New scientific paradigms typically consist of an expansion of the conceptual language with which we describe the world. Over the past decade, theoretical physics and quantum information theory have turned to category theory to model and reason about quantum protocols. This new use of categorical and algebraic tools allows a more conceptual and insightful expression of elementary events such as measurements, teleportation and entanglement operations, that were obscured in previous formalisms.
Recent work in natural language semantics has begun to use these categorical methods to relate grammatical analysis and semantic representations in a unified framework for analysing language meaning, and learning meaning from a corpus. A growing body of literature on the use of categorical methods in quantum information theory and computational linguistics shows both the need and opportunity for new research on the relation between these categorical methods and the abstract notion of information flow.
This book supplies an overview of how categorical methods are used to model information flow in both physics and linguistics. It serves as an introduction to this interdisciplinary research, and provides a basis for future research and collaboration between the different communities interested in applying category theoretic methods to their domain's open problems.

nexusstc/Quantum Physics and Linguistics: A Compositional, Diagrammatic Discourse/aa78f94af74caa161a770dce84cd567e.pdf

lgli/Quantum Physics and Linguistics - A Compositional, Diagrammatic Discourse.pdf

lgrsnf/Quantum Physics and Linguistics - A Compositional, Diagrammatic Discourse.pdf

zlib/Linguistics/Chris Heunen, Mehrnoosh Sadrzadeh, Edward Grefenstette, (eds.)/Quantum Physics and Linguistics: A Compositional, Diagrammatic Discourse_2515349.pdf

Christiaan Johan Marie Heunen; Mehrnoosh Sadrzadeh; Edward Grefenstette

ed. by Chris Heunen, Mehrnoosh Sadrzadeh, Edward Grefenstette

Heunen, Chris, Sadrzadeh, Mehrnoosh, Grefenstette, Edward

Oxford Institute for Energy Studies

German Historical Institute London

OUP Oxford

First edition, Oxford, United Kingdom, 2013

United Kingdom and Ireland, United Kingdom

Oxford University Press USA, Oxford, 2013

Oxford, cop. 2013

1, PS, 2013

0

lg1322541

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Includes bibliographical references and index.

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Preface 6
Contents 18
1 An Alternative Gospel of Structure: Order, Composition,Processes 22
1.1 Order 25
1.2 Orders and composition 30
1.3 Processes witnessing existence 36
1.4 Conclusion 42
2 Sorne Graphical Aspects of Frobenius Algebras 43
2.1 Introduction 43
2.2 Graphical calculus 45
2.3 Frobenius and Hopf algebras 52
2.4 A few pointers to further litera ture 67
3 A Graphical Approach to Measurement-BasedQuantum Computing 71
3.1 Introduction 71
3.2 The rudiments of quantum computing 75
3.3 Observables and strong complementarity 78
3.4 The zx-calculus 86
3.5 The measurement calculus 98
3.6 Determinism and flow 102
3.7 Conclusions 109
4 Quantum Groups and Braided Algebra 111
4.1 Introduction 111
4.2 Braided categories 113
4.3 Braided groups 117
4.4 Applications ofbraided groups 125
4.5 Braided-Lie algebras 130
4.6 Diagrammatic geometry 133
5 Hopf Algebras-Variant Notions and ReconstructionTheorems 136
5.1 Introduction 136
5.2 Preliminaries 138
5.3 Bialgebras and Hopf algebras in monoidal categories 140
5.4 Reconstruction theorems 148
5.5 Variations on the notion of Hopf algebra 153
6 Modular Categories 167
6.1 Introduction 167
6.2 Categories 168
6.3 Tensor categories 169
6.4 Braided tensor categories 176
6.5 Modular categories 187
6.6 Modularization of pre-modular categories: generalizations 196
6.7 The braided centre of a fusion category 198
6.8 The Witt group of modular categories 201
7 Scalars, Monads, and Categories 205
7.1 Introduction 205
7.2 Preliminaries 207
7.3 Monoids 210
7.4 Additive monads 217
7.5 Semirings and monads 226
7.6 Semirings and Lawvere theories 231
7.7 Semirings with involutions 234
7.8 Conclusions 237
8 Types and Forgetfulness in Categorical Linguistics andQuantum Mechanics 238
8.1 Introduction 238
8.2 Introducing typing to mo dels of meaning 239
8.3 What is a type? 239
8.4 Comparing words versus comparing sentences 250
8.5 Inner products, evaluation, and inverses 250
8.6 Do es evaluation preserve inner products? 251
8.7 How to type connectives? 254
8.8 Self-similarity, categorically 259
8.9 Self-similarity and lax Frobenius algebras 265
8.10 Classical structures 266
8.11 A self-similar structure familiar in logic (and linguistics) 269
Appendix 8.A Order-preserving bijections N ~ N I±J N as interiorpoints of the Cantor set 270
Appendix 8.B Isbell's argument in a general setting 272
9 From Sentence to Concept 273
9.1 Introduction 273
9.2 Notations, basic properties 275
9.3 Two-sorted first-order logic in compact closed categories 282
9.4 Semantics via pregroup grammars 288
9.5 Compositional semantics in concept spaces 295
9.6 Conclusion 302
10 ProofNets for the Lambek-Grishin Calculus 304
10.1 Background, motivation 304
10.2 Display sequent calcul us and pro of nets 309
10.3 Proof nets and focused display calculus 322
10.4 Conclusions 340
11 Algebras over a Field and Semantics for Context-BasedReasoning 342
11.1 Introduction 342
11.2 Theory of meaning 343
11.3 From logical for ms to algebra 346
11.4 Conclusion 353
12 Distributional Semantic Models 354
12.1 Introduction 354
12.2 Compositional syntax and semantics 355
12.3 Harris's notion of distribution 357
12.4 Firth's notion of collocation 359
12.5 Distributional semantics 360
12.6 Compositional distributional semantics 364
12.7 A philosophical digression 365
12.8 Experiments in compositional interpretation 368
12.9 An empirical test of distributional compositionality 374
12.10 Compositionality vs disambiguation 376
12.11 Conclusion: a new pro gram for compositionaldistributional semantics 378
13 Type-Driven Syntax and Semantics for ComposingMeaning Vectors 380
13.1 Introduction 380
13.2 Syntactic types and pregroup grammars 383
13.3 Semantic types and tensor products 386
13.4 Composition for an example sentence space 389
13.5 A real sentence space 393
13.6 Conclusion and further work 397
References 399
Index 428

An interdisciplinary attempt to bring together physicists and linguists who use the same compositional mathematical methods. Although seemingly unrelated, due to the complexity and dynamics of the compound phenomena they aim to model, and also advances in their high level methods, these fields have come to share a common mathematical structure.

2015-03-20