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Solving Algebraic Equations Using Coalgebra

Published online by Cambridge University Press:  15 January 2004

Federico De Marchi ,
Neil Ghani  and
Christoph Lüth
Federico De Marchi
Affiliation:
Mathematics and Computer Science, University of Leicester; [email protected]., [email protected].
Neil Ghani
Affiliation:
Mathematics and Computer Science, University of Leicester; [email protected]., [email protected].
Christoph Lüth
Affiliation:
FB 3 – Mathematics and Computer Science, Universität Bremen; [email protected].
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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.

Keywords

Coalgebrarecursioncategory theory.
Type
Research Article
Information
RAIRO - Theoretical Informatics and Applications , Volume 37 , Issue 4: Fixed Points in Computer Science (FICS'02) , October 2003 , pp. 301 - 314
Copyright
© EDP Sciences, 2003

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References

Aczel, P., Adámek, J., Milius, S. and Velebil, J., Infinite trees and completely iterative theories: A coalgebraic view. Theor. Comput. Sci. 300 (2003) 1-45 . CrossRef
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).
Adámek, J., Final coalgebras are ideal completions of initial algebras. J. Log. Comput. 12 (2002) 217-242. CrossRef
J. Adámek and J. Rosicky, Locally Presentable and Accessible Categories. Cambridge University Press, London Math. Soc. Lecture Notes 189 (1994).
Adámek, J., Milius, S. and Velebil, J., Free iterative theories: A coalgebraic view. Math. Struct. Comput. Sci. 13 (2003) 259-320. CrossRef
M. Barr, Terminal coalgebras for endofunctors on sets. Available from (1999).
Elgot, C.C., Bloom, S.L. and Tindell, S., On the algebraic structure of rooted trees. J. Comp. Syst. Sci. 16 (1978) 361-399. CrossRef
Courcelle, B., Fundamental properties of infinite trees. Theor. Comput. Sci. 25 (1983) 95-169 . CrossRef
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).
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).
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).
I. Guessarian, Algebraic Semantics. Springer-Verlag, Lecture Notes Comput. Sci. 99 (1979).
Kelly, G.M., A unified treatment of transfinite constructions. Bull. of Austral. Math. Soc. 22 (1980) 1-83. CrossRef
Kelly, G.M. and Power, A.J., Adjunctions whose counits are equalizers, and presentations of finitary monads. J. Pure Appl. Algebra 89 (1993) 163-179 . CrossRef
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).
F. De Marchi, Monads in Coalgebra. Ph.D. thesis, Univ. of Leicester (2003) (Submitted).
S. Milius, Final coalgebras in categories of monads (unpublished).
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).
L. Moss, The coalgebraic treatment of second-order substitution and uniinterpreted recursive program schemes. Privately circulated manuscript.
L. Moss, Parametric corecursion. Preprint, available at http://math.indiana.edu/home/moss/parametric.ps
N. Ghani and T. Uustalu, Explicit substitutions and presheafs, in Proceedings of MERLIN (2003).
E. Robinson, Variations on algebra: monadicity and generalisation of equational theories. Technical Report 6/94, Sussex Computer Science (1994).