### Abstract

The equational properties of iteration, when combined with composition and pairing, are captured by the notion of 'iteration theory'. We believe that every iterative construction satisfies at least the properties of iteration theories. Axiomatizations are given for several varieties of iteration theories which occur naturally in the semantics of programming languages, i.e., those generated by theories of trees, theories of sequacious functions, theories of partial functions, and theories of both sequacious and partial functions with distinguished predicates. We show which additional equations must be added to the axioms for iteration theories in order to obtain a set of axioms for these subvarieties. Concrete descriptions of the free theories in each variety are given.

Original language | English |
---|---|

Pages (from-to) | 939-966 |

Number of pages | 28 |

Journal | SIAM Journal on Computing |

Volume | 17 |

Issue number | 5 |

Publication status | Published - Oct 1988 |

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### ASJC Scopus subject areas

- Computational Theory and Mathematics
- Applied Mathematics
- Theoretical Computer Science

### Cite this

*SIAM Journal on Computing*,

*17*(5), 939-966.

**Varieties of iteration theories.** / Bloom, Stephen L.; Ésik, Z.

Research output: Contribution to journal › Article

*SIAM Journal on Computing*, vol. 17, no. 5, pp. 939-966.

}

TY - JOUR

T1 - Varieties of iteration theories

AU - Bloom, Stephen L.

AU - Ésik, Z.

PY - 1988/10

Y1 - 1988/10

N2 - The equational properties of iteration, when combined with composition and pairing, are captured by the notion of 'iteration theory'. We believe that every iterative construction satisfies at least the properties of iteration theories. Axiomatizations are given for several varieties of iteration theories which occur naturally in the semantics of programming languages, i.e., those generated by theories of trees, theories of sequacious functions, theories of partial functions, and theories of both sequacious and partial functions with distinguished predicates. We show which additional equations must be added to the axioms for iteration theories in order to obtain a set of axioms for these subvarieties. Concrete descriptions of the free theories in each variety are given.

AB - The equational properties of iteration, when combined with composition and pairing, are captured by the notion of 'iteration theory'. We believe that every iterative construction satisfies at least the properties of iteration theories. Axiomatizations are given for several varieties of iteration theories which occur naturally in the semantics of programming languages, i.e., those generated by theories of trees, theories of sequacious functions, theories of partial functions, and theories of both sequacious and partial functions with distinguished predicates. We show which additional equations must be added to the axioms for iteration theories in order to obtain a set of axioms for these subvarieties. Concrete descriptions of the free theories in each variety are given.

UR - http://www.scopus.com/inward/record.url?scp=0024090202&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0024090202&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0024090202

VL - 17

SP - 939

EP - 966

JO - SIAM Journal on Computing

JF - SIAM Journal on Computing

SN - 0097-5397

IS - 5

ER -