Home

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

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.

References
Hide All
Aczel, P., Adámek, J., Milius, S. and Velebil, J. (2003) Infinite trees and completely iterative algebras, a coalgebraic view. Theoretical Computer Science 300 145.
Adámek, J. (1974) Free algebras and automata realizations in the language categories. Comment. Math. Univ. Carolin. 15 589602.
Adámek, J., Börger, R., Milius, S. and Velebil, J. (2008) Iterative algebras: How iterative are they? Theory Appl. Categ. 19 6192.
Adámek, J. and Milius, S. (2006) Terminal coalgebras and free iterative theories. Inform. Comput. 204 11391172.
Adámek, J., Milius, S. and Velebil, J. (2006a) Iterative algebras at work. Mathematical Structures in Computer Science 16 10851131.
Adámek, J., Milius, S. and Velebil, J. (2006b) Elgot algebras. Logic. Meth. Comput. Sci. 2 131.
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.
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.
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.
Adámek, J., Milius, S. and Velebil, J. (2010a) Iterative reflections of monads. Mathematical Structures in Computer Science 20 (3)419452.
Adámek, J., Milius, S. and Velebil, J. (2010b) Equational properties of iteration theories. Inform. and Comput. 208 13061348.
Adámek, J. and Rosický, J. (1994) Locally Presentable and Accessible Categories, Cambridge University Press.
Barr, M. (1970) Coequalizers and free triples. Math. Z. 116 307322.
Bénabou, J. (1968) Structures algebriques dans le catégories. Cah. Topol. Géom. Différ. Catég 10 1126.
Bloom, S. L. and Ésik, Z. (1993) Iteration theories: the equational logic of iterative processes, Springer-Verlag.
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.
Carboni, A., Lack, S. and Walters, R. F. C. (1993) Introduction to extensive and distributive categories. J. Pure Appl. Algebra 84 145158.
Elgot, C. C. (1975) Monadic computation and iterative algebraic theories. In: Shepherdson, J. C. (ed.) Logic Colloquium 1973. Studies in Logic 80 174250.
Elgot, C. C., Bloom, S. L. and Tindell, R. (1978) On the algebraic structure of rooted trees. J. Comput. System Sci. 16 362399.
Ésik, Z. (1988) Independence of the Equational Axioms for Iteration Theories. J. Comput. System Sci. 36 6676.
Fiore, M., Plotkin, G. D. and Turi, D. (1999) Abstract syntax and variable binding. Proc. 14th Annual Symposium on Logic Computer Science 193–202.
Gabriel, P. and Ulmer, F. (1971) Lokal präsentierbare Kategorien. Springer-Verlag Lecture Notes in Mathematics 221.
Ginali, S. (1979) Regular trees and the free iterative theory. J. Comput. Syst. Sci. 18 228242.
Haghverdi, E. (2000) A Categorical Approach to Linear Logic, Geometry of Interaction and Full Completeness, Ph.D. thesis, University of Ottawa.
Hasegawa, M. (1999) Models of Sharing Graphs: A Categorical Semantics of let and letrec, Distinguished Dissertation Series, Springer-Verlag.
Joyal, A., Street, R. and Verity, D. (1996) Traced Monoidal Categories. Math. Proc. Cambridge Philos. Soc. 119 (3)447468.
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.
Lack, S. (1999) On the monadicity of finitary monads. J. Pure Appl. Algebra 140 6573.
Lambek, J. (1968) A Fixpoint Theorem for Complete Categories. Math. Z. 103 151161.
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.)
Mac Lane, S. (1998) Categories for the working mathematician, 2nd edition, Springer-Verlag.
Milius, S. (2005) Completely iterative algebras and completely iterative monads. Inform. Comput. 196 141.
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.
Moss, L. S. (2001) Parametric corecursion. Theoretical Computer Science 260 (1-2)139163.
Moss, L. S. (2003) Recursion and corecursion have the same equational logic. Theoretical Computer Science 294 233267.
Nelson, E (1983) Iterative algebras. Theoretical Computer Science 25 6794.
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.
Tiuryn, J. (1980) Unique fixed points vs. least fixed points. Theoretical Computer Science 12 229254.
Recommend this journal

Mathematical Structures in Computer Science
• ISSN: 0960-1295
• EISSN: 1469-8072
• URL: /core/journals/mathematical-structures-in-computer-science
Who would you like to send this to? *

×

## Full text viewsFull text views reflects the number of PDF downloads, PDFs sent to Google Drive, Dropbox and Kindle and HTML full text views.

Total number of HTML views: 0
Total number of PDF views: 0 *