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Data science workflows often integrate functionalities from a diverse set of libraries and frameworks. Tasks such as debugging require data lineage that crosses library boundaries. The problem is that the way that "lineage" is represented…

There is a strong demand for precise means for the comparison of logics in terms of expressiveness both from theoretical and from application areas. The aim of this paper is to propose a sufficiently general and reasonable formal criterion…

Several different proof translations exist between classical and intuitionistic logic (negative translations), and intuitionistic and linear logic (Girard translations). Our aims in this paper are (1) to consider extensions of…

Language models based on the Transformer architecture achieve excellent results in many language-related tasks, such as text classification or sentiment analysis. However, despite the architecture of these models being well-defined, little…

A theorem prover without an extensive library is much less useful to its potential users. Algebra, the study of algebraic structures, is a core component of such libraries. Algebraic theories also are themselves structured, the study of…

The interoperability of proof assistants and the integration of their libraries is a highly valued but elusive goal in the field of theorem proving. As a preparatory step, in previous work, we translated the libraries of multiple proof…

This paper aims to provide an analysis of what it means when we say that a pair of theories, very generously construed, are equivalent in the sense that they are interdefinable. With regard to theories articulated in first order logic, we…

Understanding the inner workings of Large Language Models (LLMs) is a critical research frontier. Prior research has shown that a single LLM's concept representations can be captured as steering vectors (SVs), enabling the control of LLM…

Biological and artificial information processing systems form representations of the world that they can use to categorize, reason, plan, navigate, and make decisions. How can we measure the similarity between the representations formed by…

Representation theorems for formal systems often take the form of an inductive translation that satisfies certain invariants, which are proved inductively. Theory morphisms and logical relations are common patterns of such inductive…

One way of proving theorems in modal logics is translating them into the predicate calculus and then using conventional resolution-style theorem provers. This approach has been regarded as inappropriate in practice, because the resulting…

The heterogeneous nature of the logical foundations used in different interactive proof assistant libraries has rendered discovery of similar mathematical concepts among them difficult. In this paper, we compare a previously proposed…

We describe the first results of a project of analyzing in which theories formal proofs can be ex- pressed. We use this analysis as the basis of interoperability between proof systems.

Are LLMs cultural technologies like photocopiers or printing presses, which transmit information but cannot create new content? A challenge for this idea, which we call bibliotechnism, is that LLMs generate novel text. We begin with a…

In real world, large language models (LLMs) can serve as the assistant to help users accomplish their jobs, and also support the development of advanced applications. For the wide application of LLMs, the inference efficiency is an…

Mechanical reasoning is a key area of research that lies at the crossroads of mathematical logic and artificial intelligence. The main aim to develop mechanical reasoning systems (also known as theorem provers) was to enable mathematicians…

LLMs increasingly serve as tools for knowledge acquisition, yet users cannot effectively specify how they want information presented. When users request that LLMs "cite reputable sources," "express appropriate uncertainty," or "include…

Since there are different ways of axiomatizing and developing a mathematical theory, knowledge about a such a theory may reside in many places and in many forms within a library of formalized mathematics. We introduce the notion of a realm…

We propose a new method for explanations in Artificial Intelligence (AI) and a tool to test its expressive power within a user interface. In order to bridge the gap between philosophy and human-computer interfaces, we show a new approach…

Benchmarking automated theorem proving (ATP) systems using standardized problem sets is a well-established method for measuring their performance. However, the availability of such libraries for non-classical logics is very limited. In this…