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Representations of First Order Function Types as Terminal Coalgebras
[chapter]
2001
Lecture Notes in Computer Science
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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
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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]
2021
arXiv
pre-print
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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]
2016
Lecture Notes in Computer Science
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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
2009
Electronical Notes in Theoretical Computer Science
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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]
2005
Lecture Notes in Computer Science
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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]
2020
arXiv
pre-print
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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]
2014
Lecture Notes in Computer Science
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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]
2003
Lecture Notes in Computer Science
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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
2001
Electronical Notes in Theoretical Computer Science
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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
2003
Theoretical Computer Science
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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
2016
Proceedings of the 28th Symposium on the Implementation and Application of Functional Programming Languages - IFL 2016
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, 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]
2001
Lecture Notes in Computer Science
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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]
2016
arXiv
pre-print
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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]
2017
arXiv
pre-print
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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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