Related papers: Formalizing Computational Paths and Fundamental Gr…
$ \ell_1 $-regularized linear inverse problems are frequently used in signal processing, image analysis, and statistics. The correct choice of the regularization parameter $ t \in \mathbb{R}_{\geq 0} $ is a delicate issue. Instead of…
We report on our experience formalizing differential geometry with mathlib, the Lean mathematical library. Our account is geared towards geometers with no knowledge of type theory, but eager to learn more about the formalization of…
Initiated by Abramsky [1994], the Proofs as Processes agenda is to establish a solid foundation for the study of concurrent languages, by researching the connection between linear logic and the $\pi$-calculus. To date, Proofs as Processes…
Given an associative, not necessarily commutative, ring R with identity, a formal matrix calculus is introduced and developed for pairs of matrices over R. This calculus subsumes the theory of homogeneous systems of linear equations with…
This paper explores formalizing Geometric (or Clifford) algebras into the Lean 3 theorem prover, building upon the substantial body of work that is the Lean mathematics library, mathlib. As we use Lean source code to demonstrate many of our…
In this paper, we investigate the computational complexity of isomorphism testing for finite groups and quasigroups, given by their multiplication tables. We crucially take advantage of their various decompositions to show the following: -…
Homotopy type theory is an interpretation of Martin-L\"of's constructive type theory into abstract homotopy theory. There results a link between constructive mathematics and algebraic topology, providing topological semantics for…
Cyclomatic complexity is an incompletely specified but mathematically principled software metric that can be usefully applied to both source and binary code. We consider the application of path homology as a stronger analogue of cyclomatic…
Homotopy type theory is a logical setting based on Martin-L\"of type theory in which geometric constructions and proofs can be carried out synthetically. Here, types can be interpreted as spaces up to homotopy, and proofs as…
Recent algorithmic advances in algebraic automata theory drew attention to semigroupoids (semicategories). These are mathematical descriptions of typed computational processes, but they have not been studied systematically in the context of…
When the theory of Leavitt path algebras was already quite advanced, it was discovered that some of the more difficult questions were susceptible to a new approach using topological groupoids. The main result that makes this possible is…
We introduce a new way of formalizing the intensional identity type based on the fact that a entity known as computational paths can be interpreted as terms of the identity type. Our approach enjoys the fact that our elimination rule is…
This is the fourth in a series of papers extending Martin-L\"of's meaning explanation of dependent type theory to higher-dimensional types. In this installment, we show how to define cubical type systems supporting a general schema of…
The strict globular $\omega$-categories formalize the execution paths of a parallel automaton and the homotopies between them. One associates to such (and any) $\omega$-category $\C$ three homology theories. The first one is called the…
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…
The signature of a path, introduced by K.T. Chen [5] in $1954$, has been extensively studied in recent years. The $2010$ paper [12] of Hambly and Lyons showed that the signature is injective on the space of continuous finite-variation paths…
Connections between homotopy theory and type theory have recently attracted a lot of attention, with Voevodsky's univalent foundations and the interpretation of Martin-Lof's identity types in Quillen model categories as some of the…
This work presents a new path classification criterion to distinguish paths geometrically and topologically from the workspace, which is divided through cell decomposition, generating a medial-axis-like skeleton structure. We use this…
In this article, we establish a mathematical framework that utilizes concepts from graph theory to formalize the parity transformation, an encoding strategy for compiling optimization problems on quantum devices. We introduce the…
We present three projects concerned with applications of proof assistants in the area of programming language theory and mathematics. The first project is about a certified compilation technique for a domain-specific programming language…