Solving algebraic equations using coalgebra
RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 37 (2003) no. 4, pp. 301-314.

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.

DOI : 10.1051/ita:2003021
Classification : 18C10, 18C35, 18C50
Mots clés : coalgebra, recursion, category theory
Marchi, Federico De  ; Ghani, Neil  ; Lüth, Christoph 1

1 FB 3 – Mathematics and Computer Science, Universität Bremen;
@article{ITA_2003__37_4_301_0,
     author = {Marchi, Federico De and Ghani, Neil and L\"uth, Christoph},
     title = {Solving algebraic equations using coalgebra},
     journal = {RAIRO - Theoretical Informatics and Applications - Informatique Th\'eorique et Applications},
     pages = {301--314},
     publisher = {EDP-Sciences},
     volume = {37},
     number = {4},
     year = {2003},
     doi = {10.1051/ita:2003021},
     mrnumber = {2053029},
     zbl = {1038.18005},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/ita:2003021/}
}
TY  - JOUR
AU  - Marchi, Federico De
AU  - Ghani, Neil
AU  - Lüth, Christoph
TI  - Solving algebraic equations using coalgebra
JO  - RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications
PY  - 2003
SP  - 301
EP  - 314
VL  - 37
IS  - 4
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/ita:2003021/
DO  - 10.1051/ita:2003021
LA  - en
ID  - ITA_2003__37_4_301_0
ER  - 
%0 Journal Article
%A Marchi, Federico De
%A Ghani, Neil
%A Lüth, Christoph
%T Solving algebraic equations using coalgebra
%J RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications
%D 2003
%P 301-314
%V 37
%N 4
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/ita:2003021/
%R 10.1051/ita:2003021
%G en
%F ITA_2003__37_4_301_0
Marchi, Federico De; Ghani, Neil; Lüth, Christoph. Solving algebraic equations using coalgebra. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 37 (2003) no. 4, pp. 301-314. doi : 10.1051/ita:2003021. http://archive.numdam.org/articles/10.1051/ita:2003021/

[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. | MR | Zbl

[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. | MR | Zbl

[4] J. Adámek and J. Rosicky, Locally Presentable and Accessible Categories. Cambridge University Press, London Math. Soc. Lecture Notes 189 (1994). | MR | Zbl

[5] J. Adámek, S. Milius and J. Velebil, Free iterative theories: A coalgebraic view. Math. Struct. Comput. Sci. 13 (2003) 259-320. | MR | Zbl

[6] M. Barr, Terminal coalgebras for endofunctors on sets. Available from ftp://www.math.mcgill.ca/pub/barr/trmclgps.zip (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. | MR | Zbl

[8] B. Courcelle, Fundamental properties of infinite trees. Theor. Comput. Sci. 25 (1983) 95-169. | MR | Zbl

[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 Series 6-8, 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). | MR | Zbl

[13] G.M. Kelly, A unified treatment of transfinite constructions. Bull. of Austral. Math. Soc. 22 (1980) 1-83. | MR | Zbl

[14] G.M. Kelly and A.J. Power, Adjunctions whose counits are equalizers, and presentations of finitary monads. J. Pure Appl. Algebra 89 (1993) 163-179. | MR | Zbl

[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). | Zbl

[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 http://math.indiana.edu/home/moss/parametric.ps | MR | Zbl

[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). | Zbl

Cité par Sources :