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Forward analysis for WSTS, part I: completions

Published online by Cambridge University Press:  19 October 2020

Alain Finkel Open the ORCID record for Alain Finkel [Opens in a new window]  and
Jean Goubault-Larrecq Open the ORCID record for Jean Goubault-Larrecq [Opens in a new window]
Alain Finkel
Affiliation:
Université Paris-Saclay, ENS Paris-Saclay, CNRS, LSV, Institut Universitaire de France, 91190 Gif-sur-Yvette, France
Jean Goubault-Larrecq*
Affiliation:
Université Paris-Saclay, ENS Paris-Saclay, CNRS, LSV, 91190 Gif-sur-Yvette, France
*
*Corresponding author. Email: goubault@ens-paris-saclay.fr
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Abstract

We define representations for downward-closed subsets of a rich family of well-quasi-orders, and more generally for closed subsets of an even richer family of Noetherian topological spaces. This includes the cases of finite words, of multisets, of finite trees, notably. Those representations are given as finite unions of ideals, or more generally of irreducible closed subsets. All the representations we explore are computable, in the sense that we exhibit algorithms that decide inclusion, and compute finite unions and finite intersections. The origin of this work lies in the need for computing finite representations of sets of successors of the downward closure of one state, or more generally of a downward-closed set of states, in a well-structured transition system, and this is where we start: we define adequate notions of completions of well-quasi-orders, and more generally, of Noetherian spaces. For verification purposes, we argue that the required completions must be ideal completions, or more generally sobrifications, that is, spaces of irreducible closed subsets.

Keywords

Well-structured transition systemswell-quasi-orderingsdownwards-closed subsetsNoetherian topological spacesfinite computable representationsideal completions
Type
Paper
Information
Mathematical Structures in Computer Science , Volume 30 , Issue 7 , August 2020 , pp. 752 - 832
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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Footnotes

†

An extended abstract already appeared in Proc. 26th International Symposium on Theoretical Aspects of Computer Science (STACS'09).

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