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Computing Least and Greatest Fixed Points in Absorptive Semirings

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Relational and Algebraic Methods in Computer Science (RAMiCS 2021)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 13027))

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Abstract

We present two methods to algorithmically compute both least and greatest solutions of polynomial equation systems over absorptive semirings (with certain completeness and continuity assumptions), such as the tropical semiring. Both methods require a polynomial number of semiring operations, including semiring addition, multiplication and an infinitary power operation.

Our main result is a closed-form solution for least and greatest fixed points based on the fixed-point iteration. The proof builds on the notion of (possibly infinite) derivation trees; a careful analysis of the shape of these trees allows us to collapse the fixed-point iteration to a linear number of steps. The second method is an iterative symbolic computation in the semiring of generalized absorptive polynomials, largely based on results on Kleene algebras.

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Notes

  1. 1.

    We use \(+_\mathbb {R}\) for the addition on \(\mathbb {R}\) to distinguish it from the semiring operation \(+\).

  2. 2.

    In idempotent semirings, it is equivalent to only assume suprema/infima of chains [4].

  3. 3.

    \({\mathbb S}^{\infty }[{\boldsymbol{X}}]\) generalizes the semiring \(\mathrm {Sorp}({\boldsymbol{X}})\) of absorptive polynomials in [5] by adding the exponent \(\infty \) (which is needed to have fully-continuous homomorphisms in Theorem 9). We only use \({\mathbb S}^{\infty }[{\boldsymbol{X}}]\) in this paper and hence drop generalized in the following.

  4. 4.

    We note that absorptive, fully-continuous semirings also satisfy the additional axioms of Cohen’s \(\omega \)-algebras [2] by setting \(a^* = 1\) and \(a^\omega = a^\infty \). However, these axioms seem too weak to axiomatize the operation \(a^\infty \); we discuss an alternative in [16].

References

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Acknowledgements

I would like to thank the anonymous reviewers for their helpful comments and for suggesting related concepts, in particular [2, 8] and Remark 31.

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Authors and Affiliations

  1. RWTH Aachen University, Aachen, Germany

    Matthias Naaf

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  1. Matthias Naaf
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Correspondence to Matthias Naaf .

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Editors and Affiliations

  1. École polytechnique, Palaiseau, France

    Uli Fahrenberg

  2. Université Côte d’Azur, Nice, France

    Mai Gehrke

  3. Aix-Marseille University, Marseille, France

    Luigi Santocanale

  4. Brock University, St Catharines, ON, Canada

    Michael Winter

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Naaf, M. (2021). Computing Least and Greatest Fixed Points in Absorptive Semirings. In: Fahrenberg, U., Gehrke, M., Santocanale, L., Winter, M. (eds) Relational and Algebraic Methods in Computer Science. RAMiCS 2021. Lecture Notes in Computer Science(), vol 13027. Springer, Cham. https://doi.org/10.1007/978-3-030-88701-8_21

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  • DOI: https://doi.org/10.1007/978-3-030-88701-8_21

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-88700-1

  • Online ISBN: 978-3-030-88701-8

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