|
DisCoCat
DisCoCat (Categorical Compositional Distributional) is a mathematical framework for natural language processing which uses category theory to unify distributional semantics with the principle of compositionality. The grammatical derivations in a categorial grammar (usually a pregroup grammar) are interpreted as linear maps acting on the tensor product of word vectors to produce the meaning of a sentence or a piece of text. String diagrams are used to visualise information flow and reason about natural language semantics. History The framework was first introduced by Bob Coecke, Mehrnoosh Sadrzadeh, and Stephen Clark as an application of categorical quantum mechanics to natural language processing. It started with the observation that pregroup grammars and quantum processes shared a common mathematical structure: they both form a rigid category (also known as a non-symmetric compact closed category). As such, they both benefit from a graphical calculus, which allows a purely diagr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bob Coecke
Bob Coecke (born 23 July 1968) is a Belgian theoretical physicist and logician who was professor of Quantum Foundations, Logics and Structures at Oxford University until 2020, when he became Chief Scientist of Cambridge Quantum Computing, and after the merger with Honeywell Quantum Systems, Chief Scientist of Quantinuum. He pioneered categorical quantum mechanics (entry 18M40 in Mathematics Subject Classification 2020), Quantum Picturalism, ZX-calculus, DisCoCat model for natural language, and quantum natural language processing (QNLP). He is a founder of the Quantum Physics and Logic community and conference series, and of the applied category theory community, conference series, and diamond-open-access journal ''Compositionality''. Education and career Coecke obtained his Doctorate in Sciences at the Vrije Universiteit Brussel in 1996, and performed postdoctoral work in the Theoretical Physics Group of Imperial College, London in the Category Theory Group of the Mathematic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Natural Language Processing
Quantum natural language processing (QNLP) is the application of quantum computing to natural language processing (NLP). It computes word embeddings as parameterised quantum circuits that can solve NLP tasks faster than any classical computer. It is inspired by categorical quantum mechanics and the DisCoCat framework, making use of string diagrams to translate from grammatical structure to quantum processes. Theory The first quantum algorithm for natural language processing used the DisCoCat framework and Grover's algorithm to show a quadratic quantum speedup for a text classification task. It was later shown that quantum language processing is BQP-Complete, i.e. quantum language models are more expressive than their classical counterpart, unless quantum mechanics can be efficiently simulated by classical computers. These two theoretical results assume fault-tolerant quantum computation and a QRAM, i.e. an efficient way to load classical data on a quantum computer. Thus, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
String Diagrams
String diagrams are a formal graphical language for representing morphisms in monoidal categories, or more generally 2-cells in 2-categories. They are a prominent tool in applied category theory. When interpreted in the monoidal category of vector spaces and linear maps with the tensor product, string diagrams are called tensor networks or Penrose graphical notation. This has led to the development of categorical quantum mechanics where the axioms of quantum theory are expressed in the language of monoidal categories. History Günter Hotz gave the first mathematical definition of string diagrams in order to formalise electronic circuits, but the article remained confidential because of the absence of an English translation. The invention of string diagrams is usually credited to Roger Penrose with Feynman diagrams also described as a precursor. They were later characterised as the arrows of free monoidal categories in a seminal article by André Joyal and Ross Street. While the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Categorical Quantum Mechanics
Categorical quantum mechanics is the study of quantum foundations and quantum information using paradigms from mathematics and computer science, notably monoidal category theory. The primitive objects of study are physical processes, and the different ways that these can be composed. It was pioneered in 2004 by Samson Abramsky and Bob Coecke. Categorical quantum mechanics is entry 18M40 in MSC2020. Mathematical setup Mathematically, the basic setup is captured by a dagger symmetric monoidal category: composition of morphisms models sequential composition of processes, and the tensor product describes parallel composition of processes. The role of the dagger is to assign to each state a corresponding test. These can then be adorned with more structure to study various aspects. For instance: * A dagger compact category allows one to distinguish between an "input" and "output" of a process. In the diagrammatic calculus, it allows wires to be bent, allowing for a less restrict ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
String Diagrams
String diagrams are a formal graphical language for representing morphisms in monoidal categories, or more generally 2-cells in 2-categories. They are a prominent tool in applied category theory. When interpreted in the monoidal category of vector spaces and linear maps with the tensor product, string diagrams are called tensor networks or Penrose graphical notation. This has led to the development of categorical quantum mechanics where the axioms of quantum theory are expressed in the language of monoidal categories. History Günter Hotz gave the first mathematical definition of string diagrams in order to formalise electronic circuits, but the article remained confidential because of the absence of an English translation. The invention of string diagrams is usually credited to Roger Penrose with Feynman diagrams also described as a precursor. They were later characterised as the arrows of free monoidal categories in a seminal article by André Joyal and Ross Street. While the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mehrnoosh Sadrzadeh
Mehrnoosh Sadrzadeh is an Iranian British academic who is a professor at University College London. She was awarded a senior research fellowship at the Royal Academy of Engineering in 2022. Early life and education Sadrzadeh is from Iran. She received her undergraduate and masters degrees at Sharif University of Technology. After earning her master's degree Sadrzadeh moved to Canada. She was a doctoral researcher first at the University of Ottawa, where she was awarded an Ontario Graduate Scholarship, a University of Ottawa Excellence Scholarship, and a Canada Female Doctoral Student Award, and then at the Université du Québec à Montréal. Her research considered epistemic logic. Alongside earning her doctorate, Sadrzadeh moved to the University of Oxford as an EPSRC postdoctoral fellow. Research and career In 2011, Sadrzadeh was awarded an Engineering and Physical Sciences Research Council Career Acceleration Fellowship. She was appointed to the faculty at Queen Mary Univ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
FinVect
In the mathematical field of category theory, FinVect (or FdVect) is the category whose objects are all finite-dimensional vector spaces and whose morphisms are all linear maps between them. Properties FinVect has two monoidal products: * the direct sum of vector spaces, which is both a categorical product and a coproduct, * the tensor product, which makes FinVect a compact closed category. Examples Tensor networks are string diagrams interpreted in FinVect. Group representations are functors from groups, seen as one-object categories, into FinVect. DisCoCat models are monoidal functors from a pregroup grammar to FinVect. See also * FinSet * ZX-calculus * category of modules In algebra, given a ring ''R'', the category of left modules over ''R'' is the category whose objects are all left modules over ''R'' and whose morphisms are all module homomorphisms between left ''R''-modules. For example, when ''R'' is the ring o ... References {{reflist Categories in categor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vector Space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can be complex numbers or, more generally, elements of any field. The operations of vector addition and scalar multiplication must satisfy certain requirements, called ''vector axioms''. The terms real vector space and complex vector space are often used to specify the nature of the scalars: real coordinate space or complex coordinate space. Vector spaces generalize Euclidean vectors, which allow modeling of physical quantities, such as forces and velocity, that have not only a magnitude, but also a direction. The concept of vector spaces is fundamental for linear algebra, together with the concept of matrix, which allows computing in vector spaces. This provides a concise and synthetic way for manipulating and studying systems of linear eq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Strong Monoidal Functor
In category theory, monoidal functors are functors between monoidal categories which preserve the monoidal structure. More specifically, a monoidal functor between two monoidal categories consists of a functor between the categories, along with two ''coherence maps''—a natural transformation and a morphism that preserve monoidal multiplication and unit, respectively. Mathematicians require these coherence maps to satisfy additional properties depending on how strictly they want to preserve the monoidal structure; each of these properties gives rise to a slightly different definition of monoidal functors * The coherence maps of lax monoidal functors satisfy no additional properties; they are not necessarily invertible. * The coherence maps of strong monoidal functors are invertible. * The coherence maps of strict monoidal functors are identity maps. Although we distinguish between these different definitions here, authors may call any one of these simply monoidal functors. Defi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Adjoint Functors
In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are known as adjoint functors, one being the left adjoint and the other the right adjoint. Pairs of adjoint functors are ubiquitous in mathematics and often arise from constructions of "optimal solutions" to certain problems (i.e., constructions of objects having a certain universal property), such as the construction of a free group on a set in algebra, or the construction of the Stone–Čech compactification of a topological space in topology. By definition, an adjunction between categories \mathcal and \mathcal is a pair of functors (assumed to be covariant) :F: \mathcal \rightarrow \mathcal and G: \mathcal \rightarrow \mathcal and, for all objects X in \mathcal and Y in \mathcal a bijection between the respective morphism s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tensor Network
Tensor networks or tensor network states are a class of variational wave functions used in the study of many-body quantum systems. Tensor networks extend one-dimensional matrix product states to higher dimensions while preserving some of their useful mathematical properties. The wave function is encoded as a tensor contraction of a network of individual tensors. The structure of the individual tensors can impose global symmetries on the wave function (such as antisymmetry under exchange of fermions) or restrict the wave function to specific quantum numbers, like total charge, angular momentum, or spin. It is also possible to derive strict bounds on quantities like entanglement and correlation length using the mathematical structure of the tensor network. This has made tensor networks useful in theoretical studies of quantum information in many-body systems. They have also proved useful in variational studies of ground states, excited states, and dynamics of strongly corr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Categorical Product
In category theory, the product of two (or more) objects in a category is a notion designed to capture the essence behind constructions in other areas of mathematics such as the Cartesian product of sets, the direct product of groups or rings, and the product of topological spaces. Essentially, the product of a family of objects is the "most general" object which admits a morphism to each of the given objects. Definition Product of two objects Fix a category C. Let X_1 and X_2 be objects of C. A product of X_1 and X_2 is an object X, typically denoted X_1 \times X_2, equipped with a pair of morphisms \pi_1 : X \to X_1, \pi_2 : X \to X_2 satisfying the following universal property: * For every object Y and every pair of morphisms f_1 : Y \to X_1, f_2 : Y \to X_2, there exists a unique morphism f : Y \to X_1 \times X_2 such that the following diagram commutes: *: Whether a product exists may depend on C or on X_1 and X_2. If it does exist, it is unique up to canonical is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
