Inductive *-semirings

Zoltán Ésik, Werner Kuich

Research output: Contribution to journalConference article

28 Citations (Scopus)


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 languageEnglish
Pages (from-to)3-33
Number of pages31
JournalTheoretical Computer Science
Issue number1
Publication statusPublished - Sep 16 2004
EventWords, Languages and Combinatorics - Kyoto, Japan
Duration: Mar 14 2000Mar 18 2000


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

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computer Science(all)

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