Solving Algebraic Equations Using Coalgebra

Federico De Marchi; Neil Ghani; Christoph Lüth

RAIRO - Theoretical Informatics and Applications (2010)

  • Volume: 37, Issue: 4, page 301-314
  • ISSN: 0988-3754

Abstract

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Algebraic systems of equations define functions using recursion where parameter passing is permitted. This generalizes the notion of a rational system of equations where parameter passing is prohibited. It has been known for some time that algebraic systems in Greibach Normal Form have unique solutions. This paper presents a categorical approach to algebraic systems of equations which generalizes the traditional approach in two ways i) we define algebraic equations for locally finitely presentable categories rather than just Set; and ii) we define algebraic equations to allow right-hand sides which need not consist of finite terms. We show these generalized algebraic systems of equations have unique solutions by replacing the traditional metric-theoretic arguments with coalgebraic arguments.

How to cite

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De Marchi, Federico, Ghani, Neil, and Lüth, Christoph. "Solving Algebraic Equations Using Coalgebra." RAIRO - Theoretical Informatics and Applications 37.4 (2010): 301-314. <http://eudml.org/doc/92725>.

@article{DeMarchi2010,
abstract = { Algebraic systems of equations define functions using recursion where parameter passing is permitted. This generalizes the notion of a rational system of equations where parameter passing is prohibited. It has been known for some time that algebraic systems in Greibach Normal Form have unique solutions. This paper presents a categorical approach to algebraic systems of equations which generalizes the traditional approach in two ways i) we define algebraic equations for locally finitely presentable categories rather than just Set; and ii) we define algebraic equations to allow right-hand sides which need not consist of finite terms. We show these generalized algebraic systems of equations have unique solutions by replacing the traditional metric-theoretic arguments with coalgebraic arguments. },
author = {De Marchi, Federico, Ghani, Neil, Lüth, Christoph},
journal = {RAIRO - Theoretical Informatics and Applications},
keywords = {Coalgebra; recursion; category theory.; algebraic system of equations; monad; coalgebra; locally finitely presented category; natural transformation},
language = {eng},
month = {3},
number = {4},
pages = {301-314},
publisher = {EDP Sciences},
title = {Solving Algebraic Equations Using Coalgebra},
url = {http://eudml.org/doc/92725},
volume = {37},
year = {2010},
}

TY - JOUR
AU - De Marchi, Federico
AU - Ghani, Neil
AU - Lüth, Christoph
TI - Solving Algebraic Equations Using Coalgebra
JO - RAIRO - Theoretical Informatics and Applications
DA - 2010/3//
PB - EDP Sciences
VL - 37
IS - 4
SP - 301
EP - 314
AB - Algebraic systems of equations define functions using recursion where parameter passing is permitted. This generalizes the notion of a rational system of equations where parameter passing is prohibited. It has been known for some time that algebraic systems in Greibach Normal Form have unique solutions. This paper presents a categorical approach to algebraic systems of equations which generalizes the traditional approach in two ways i) we define algebraic equations for locally finitely presentable categories rather than just Set; and ii) we define algebraic equations to allow right-hand sides which need not consist of finite terms. We show these generalized algebraic systems of equations have unique solutions by replacing the traditional metric-theoretic arguments with coalgebraic arguments.
LA - eng
KW - Coalgebra; recursion; category theory.; algebraic system of equations; monad; coalgebra; locally finitely presented category; natural transformation
UR - http://eudml.org/doc/92725
ER -

References

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  1. P. Aczel, J. Adámek, S. Milius and J. Velebil, Infinite trees and completely iterative theories: A coalgebraic view. Theor. Comput. Sci.300 (2003) 1-45 .  
  2. P. Aczel, J. Adámek and J. Velebil, A coalgebraic view of infinite trees and iteration, in Proceedings 4th Workshop on Coalgebraic Methods in Computer Science, CMCS'01, edited by A. Corradini, M. Lenisa and U. Montanari, Genova, Italy, 6-7 April 2001. Elsevier, Electronics Notes Theor. Comput. Sci.44 (2001).  
  3. J. Adámek, Final coalgebras are ideal completions of initial algebras. J. Log. Comput.12 (2002) 217-242.  
  4. J. Adámek and J. Rosicky, Locally Presentable and Accessible Categories. Cambridge University Press, London Math. Soc. Lecture Notes189 (1994).  
  5. J. Adámek, S. Milius and J. Velebil, Free iterative theories: A coalgebraic view. Math. Struct. Comput. Sci.13 (2003) 259-320.  
  6. M. Barr, Terminal coalgebras for endofunctors on sets. Available from (1999).  
  7. C.C. Elgot, S.L. Bloom and S. Tindell, On the algebraic structure of rooted trees. J. Comp. Syst. Sci.16 (1978) 361-399.  
  8. B. Courcelle, Fundamental properties of infinite trees. Theor. Comput. Sci.25 (1983) 95-169 .  
  9. N. Ghani, C. Lüth and F. De Marchi, Coalgebraic approaches to algebraic terms, in Fixed Points in Computer Science, edited by Z. Ésik and A. Ingólfsdóttir. BRICS Notes Series6-8, July 20-21 NS-02-2 (2002).  
  10. N. Ghani, C. Lüth and F. De Marchi, Coalgebraic monads, in Proc. 5th Workshop on Coalgebraic Methods in Computer Science, edited by L.M. Moss, Grenoble, France, 6-7 April (2002).  
  11. N. Ghani, C. Lüth, F. De Marchi and J. Power, Algebras, coalgebras, monads and comonads, in Proceedings 4th Workshop on Coalgebraic Methods in Computer Science, edited by A. Corradini, M. Lenisa and U. Montanari, Genova, Italy, 6-7 April 2001. Elsevier, Electronics Notes Theor. Comput. Sci.44 (2001).  
  12. I. Guessarian, Algebraic Semantics. Springer-Verlag, Lecture Notes Comput. Sci.99 (1979).  
  13. G.M. Kelly, A unified treatment of transfinite constructions. Bull. of Austral. Math. Soc.22 (1980) 1-83.  
  14. G.M. Kelly and A.J. Power, Adjunctions whose counits are equalizers, and presentations of finitary monads. J. Pure Appl. Algebra89 (1993) 163-179 .  
  15. C. Lüth and N. Ghani, Monads and modular term rewriting, in Proc. 7th Int. Conf. on Category Theory and Computer Science, edited by E. Moggi and G. Rosolini, Santa Margherita Ligure, Italy, 4-6 September 1997. Springer-Verlag, Lecture Notes Comput. Sci.1290 69-86 (1997).  
  16. F. De Marchi, Monads in Coalgebra. Ph.D. thesis, Univ. of Leicester (2003) (Submitted).  
  17. S. Milius, Final coalgebras in categories of monads (unpublished).  
  18. S. Milius, Free iterative theories: a coalgebraic view (extended abstract), presented at FICS 2001 - Fixed Points in Computer Science, 7-8 September, Florence, Italy (2001).  
  19. L. Moss, The coalgebraic treatment of second-order substitution and uniinterpreted recursive program schemes. Privately circulated manuscript.  
  20. L. Moss, Parametric corecursion. Preprint, available at  URIhttp://math.indiana.edu/home/moss/parametric.ps
  21. N. Ghani and T. Uustalu, Explicit substitutions and presheafs, in Proceedings of MERLIN (2003).  
  22. E. Robinson, Variations on algebra: monadicity and generalisation of equational theories. Technical Report 6/94, Sussex Computer Science (1994).  

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