mirror of
https://git8.cs.fau.de/theses/bsc-leon-vatthauer.git
synced 2024-05-31 07:28:34 +02:00
258 lines
15 KiB
BibTeX
258 lines
15 KiB
BibTeX
@inproceedings{agda,
|
||
title = {Towards a practical programming language based on dependent type theory},
|
||
author = {Ulf Norell},
|
||
year = {2007},
|
||
url = {https://api.semanticscholar.org/CorpusID:118357515}
|
||
}
|
||
|
||
@inproceedings{agda-categories,
|
||
author = {Hu, Jason Z. S. and Carette, Jacques},
|
||
title = {Formalizing Category Theory in Agda},
|
||
year = {2021},
|
||
isbn = {9781450382991},
|
||
publisher = {Association for Computing Machinery},
|
||
address = {New York, NY, USA},
|
||
url = {https://doi.org/10.1145/3437992.3439922},
|
||
doi = {10.1145/3437992.3439922},
|
||
abstract = {The generality and pervasiveness of category theory in modern mathematics makes it a frequent and useful target of formalization. It is however quite challenging to formalize, for a variety of reasons. Agda currently (i.e. in 2020) does not have a standard, working formalization of category theory. We document our work on solving this dilemma. The formalization revealed a number of potential design choices, and we present, motivate and explain the ones we picked. In particular, we find that alternative definitions or alternative proofs from those found in standard textbooks can be advantageous, as well as "fit" Agda's type theory more smoothly. Some definitions regarded as equivalent in standard textbooks turn out to make different "universe level" assumptions, with some being more polymorphic than others. We also pay close attention to engineering issues so that the library integrates well with Agda's own standard library, as well as being compatible with as many of supported type theories in Agda as possible.},
|
||
booktitle = {Proceedings of the 10th ACM SIGPLAN International Conference on Certified Programs and Proofs},
|
||
pages = {327–342},
|
||
numpages = {16},
|
||
keywords = {formal mathematics, Agda, category theory},
|
||
location = {Virtual, Denmark},
|
||
series = {CPP 2021}
|
||
}
|
||
|
||
@incollection{Altenkirch_2017,
|
||
doi = {10.1007/978-3-662-54458-7_31},
|
||
url = {https://doi.org/10.1007%2F978-3-662-54458-7_31},
|
||
year = {2017},
|
||
publisher = {Springer Berlin Heidelberg},
|
||
pages = {534--549},
|
||
author = {Thorsten Altenkirch and Nils Anders Danielsson and Nicolai Kraus},
|
||
title = {Partiality, Revisited},
|
||
booktitle = {Lecture Notes in Computer Science}
|
||
}
|
||
|
||
@article{delay,
|
||
author = {Venanzio Capretta},
|
||
title = {General Recursion via Coinductive Types},
|
||
journal = {CoRR},
|
||
volume = {abs/cs/0505037},
|
||
year = {2005},
|
||
url = {http://arxiv.org/abs/cs/0505037},
|
||
eprinttype = {arXiv},
|
||
eprint = {cs/0505037},
|
||
timestamp = {Mon, 13 Aug 2018 16:46:14 +0200},
|
||
biburl = {https://dblp.org/rec/journals/corr/abs-cs-0505037.bib},
|
||
bibsource = {dblp computer science bibliography, https://dblp.org}
|
||
}
|
||
|
||
@article{elgotalgebras,
|
||
author = {Jir{\'{\i}} Ad{\'{a}}mek and
|
||
Stefan Milius and
|
||
Jir{\'{\i}} Velebil},
|
||
title = {Elgot Algebras},
|
||
journal = {CoRR},
|
||
volume = {abs/cs/0609040},
|
||
year = {2006},
|
||
url = {http://arxiv.org/abs/cs/0609040},
|
||
eprinttype = {arXiv},
|
||
eprint = {cs/0609040},
|
||
timestamp = {Mon, 04 Sep 2023 12:29:24 +0200},
|
||
biburl = {https://dblp.org/rec/journals/corr/abs-cs-0609040.bib},
|
||
bibsource = {dblp computer science bibliography, https://dblp.org}
|
||
}
|
||
|
||
@article{elgotmonad,
|
||
title = {Elgot theories: a new perspective on the equational properties of iteration},
|
||
volume = {21},
|
||
doi = {10.1017/S0960129510000496},
|
||
number = {2},
|
||
journal = {Mathematical Structures in Computer Science},
|
||
publisher = {Cambridge University Press},
|
||
author = {Jir{\'{\i}} Ad{\'{a}}mek and
|
||
Stefan Milius and
|
||
Jir{\'{\i}} Velebil},
|
||
year = {2011},
|
||
pages = {417–480}
|
||
}
|
||
|
||
@article{eqlm,
|
||
author = {Bucalo, Anna and F\"{u}hrmann, Carsten and Simpson, Alex},
|
||
title = {An Equational Notion of Lifting Monad},
|
||
year = {2003},
|
||
issue_date = {15 February 2003},
|
||
publisher = {Elsevier Science Publishers Ltd.},
|
||
address = {GBR},
|
||
volume = {294},
|
||
number = {1–2},
|
||
issn = {0304-3975},
|
||
url = {https://doi.org/10.1016/S0304-3975(01)00243-2},
|
||
doi = {10.1016/S0304-3975(01)00243-2},
|
||
abstract = {We introduce the notion of an equational lifting monad: a commutative strong monad satisfying one additional equation (valid for monads arising from partial map classifiers). We prove that any equational lifting monad has a representation by a partial map classifier such that the Kleisli category of the former fully embeds in the partial category of the latter. Thus, equational lifting monads precisely capture the equational properties of partial maps as induced by partial map classifiers. The representation theorem also provides a tool for transferring nonequational properties of partial map classifiers to equational lifting monads. It is proved using a direct axiomatization of Kleisli categories of equational lifting monads. This axiomatization is of interest in its own right.},
|
||
journal = {Theor. Comput. Sci.},
|
||
month = {2},
|
||
pages = {31–60},
|
||
numpages = {30},
|
||
keywords = {premonoidal categories, categories, partiality and partial categories, abstract Kleish, commutative strong monads}
|
||
}
|
||
|
||
@book{inductive,
|
||
title = {Categorical programming with inductive and coinductive types},
|
||
author = {Vene, Varmo},
|
||
year = {2000},
|
||
publisher = {Citeseer}
|
||
}
|
||
|
||
@inproceedings{Lane1971,
|
||
title = {Categories for the Working Mathematician},
|
||
author = {Saunders Mac Lane},
|
||
year = {1971},
|
||
url = {https://api.semanticscholar.org/CorpusID:122892655}
|
||
}
|
||
|
||
@article{moggi,
|
||
author = {Moggi, Eugenio},
|
||
title = {Notions of Computation and Monads},
|
||
year = {1991},
|
||
issue_date = {July 1991},
|
||
publisher = {Academic Press, Inc.},
|
||
address = {USA},
|
||
volume = {93},
|
||
number = {1},
|
||
issn = {0890-5401},
|
||
url = {https://doi.org/10.1016/0890-5401(91)90052-4},
|
||
doi = {10.1016/0890-5401(91)90052-4},
|
||
journal = {Inf. Comput.},
|
||
month = {7},
|
||
pages = {55–92},
|
||
numpages = {38}
|
||
}
|
||
|
||
@online{nad-delay,
|
||
author = {Nils Anders Danielsson},
|
||
title = {The delay monad, defined coinductively},
|
||
url = {https://www.cse.chalmers.se/~nad/listings/delay-monad/Delay-monad.html},
|
||
urlday = {15},
|
||
urlmonth = {02},
|
||
urlyear = {2024}
|
||
}
|
||
|
||
@article{param-corec,
|
||
title = {Parametric corecursion},
|
||
journal = {Theoretical Computer Science},
|
||
volume = {260},
|
||
number = {1},
|
||
pages = {139-163},
|
||
year = {2001},
|
||
note = {Coalgebraic Methods in Computer Science 1998},
|
||
issn = {0304-3975},
|
||
doi = {https://doi.org/10.1016/S0304-3975(00)00126-2},
|
||
url = {https://www.sciencedirect.com/science/article/pii/S0304397500001262},
|
||
author = {Lawrence S. Moss},
|
||
abstract = {This paper gives a treatment of substitution for “parametric” objects in final coalgebras, and also presents principles of definition by corecursion for such objects. The substitution results are coalgebraic versions of well-known consequences of initiality, and the work on corecursion is a general formulation which allows one to specify elements of final coalgebras using systems of equations. One source of our results is the theory of hypersets, and at the end of this paper we sketch a development of that theory which calls upon the general work of this paper to a very large extent and particular facts of elementary set theory to a much smaller extent.}
|
||
}
|
||
|
||
@inproceedings{quotienting,
|
||
author = {Chapman, James and Uustalu, Tarmo and Veltri, Niccol\`{o}},
|
||
title = {Quotienting the Delay Monad by Weak Bisimilarity},
|
||
year = {2015},
|
||
isbn = {9783319251493},
|
||
publisher = {Springer-Verlag},
|
||
address = {Berlin, Heidelberg},
|
||
url = {https://doi.org/10.1007/978-3-319-25150-9_8},
|
||
doi = {10.1007/978-3-319-25150-9_8},
|
||
abstract = {The delay datatype was introduced by Capretta [3] as a means to deal with partial functions as in computability theory in Martin-L\"{o}f type theory. It is a monad and it constitutes a constructive alternative to the maybe monad. It is often desirable to consider two delayed computations equal, if they terminate with equal values, whenever one of them terminates. The equivalence relation underlying this identification is called weak bisimilarity. In type theory, one commonly replaces quotients with setoids. In this approach, the delay monad quotiented by weak bisimilarity is still a monad. In this paper, we consider Hofmann's alternative approach [6] of extending type theory with inductive-like quotient types. In this setting, it is difficult to define the intended monad multiplication for the quotiented datatype. We give a solution where we postulate some principles, crucially proposition extensionality and the semi-classical axiom of countable choice. We have fully formalized our results in the Agda dependently typed programming language.},
|
||
booktitle = {Proceedings of the 12th International Colloquium on Theoretical Aspects of Computing - ICTAC 2015 - Volume 9399},
|
||
pages = {110–125},
|
||
numpages = {16}
|
||
}
|
||
|
||
@article{restriction,
|
||
author = {Cockett, J. R. B. and Lack, Stephen},
|
||
title = {Restriction Categories I: Categories of Partial Maps},
|
||
year = {2002},
|
||
issue_date = {January},
|
||
publisher = {Elsevier Science Publishers Ltd.},
|
||
address = {GBR},
|
||
volume = {270},
|
||
number = {1–2},
|
||
issn = {0304-3975},
|
||
url = {https://doi.org/10.1016/S0304-3975(00)00382-0},
|
||
doi = {10.1016/S0304-3975(00)00382-0},
|
||
abstract = {Given a category with a stable system of monics, one can form the corresponding category of partial maps. To each map in this category there is, on the domain of the map, an associated idempotent, which measures the degree of partiality. This structure is captured abstractly by the notion of a restriction category, in which every arrow is required to have such an associated idempotent. Categories with a stable system of monics, functors preserving this structure, and natural transformations which are cartesian with respect to the chosen monics, form a 2-category which we call MCat. The construction of categories of partial maps provides a 2-functor Par:Mcat→Cat. We show that Par can be made into an equivalence of 2-categories between MCat and a 2-category of restriction categories. The underlying ordinary functor Par&r0:Mcat&0 → Ca t0 of the above 2-functor Par turns out to be monadic, and, from this, we deduce the completeness and cocompleteness of the 2-categories of M-categories and of restriction categories. We also consider the problem of how to turn a formal system of subobjects into an actual system of subobjects. A formal system of subobjects is given by a functor into the category sLat of semilattices. This structure gives rise to a restriction category which, via the above equivalence of 2-categories, gives an M-category. This M-category contains the universal realization of the given formal subobjects as actual subobjects.},
|
||
journal = {Theor. Comput. Sci.},
|
||
month = {1},
|
||
pages = {223–259},
|
||
numpages = {37}
|
||
}
|
||
|
||
@article{setoids,
|
||
author = {Barthe, Gilles and Capretta, Venanzio and Pons, Olivier},
|
||
title = {Setoids in type theory},
|
||
year = {2003},
|
||
issue_date = {March 2003},
|
||
publisher = {Cambridge University Press},
|
||
address = {USA},
|
||
volume = {13},
|
||
number = {2},
|
||
issn = {0956-7968},
|
||
url = {https://doi.org/10.1017/S0956796802004501},
|
||
doi = {10.1017/S0956796802004501},
|
||
abstract = {Formalising mathematics in dependent type theory often requires to represent sets as setoids, i.e. types with an explicit equality relation. This paper surveys some possible definitions of setoids and assesses their suitability as a basis for developing mathematics. According to whether the equality relation is required to be reflexive or not we have total or partial setoid, respectively. There is only one definition of total setoid, but four different definitions of partial setoid, depending on four different notions of setoid function. We prove that one approach to partial setoids in unsuitable, and that the other approaches can be divided in two classes of equivalence. One class contains definitions of partial setoids that are equivalent to total setoids; the other class contains an inherently different definition, that has been useful in the modeling of type systems. We also provide some elements of discussion on the merits of each approach from the viewpoint of formalizing mathematics. In particular, we exhibit a difficulty with the common definition of subsetoids in the partial setoid approach.},
|
||
journal = {J. Funct. Program.},
|
||
month = {mar},
|
||
pages = {261–293},
|
||
numpages = {33}
|
||
}
|
||
|
||
@article{sol-thm,
|
||
author = {Aczel, Peter and Ad\'{a}mek, Jir\'{\i} and Milius, Stefan and Velebil, Jir\'{\i}},
|
||
title = {Infinite trees and completely iterative theories: a coalgebraic view},
|
||
year = {2003},
|
||
issue_date = {07 May 2003},
|
||
publisher = {Elsevier Science Publishers Ltd.},
|
||
address = {GBR},
|
||
volume = {300},
|
||
number = {1–3},
|
||
issn = {0304-3975},
|
||
url = {https://doi.org/10.1016/S0304-3975(02)00728-4},
|
||
doi = {10.1016/S0304-3975(02)00728-4},
|
||
abstract = {Infinite trees form a free completely iterative theory over any given signature--this fact, proved by Elgot, Bloom and Tindell, turns out to be a special case of a much more general categorical result exhibited in the present paper. We prove that whenever an endofunctor H of a category has final coalgebras for all functors H(-) + X, then those coalgebras, TX, form a monad. This monad is completely iterative, i.e., every guarded system of recursive equations has a unique solution. And it is a free completely iterative monad on H. The special case of polynomial endofunctors of the category Set is the above mentioned theory, or monad, of infinite trees.This procedure can be generalized to monoidal categories satisfying a mild side condition: if, for an object H, the endofunctor H ⊗ _ + I has a final coalgebra, T, then T is a monoid. This specializes to the above case for the monoidal category of all endofunctors.},
|
||
journal = {Theor. Comput. Sci.},
|
||
month = {5},
|
||
pages = {1–45},
|
||
numpages = {45},
|
||
keywords = {coalgebra, completely iterative theory, monad, monoidal category, solution theorem}
|
||
}
|
||
|
||
@article{uniformelgot,
|
||
author = {Sergey Goncharov},
|
||
title = {Uniform Elgot Iteration in Foundations},
|
||
journal = {CoRR},
|
||
volume = {abs/2102.11828},
|
||
year = {2021},
|
||
url = {https://arxiv.org/abs/2102.11828},
|
||
eprinttype = {arXiv},
|
||
eprint = {2102.11828},
|
||
timestamp = {Fri, 26 Feb 2021 14:31:25 +0100},
|
||
biburl = {https://dblp.org/rec/journals/corr/abs-2102-11828.bib},
|
||
bibsource = {dblp computer science bibliography, https://dblp.org}
|
||
}
|
||
|
||
@article{while,
|
||
author = {Sergey Goncharov and
|
||
Lutz Schr{\"{o}}der and
|
||
Christoph Rauch},
|
||
title = {(Co-)Algebraic Foundations for Effect Handling and Iteration},
|
||
journal = {CoRR},
|
||
volume = {abs/1405.0854},
|
||
year = {2014},
|
||
url = {http://arxiv.org/abs/1405.0854},
|
||
eprinttype = {arXiv},
|
||
eprint = {1405.0854},
|
||
timestamp = {Mon, 13 Aug 2018 16:47:19 +0200},
|
||
biburl = {https://dblp.org/rec/journals/corr/GoncharovSR14.bib},
|
||
bibsource = {dblp computer science bibliography, https://dblp.org}
|
||
}
|