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Up-directed rough sets are introduced and studied by the present author in earlier papers. This is extended by her in two different granular directions in this research, with a surprising algebraic semantics. The granules are based on ideas…
A formal context consists of objects, properties, and the incidence relation between them. Various notions of concepts defined with respect to formal contexts and their associated algebraic structures have been studied extensively,…
A quantitative model of concurrent interaction is introduced. The basic objects are linear combinations of partial order relations, acted upon by a group of permutations that represents potential non-determinism in synchronisation. This…
An abstract mathematical framework is presented in this paper as a unification of several deformed or generalized algebra proposed recently in the context of generalized statistical theories intended to treat certain complex thermodynamic…
Current language models have demonstrated their capability to develop basic reasoning, but struggle in more complicated reasoning tasks that require a combination of atomic skills, such as math word problem requiring skills like arithmetic…
A finite-dimensional unital and associative algebra over $\mathbb{R}$, or what we shall call simply "an algebra" in this paper for short, generalities the construction by which we derive the complex numbers by "adjoining an element $i$" to…
This paper is about conceptual modeling of aggregates in software engineering. An aggregate is a cluster of domain objects that can be treated as a single unit. In UML, an aggregation is a type of association in which objects are configured…
This paper presents a Fuzzy Cognitive Map model to quantify implicit bias in structured datasets where features can be numeric or discrete. In our proposal, problem features are mapped to neural concepts that are initially activated by…
Rough set theory is a new mathematical approach to imperfect knowledge. The notion of rough sets is generalized by using an arbitrary binary relation on attribute values in information systems, instead of the trivial equality relation. The…
While LLMs have emerged as performant architectures for reasoning tasks, their compositional generalization capabilities have been questioned. In this work, we introduce a Compositional Generalization Challenge for Graph-based Commonsense…
This work contributes to the theory of judgment aggregation by discussing a number of significant non-classical logics. After adapting the standard framework of judgment aggregation to cope with non-classical logics, we discuss in…
We study atom canonicity for several varieties of cylindric like algebras that contain properly the variety of representable algebras. The algebras in such varieties have relativized representations, and we thereby obtain many omitting…
The ability to automatically generalise (interactive) proofs and use such generalisations to discharge related conjectures is a very hard problem which remains unsolved. Here, we develop a notion of goal types to capture key properties of…
We develop two algebraic semantics for bitten rough set theory (\cite{SW}) over similarity spaces and their abstract granular versions. Connections with choice based generalized rough semantics developed in \cite{AM69} by the present author…
A longstanding question in cognitive science concerns the learning mechanisms underlying compositionality in human cognition. Humans can infer the structured relationships (e.g., grammatical rules) implicit in their sensory observations…
One of the weaknesses of classical (fuzzy) rough sets is their sensitivity to noise, which is particularly undesirable for machine learning applications. One approach to solve this issue is by making use of fuzzy quantifiers, as done by the…
Abstraction is key to human and artificial intelligence as it allows one to see common structure in otherwise distinct objects or situations and as such it is a key element for generality in AI. Anti-unification (or generalization) is…
Algebraic model counting unifies many inference tasks on logic formulas by exploiting semirings. Rather than focusing on inference, we consider learning, especially in statistical-relational and neurosymbolic AI, which combine logical,…
Statistics and Optimization are foundational to modern Machine Learning. Here, we propose an alternative foundation based on Abstract Algebra, with mathematics that facilitates the analysis of learning. In this approach, the goal of the…
Pre-trained language models (LMs) are able to perform complex reasoning without explicit fine-tuning. To understand how pre-training with a next-token prediction objective contributes to the emergence of such reasoning capability, we…