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Effective category and measure in abstract complexity theory

Cristian Calude, Marius Zimand
1996 Theoretical Computer Science  
Strong variants of the Operator Speed-up Theorem, Operator Gap Theorem and Compression Theorem are obtained using an effective Version of Baire Category Theorem.  ...  It is also shown that all complexity classes of recursive predicates have effective measure zero in the space of recursive predicates and, on the other hand, the class of predicates with almost everywhere  ...  Esch set Computer Science 154 (1996) 307-327 G u A(i,+ i,k>O  ... 
doi:10.1016/0304-3975(95)00066-6 fatcat:b277wpq37fe7ja5c5l44wer654

Constructing categories and setoids of setoids in type theory

Erik Palmgren, Olov Wilander, Martín Escardó
2014 Logical Methods in Computer Science  
In this paper we consider the problem of building rich categories of setoids, in standard intensional Martin-L\"of type theory (MLTT), and in particular how to handle the problem of equality on objects  ...  A very large family F may be obtained from Aczel's model construction of CZF in type theory. It is proved that the category so obtained is isomorphic to the internal category of sets in this model.  ...  of families of setoids, and type theory has a more direct computational interpretation.  ... 
doi:10.2168/lmcs-10(3:25)2014 fatcat:4cp4t376anhftajgxuyi5d4puu

Interfacing biology, category theory and mathematical statistics

Dominique Pastor, Erwan Beurier, Andrée Ehresmann, Roger Waldeck
2020 Electronic Proceedings in Theoretical Computer Science  
Motivated by the concept of degeneracy in biology (Edelman, Gally 2001), we establish a first connection between the Multiplicity Principle (Ehresmann, Vanbremeersch 2007) and mathematical statistics.  ...  Acknowledgements The authors are very grateful to the reviewers for their strong encouragements and insightful remarks that help improve the readiness of this paper.  ...  However, to ease the reading of a paper at the interface between category theory and mathematical statistics, we consider a simplified version of this problem. The paper is organized as follows.  ... 
doi:10.4204/eptcs.323.9 fatcat:drwajt635nhofgejpmlv337nyi

Using category theory to design implicit conversions and generic operators [chapter]

John C. Reynolds
1980 Lecture Notes in Computer Science  
A generalization of many-sorted algebras, called categorysorted algebras, is defined and applied to the language-design problem of avoiding anomalies in the interaction of implicit conversions and generic  ...  This is a clear illustration of what we mean by applying category theory to language definition.  ...  Our intention is not to use any deep theorems of category theory, but merely to employ the basic concepts of this field as organizing principles.  ... 
doi:10.1007/3-540-10250-7_24 fatcat:c5nyy6mvszgbng6dfbddkk2tcy

Nominal Lawvere Theories: A category theoretic account of equational theories with names

Ranald Clouston
2014 Journal of computer and system sciences (Print)  
This paper gives a category theoretic account of NEL theories, by analogy with Lawvere's classic correspondence between equational theories and small categories with finite products.  ...  Names, or object-level variables, are a ubiquitous feature in programming languages and other computational applications.  ...  [22, Chapter VI]), as monads relate to the fundamental category theoretic notion of adjoint functors, and have found computer science application for modelling side effects [23] .  ... 
doi:10.1016/j.jcss.2014.04.002 fatcat:62xbj3kqm5cz5oq7nnkobntztq

Classical Control and Quantum Circuits in Enriched Category Theory

Mathys Rennela, Sam Staton
2018 Electronical Notes in Theoretical Computer Science  
Our categorical semantics for the host language is standard, and involves cartesian closed categories and monads.  ...  As an extended example, we recall an earlier result that the category of W*-algebras is dcpo-enriched, and we use this model to extend the circuit language with some recursive types.  ...  Acknowledgement The authors would like to thank Bart Jacobs, Michele Pagani and Shane Mansfield for helpful discussions, but also Bert Lindenhovius, Michael Mislove and Vladimir Zamdzhiev for hosting the  ... 
doi:10.1016/j.entcs.2018.03.027 fatcat:lknzyacpbzbwtdjhkj34ypaade

On generalized algebraic theories and categories with families

Marc Bezem, Thierry Coquand, Peter Dybjer, Martín Escardó
2021 Mathematical Structures in Computer Science  
We provide examples of generalized algebraic theories for monoids, categories, categories with families, and categories with families with extra structure for some type formers of Martin-Löf type theory  ...  To this end, we simultaneously define the notion of a presentation Σ of a generalized algebraic theory and the associated category CwFΣ of small cwfs with a Σ-structure and cwf-morphisms that preserve  ...  We are grateful to the anonymous referees for constructive criticism and pointers to related work. We would also like to thank Andrej Bauer, John Cartmell, and Christian Sattler for useful comments.  ... 
doi:10.1017/s0960129521000268 fatcat:7ep7rxao4vcbvg6zrjxesu4e7i

The biequivalence of locally cartesian closed categories and Martin-Löf type theories

PIERRE CLAIRAMBAULT, PETER DYBJER
2014 Mathematical Structures in Computer Science  
category of Martin-L\"of type theories with Pi-types, Sigma-types and extensional identity types.  ...  Seely's paper "Locally cartesian closed categories and type theory" contains a well-known result in categorical type theory: that the category of locally cartesian closed categories is equivalent to the  ...  We would also like to acknowledge the support of the (UK) EPSRC grant RC-CM1025 for the first author and of the (Swedish) Vetenskapsrådet grant "Types for Proofs and Programs" for the second author.  ... 
doi:10.1017/s0960129513000881 fatcat:3j4tshh3ufdg3chq2tqyajbnaq

The Biequivalence of Locally Cartesian Closed Categories and Martin-Löf Type Theories [chapter]

Pierre Clairambault, Peter Dybjer
2011 Lecture Notes in Computer Science  
category of Martin-Löf type theories with Π, Σ, and extensional identity types.  ...  Seely's paper Locally cartesian closed categories and type theory contains a well-known result in categorical type theory: that the category of locally cartesian closed categories is equivalent to the  ...  We would also like to acknowledge the support of the (UK) EPSRC grant RC-CM1025 for the first author and of the (Swedish) Vetenskapsrådet grant "Types for Proofs and Programs" for the second author.  ... 
doi:10.1007/978-3-642-21691-6_10 fatcat:o2jreyvcgfbutb2jriduntpehi

A duality theory for decomposable systems in a category

G. Naudé
1980 Journal of computer and system sciences (Print)  
The Arbib-Manes duality theory for decomposable systems in a category is generalized, making it possible to apply the theory to infinite-dimensional linear systems in reflexive Banach spaces. q(t + 1)  ...  We assume that our category J is equipped with such an image-factorization system. In [l] a categorical duality theory is developed for decomposable systems in a category &'.  ...  The resulting systems duality is obtained by takingd=X=K-LPP', 29=Z=cK--9%9', etc. (4) Linear Systems over Rings An example where the categories J/ and 9 in our duality theory are not the same is obtained  ... 
doi:10.1016/0022-0000(80)90025-2 fatcat:ttfnhl72j5h5roy2alglh22wb4

Modular correspondence between dependent type theories and categories including pretopoi and topoi

MARIA EMILIA MAIETTI
2005 Mathematical Structures in Computer Science  
Starting from lex categories, through regular ones we provide internal languages of pretopoi and topoi and some variations of them, like for example Heyting pretopoi.  ...  MSC 2000: 03G30 03B15 18C50 Keywords: Categorical logic, topoi, Higher-order logic and type theory, Categorical semantics of formal languages. † In previous versions of this paper or in (Maietti 2001)  ...  ; -viceversa, to import in type theory categorical proofs; -to study the computational contents of internal languages of categories by employing type-theoretic methods; -to relate the internal dependent  ... 
doi:10.1017/s0960129505004962 fatcat:h7uwd6voy5etzklb5iunkxnpqm

Classical Control, Quantum Circuits and Linear Logic in Enriched Category Theory

Mathys Rennela, Sam Staton
2017 Logical Methods in Computer Science  
Our categorical semantics for the host language is standard, and involves cartesian closed categories and monads.  ...  As an extended example, we recall an earlier result that the category of W*-algebras is dcpo-enriched, and we use this model to extend the circuit language with some recursive types.  ...  , and Vladimir Zamdzhiev for hosting the first author at Tulane University during the early stage of the elaboration of this work, and Jennifer Paykin and Robert Rand for introducing us to the subtle aspects  ... 
doi:10.23638/lmcs-16(1:30)2020 fatcat:mzdtexumm5ayvjdk3vkxrx25k4

The Category Theoretic Understanding of Universal Algebra: Lawvere Theories and Monads

Martin Hyland, John Power
2007 Electronical Notes in Theoretical Computer Science  
Lawvere theories and monads have been the two main category theoretic formulations of universal algebra, Lawvere theories arising in 1963 and the connection with monads being established a few years later  ...  interest in them in a computer science setting might develop in future.  ...  A year later, at the Constructive Logic and Category Theory workshop at the Isle of Thorns, Peter Freyd, who was the leading North American category theorist connected with computer science at the time  ... 
doi:10.1016/j.entcs.2007.02.019 fatcat:5pmik2frhzf3rdmcbd3xkjvgsm

Category Theory and Model-Driven Engineering: From Formal Semantics to Design Patterns and Beyond

Zinovy Diskin, Tom Maibaum
2012 Electronic Proceedings in Theoretical Computer Science  
Moreover, the paper aims to elucidate a claim that relationships between CT and MDE are more complex and richer than is normally assumed for "applied mathematics".  ...  CT provides a toolbox of design patterns and structural principles of real practical value for MDE.  ...  Ideas presented in the paper have been discussed at NECSIS seminars at McMaster and the University of Waterloo; we are grateful to all their participants, and especially to Michał Antkiewicz and Krzysztof  ... 
doi:10.4204/eptcs.93.1 fatcat:ouc7yan3pndw3ow4rqxdfemaze

Using Failures and Category Theory to Verify Process Communications between Design and Implementation of Concurrent Systems

Ming Zhu, Peter Grogono, Olga Ormandjieva, Heng Kuang
2016 Procedia Computer Science  
In this framework, we use Failures in Communicating Sequential Processes (CSP), Erasmus and Category Theory as the foundation. The framework is illustrated by using a running example.  ...  The process-oriented design and implementation of concurrent systems have important advantages.  ...  Category Theory Due to its abstractness and generality, category theory has led to its use as a conceptual framework in many areas of computer science 12 and software engineering 13 .  ... 
doi:10.1016/j.procs.2016.04.155 fatcat:chzo2ddgxfh6rag2v3evbhojuy
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