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Effective category and measure in abstract complexity theory

1996
*
Theoretical Computer Science
*

Strong variants of the Operator Speed-up Theorem, Operator Gap Theorem

doi:10.1016/0304-3975(95)00066-6
fatcat:b277wpq37fe7ja5c5l44wer654
*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 ...##
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Constructing categories and setoids of setoids in type theory

2014
*
Logical Methods in Computer Science
*

In this paper we consider the problem of building rich

doi:10.2168/lmcs-10(3:25)2014
fatcat:4cp4t376anhftajgxuyi5d4puu
*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. ...##
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Interfacing biology, category theory and mathematical statistics

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)

doi:10.4204/eptcs.323.9
fatcat:drwajt635nhofgejpmlv337nyi
*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. ...##
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Using category theory to design implicit conversions and generic operators
[chapter]

1980
*
Lecture Notes in Computer Science
*

A generalization of many-sorted algebras, called categorysorted algebras, is defined

doi:10.1007/3-540-10250-7_24
fatcat:c5nyy6mvszgbng6dfbddkk2tcy
*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. ...##
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Nominal Lawvere Theories: A category theoretic account of equational theories with names

2014
*
Journal of computer and system sciences (Print)
*

This paper gives a

doi:10.1016/j.jcss.2014.04.002
fatcat:62xbj3kqm5cz5oq7nnkobntztq
*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] . ...##
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Classical Control and Quantum Circuits in Enriched Category Theory

2018
*
Electronical Notes in Theoretical Computer Science
*

Our categorical semantics for the host language is standard,

doi:10.1016/j.entcs.2018.03.027
fatcat:lknzyacpbzbwtdjhkj34ypaade
*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 ...##
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On generalized algebraic theories and categories with families

2021
*
Mathematical Structures in Computer Science
*

We provide examples of generalized algebraic

doi:10.1017/s0960129521000268
fatcat:7ep7rxao4vcbvg6zrjxesu4e7i
*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. ...##
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The biequivalence of locally cartesian closed categories and Martin-Löf type theories

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. ...

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

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. ...

##
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A duality theory for decomposable systems in a category

1980
*
Journal of computer and system sciences (Print)
*

The Arbib-Manes duality

doi:10.1016/0022-0000(80)90025-2
fatcat:ttfnhl72j5h5roy2alglh22wb4
*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 ...##
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Modular correspondence between dependent type theories and categories including pretopoi and topoi

2005
*
Mathematical Structures in Computer Science
*

Starting from lex

doi:10.1017/s0960129505004962
fatcat:h7uwd6voy5etzklb5iunkxnpqm
*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 ...##
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Classical Control, Quantum Circuits and Linear Logic in Enriched Category Theory

2017
*
Logical Methods in Computer Science
*

Our categorical semantics for the host language is standard,

doi:10.23638/lmcs-16(1:30)2020
fatcat:mzdtexumm5ayvjdk3vkxrx25k4
*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 ...##
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The Category Theoretic Understanding of Universal Algebra: Lawvere Theories and Monads

2007
*
Electronical Notes in Theoretical Computer Science
*

Lawvere

doi:10.1016/j.entcs.2007.02.019
fatcat:5pmik2frhzf3rdmcbd3xkjvgsm
*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 ...##
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Category Theory and Model-Driven Engineering: From Formal Semantics to Design Patterns and Beyond

2012
*
Electronic Proceedings in Theoretical Computer Science
*

Moreover, the paper aims to elucidate a claim that relationships between CT

doi:10.4204/eptcs.93.1
fatcat:ouc7yan3pndw3ow4rqxdfemaze
*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 ...##
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Using Failures and Category Theory to Verify Process Communications between Design and Implementation of Concurrent Systems

2016
*
Procedia Computer Science
*

In this framework, we use Failures in Communicating Sequential Processes (CSP), Erasmus

doi:10.1016/j.procs.2016.04.155
fatcat:chzo2ddgxfh6rag2v3evbhojuy
*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 . ...
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