Hostname: page-component-8448b6f56d-42gr6 Total loading time: 0 Render date: 2024-04-18T12:26:28.362Z Has data issue: false hasContentIssue false
  • English
  • Français

Dagger extension theorem†

Published online by Cambridge University Press:  22 August 2011

Z. ÉSIK  and
T. HAJGATÓ
Z. ÉSIK
Affiliation:
Department of Computer Science, University of Szeged, Hungary Email: ze@inf.u-szeged.hu; hajgato@inf.u-szeged.hu
T. HAJGATÓ
Affiliation:
Department of Computer Science, University of Szeged, Hungary Email: ze@inf.u-szeged.hu; hajgato@inf.u-szeged.hu
Get access Rights & Permissions [Opens in a new window]

Abstract

Partial iterative theories are algebraic theories such that for certain morphisms f the equation ξ = f ⋅ 〈ξ, 1p〉 has a unique solution. Iteration theories are algebraic theories satisfying a certain set of identities. We investigate some similarities between partial iterative theories and iteration theories.

In our main result, we give a sufficient condition ensuring that the partially defined dagger operation of a partial iterative theory can be extended to a totally defined operation so that the resulting theory becomes an iteration theory. We show that this general extension theorem can be instantiated to prove that every Elgot iterative theory with at least one constant morphism 1 → 0 can be extended to an iteration theory. We also apply our main result to theories equipped with an additive structure.

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 21 , Issue 5 , October 2011 , pp. 1035 - 1066
Copyright
Copyright © Cambridge University Press 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

de Bakker, J. W. and Scott, D. (1969) A theory of programs, IBM Seminar, Vienna.Google Scholar
Bekić, H. (1984) Definable operations in general algebras and the theory of automata and flowcharts. Springer-Verlag Lecture Notes in Computer Science 177 30–55.CrossRefGoogle Scholar
Bloom, S. L., Elgot, C. C. and Wright, J. B. (1980a) Solutions of the iteration equation and extension of the scalar iteration operation. SIAM J. Computing 9 26–45.CrossRefGoogle Scholar
Bloom, S. L., Elgot, C. C. and Wright, J. B. (1980b) Vector iteration in pointed iterative theories. SIAM J. Computing 9 525–540.CrossRefGoogle Scholar
Bloom, S. L. and Ésik, Z. (1993) Iteration Theories, Springer-Verlag.CrossRefGoogle Scholar
Bloom, S. L. and Ésik, Z. (2003) An extension theorem with an application to formal tree series. J. of Automata, Languages and Combinatorics 8 (2)145–185.Google Scholar
Bloom, S. L. and Ésik, Z. (2009) Axiomatizing rational power series over natural numbers. Information and Computation 207 793–811.CrossRefGoogle Scholar
Bloom, S. L., Ginali, S. and Rutledge, J. D. (1977) Scalar and vector iteration. J. Comput. System Sci. 14 251–256.CrossRefGoogle Scholar
Elgot, C. C. (1976) Matricial theories. J. Algebra 42 391–421.CrossRefGoogle Scholar
Elgot, C. C. (1973) Monadic computation and iterative algebraic theories. Logic Colloquium '73, Studies in Logic and the Foundations of Mathematics 80, North-Holland175–230.Google Scholar
Ésik, Z. (1980) Identities in iterative and rational algebraic theories. Computational Linguistics and Computer Languages XIV 183–207.Google Scholar
Ésik, Z. (1982) On generalized iterative algebraic theories. Computational Linguistics and Computer Languages XV 95–110.Google Scholar
Ésik, Z. (1997) Completeness of Park induction. Theoret. Comput. Sci. 177 217–283.CrossRefGoogle Scholar
Ésik, Z. (1999) Group axioms for iteration. Information and Computation 148 131–180.CrossRefGoogle Scholar
Ésik, Z. and Kuich, W. (2003) Formal tree series. J. Autom. Lang. Comb. 8 219–285.Google Scholar
Fokkink, W. (2007) Introduction to Process Algebra, Computer Science – Monograph, Springer-Verlag.Google Scholar
Golan, J. (1999) Semirings and Their Applications, Kluwer Academic Publishers.CrossRefGoogle Scholar
Lawvere, F. W. (1963) Functorial semantics of algebraic theories. Proc. Nat. Acad. Sci. U.S.A. 50 869–872.CrossRefGoogle ScholarPubMed
Niwinski, D. (1985) Equational μ-calculus. In: Skowron, A. (ed.) Computation theory – Proceedings, Fifth Symposium, Zaborów, Poland. Springer-Verlag Lecture Notes in Computer Science 208 169–176.CrossRefGoogle Scholar
Niwinski, D. (1986) On fixed-point clones (extended abstract). In: Kott, L. (ed.) Automata, languages and programming – Proceedings 13th International Colloquium. Springer-Verlag Lecture Notes in Computer Science 226 464–473.CrossRefGoogle Scholar
Park, D. M. R. (1970) Fixed point induction and proofs of program properties. In: Michie, D. and Meltzer, B. (eds.) Machine Intelligence 5, Edinburgh University Press 59–78.Google Scholar
Plotkin, G. (1983) Domains, University of Edinburgh.Google Scholar
Simpson, A. and Plotkin, G. (2000) Complete axioms for categorical fixed-point operators. 15th Annual IEEE Symposium on Logic in Computer Science (Santa Barbara, CA, 2000), IEEE Computer Society Press 30–41.Google Scholar
Wright, J. B., Thatcher, J. W., Wagner, E. G. and Goguen, J. A. (1976) Rational algebraic theories and fixed-point solutions, 17th Annual Symposium on Foundations of Computer Science, IEEE Computer Society 147–158.Google Scholar