Substructural type systems, such as affine (and linear) type systems, are type systems which impose restrictions on copying (and discarding) of variables, and they have found many applications in computer science, including quantum programming. We describe one linear and one affine type systems and we formulate abstract categorical models for both of them which are sound and computationally adequate. We also show, under basic assumptions, that interpreting lambda abstractions via a monoidal closed structure (a popular method for linear type systems) necessarily leads to degenerate and inadequate models for call-by-value affine type systems, so we avoid doing this in our categorical treatment, where a solution to this problem is clearly identified. Our categorical models are more general than linear/non-linear models used to study linear logic and we present a homogeneous categorical account of both linear and affine type systems in a call-by-value setting. We also give examples with many concrete models, including classical and quantum ones.
{"title":"Computational Adequacy for Substructural Lambda Calculi","authors":"Vladimir Zamdzhiev","doi":"10.4204/EPTCS.333.22","DOIUrl":"https://doi.org/10.4204/EPTCS.333.22","url":null,"abstract":"Substructural type systems, such as affine (and linear) type systems, are type systems which impose restrictions on copying (and discarding) of variables, and they have found many applications in computer science, including quantum programming. We describe one linear and one affine type systems and we formulate abstract categorical models for both of them which are sound and computationally adequate. We also show, under basic assumptions, that interpreting lambda abstractions via a monoidal closed structure (a popular method for linear type systems) necessarily leads to degenerate and inadequate models for call-by-value affine type systems, so we avoid doing this in our categorical treatment, where a solution to this problem is clearly identified. Our categorical models are more general than linear/non-linear models used to study linear logic and we present a homogeneous categorical account of both linear and affine type systems in a call-by-value setting. We also give examples with many concrete models, including classical and quantum ones.","PeriodicalId":11810,"journal":{"name":"essentia law Merchant Shipping Act 1995","volume":"47 1","pages":"322-334"},"PeriodicalIF":0.0,"publicationDate":"2020-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77577390","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Matthew Wilson, James Hefford, G. Boisseau, Vincent Wang
We build upon our recently introduced concept of an update structure to show that they are a generalisation of very-well-behaved lenses, that is, there is a bijection between a strict subset of update structures and vwb lenses in cartesian categories. We then begin to investigate the zoo of possible update structures. We show that update structures survive decoherence and are sufficiently general to capture quantum observables, pinpointing the additional assumptions required to make the two coincide. In doing so, we shift the focus from dagger-special commutative Frobenius algebras to interacting (co)magma (co)module pairs, showing that the algebraic properties of the (co)multiplication arise from the module-comodule interaction, rather than direct assumptions about the magma comagma pair. Thus this work is of foundational interest as update structures form a strictly more general class of algebraic objects, the taming of which promises to illuminate novel relationships between separately studied mathematical structures.
{"title":"The Safari of Update Structures: Visiting the Lens and Quantum Enclosures","authors":"Matthew Wilson, James Hefford, G. Boisseau, Vincent Wang","doi":"10.4204/EPTCS.333.1","DOIUrl":"https://doi.org/10.4204/EPTCS.333.1","url":null,"abstract":"We build upon our recently introduced concept of an update structure to show that they are a generalisation of very-well-behaved lenses, that is, there is a bijection between a strict subset of update structures and vwb lenses in cartesian categories. We then begin to investigate the zoo of possible update structures. We show that update structures survive decoherence and are sufficiently general to capture quantum observables, pinpointing the additional assumptions required to make the two coincide. In doing so, we shift the focus from dagger-special commutative Frobenius algebras to interacting (co)magma (co)module pairs, showing that the algebraic properties of the (co)multiplication arise from the module-comodule interaction, rather than direct assumptions about the magma comagma pair. Thus this work is of foundational interest as update structures form a strictly more general class of algebraic objects, the taming of which promises to illuminate novel relationships between separately studied mathematical structures.","PeriodicalId":11810,"journal":{"name":"essentia law Merchant Shipping Act 1995","volume":"76 1","pages":"1-18"},"PeriodicalIF":0.0,"publicationDate":"2020-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80713427","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Morphisms in a monoidal category are usually interpreted as processes, and graphically depicted as square boxes. In practice, we are faced with the problem of interpreting what non-square boxes ought to represent in terms of the monoidal category and, more importantly, how should they be composed. Examples of this situation include lenses or learners. We propose a description of these non-square boxes, which we call open diagrams, using the monoidal bicategory of profunctors. A graphical coend calculus can then be used to reason about open diagrams and their compositions.
{"title":"Open Diagrams via Coend Calculus","authors":"Mario Rom'an","doi":"10.4204/EPTCS.333.5","DOIUrl":"https://doi.org/10.4204/EPTCS.333.5","url":null,"abstract":"Morphisms in a monoidal category are usually interpreted as processes, and graphically depicted as square boxes. In practice, we are faced with the problem of interpreting what non-square boxes ought to represent in terms of the monoidal category and, more importantly, how should they be composed. Examples of this situation include lenses or learners. We propose a description of these non-square boxes, which we call open diagrams, using the monoidal bicategory of profunctors. A graphical coend calculus can then be used to reason about open diagrams and their compositions.","PeriodicalId":11810,"journal":{"name":"essentia law Merchant Shipping Act 1995","volume":"66 1","pages":"65-78"},"PeriodicalIF":0.0,"publicationDate":"2020-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87115479","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
F. Genovese, A. Gryzlov, Jelle Herold, A. Knispel, Marco Perone, Erik Post, André Videla
We introduce idris-ct, a Idris library providing verified type definitions of categorical concepts. idris-ct strives to be a bridge between academy and industry, catering both to category theorists who want to implement and try their ideas in a practical environment and to businesses and engineers who care about formalization with category theory: It is inspired by similar libraries developed for theorem proving but remains very practical, being aimed at software production in business. Nevertheless, the use of dependent types allows for a formally correct implementation of categorical concepts, so that guarantees can be made on software properties.
{"title":"idris-ct: A Library to do Category Theory in Idris","authors":"F. Genovese, A. Gryzlov, Jelle Herold, A. Knispel, Marco Perone, Erik Post, André Videla","doi":"10.4204/EPTCS.323.16","DOIUrl":"https://doi.org/10.4204/EPTCS.323.16","url":null,"abstract":"We introduce idris-ct, a Idris library providing verified type definitions of categorical concepts. idris-ct strives to be a bridge between academy and industry, catering both to category theorists who want to implement and try their ideas in a practical environment and to businesses and engineers who care about formalization with category theory: It is inspired by similar libraries developed for theorem proving but remains very practical, being aimed at software production in business. Nevertheless, the use of dependent types allows for a formally correct implementation of categorical concepts, so that guarantees can be made on software properties.","PeriodicalId":11810,"journal":{"name":"essentia law Merchant Shipping Act 1995","volume":"1 1","pages":"246-254"},"PeriodicalIF":0.0,"publicationDate":"2019-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80425975","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
John S. Nolan, Blake S. Pollard, Spencer Breiner, D. Anand, E. Subrahmanian
The problem of integrating multiple overlapping models and data is pervasive in engineering, though often implicit. We consider this issue of model management in the context of the electrical power grid as it transitions towards a modern 'Smart Grid.' We present a methodology for specifying, managing, and reasoning within multiple models of distributed energy resources (DERs), entities which produce, consume, or store power, using categorical databases and symmetric monoidal categories. Considering the problem of distributing power on the grid in the presence of DERs, we show how to connect a generic problem specification with implementation-specific numerical solvers using the paradigm of categorical databases.
{"title":"Compositional Models for Power Systems","authors":"John S. Nolan, Blake S. Pollard, Spencer Breiner, D. Anand, E. Subrahmanian","doi":"10.4204/EPTCS.323.10","DOIUrl":"https://doi.org/10.4204/EPTCS.323.10","url":null,"abstract":"The problem of integrating multiple overlapping models and data is pervasive in engineering, though often implicit. We consider this issue of model management in the context of the electrical power grid as it transitions towards a modern 'Smart Grid.' We present a methodology for specifying, managing, and reasoning within multiple models of distributed energy resources (DERs), entities which produce, consume, or store power, using categorical databases and symmetric monoidal categories. Considering the problem of distributing power on the grid in the presence of DERs, we show how to connect a generic problem specification with implementation-specific numerical solvers using the paradigm of categorical databases.","PeriodicalId":11810,"journal":{"name":"essentia law Merchant Shipping Act 1995","volume":"24 1","pages":"149-160"},"PeriodicalIF":0.0,"publicationDate":"2019-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84750460","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
D. Pastor, Erwan Beurier, A. Ehresmann, R. Waldeck
Motivated by the concept of degeneracy in biology [3], we establish a first connection between the Multiplicity Principle [4, 5] and mathematical statistics. Specifically , we exhibit two families of tests that satisfy this principle to achieve the detection of a signal in noise.
{"title":"Interfacing biology, category theory and mathematical statistics","authors":"D. Pastor, Erwan Beurier, A. Ehresmann, R. Waldeck","doi":"10.4204/EPTCS.323.9","DOIUrl":"https://doi.org/10.4204/EPTCS.323.9","url":null,"abstract":"Motivated by the concept of degeneracy in biology [3], we establish a first connection between the Multiplicity Principle [4, 5] and mathematical statistics. Specifically , we exhibit two families of tests that satisfy this principle to achieve the detection of a signal in noise.","PeriodicalId":11810,"journal":{"name":"essentia law Merchant Shipping Act 1995","volume":"133 1","pages":"136-148"},"PeriodicalIF":0.0,"publicationDate":"2019-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73655371","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Micah Halter, Christine Herlihy, James P. Fairbanks
Scientists construct and analyze computational models to understand the world. That understanding comes from efforts to augment, combine, and compare models of related phenomena. We propose SemanticModels.jl, a system that leverages techniques from static and dynamic program analysis to process executable versions of scientific models to perform such metamodeling tasks. By framing these metamodeling tasks as metaprogramming problems, SemanticModels.jl enables writing programs that generate and expand models. To this end, we present a category theory-based framework for defining metamodeling tasks, and extracting semantic information from model implementations, and show how this framework can be used to enhance scientific workflows in a working case study.
{"title":"A Compositional Framework for Scientific Model Augmentation","authors":"Micah Halter, Christine Herlihy, James P. Fairbanks","doi":"10.4204/EPTCS.323.12","DOIUrl":"https://doi.org/10.4204/EPTCS.323.12","url":null,"abstract":"Scientists construct and analyze computational models to understand the world. That understanding comes from efforts to augment, combine, and compare models of related phenomena. We propose SemanticModels.jl, a system that leverages techniques from static and dynamic program analysis to process executable versions of scientific models to perform such metamodeling tasks. By framing these metamodeling tasks as metaprogramming problems, SemanticModels.jl enables writing programs that generate and expand models. To this end, we present a category theory-based framework for defining metamodeling tasks, and extracting semantic information from model implementations, and show how this framework can be used to enhance scientific workflows in a working case study.","PeriodicalId":11810,"journal":{"name":"essentia law Merchant Shipping Act 1995","volume":"43 1","pages":"172-182"},"PeriodicalIF":0.0,"publicationDate":"2019-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75563393","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A compact T-algebra is an initial T-algebra whose inverse is a final T-coalgebra. Functors with this property are said to be algebraically compact. This is a very strong property used in programming semantics which allows one to interpret recursive datatypes involving mixed-variance functors, such as function space. The construction of compact algebras is usually done in categories with a zero object where some form of a limit-colimit coincidence exists. In this paper we consider a more abstract approach and show how one can construct compact algebras in categories which have neither a zero object, nor a (standard) limit-colimit coincidence by reflecting the compact algebras from categories which have both. In doing so, we provide a constructive description of a large class of algebraically compact functors (satisfying a compositionality principle) and show our methods compare quite favorably to other approaches from the literature.
{"title":"Reflecting Algebraically Compact Functors","authors":"Vladimir Zamdzhiev","doi":"10.4204/EPTCS.323.2","DOIUrl":"https://doi.org/10.4204/EPTCS.323.2","url":null,"abstract":"A compact T-algebra is an initial T-algebra whose inverse is a final T-coalgebra. Functors with this property are said to be algebraically compact. This is a very strong property used in programming semantics which allows one to interpret recursive datatypes involving mixed-variance functors, such as function space. The construction of compact algebras is usually done in categories with a zero object where some form of a limit-colimit coincidence exists. In this paper we consider a more abstract approach and show how one can construct compact algebras in categories which have neither a zero object, nor a (standard) limit-colimit coincidence by reflecting the compact algebras from categories which have both. In doing so, we provide a constructive description of a large class of algebraically compact functors (satisfying a compositionality principle) and show our methods compare quite favorably to other approaches from the literature.","PeriodicalId":11810,"journal":{"name":"essentia law Merchant Shipping Act 1995","volume":"6 1","pages":"15-23"},"PeriodicalIF":0.0,"publicationDate":"2019-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86193198","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We present a complete categorical axiomatization of a wide class of dataflow programs. This gives a three-dimensional diagrammatic language for workflows, more expressive than the directed acyclic graphs generally used for this purpose. This calls for an implementation of these representations in data transformation tools.
{"title":"A complete language for faceted dataflow programs","authors":"Antonin Delpeuch","doi":"10.4204/EPTCS.323.1","DOIUrl":"https://doi.org/10.4204/EPTCS.323.1","url":null,"abstract":"We present a complete categorical axiomatization of a wide class of dataflow programs. This gives a three-dimensional diagrammatic language for workflows, more expressive than the directed acyclic graphs generally used for this purpose. This calls for an implementation of these representations in data transformation tools.","PeriodicalId":11810,"journal":{"name":"essentia law Merchant Shipping Act 1995","volume":"211 1","pages":"1-14"},"PeriodicalIF":0.0,"publicationDate":"2019-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88672091","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Taking advantage of a recently discovered associativity property of rule compositions, we extend the classical concurrency theory for rewriting systems over adhesive categories. We introduce the notion of tracelets, which are defined as minimal derivation traces that universally encode sequential compositions of rewriting rules. Tracelets are compositional, capture the causality of equivalence classes of traditional derivation traces, and intrinsically suggest a clean mathematical framework for the definition of various notions of abstractions of traces. We illustrate these features by introducing a first prototype for a framework of tracelet analysis, which as a key application permits to formulate a first-of-its-kind algorithm for the static generation of minimal derivation traces with prescribed terminal events.
{"title":"Tracelets and Tracelet Analysis Of Compositional Rewriting Systems","authors":"Nicolas Behr","doi":"10.4204/EPTCS.323.4","DOIUrl":"https://doi.org/10.4204/EPTCS.323.4","url":null,"abstract":"Taking advantage of a recently discovered associativity property of rule compositions, we extend the classical concurrency theory for rewriting systems over adhesive categories. We introduce the notion of tracelets, which are defined as minimal derivation traces that universally encode sequential compositions of rewriting rules. Tracelets are compositional, capture the causality of equivalence classes of traditional derivation traces, and intrinsically suggest a clean mathematical framework for the definition of various notions of abstractions of traces. We illustrate these features by introducing a first prototype for a framework of tracelet analysis, which as a key application permits to formulate a first-of-its-kind algorithm for the static generation of minimal derivation traces with prescribed terminal events.","PeriodicalId":11810,"journal":{"name":"essentia law Merchant Shipping Act 1995","volume":"102 1","pages":"44-71"},"PeriodicalIF":0.0,"publicationDate":"2019-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80569547","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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