Georgios Bakirtzis, Christina N. Vasilakopoulou, C. Fleming
Assuring the correct behavior of cyber-physical systems requires significant modeling effort, particularly during early stages of the engineering and design process when a system is not yet available for testing or verification of proper behavior. A primary motivation for `getting things right' in these early design stages is that altering the design is significantly less costly and more effective than when hardware and software have already been developed. Engineering cyber-physical systems requires the construction of several different types of models, each representing a different view, which include stakeholder requirements, system behavior, and the system architecture. Furthermore, each of these models can be represented at different levels of abstraction. Formal reasoning has improved the precision and expanded the available types of analysis in assuring correctness of requirements, behaviors, and architectures. However, each is usually modeled in distinct formalisms and corresponding tools. Currently, this disparity means that a system designer must manually check that the different models are in agreement. Manually editing and checking models is error prone, time consuming, and sensitive to any changes in the design of the models themselves. Wiring diagrams and related theory provide a means for formally organizing these different but related modeling views, resulting in a compositional modeling language for cyber-physical systems. Such a categorical language can make concrete the relationship between different model views, thereby managing complexity, allowing hierarchical decomposition of system models, and formally proving consistency between models.
{"title":"Compositional Cyber-Physical Systems Modeling","authors":"Georgios Bakirtzis, Christina N. Vasilakopoulou, C. Fleming","doi":"10.4204/EPTCS.333.9","DOIUrl":"https://doi.org/10.4204/EPTCS.333.9","url":null,"abstract":"Assuring the correct behavior of cyber-physical systems requires significant modeling effort, particularly during early stages of the engineering and design process when a system is not yet available for testing or verification of proper behavior. A primary motivation for `getting things right' in these early design stages is that altering the design is significantly less costly and more effective than when hardware and software have already been developed. Engineering cyber-physical systems requires the construction of several different types of models, each representing a different view, which include stakeholder requirements, system behavior, and the system architecture. Furthermore, each of these models can be represented at different levels of abstraction. Formal reasoning has improved the precision and expanded the available types of analysis in assuring correctness of requirements, behaviors, and architectures. However, each is usually modeled in distinct formalisms and corresponding tools. Currently, this disparity means that a system designer must manually check that the different models are in agreement. Manually editing and checking models is error prone, time consuming, and sensitive to any changes in the design of the models themselves. Wiring diagrams and related theory provide a means for formally organizing these different but related modeling views, resulting in a compositional modeling language for cyber-physical systems. Such a categorical language can make concrete the relationship between different model views, thereby managing complexity, allowing hierarchical decomposition of system models, and formally proving consistency between models.","PeriodicalId":11810,"journal":{"name":"essentia law Merchant Shipping Act 1995","volume":"57 1","pages":"125-138"},"PeriodicalIF":0.0,"publicationDate":"2021-01-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89144432","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 consider certain decision problems for the free model of the theory of Cartesian monoids. We introduce a model of computation based on the notion of a single stack one-way PDA due to Ginsburg, Greibach and Harrison. This model allows us to solve problems such as (1) Given a finite set B of elements and an element F, is F a product of members of B? (2) Is the submonoid generated by the finite set B infinite? for certain fragments of the free Cartesian monoid. These fragments include the submonoid of right invertible elements and so our results apply to the Thompson-Higman groups.
{"title":"Products in a Category with Only One Object","authors":"R. Statman","doi":"10.4204/EPTCS.333.24","DOIUrl":"https://doi.org/10.4204/EPTCS.333.24","url":null,"abstract":"We consider certain decision problems for the free model of the theory of Cartesian monoids. We introduce a model of computation based on the notion of a single stack one-way PDA due to Ginsburg, Greibach and Harrison. This model allows us to solve problems such as (1) Given a finite set B of elements and an element F, is F a product of members of B? (2) Is the submonoid generated by the finite set B infinite? for certain fragments of the free Cartesian monoid. These fragments include the submonoid of right invertible elements and so our results apply to the Thompson-Higman groups.","PeriodicalId":11810,"journal":{"name":"essentia law Merchant Shipping Act 1995","volume":"39 1","pages":"347-353"},"PeriodicalIF":0.0,"publicationDate":"2021-01-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88523594","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 point process on a space is a random bag of elements of that space. In this paper we explore programming with point processes in a monadic style. To this end we identify point processes on a space X with probability measures of bags of elements in X. We describe this view of point processes using the composition of the Giry and bag monads on the category of measurable spaces and functions and prove that this composition also forms a monad using a distributive law for monads. Finally, we define a morphism from a point process to its intensity measure, and show that this is a monad morphism. A special case of this monad morphism gives us Wald's Lemma, an identity used to calculate the expected value of the sum of a random number of random variables. Using our monad we define a range of point processes and point process operations and compositionally compute their corresponding intensity measures using the monad morphism.
{"title":"A Monad for Probabilistic Point Processes","authors":"Swaraj Dash, S. Staton","doi":"10.4204/EPTCS.333.2","DOIUrl":"https://doi.org/10.4204/EPTCS.333.2","url":null,"abstract":"A point process on a space is a random bag of elements of that space. In this paper we explore programming with point processes in a monadic style. To this end we identify point processes on a space X with probability measures of bags of elements in X. We describe this view of point processes using the composition of the Giry and bag monads on the category of measurable spaces and functions and prove that this composition also forms a monad using a distributive law for monads. Finally, we define a morphism from a point process to its intensity measure, and show that this is a monad morphism. A special case of this monad morphism gives us Wald's Lemma, an identity used to calculate the expected value of the sum of a random number of random variables. Using our monad we define a range of point processes and point process operations and compositionally compute their corresponding intensity measures using the monad morphism.","PeriodicalId":11810,"journal":{"name":"essentia law Merchant Shipping Act 1995","volume":"20 1","pages":"19-32"},"PeriodicalIF":0.0,"publicationDate":"2021-01-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84021069","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}
With the increased interest in machine learning, and deep learning in particular, the use of automatic differentiation has become more wide-spread in computation. There have been two recent developments to provide the theoretical support for this types of structure. One approach, due to Abadi and Plotkin, provides a simple differential programming language. Another approach is the notion of a reverse differential category. In the present paper we bring these two approaches together. In particular, we show how an extension of reverse derivative categories models Abadi and Plotkin's language, and describe how this categorical model allows one to consider potential improvements to the operational semantics of the language.
{"title":"Categorical semantics of a simple differential programming language","authors":"G. Cruttwell, J. Gallagher, D. Pronk","doi":"10.4204/EPTCS.333.20","DOIUrl":"https://doi.org/10.4204/EPTCS.333.20","url":null,"abstract":"With the increased interest in machine learning, and deep learning in particular, the use of automatic differentiation has become more wide-spread in computation. There have been two recent developments to provide the theoretical support for this types of structure. One approach, due to Abadi and Plotkin, provides a simple differential programming language. Another approach is the notion of a reverse differential category. In the present paper we bring these two approaches together. In particular, we show how an extension of reverse derivative categories models Abadi and Plotkin's language, and describe how this categorical model allows one to consider potential improvements to the operational semantics of the language.","PeriodicalId":11810,"journal":{"name":"essentia law Merchant Shipping Act 1995","volume":"5 1","pages":"289-310"},"PeriodicalIF":0.0,"publicationDate":"2021-01-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86532788","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 introduce Reverse Derivative Ascent: a categorical analogue of gradient based methods for machine learning. Our algorithm is defined at the level of so-called reverse differential categories. It can be used to learn the parameters of models which are expressed as morphisms of such categories. Our motivating example is boolean circuits: we show how our algorithm can be applied to such circuits by using the theory of reverse differential categories. Note our methodology allows us to learn the parameters of boolean circuits directly, in contrast to existing binarised neural network approaches. Moreover, we demonstrate its empirical value by giving experimental results on benchmark machine learning datasets.
{"title":"Reverse Derivative Ascent: A Categorical Approach to Learning Boolean Circuits","authors":"Paul W. Wilson, F. Zanasi","doi":"10.4204/EPTCS.333.17","DOIUrl":"https://doi.org/10.4204/EPTCS.333.17","url":null,"abstract":"We introduce Reverse Derivative Ascent: a categorical analogue of gradient based methods for machine learning. Our algorithm is defined at the level of so-called reverse differential categories. It can be used to learn the parameters of models which are expressed as morphisms of such categories. Our motivating example is boolean circuits: we show how our algorithm can be applied to such circuits by using the theory of reverse differential categories. Note our methodology allows us to learn the parameters of boolean circuits directly, in contrast to existing binarised neural network approaches. Moreover, we demonstrate its empirical value by giving experimental results on benchmark machine learning datasets.","PeriodicalId":11810,"journal":{"name":"essentia law Merchant Shipping Act 1995","volume":"50 1","pages":"247-260"},"PeriodicalIF":0.0,"publicationDate":"2021-01-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80706755","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}
Mereology is the study of parts and the relationships that hold between them. We introduce a behavioral approach to mereology, in which systems and their parts are known only by the types of behavior they can exhibit. Our discussion is formally topos-theoretic, and agnostic to the topos, providing maximal generality; however, by using only its internal logic we can hide the details and readers may assume a completely elementary set-theoretic discussion. We consider the relationship between various parts of a whole in terms of how behavioral constraints are passed between them, and give an inter-modal logic that generalizes the usual alethic modalities in the setting of symmetric accessibility.
{"title":"Behavioral Mereology: A Modal Logic for Passing Constraints","authors":"Brendan Fong, D. J. Myers, David I. Spivak","doi":"10.4204/EPTCS.333.19","DOIUrl":"https://doi.org/10.4204/EPTCS.333.19","url":null,"abstract":"Mereology is the study of parts and the relationships that hold between them. We introduce a behavioral approach to mereology, in which systems and their parts are known only by the types of behavior they can exhibit. Our discussion is formally topos-theoretic, and agnostic to the topos, providing maximal generality; however, by using only its internal logic we can hide the details and readers may assume a completely elementary set-theoretic discussion. We consider the relationship between various parts of a whole in terms of how behavioral constraints are passed between them, and give an inter-modal logic that generalizes the usual alethic modalities in the setting of symmetric accessibility.","PeriodicalId":11810,"journal":{"name":"essentia law Merchant Shipping Act 1995","volume":"8 1","pages":"276-288"},"PeriodicalIF":0.0,"publicationDate":"2021-01-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85863269","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 define a categorical notion of cybernetic system as a dynamical realisation of a generalized open game, along with a coherence condition. We show that this notion captures a wide class of cybernetic systems in computational neuroscience and statistical machine learning, exposes their compositional structure, and gives an abstract justification for the bidirectional structure empirically observed in cortical circuits. Our construction is built on the observation that Bayesian updates compose optically, a fact which we prove along the way, via a fibred category of state-dependent stochastic channels.
{"title":"Cyber Kittens, or Some First Steps Towards Categorical Cybernetics","authors":"T. S. C. Smithe","doi":"10.4204/EPTCS.333.8","DOIUrl":"https://doi.org/10.4204/EPTCS.333.8","url":null,"abstract":"We define a categorical notion of cybernetic system as a dynamical realisation of a generalized open game, along with a coherence condition. We show that this notion captures a wide class of cybernetic systems in computational neuroscience and statistical machine learning, exposes their compositional structure, and gives an abstract justification for the bidirectional structure empirically observed in cortical circuits. Our construction is built on the observation that Bayesian updates compose optically, a fact which we prove along the way, via a fibred category of state-dependent stochastic channels.","PeriodicalId":11810,"journal":{"name":"essentia law Merchant Shipping Act 1995","volume":"53 1","pages":"108-124"},"PeriodicalIF":0.0,"publicationDate":"2021-01-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82222984","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}
It is argued in (Eklund et al., 2018) that the quantale [L,L] of sup-preserving endomaps of a complete lattice L is a Girard quantale exactly when L is completely distributive. We have argued in (Santocanale, 2020) that this Girard quantale structure arises from the dual quantale of inf-preserving endomaps of L via Raney's transforms and extends to a Girard quantaloid structure on the full subcategory of SLatt (the category of complete lattices and sup-preserving maps) whose objects are the completely distributive lattices. It is the goal of this talk to illustrate further this connection between the quantale structure, Raney's transforms, and complete distributivity. Raney's transforms are indeed mix maps in the isomix category SLatt and most of the theory can be developed relying on naturality of these maps. We complete then the remarks on cyclic elements of [L,L] developed in (Santocanale, 2020) by investigating its dualizing elements. We argue that if [L,L] has the structure a Frobenius quantale, that is, if it has a dualizing element, not necessarily a cyclic one, then L is once more completely distributive. It follows then from a general statement on involutive residuated lattices that there is a bijection between dualizing elements of [L,L] and automorphisms of L. Finally, we also argue that if L is finite and [L,L] is autodual, then L is distributive.
(Eklund et al.， 2018)认为，完全格L的超保持内映射的量子[L,L]恰好在L完全分布时是吉拉德量子。我们在(Santocanale, 2020)中提出，这种吉拉德量子结构是由L的保内映射的对偶量子通过Raney变换产生的，并扩展到SLatt的满子范畴(完全格和超保映射的范畴)上的吉拉德量子样结构，其对象是完全分布格。这次演讲的目的是进一步说明量子结构、兰尼变换和完全分布性之间的联系。拉尼变换实际上是等分混合映射，大部分理论都可以依靠这些映射的自然性来发展。然后，我们通过研究[L,L]的二元元素来完成(Santocanale, 2020)中关于[L,L]的循环元素的注释。我们论证了如果[L,L]具有Frobenius量子化结构，也就是说，如果它有一个对偶元，而不一定是一个循环元，那么L又是完全分布的。由对合剩格的一般论述可知[L,L]的对偶元与L的自同构之间存在双射。最后，我们还论证了如果L是有限的，且[L,L]是自对偶的，则L是分配的。
{"title":"Dualizing sup-preserving endomaps of a complete lattice","authors":"L. Santocanale","doi":"10.4204/EPTCS.333.23","DOIUrl":"https://doi.org/10.4204/EPTCS.333.23","url":null,"abstract":"It is argued in (Eklund et al., 2018) that the quantale [L,L] of sup-preserving endomaps of a complete lattice L is a Girard quantale exactly when L is completely distributive. We have argued in (Santocanale, 2020) that this Girard quantale structure arises from the dual quantale of inf-preserving endomaps of L via Raney's transforms and extends to a Girard quantaloid structure on the full subcategory of SLatt (the category of complete lattices and sup-preserving maps) whose objects are the completely distributive lattices. It is the goal of this talk to illustrate further this connection between the quantale structure, Raney's transforms, and complete distributivity. Raney's transforms are indeed mix maps in the isomix category SLatt and most of the theory can be developed relying on naturality of these maps. We complete then the remarks on cyclic elements of [L,L] developed in (Santocanale, 2020) by investigating its dualizing elements. We argue that if [L,L] has the structure a Frobenius quantale, that is, if it has a dualizing element, not necessarily a cyclic one, then L is once more completely distributive. It follows then from a general statement on involutive residuated lattices that there is a bijection between dualizing elements of [L,L] and automorphisms of L. Finally, we also argue that if L is finite and [L,L] is autodual, then L is distributive.","PeriodicalId":11810,"journal":{"name":"essentia law Merchant Shipping Act 1995","volume":"23 1","pages":"335-346"},"PeriodicalIF":0.0,"publicationDate":"2021-01-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90885553","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}
When designing plans in engineering, it is often necessary to consider attributes associated to objects, e.g. the location of a robot. Our aim in this paper is to incorporate attributes into existing categorical formalisms for planning, namely those based on symmetric monoidal categories and string diagrams. To accomplish this, we define a notion of a"symmetric monoidal category with attributes."This is a symmetric monoidal category in which objects are equipped with retrievable information and where the interactions between objects and information are governed by an"attribute structure."We discuss examples and semantics of such categories in the context of robotics to illustrate our definition.
{"title":"Symmetric Monoidal Categories with Attributes","authors":"Spencer Breiner, John S. Nolan","doi":"10.4204/EPTCS.333.3","DOIUrl":"https://doi.org/10.4204/EPTCS.333.3","url":null,"abstract":"When designing plans in engineering, it is often necessary to consider attributes associated to objects, e.g. the location of a robot. Our aim in this paper is to incorporate attributes into existing categorical formalisms for planning, namely those based on symmetric monoidal categories and string diagrams. To accomplish this, we define a notion of a\"symmetric monoidal category with attributes.\"This is a symmetric monoidal category in which objects are equipped with retrievable information and where the interactions between objects and information are governed by an\"attribute structure.\"We discuss examples and semantics of such categories in the context of robotics to illustrate our definition.","PeriodicalId":11810,"journal":{"name":"essentia law Merchant Shipping Act 1995","volume":"119 10 1","pages":"33-48"},"PeriodicalIF":0.0,"publicationDate":"2021-01-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90252496","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}
Applications of category theory often involve symmetric monoidal categories (SMCs), in which abstract processes or operations can be composed in series and parallel. However, in 2020 there remains a dearth of computational tools for working with SMCs. We present an "unbiased" approach to implementing symmetric monoidal categories, based on an operad of directed, acyclic wiring diagrams. Because the interchange law and other laws of a SMC hold identically in a wiring diagram, no rewrite rules are needed to compare diagrams. We discuss the mathematics of the operad of wiring diagrams, as well as its implementation in the software package Catlab.
{"title":"Wiring diagrams as normal forms for computing in symmetric monoidal categories","authors":"Evan Patterson, David I. Spivak, D. Vagner","doi":"10.4204/EPTCS.333.4","DOIUrl":"https://doi.org/10.4204/EPTCS.333.4","url":null,"abstract":"Applications of category theory often involve symmetric monoidal categories (SMCs), in which abstract processes or operations can be composed in series and parallel. However, in 2020 there remains a dearth of computational tools for working with SMCs. We present an \"unbiased\" approach to implementing symmetric monoidal categories, based on an operad of directed, acyclic wiring diagrams. Because the interchange law and other laws of a SMC hold identically in a wiring diagram, no rewrite rules are needed to compare diagrams. We discuss the mathematics of the operad of wiring diagrams, as well as its implementation in the software package Catlab.","PeriodicalId":11810,"journal":{"name":"essentia law Merchant Shipping Act 1995","volume":"30 1","pages":"49-64"},"PeriodicalIF":0.0,"publicationDate":"2021-01-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83338807","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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