### Abstract

One of the most well-known induction principles in computer science is the fixed point induction rule, or least pre-fixed point rule. Inductive *-semirings are partially ordered semirings equipped with a star operation satisfying the fixed point equation and the fixed point induction rule for linear terms. Inductive *-semirings are extensions of continuous semirings and the Kleene algebras of Conway and Kozen. We develop, in a systematic way, the rudiments of the theory of inductive *-semirings in relation to automata, languages and power series. In particular, we prove that if S is an inductive *-semiring, then so is the semiring of matrices S ^{n×n}, for any integer n≥0, and that if S is an inductive *-semiring, then so is any semiring of power series S〈〈 A*〉〉. As shown by Kozen, the dual of an inductive *-semiring may not be inductive. In contrast, we show that the dual of an iteration semiring is an iteration semiring. Kuich proved a general Kleene theorem for continuous semirings, and Bloom and Ésik proved a Kleene theorem for all Conway semirings. Since any inductive *-semiring is a Conway semiring and an iteration semiring, as we show, there results a Kleene theorem applicable to all inductive *-semirings. We also describe the structure of the initial inductive *-semiring and conjecture that any free inductive *-semiring may be given as a semiring of rational power series with coefficients in the initial inductive *-semiring. We relate this conjecture to recent axiomatization results on the equational theory of the regular sets.

Original language | English |
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Pages (from-to) | 3-33 |

Number of pages | 31 |

Journal | Theoretical Computer Science |

Volume | 324 |

Issue number | 1 |

DOIs | |

Publication status | Published - Sep 16 2004 |

Event | Words, Languages and Combinatorics - Kyoto, Japan Duration: Mar 14 2000 → Mar 18 2000 |

### Keywords

- Equational logic
- Least fixed point
- Power series
- Semiring

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)

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## Cite this

*Theoretical Computer Science*,

*324*(1), 3-33. https://doi.org/10.1016/j.tcs.2004.03.050