Related papers: Biform Theories: Project Description
"Theory figures" are a staple of theoretical visualization research. Common shapes such as Cartesian planes and flowcharts can be used not only to explain conceptual contributions, but to think through and refine the contribution itself.…
Large Language Models (LLMs) have demonstrated impressive capabilities in structured reasoning and symbolic tasks, with coding emerging as a particularly successful application. This progress has naturally motivated efforts to extend these…
This book explores an alternative to the current dominant paradigm where a discrete computer model is constructed as an attempt to approximate some continuum theory. We focus on a class of discrete computer models that are based on simple…
"Clarithmetic" is a generic name for formal number theories similar to Peano arithmetic, but based on computability logic (see http://www.cis.upenn.edu/~giorgi/cl.html) instead of the more traditional classical or intuitionistic logics.…
Hyperdimensional (HD) computing is a set of neurally inspired methods for obtaining high-dimensional, low-precision, distributed representations of data. These representations can be combined with simple, neurally plausible algorithms to…
Machine reading comprehension has aroused wide concerns, since it explores the potential of model for text understanding. To further equip the machine with the reasoning capability, the challenging task of logical reasoning is proposed.…
Bigraphs are a universal computational modelling formalism for the spatial and temporal evolution of a system in which entities can be added and removed. We extend bigraphs to probablistic bigraphs, and then again to action bigraphs, which…
Bigraph reactive systems offer a powerful and flexible mathematical framework for modelling both spatial and non-spatial relationships between agents, with practical applications in domains such as smart technologies, networks, sensor…
The framework of algorithmic knowledge assumes that agents use algorithms to compute the facts they explicitly know. In many cases of interest, a deductive system, rather than a particular algorithm, captures the formal reasoning used by…
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…
The theory of stratified institutions is a general axiomatic approach to model theories where the satisfaction is parameterised by states of the models. In this paper we further develop this theory by introducing a new technique for…
This essay considers the special character of mathematical reasoning, and draws on observations from interactive theorem proving and the history of mathematics to clarify the nature of formal and informal mathematical language. It proposes…
In this paper we introduce a novel notion of probabilistic bisimulation for quantum processes and prove that it is congruent with respect to various process algebra combinators including parallel composition even when both classical and…
We show that any multiple-valued function can be represented by a linear lambda term typed in a second-order polymorphic type system, using two distinct styles. The first is a circuit style, which mimics combinational circuits in switching…
Recently, biclustering is one of the hot topics in bioinformatics and takes the attention of authors from several different disciplines. Hence, many different methodologies from a variety of disciplines are proposed as a solution to the…
Automated theorem provers and formal proof assistants are general reasoning systems that are in theory capable of proving arbitrarily hard theorems, thus solving arbitrary problems reducible to mathematics and logical reasoning. In…
We compare different epistemic notions in the presence of awareness of propositional variables: the logics of implicit knowledge (in which explicit knowledge is definable), explicit knowledge, and speculative knowledge. Different notions of…
Analysing several characteristic mathematical models: natural and real numbers, Euclidean geometry, group theory, and set theory, I argue that a mathematical model in its final form is a junction of a set of axioms and an internal partial…
The article introduces the concept of uniformity, which is formulated as a scheme of axioms. The connection of this concept with ordered sets is studied. The effectiveness of using axiom schemes as a convenient and short way of replacing…
Field theory is an area in physics with a deceptively compact notation. Although general purpose computer algebra systems, built around generic list-based data structures, can be used to represent and manipulate field-theory expressions,…