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A Diagrammatic Calculus for Algebraic Effects
[article]
2020
arXiv
pre-print
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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]
2022
arXiv
pre-print
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2022 pre-print arXiv:2201.13250v1 fatcat:vf42dcv2hjarhh7lkpsv5lbbsi
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]
2021
arXiv
pre-print
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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]
2015
arXiv
pre-print
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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]
2009
arXiv
pre-print
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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
2020
International Colloquium on Automata, Languages and Programming
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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]
2013
arXiv
pre-print
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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
2005
Electronical Notes in Theoretical Computer Science
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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
2020
Procedia Computer Science
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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
2018
International Conference on Rewriting Techniques and Applications
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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
2010
Contemporary physics (Print)
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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]
2021
arXiv
pre-print
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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]
2019
arXiv
pre-print
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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
2011
Electronic Proceedings in Theoretical Computer Science
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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
1935
Isis
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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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