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Reverse-mode differentiation is used for optimization, but it introduces references, which break the purity of the underlying programs, making them notoriously harder to optimize. We present a reverse-mode differentiation on a purely…
The program Reverse Mathematics (RM for short) seeks to identify the axioms necessary to prove theorems of ordinary mathematics, usually working in the language of second-order arithmetic $L_{2}$. A major theme in RM is therefore the study…
The reversible logic can be used in various research areas, e.g. quantum computation, cryptography and signal processing. In the paper we study reversible logic circuits with additional inputs, which consist of NOT, CNOT and…
Many natural program correctness properties can be stated in terms of symmetries, but existing formal methods have little support for reasoning about such properties. We consider how to formally verify a broad class of symmetry properties…
We construct a $\mathcal{PT}$-symmetric Richardson--Gaudin models for spin-$\tfrac{1}{2}$ systems by deforming the closed integrable Hamiltonian through complex-valued transverse magnetic fields and coupling constants. By defining parity as…
Symmetry groups are projectively represented in quantum mechanics, and crystalline symmetries are fundamental in condensed matter physics. Here, we systematically present a unified theory of quantum mechanical space groups from two…
We describe a mathematical framework for equational reasoning about infinite families of string diagrams which is amenable to computer automation. The framework is based on context-free families of string diagrams which we represent using…
We establish a connection between problems studied in rigidity theory and matroids arising from linear algebraic constructions like tensor products and symmetric products. A special case of this correspondence identifies the problem of…
In this paper, we discuss the computational power of parallel communicating finite automata system with 1-way reversible finite automaton as components. We show that unlike the multi-head one way reversible finite automata model (where we…
The toric code can be constructed as a gauge theory of finite groups on oriented two dimensional lattices. Here we construct analogous models with the gauge fields belonging to groupoids, which are categories where every morphism has an…
The main objective of this work is to study mathematical properties of computational paths. Originally proposed by de Queiroz \& Gabbay (1994) as `sequences of rewrites', computational paths can be seen as the grounds on which the…
The study of finite automata and regular languages is a privileged meeting point of algebra and logic. Since the work of Buchi, regular languages have been classified according to their descriptive complexity, i.e. the type of logical…
Quantum error correction is believed to be essential for scalable quantum computation, but its implementation is challenging due to its considerable space-time overhead. Motivated by recent experiments demonstrating efficient manipulation…
Program verification is to develop the program's proof system, and to prove the proof system soundness with respect to a trusted operational semantics of the program. However, many practical program verifiers are not based on operational…
Let $K$ be a totally real field and $\pi$ be a regular algebraic polarized cuspidal automorphic representation of $\mathrm{GL}_n(\mathbb A_K)$. Let $\{\rho_{\pi,\lambda}:\mathrm{Gal}_K\to\mathrm{GL}_n(\overline E_\lambda)\}_\lambda$ be the…
In 2021, Augot, Couvreur, Lavauzelle and Neri introduced a new class of rank metric codes which can be regarded as rank metric counterparts of Reed-Muller codes. Given a finite Galois extension $\mathbb{L} / \mathbb{K}$, these codes are…
With the increased interest in machine learning, and deep learning in particular, the use of automatic differentiation has become more wide-spread in computation. There have been two recent developments to provide the theoretical support…
We extend some fundamental definitions and constructions in the established generalisation of Lie theory involving Lie groupoids by reformulating them in terms of groupoids internal to a well-adapted model of synthetic differential…
We propose a novel geometric approach for learning bilingual mappings given monolingual embeddings and a bilingual dictionary. Our approach decouples learning the transformation from the source language to the target language into (a)…
In this work, we propose to study noncommutative geometry using the language of categories of sheaves of algebras with polynomial identities and their properties, introducing new (graded) noncommutative geometries. These include, for…