Related papers: Tutte's theorem as an educational formalization pr…
In this paper, we utilize Isabelle/HOL to develop a formal framework for the basic theory of double-pushout graph transformation. Our work includes defining essential concepts like graphs, morphisms, pushouts, and pullbacks, and…
Mathematical proofs are often said to justify their conclusions by indicating the existence of a corresponding formal derivation. We argue that this widespread view relies on an under-examined notion of correspondence, or what it means for…
Theory of choreographic languages typically includes a number of complex results that are proved by structural induction. The high number of cases and the subtle details in some of them lead to long reviewing processes, and occasionally to…
As automated reasoning systems advance rapidly, there is a growing need for research-level formal mathematical problems to accurately evaluate their capabilities. To address this, we present Formal Conjectures, an evolving benchmark of…
Identities obtained by elementary finite Fourier analysis are used to derive a variety of evaluations of the Tutte polynomial of a graph G at certain points (a,b) where (a-1)(b-1) equals 2 or 4. These evaluations are expressed in terms of…
This document reports on the use of an algebraic, visual, formal approach to the specification of patterns for the formalization of the GoF design patterns. The approach is based on graphs, morphisms and operations from category theory and…
The ongoing development of Lean 4's Mathlib has produced a macroscopic structural complexity that interweaves logical, mathematical, and infrastructural dependencies. We present a network analysis of this library, extracting its dependency…
A plausible future mathematical claim must satisfy two constraints: it should follow the direction of prior work and respect the formal dependencies that constrain what can validly follow. Existing approaches typically model only one of…
Knowledge graphs are useful for many artificial intelligence (AI) tasks. However, knowledge graphs often have missing facts. To populate the graphs, knowledge graph embedding models have been developed. Knowledge graph embedding models map…
In this paper we develop a structure called Link Algebra, in which we present a Set with two binary operations and an axiom system developed from the study of graph theory and set/antiset theory, sowing main theorems and definitions. Once…
Tutte has described in the book "Connectivity in graphs" a canonical decomposition of any graph into 3-connected components. In this article we translate (using the language of symbolic combinatorics) Tutte's decomposition into a general…
We perform a thorough analysis of the formal and informal statements in the miniF2F benchmark from the perspective of an AI system that is tasked to participate in a math Olympiad consisting of the problems in miniF2F. In such setting, the…
In this paper we formalize some foundation concepts and theorems of group theory in a variant of type theory called the Calculus of Constructions with Definitions. In this theory we introduce definition of a group, which is both general and…
Topological models of empirical and formal inquiry are increasingly prevalent. They have emerged in such diverse fields as domain theory [1, 16], formal learning theory [18], epistemology and philosophy of science [10, 15, 8, 9, 2],…
Visual rendering of graphs is a key task in the mapping of complex network data. Although most graph drawing algorithms emphasize aesthetic appeal, certain applications such as travel-time maps place more importance on visualization of…
There are two methods to study families of conformal theories in the operator formalism. In the first method we begin with a theory and a family of deformed theories is defined in the state space of the original theory. In the other there…
Tame geometry originated in mathematical logic and implements strong finiteness properties by defining the notion of tame sets and functions. In part I we argued that observables in a wide class of quantum field theories are tame functions…
As the prototypical category, $\mathbf{Set}$ has many properties which make it special amongst categories. From the point of view of mathematical logic, one such property is that $\mathbf{Set}$ has enough structure to "properly" formalise…
This is a graduate-level introduction to graph theory, corresponding to a quarter-long course. It covers simple graphs, multigraphs as well as their directed analogues, and more restrictive classes such as tournaments, trees and…
The modeling of probability distributions, specifically generative modeling and density estimation, has become an immensely popular subject in recent years by virtue of its outstanding performance on sophisticated data such as images and…