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Representations of First Order Function Types as Terminal Coalgebras [chapter]

Thorsten Altenkirch
2001 Lecture Notes in Computer Science  
We show that function types which have only initial algebras for regular functors in the domains, i.e. first order function types, can be represented by terminal coalgebras for certain nested functors.  ...  The representation exploits properties of ω op -limits and local ω-colimits.  ...  In programming and type theory the universe of types can be divided as follows: -function types (cartesian closure) -algebraic types • inductive types (initial algebras) • coinductive types (terminal coalgebras  ... 
doi:10.1007/3-540-45413-6_5 fatcat:i6pqmreunvaebfzpgthu6ebyfe

Constructive analysis, types and exact real numbers

2007 Mathematical Structures in Computer Science  
We then put these models in a uniform framework using realisability, opening the door for the use of type theoretic and coalgebraic constructions both in computing and reasoning about these computations  ...  We will present some of the problems and solutions of exact real arithmetic varying from concrete implementations, representation and algorithms to various models for real computation.  ...  Instead of using the ('first order') data-type of intervals one then uses functions that map desired precisions to intervals.  ... 
doi:10.1017/s0960129506005834 fatcat:kdehtcealnbszcfuszmcbzpdhu

Notes on Coalgebras in Stylometry [article]

Joël A. Doat
2021 arXiv   pre-print
In this paper, we discuss how coalgebras are used to formalise the notion of behaviour by embedding syntactic features of a given text into probabilistic transition systems.  ...  The syntactic behaviour of texts can highly vary depending on their contexts (e.g. author, genre, etc.). From the standpoint of stylometry, it can be helpful to objectively measure this behaviour.  ...  Therefore, states in the terminal coalgebra can be seen as behavioural realisations of transitions systems.  ... 
arXiv:2010.02733v2 fatcat:jxjrcdg3pjet5gmyvx732kdmgy

A Coalgebraic View of Bar Recursion and Bar Induction [chapter]

Venanzio Capretta, Tarmo Uustalu
2016 Lecture Notes in Computer Science  
One application of the principle is the tabulation of continuous functions: Ghani, Hancock and Pattinson defined a type of wellfounded trees that represent continuous functions on streams.  ...  It is a framework to characterize valid forms of recursion for terminating functional programs.  ...  Capretta is grateful to the School of Computer Science that gave him a sabbatical semester.  ... 
doi:10.1007/978-3-662-49630-5_6 fatcat:lbmygodc2jdz5id5crmgodb2jq

Continuous Functions on Final Coalgebras

Neil Ghani, Peter Hancock, Dirk Pattinson
2009 Electronical Notes in Theoretical Computer Science  
To program and reason about representations of continuous functions requires a language whose type system incorporates the dependent function and pair types, inductive definitions at types Set, I → Set  ...  We also defined a combinator on the representations of such continuous functions that reflects composition. Streams are one of the simplest examples of non-trivial final coalgebras.  ...  In order to represent functions of type ν(S ¡ P ) −→ ν(Q ¡ R) we need to devise a representation for an entire doubly indexed family of such functions, namely (P (s) → ν(S ¡ P )) −→ (R (q) → ν(Q ¡ R))  ... 
doi:10.1016/j.entcs.2009.07.081 fatcat:rj33jvuzknaszcqwb3a3iyjczy

Formalising Exact Arithmetic in Type Theory [chapter]

Milad Niqui
2005 Lecture Notes in Computer Science  
We show examples of how infinite objects such as streams and expression trees can be formalised as coinductive types.  ...  However, if some information about the complexity of a function is provided, one may be able to show the productivity of that function.  ...  Thus NAlg can not be formalised as above in the type theory of CIC extended with coinductive types. First, there is a type checking problem in the above presentation of NAlg.  ... 
doi:10.1007/11494645_46 fatcat:alsonnqjonfebi7b3ickdcuzru

Non-wellfounded sets in homotopy type theory [article]

Håkon Robbestad Gylterud, Elisabeth Bonnevier
2020 arXiv   pre-print
The results presented here are being formalised in Agda as part of a bigger project of formalising set theory on the basis of homotopy type theory.  ...  We show that this model is dual to the model previously described in the HoTT book -- being the final coalgebra, as opposed to the initial algebra, of the powerset operation.  ...  ZF is a formal theory in the language of first-order logic, intended as a formal foundation for mathematics.  ... 
arXiv:2001.06696v2 fatcat:t5k7p2afc5en3kgvqrqa5qpdwy

A Coalgebraic Foundation for Coinductive Union Types [chapter]

Marcello Bonsangue, Jurriaan Rot, Davide Ancona, Frank de Boer, Jan Rutten
2014 Lecture Notes in Computer Science  
This paper introduces a coalgebraic foundation for coinductive types, interpreted as sets of values and extended with set theoretic union.  ...  We obtain inclusion of tree languages as a sound and complete method to show semantic subtyping of recursive types with basic types, product and union, interpreted coinductively.  ...  Given sets A and B, coalgebras for the functor LX = B + (A × X) are representations of infinite lists over A and finite lists over A with termination in B.  ... 
doi:10.1007/978-3-662-43951-7_6 fatcat:dxksvgigpne67kbrfqjtjiyiim

From Co-algebraic Specifications to Implementation: The Mihda Toolkit [chapter]

Gianluigi Ferrari, Ugo Montanari, Roberto Raggi, Emilio Tuosto
2003 Lecture Notes in Computer Science  
The direct correspondence between the coalgebraic specification and the implementation structure facilitates the proof of correctness of the implementation.  ...  HD-automata have been specifically designed to allocate and garbage collect names and they provide faithful finite state representations of the behaviours of π-calculus processes.  ...  HD-automata have a natural representation as coalgebras on a category of named sets and named functions.  ... 
doi:10.1007/978-3-540-39656-7_13 fatcat:gjfcuecxn5aqfnhydm4u4w25xi

Modal Languages for Coalgebras in a Topological Setting

Dirk Pattinson
2001 Electronical Notes in Theoretical Computer Science  
This suggests a coalgebraic approach to obtain a logical representation of the observable properties of Z.  ...  This can be seen as abs1.tex a first step towards a final perspective on Abramsky's domain theory in logical form.  ...  Taking Scott-open subsets o ⊆ Z as predicates on Z, we can use the methods of coalgebraic modal logic in order to obtain a syntactical representation of the underlying set of the frame O(Z) of Scott-open  ... 
doi:10.1016/s1571-0661(04)80913-7 fatcat:7zfaz2qtfvcl5hqdozjdrnbuue

Coalgebras and monads in the semantics of Java

Bart Jacobs, Erik Poll
2003 Theoretical Computer Science  
This paper describes the basic structures in the denotational and axiomatic semantics of sequential Java, both from a monadic and a coalgebraic perspective.  ...  This semantics is an abstraction of the one used for the veriÿcation of (sequential) Java programs using proof tools in the LOOP project at the University of Nijmegen.  ...  Acknowledgements Thanks are due to the anonymous referees whose sharp comments greatly improved the quality of the paper.  ... 
doi:10.1016/s0304-3975(02)00366-3 fatcat:wz6qnbh6hzdq3joctxeuy2i7gm

Contractive Functions on Infinite Data Structures

Venanzio Capretta, Graham Hutton, Mauro Jaskelioff
2016 Proceedings of the 28th Symposium on the Implementation and Application of Functional Programming Languages - IFL 2016  
, first to infinite binary trees, then to final coalgebras of container functors.  ...  Coinductive data structures, such as streams or infinite lists, have many applications in functional programming and type theory, and are naturally defined using recursive equations.  ...  We can generalize the representation of contractive functions by using any final coalgebra as codomain, in place of BTree B.  ... 
doi:10.1145/3064899.3064900 dblp:conf/ifl/CaprettaHJ16 fatcat:pl5qdyphhbeexbfh7pde4hwkoi

A Formalisation of Java's Exception Mechanism [chapter]

Bart Jacobs
2001 Lecture Notes in Computer Science  
This formalisation uses so-called coalgebras for modeling Java statements and expressions, thus providing a convenient setting for handling the various termination options that may arise in exception handling  ...  This paper examines Java's exception mechanism, and formalises its main operations (throw, try-catch and try-catch-finally) in a type-theoretic setting.  ...  Joachim van den Berg, Marieke Huisman, Hans Meijer and Erik Poll provided useful feedback on a first draft.  ... 
doi:10.1007/3-540-45309-1_19 fatcat:fa2aufjb4vcgdf4abdolb6ibza

Endofunctors modelling higher-order behaviours [article]

Marco Peressotti
2016 arXiv   pre-print
In this paper we show how the abstract behaviours of higher-order systems can be modelled as final coalgebras of suitable behavioural functors.  ...  To achieve this property, we shift from term passing to behaviour passing: in the former higher-order is expressed by passing around syntactic objects (such as terms or processes) as representations of  ...  As a direct application of the theory of coalgebras, we are now able to define a canonical higher-order coalgebraic bisimulation and show it to coincide with coalgebraic bisimulation for value-passing  ... 
arXiv:1602.06221v2 fatcat:fudx2s3ckrbvpmupf22wsc4fke

Formal Languages, Formally and Coinductively [article]

Dmitriy Traytel
2017 arXiv   pre-print
More recently, the alternative coalgebraic or coinductive representation as infinite tries, i.e., prefix trees branching over the alphabet, has been used to obtain compact and elegant proofs of classic  ...  Traditionally, formal languages are defined as sets of words.  ...  Jurriaan Rot suggested important related work and recommended proving inequalities of tries by coinduction on ≤ rather than on =.  ... 
arXiv:1611.09633v2 fatcat:r5wup2poajd53dd2i4vw7oftem
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