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A Diagrammatic Calculus for Algebraic Effects [article]

Ugo Dal Lago, Francesco Gavazzo
2020 arXiv   pre-print
Additionally, we show a number of algebraic and order-theoretic laws on diagrams, this way laying the foundations for a diagrammatic calculus of algebraic effects.  ...  We introduce a new diagrammatic notation for representing the result of (algebraic) effectful computations.  ...  We have introduced a diagrammatic notation and calculus for countable monads with algebraic operations. We have shown by means of examples and general results some of advantages of our notation.  ... 
arXiv:2001.01337v2 fatcat:xql67jnewrdk5pihzbcrx5hgua

Differentiating and Integrating ZX Diagrams [article]

Quanlong Wang, Richie Yeung
2022 arXiv   pre-print
ZX-calculus has proved to be a useful tool for quantum technology with a wide range of successful applications. Most of these applications are of an algebraic nature.  ...  We explicitly illustrate the new analytic framework of ZX-calculus by applying it in context of quantum machine learning.  ...  Acknowledgements We would like to thank Bob Coecke and Konstantinos Meichanetzidis for useful discussions and valuable comments.  ... 
arXiv:2201.13250v2 fatcat:g42aeofvifc6ne2kttaw47u2ya

Fast Automated Reasoning over String Diagrams using Multiway Causal Structure [article]

Jonathan Gorard, Manojna Namuduri, Xerxes D. Arsiwalla
2021 arXiv   pre-print
with the causal structure of the underlying diagrammatic calculus.  ...  semantics for the full multiway evolution causal graph of a generic Wolfram model hypergraph rewriting system.  ...  Spivak and Stephen Wolfram for useful conversations and/or suggestions.  ... 
arXiv:2105.04057v1 fatcat:zjfepnkfubbu5pk3fmng6tibge

A Diagrammatic Axiomatisation for Qubit Entanglement [article]

Amar Hadzihasanovic
2015 arXiv   pre-print
This refines a version of the preexisting ZX calculus, while keeping its most desirable characteristics: undirectedness, a large degree of symmetry, and an algebraic underpinning.  ...  In the context of categorical quantum mechanics, Coecke and Kissinger suggested that two 3-qubit states, GHZ and W, may be used as the building blocks of a new graphical calculus, aimed at a diagrammatic  ...  Many thanks to Bob Coecke, Stefano Gogioso, Aleks Kissinger, Jamie Vicary, and Linde Wester for useful discussions and suggestions.  ... 
arXiv:1501.07082v1 fatcat:ps63nn7s2zgdvh5zlhzgoa2ywq

Temperley-Lieb Algebra: From Knot Theory to Logic and Computation via Quantum Mechanics [article]

Samson Abramsky
2009 arXiv   pre-print
We shall introduce a "planar lambda-calculus" for this purpose, and show how it can be interpreted in the Temperley-Lieb category.  ...  "composition by feedback") preserves planarity. - We shall also show how the Temperley-Lieb algebra provides a natural setting in which computation can be performed diagrammatically as geometric simplification  ...  Moreover, this categorical axiomatics can be presented in terms of a diagrammatic calculus which is both intuitive and effective, and can replace low-level computation with matrices by much more conceptual  ... 
arXiv:0910.2737v1 fatcat:egobh6ffxnamlidvfe5by45cba

A Recipe for Quantum Graphical Languages

Titouan Carette, Emmanuel Jeandel, Emanuela Merelli, Anuj Dawar, Artur Czumaj
2020 International Colloquium on Automata, Languages and Programming  
All those language share a common core structure we call Z^*-algebras.  ...  We do the same for linear relations and show that the calculus of [Bonchi et al., 2017] is essentially the unique one.  ...  for quantum computing up to isomorphism: the ZX-calculus, the ZW-calculus, the ZW-calculus, and the (trivial) ZZ-calculus.  ... 
doi:10.4230/lipics.icalp.2020.118 dblp:conf/icalp/CaretteJ20 fatcat:bv2gytb6ozfnznxwn2tbpssrgq

GLC actors, artificial chemical connectomes, topological issues and knots [article]

Marius Buliga, Louis H. Kauffman
2013 arXiv   pre-print
may graft lambda calculus and knot diagrammatics.  ...  Based on graphic lambda calculus, we propose a program for a new model of asynchronous distributed computing, inspired from Hewitt Actor Model, as well as several investigation paths, concerning how one  ...  from PSS for finding ways to make this type of computing real.  ... 
arXiv:1312.4333v1 fatcat:tp4odm624bgtdn5a7pajlgouje

Diagram Chase in Relational System Development

Michael Ebert, Georg Struth
2005 Electronical Notes in Theoretical Computer Science  
Diagrams are endowed with a simple algebraic semantics that imposes a convenient balance between expressive and algorithmic power.  ...  We propose diagrammatic techniques for visualizing relational reasoning in formal methods like B or Z; in particular for induction and coinduction.  ...  Acknowledgment: The authors would like to thank the anonymous referees and the participants of the VLFM'04 for valuable discussions.  ... 
doi:10.1016/j.entcs.2004.08.049 fatcat:gwar6lxgabhinar2m2yq3zgmla

Natural logic is diagrammatic reasoning about mental models

John F. Sowa
2020 Procedia Computer Science  
A combination of Peirce's graph logic, Polya's heuristics, and Euclid's diagrams is a better candidate for a natural logic than any algebraic formalism.  ...  For over two millennia, Euclid's diagrammatic methods set the standard for mathematical rigor.  ...  They establish the feasibility of a diagrammatic system of reasoning equivalent to the first-order predicate calculus.  ... 
doi:10.1016/j.procs.2020.02.217 fatcat:bco2uoijjje5tdtwgayeyv4fu4

A Diagrammatic Axiomatisation of Fermionic Quantum Circuits

Amar Hadzihasanovic, Giovanni De Felice, Kang Feng Ng, Michael Wagner
2018 International Conference on Rewriting Techniques and Applications  
We introduce the fermionic ZW calculus, a string-diagrammatic language for fermionic quantum computing (FQC).  ...  We conclude by showing, as an example, how the statistics of a fermionic Mach-Zehnder interferometer can be calculated in the diagrammatic language.  ...  Introduction The ZW calculus is a string-diagrammatic language for qubit quantum computing, introduced by the first author in [16] .  ... 
doi:10.4230/lipics.fscd.2018.17 dblp:conf/rta/HadzihasanovicF18 fatcat:vndkk7qcufbnbfij4japvw56jm

Quantum picturalism

Bob Coecke
2010 Contemporary physics (Print)  
In this review we present steps towards a diagrammatic 'high-level' alternative for the Hilbert space formalism, one which appeals to our intuition.  ...  The underlying mathematical foundation of this high-level diagrammatic formalism relies on so-called monoidal categories, a product of a fairly recent development in mathematics.  ...  Formalism: Categories as a Canvas for Quantum Foundations', and by an EC-FP6-STREP entitled 'Foundational Structures in Quantum Information and Computation'.  ... 
doi:10.1080/00107510903257624 fatcat:fhnyqqrj5fatpoikyfaykyrot4

Anaphora and Ellipsis in Lambek Calculus with a Relevant Modality: Syntax and Semantics [article]

Lachlan McPheat, Gijs Wijnholds, Mehrnoosh Sadrzadeh, Adriana Correia, Alexis Toumi
2021 arXiv   pre-print
What all these calculi add to Lambek calculus is a copying and moving behaviour.  ...  Lambek calculus with a relevant modality !𝐋^* of arXiv:1601.06303 syntactically resolves parasitic gaps in natural language.  ...  L * through the use of Fermionic tensor algebras and in effect infinitely many linear copying maps.  ... 
arXiv:2110.10641v1 fatcat:ympbsfnr4rbdpdsnmpsednj52e

A diagrammatic calculus of fermionic quantum circuits [article]

Giovanni de Felice, Amar Hadzihasanovic, Kang Feng Ng
2019 arXiv   pre-print
We introduce the fermionic ZW calculus, a string-diagrammatic language for fermionic quantum computing (FQC).  ...  We conclude by giving a diagrammatic treatment of the dual-rail encoding, a standard method in optical quantum computing used to perform universal quantum computation.  ...  Introduction The ZW calculus is a string-diagrammatic language for qubit quantum computing, introduced by the second author in [Had15] .  ... 
arXiv:1801.01231v4 fatcat:s5hq74tsqbeyfj7yjbs6dlvuly

The GHZ/W-calculus contains rational arithmetic

Bob Coecke, Aleks Kissinger, Alex Merry, Shibdas Roy
2011 Electronic Proceedings in Theoretical Computer Science  
Graphical calculi for representing interacting quantum systems serve a number of purposes: compositionally, intuitive graphical reasoning, and a logical underpinning for automation.  ...  One such calculus takes the GHZ and W states as its basic generators.  ...  calculus [4] via the Penrose-Joyal-Street diagrammatic calculus for symmetric monoidal categories [18, 15] , augmented with Kelly and Laplaza's coherence result for compact categories, and Lack's work  ... 
doi:10.4204/eptcs.52.4 fatcat:omuthim3gfanpacjj4ayr425km

Collected Papers of Charles Sanders Peirce.--Volume IV: The Simplest Mathematics. Charles Hartshorne , Paul Weiss

W. V. Quine
1935 Isis  
Under the title “‘ The Simplest Mathematics,” this volume assembles a variety of writings, mostly hitherto unpublished, in whose motley subject matter are represented algebraic and diagrammatic logic,  ...  He defines a cyclic system in effect as a couple (a,R) such that (i) every member of the class a has the relation R to some member of a, but (ii) no proper part REVIEWS 553 a of a fulfills (i).  ... 
doi:10.1086/346931 fatcat:cgjrkune2fgsvcburqx7idktwa
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