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Newton Series, Coinductively [chapter]

Henning Basold, Helle Hvid Hansen, Jean-Éric Pin, Jan Rutten
2015 Lecture Notes in Computer Science  
Exploiting the fact that the set of weighted languages is a final coalgebra, we use coinduction to prove that a classical operator from difference calculus in mathematics: the Newton transform, generalises  ...  Newton transform of a language, together with concrete calculation rules for computing them.  ...  Theorem 53 (Newton series for languages, 1st).  ... 
doi:10.1007/978-3-319-25150-9_7 fatcat:bdiougf6ijdihdvjeknlm3olhm

Newton series, coinductively: a comparative study of composition

HENNING BASOLD, HELLE HVID HANSEN, JEAN-ÉRIC PIN, JAN RUTTEN
2017 Mathematical Structures in Computer Science  
Exploiting the fact that the set of weighted languages is a final coalgebra, we use coinduction to prove that an operator of the classical difference calculus, the Newton transform, generalises from infinite  ...  Newton transform of a language, together with concrete calculation rules for computing them.  ...  The set of weighted languages, together Newton series, coinductively 3 its stream derivative by σ = (σ(1), σ(2), σ(3), . . . ).  ... 
doi:10.1017/s0960129517000159 fatcat:4l6trke2hvethpppsdhnwps6py

Preface to special issue: ICTAC 2015

MARTIN LEUCKER, JORGE A. PÉREZ, CAMILO RUEDA, FRANK D. VALENCIA
2018 Mathematical Structures in Computer Science  
The paper by Basold, Hansen, Pin and Rutten (Newton Series, Coinductively: A Comparative Study of Composition) uses a coinductive calculus of infinite streams by one of the authors to give an analysis  ...  The authors use coinduction to prove that an operator of the classical difference calculus, the Newton transform, generalizes from infinite sequences to weighted languages.  ... 
doi:10.1017/s0960129518000130 fatcat:7e2hvlpvnjdkladbg6bzzfuzoa

Rationality and Escalation in Infinite Extensive Games [article]

Pierre Lescanne
2012 arXiv   pre-print
The reasoning is based on the concept of coinduction conceived by computer scientists to model infinite computations and used by economic agents unknowingly.  ...  When used consciously, this concept is not as simple as induction and we could paraphrase Newton: "Modeling the madness of people is more difficult than modeling the motion of planets".  ...  Like Weierstrass' discovery led to the development of function series, logicians have devised methods for correct deductions on infinite structures.  ... 
arXiv:1112.1185v3 fatcat:zxkeb6w3argbvgjptf6wfv22ry

Scientific Modelling with Coalgebra-Algebra Homomorphisms [article]

Baltasar Trancón y Widemann, Michael Hauhs
2015 arXiv   pre-print
Kleisli Coinduction e † : X → T Y is a Kleisli-coinductive solution, then so is T h • e † : X → T Z.  ...  In a Newton-style modelling approach, this system is specified by a second-order linear differential equationẍ + k m x = 0, which simply states that all acceleration is due to the restoring force.  ... 
arXiv:1506.07290v1 fatcat:ogbblkrosncxbnvqymysxgic3m

In Praise of Sequence (Co-)Algebra and its implementation in Haskell [article]

Kieran Clenaghan
2019 arXiv   pre-print
Introduction Consider these titles: Formal Power Series [64] , Power Series, Power Serious [57] , A Coinductive Calculus of Streams [72] , and Concrete Stream Calculus [39] .  ...  A proof by coinduction is presented for contrast in item Q of the next section. Let f ∈ S, g ∈ S 0 , then, 2. (af + bg) = a f + b g linear 3. (n + 1)[x n+1 ] = [x n ] power series 4.  ... 
arXiv:1812.05878v2 fatcat:qu5f7ne6nrcy7hd4zq5xg5tzcq

Stream Differential Equations: Specification Formats and Solution Methods [article]

Helle Hvid Hansen, Clemens Kupke, Jan Rutten
2016 arXiv   pre-print
Stream differential equations are a coinductive method for specifying streams and stream operations, and their theory has been developed in many papers over the past two decades.  ...  facts about formal power series.  ...  More recently, the Newton transform between the ∆-and tail-structures has been studied in [9] .  ... 
arXiv:1609.08367v1 fatcat:pwvidt3ubbfxdpsr2qqyyxs44m

Les crashs sont rationnels [article]

Pierre Lescanne
2012 arXiv   pre-print
Among those new paths, coinduction is the basis of our reasoning in infinite games.  ...  L'escalade consisteà prendre, sans se donner de borne, une série de décisions de plus en plus lourdes de conséquences.  ...  La base de celui-ci, qui aété conçu pour s'adapter aux objets infinis s'appelle la coinduction. On fait donc de la coinduction rétrograde . En fait, il y a au moins deuxéquilibres : 1.  ... 
arXiv:1111.7299v5 fatcat:bxpfrmdutzddljhyxrb5kwtsee

On stratification for spaces with Noetherian mod p cohomology [article]

Tobias Barthel, Natalia Castellana, Drew Heard, Gabriel Valenzuela
2021 arXiv   pre-print
The first and second author would furthermore like to thank the Isaac Newton Institute for Mathematical Sciences, Cambridge, for support and hospitality during the programme Homotopy Harnessing Higher  ...  In a series of papers [BIK08, BIK11a] culminating in [BIK11b] , Benson, Iyengar, and Krause classified all modular representations of a finite group G up to a certain equivalence relation: Two representations  ...  By Remark 2.15 coinduction along f is also conservative.  ... 
arXiv:1904.12841v2 fatcat:jxmjfasywnaoznkznar3ti7moa

Formal Verification of Exact Computations Using Newton's Method [chapter]

Nicolas Julien, Ioana Paşca
2009 Lecture Notes in Computer Science  
CoInductive represents (β : Z) : stream digit → R → Prop := | rep : ∀ s r k, −β < k < β → −1 ≤ r ≤ 1 → represents β s r → represents β (Cons k s) k+r β .  ...  , Y 0 n + 2 µ n B n ] ⊆ [Y 0 0 − 3B 0 , Y 0 0 + 3B 0 ] ⊂]a, b [ We do not discuss all the details as they are elementary reasoning steps concerning inequalities, second degree equations or geometric series  ...  CoFixpoint exact newton (g: stream digit → stream digit) ex0 n:= match (make digit (EXn g ex0 n) with |d1::x' ⇒ d1::exact newton (fun x ⇒ (β ⊙ g (d1::x))) x' n end.  ... 
doi:10.1007/978-3-642-03359-9_28 fatcat:26eeozkikre3jb66lhyxgnt324

Algebra and coalgebra of stream products [article]

Michele Boreale, Daniele Gorla
2021 arXiv   pre-print
Below, we consider n × n matrices of power series; power series in z with scalar matrices as coefficients are interpreted element-wise, that is as defining matrices of power series.  ...  We also mention [7, 10] , that adopt a coinductive approach to reason on polynomial ODEs.  ... 
arXiv:2107.04455v1 fatcat:7t7nylwl7ffalayfvddrqpj7p4

Relational Interpretations of Recursive Types in an Operational Setting

Lars Birkedal, Robert Harper
1999 Information and Computation  
, we derive a notion of logical equivalence for expressions of the language that we show coincides with contextual equivalence and which, by virtue of its construction, validates useful induction and coinduction  ...  We are grateful to the Newton Institute and the organizers of the program on Semantics of Computation for their support.  ...  The work described here was carried out in part at the Isaac Newton Institute for Mathematical Sciences of Cambridge University in the autumn of 1995.  ... 
doi:10.1006/inco.1999.2828 fatcat:ft3r73aakff2rouohdlguyiwsu

Un acercamiento coinductivo al análisis real
Español

Guillermo Ortiz Rico, Lina Isabel Triviño Viera
2017 Revista Integración  
Our aim will be to use the framework of stream automata to illustrate the coinductive character of real analysis through classical results as the fundamental theorem of calculus, Taylor series and the  ...  Coinduction is a dual concept to induction; it has been discovered and studied recently.  ...  Serie de Taylor Considere el conjunto A = {f : R −→ R | f es analítica en 0}.  ... 
doi:10.18273/revint.v35n1-2017007 fatcat:nxl2l6dvrjhx7hujoxmm67okra

A monadic, functional implementation of real numbers

RUSSELL O'CONNOR
2007 Mathematical Structures in Computer Science  
It can be seen as a dependently typed functional programming language with inductive and coinductive data types.  ...  More Efficient Power Series. A power series converges faster for values closer to 0.  ...  ((1+n)/2, 1/2) newton (x,err) = (approxBase x' e1, e1*2) where x' = (n+x^2)/(2*x) e1 = err^2/2 sqrtCts :: UniformCts Base (Complete Base) sqrtCts = UniformCts (^2) (approx . rationalSqrt) realSqrt ::  ... 
doi:10.1017/s0960129506005871 fatcat:4kddlvemijbr3huancudhdpjzi

Continuous Functions on Final Coalgebras

Neil Ghani, Peter Hancock, Dirk Pattinson
2009 Electronical Notes in Theoretical Computer Science  
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 and (Σ I : Set) Set I , coinductive  ...  It has 0 and +, 1 and ×, ·, Σ f , Π f , Δ f , μ and ν, not to mention some linear logic connectives and connections with the Newton-Leibnitz differential calculus.  ...  Working with container technology seems to be much cleaner than the alternative of working with power-series functors.  ... 
doi:10.1016/j.entcs.2009.07.081 fatcat:rj33jvuzknaszcqwb3a3iyjczy
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