usual associativity given by the move of parentheses in the relevant expression) for operations of high arity is introduced. It is proved that the…
In many statistical problems, a more coarse-grained model may be suitable for population-level behaviour, whereas a more detailed model is appropriate for accurate modelling of individual behaviour. This raises the question of how to…
Classical algebraic structures require exact satisfaction of their defining axioms. We propose similarity algebra, a framework extending algebraic and Lie structures to settings where operations satisfy quantitative bounds up to a tolerance…
This Article introduces the generative reasonable person, a new tool for estimating how ordinary people judge reasonableness. As claims about AI capabilities often outpace evidence, the Article proceeds empirically: adapting randomized…
With recent dramatic increases in AI system capabilities, there has been growing interest in utilizing machine learning for reasoning-heavy, quantitative tasks, particularly mathematics. While there are many resources capturing mathematics…
Recent work (Ross et al., 2025, 2024) has argued that the ability of humans and LLMs respectively to generalize to novel adjective-noun combinations shows that they each have access to a compositional mechanism to determine the phrase's…
In tasks like semantic parsing, instruction following, and question answering, standard deep networks fail to generalize compositionally from small datasets. Many existing approaches overcome this limitation with model architectures that…
Regularized regression models are well studied and, under appropriate conditions, offer fast and statistically interpretable results. However, large data in many applications are heterogeneous in the sense of harboring distributional…
Human cognition exhibits systematic compositionality, the algebraic ability to generate infinite novel combinations from finite learned components, which is the key to understanding and reasoning about complex logic. In this work, we…
In dealing with veracity of data analytics, fuzzy methods are more and more relying on probabilistic and statistical techniques to underpin their applicability. Conversely, standard statistical models usually disregard to take into account…
Two major areas of interest in the era of Large Language Models regard questions of what do LLMs know, and if and how they may be able to reason (or rather, approximately reason). Since to date these lines of work progressed largely in…
An increasingly common use case for machine learning models is augmenting the abilities of human decision makers. For classification tasks where neither the human or model are perfectly accurate, a key step in obtaining high performance is…
We define a new notion of an algebraic model structure, in which the cofibrations and fibrations are retracts of coalgebras for comonads and algebras for monads, and prove "algebraic" analogs of classical results. Using a modified version…
We provide a new foundational approach to the generalization of terms up to equational theories. We interpret generalization problems in a universal-algebraic setting making a key use of projective and exact algebras in the variety…
We show that the theories of partially ordered sets, lattices, semilattices, Boolean algebras, Heyting algebras with a further coarser partial order, or a linearization, or an auxiliary relation have the strong amalgamation property,…
Collective versions of order convergences and corresponding types of collectively qualified sets of operators in vector lattices are investigated. It is proved that collectively order to norm bounded sets are bounded in the operator norm…
New concepts of rough natural number systems are introduced in this research paper from both formal and less formal perspectives. These are used to improve most rough set-theoretical measures in general Rough Set theory (\textsf{RST}) and…
How should one combine noisy information from diverse sources to make an inference about an objective ground truth? This frequently recurring, normative question lies at the core of statistics, machine learning, policy-making, and everyday…