Related papers: Building on the Diamonds between Theories: Theory …
Developing a 21st Century Global Library for Mathematics Research discusses how information about what the mathematical literature contains can be formalized and made easier to express, encode, and explore. Many of the tools necessary to…
Sharing of notations and theories across an inheritance hierarchy of mathematical structures, e.g., groups and rings, is important for productivity when formalizing mathematics in proof assistants. The packed classes methodology is a…
In previous papers on this project a general static logical framework for formalizing and mechanizing set theories of different strength was suggested, and the power of some predicatively acceptable theories in that framework was explored.…
In combinatorics, the probabilistic method is a very powerful tool to prove the existence of combinatorial objects with interesting and useful properties. Explicit constructions of objects with such properties are often very difficult, or…
In this paper we present new results for the combinatorics of web diagrams and web worlds. These are discrete objects that arise in the physics of calculating scattering amplitudes in non-abelian gauge theories. Web-colouring and web-mixing…
The increasing power of computer technology does not dispense with the need to extract meaningful in- formation out of data sets of ever growing size, and indeed typically exacerbates the complexity of this task. To tackle this general…
This paper describes mathlib, a community-driven effort to build a unified library of mathematics formalized in the Lean proof assistant. Among proof assistant libraries, it is distinguished by its dependently typed foundations, focus on…
We construct a realizability model of linear dependent type theory from a linear combinatory algebra. Our model motivates a number of additions to the type theory. In particular, we add a universe with two decoding operations: one takes…
To adequately model mathematical arguments the analyst must be able to represent the mathematical objects under discussion and the relationships between them, as well as inferences drawn about these objects and relationships as the…
I explore the concept of growth being rooted in the recombination of existing technology as an explanation for the remarkable growth witnessed during the Industrial Revolution as it was recently proposed by Koppl et al.(2023). I adapt their…
We describe Active Libraries, which take an active role in compilation. Unlike traditional libraries which are passive collections of functions and objects, Active Libraries may generate components, specialize algorithms, optimize code,…
Contemporary Large Language Models (LLMs) exhibit a high degree of code generation and comprehension capability. A particularly promising area is their ability to interpret code modules from unfamiliar libraries for solving user-instructed…
Fast numerical libraries have been a cornerstone of scientific computing for decades, but this comes at a price. Programs may be tied to vendor specific software ecosystems resulting in polluted, non-portable code. As we enter an era of…
Predictor combination aims to improve a (target) predictor of a learning task based on the (reference) predictors of potentially relevant tasks, without having access to the internals of individual predictors. We present a new predictor…
The saturation-based reasoning methods are among the most theoretically developed ones and are used by most of the state-of-the-art first-order logic reasoners. In the last decade there was a sharp increase in performance of such systems,…
Recent research has explored using Large Language Models for recommendation tasks by transforming user interaction histories and item metadata into text prompts, then having the LLM produce rankings or recommendations. A promising approach…
Multi-AI collaboration, such as ensembling or debating large language models (LLMs), is a promising paradigm for aggregating information and boosting performance. A foundational step in these pipelines is to feed the responses of several…
Theory revision integrates inductive learning and background knowledge by combining training examples with a coarse domain theory to produce a more accurate theory. There are two challenges that theory revision and other theory-guided…
A hybrid model involves the cooperation of an interpretable model and a complex black box. At inference, any input of the hybrid model is assigned to either its interpretable or complex component based on a gating mechanism. The advantages…
This thesis proposes a combinatorial generalization of a nilpotent operator on a vector space. The resulting object is highly natural, with basic connections to a variety of fields in pure mathematics, engineering, and the sciences. For the…