### Abstract

The (in)equational properties of iteration, i.e., least (pre-)fixed point solutions over cpo's, are captured by the axioms of iteration theories. All known axiomatizations of iteration theories consist of the Conway identities and a complicated equation scheme, the commutative identity. The results of this paper show that the commutative identity is implied by the Conway identities and a weak form of the Park induction principle. Hence, we obtain a simple first order axiomatization of the (in)equational theory of iteration. It follows that a few simple identities and a weak form of the Scott induction principle, formulated to involve only inequations, are also complete. We also show that the Conway identities and the Park induction principle are not complete for the universal Horn theory of iteration.

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

Pages (from-to) | 217-283 |

Number of pages | 67 |

Journal | Theoretical Computer Science |

Volume | 177 |

Issue number | 1 |

Publication status | Published - Apr 30 1997 |

### Fingerprint

### ASJC Scopus subject areas

- Computational Theory and Mathematics

### Cite this

**Completeness of Park induction.** / Ésik, Z.

Research output: Contribution to journal › Article

*Theoretical Computer Science*, vol. 177, no. 1, pp. 217-283.

}

TY - JOUR

T1 - Completeness of Park induction

AU - Ésik, Z.

PY - 1997/4/30

Y1 - 1997/4/30

N2 - The (in)equational properties of iteration, i.e., least (pre-)fixed point solutions over cpo's, are captured by the axioms of iteration theories. All known axiomatizations of iteration theories consist of the Conway identities and a complicated equation scheme, the commutative identity. The results of this paper show that the commutative identity is implied by the Conway identities and a weak form of the Park induction principle. Hence, we obtain a simple first order axiomatization of the (in)equational theory of iteration. It follows that a few simple identities and a weak form of the Scott induction principle, formulated to involve only inequations, are also complete. We also show that the Conway identities and the Park induction principle are not complete for the universal Horn theory of iteration.

AB - The (in)equational properties of iteration, i.e., least (pre-)fixed point solutions over cpo's, are captured by the axioms of iteration theories. All known axiomatizations of iteration theories consist of the Conway identities and a complicated equation scheme, the commutative identity. The results of this paper show that the commutative identity is implied by the Conway identities and a weak form of the Park induction principle. Hence, we obtain a simple first order axiomatization of the (in)equational theory of iteration. It follows that a few simple identities and a weak form of the Scott induction principle, formulated to involve only inequations, are also complete. We also show that the Conway identities and the Park induction principle are not complete for the universal Horn theory of iteration.

UR - http://www.scopus.com/inward/record.url?scp=0031122739&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0031122739&partnerID=8YFLogxK

M3 - Article

VL - 177

SP - 217

EP - 283

JO - Theoretical Computer Science

JF - Theoretical Computer Science

SN - 0304-3975

IS - 1

ER -