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Elgot theories: a new perspective on the equational properties of iteration

Published online by Cambridge University Press:  25 March 2011

JIŘÍ ADÁMEK
Affiliation:
Institut für Theoretische Informatik, Technische Universität Braunschweig, Germany E-mail: adamek@iti.cs.tu-bs.de, mail@stefan-milius.eu
STEFAN MILIUS
Affiliation:
Institut für Theoretische Informatik, Technische Universität Braunschweig, Germany E-mail: adamek@iti.cs.tu-bs.de, mail@stefan-milius.eu
JIŘÍ VELEBIL
Affiliation:
Faculty of Electrical Engineering, Czech Technical University in Prague, Prague, Czech Republic E-mail: velebil@math.feld.cvut.cz

Abstract

Bloom and Ésik's concept of iteration theory summarises all equational properties that iteration has in common applications, for example, in domain theory, where to every system of recursive equations, the least solution is assigned. This paper shows that in the coalgebraic approach to iteration, the more appropriate concept is that of a functorial iteration theory (called Elgot theory). These theories have a particularly simple axiomatisation, and all well-known examples of iteration theories are functorial. Elgot theories are proved to be monadic over the category of sets in context (or, more generally, the category of finitary endofunctors of a locally finitely presentable category). This demonstrates that functoriality is an equational property from the perspective of sets in context. In contrast, Bloom and Ésik worked in the base category of signatures rather than sets in context, and there iteration theories are monadic but Elgot theories are not. This explains why functoriality was not included in the definition of iteration theories.

Type
Paper
Copyright
Copyright © Cambridge University Press 2011

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References

Aczel, P., Adámek, J., Milius, S. and Velebil, J. (2003) Infinite trees and completely iterative algebras, a coalgebraic view. Theoretical Computer Science 300 145.CrossRefGoogle Scholar
Adámek, J. (1974) Free algebras and automata realizations in the language categories. Comment. Math. Univ. Carolin. 15 589602.Google Scholar
Adámek, J., Börger, R., Milius, S. and Velebil, J. (2008) Iterative algebras: How iterative are they? Theory Appl. Categ. 19 6192.Google Scholar
Adámek, J. and Milius, S. (2006) Terminal coalgebras and free iterative theories. Inform. Comput. 204 11391172.CrossRefGoogle Scholar
Adámek, J., Milius, S. and Velebil, J. (2006a) Iterative algebras at work. Mathematical Structures in Computer Science 16 10851131.CrossRefGoogle Scholar
Adámek, J., Milius, S. and Velebil, J. (2006b) Elgot algebras. Logic. Meth. Comput. Sci. 2 131.Google Scholar
Adámek, J., Milius, S. and Velebil, J. (2007) What are iteration theories? In: Kučera, L. and Kučera, A. (eds.) Proc. MFCS 2007. Springer-Verlag Lecture Notes in Computer Science 4708 240252.CrossRefGoogle Scholar
Adámek, J., Milius, S. and Velebil, J. (2009a) Elgot theories: a new perspective of iteration theories (extended abstract). Proceedings, Mathematical Foundations of Computer Science (MFPS 25). Electronic Notes in Theoretical Computer Science 249 407427.CrossRefGoogle Scholar
Adámek, J., Milius, S. and Velebil, J. (2009b) Semantics of higher-order recursion schemes. In: Kurz, A., Lenisa, M. and Tarlecki, A. (eds.) Proc. CALCO 2009. Springer-Verlag Lecture Notes in Computer Science 5728 4963.CrossRefGoogle Scholar
Adámek, J., Milius, S. and Velebil, J. (2010a) Iterative reflections of monads. Mathematical Structures in Computer Science 20 (3)419452.CrossRefGoogle Scholar
Adámek, J., Milius, S. and Velebil, J. (2010b) Equational properties of iteration theories. Inform. and Comput. 208 13061348.CrossRefGoogle Scholar
Adámek, J. and Rosický, J. (1994) Locally Presentable and Accessible Categories, Cambridge University Press.CrossRefGoogle Scholar
Barr, M. (1970) Coequalizers and free triples. Math. Z. 116 307322.CrossRefGoogle Scholar
Bénabou, J. (1968) Structures algebriques dans le catégories. Cah. Topol. Géom. Différ. Catég 10 1126.Google Scholar
Bloom, S. L. and Ésik, Z. (1993) Iteration theories: the equational logic of iterative processes, Springer-Verlag.CrossRefGoogle Scholar
Bonsangue, M., Rutten, J. J. M. M. and Silva, A. (2009) An Algebra for Kripke Polynomial Coalgebras. In: Proc. 24th Annual Symposium on Logic in Computer Science (LICS'09), IEEE Computer Society 4958.Google Scholar
Carboni, A., Lack, S. and Walters, R. F. C. (1993) Introduction to extensive and distributive categories. J. Pure Appl. Algebra 84 145158.CrossRefGoogle Scholar
Elgot, C. C. (1975) Monadic computation and iterative algebraic theories. In: Shepherdson, J. C. (ed.) Logic Colloquium 1973. Studies in Logic 80 174250.Google Scholar
Elgot, C. C., Bloom, S. L. and Tindell, R. (1978) On the algebraic structure of rooted trees. J. Comput. System Sci. 16 362399.CrossRefGoogle Scholar
Ésik, Z. (1988) Independence of the Equational Axioms for Iteration Theories. J. Comput. System Sci. 36 6676.CrossRefGoogle Scholar
Fiore, M., Plotkin, G. D. and Turi, D. (1999) Abstract syntax and variable binding. Proc. 14th Annual Symposium on Logic Computer Science 193–202.CrossRefGoogle Scholar
Gabriel, P. and Ulmer, F. (1971) Lokal präsentierbare Kategorien. Springer-Verlag Lecture Notes in Mathematics 221.Google Scholar
Ginali, S. (1979) Regular trees and the free iterative theory. J. Comput. Syst. Sci. 18 228242.CrossRefGoogle Scholar
Haghverdi, E. (2000) A Categorical Approach to Linear Logic, Geometry of Interaction and Full Completeness, Ph.D. thesis, University of Ottawa.Google Scholar
Hasegawa, M. (1999) Models of Sharing Graphs: A Categorical Semantics of let and letrec, Distinguished Dissertation Series, Springer-Verlag.CrossRefGoogle Scholar
Joyal, A., Street, R. and Verity, D. (1996) Traced Monoidal Categories. Math. Proc. Cambridge Philos. Soc. 119 (3)447468.CrossRefGoogle Scholar
Kelly, G. M. and Power, A. J. (1993) Adjunctions whose counits are coequalizers, and presentations of finitary enriched monads. J. Pure Appl. Algebra 89 163179.CrossRefGoogle Scholar
Lack, S. (1999) On the monadicity of finitary monads. J. Pure Appl. Algebra 140 6573.CrossRefGoogle Scholar
Lambek, J. (1968) A Fixpoint Theorem for Complete Categories. Math. Z. 103 151161.CrossRefGoogle Scholar
Lawvere, F. W. (1963) Functorial semantics of algebraic theories, Ph.D. thesis, Columbia University. (Republished in 2004 in Reprints in Theory Appl. Categ. 5 1–121.)Google Scholar
Mac Lane, S. (1998) Categories for the working mathematician, 2nd edition, Springer-Verlag.Google Scholar
Milius, S. (2005) Completely iterative algebras and completely iterative monads. Inform. Comput. 196 141.CrossRefGoogle Scholar
Milius, S. (2010) A Sound and Complete Calculus for finite Stream Circuits. Proc. 25th Annual Symposium on Logic in Computer Science (LICS'10), IEEE Computer Society 449458.Google Scholar
Moss, L. S. (2001) Parametric corecursion. Theoretical Computer Science 260 (1-2)139163.CrossRefGoogle Scholar
Moss, L. S. (2003) Recursion and corecursion have the same equational logic. Theoretical Computer Science 294 233267.CrossRefGoogle Scholar
Nelson, E (1983) Iterative algebras. Theoretical Computer Science 25 6794.CrossRefGoogle Scholar
Simpson, A. and Plotkin, G. D. (2000) Complete axioms for categorical fixed-point operators. Proc. 15th Symposium on Logic in Computer Science LICS 2000 30–41.Google Scholar
Tiuryn, J. (1980) Unique fixed points vs. least fixed points. Theoretical Computer Science 12 229254.CrossRefGoogle Scholar
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