Abstract
A set of configurations H is a homespace for a set of configurations X of a Petri net if every configuration reachable from (any configuration in) X can reach (some configuration in) H. The semilinear homespace problem for Petri nets asks, given a Petri net and semilinear sets of configurations X, H, if H is a homespace for X. In 1989, David de Frutos Escrig and Colette Johnen proved that the problem is decidable when X is a singleton and H is a finite union of linear sets with the same periods. In this paper, we show that the general (semilinear) problem is decidable. This result is obtained by proving a duality between the reachability problem and the nonhomespace problem. In particular, we prove that for any Petri net and any linear set of configurations L we can effectively compute a semilinear set C of configurations, called a nonreachability core for L, such that for every set X the set L is not a homespace for X if, and only if, C is reachable from X. We show that the established relation to the reachability problem yields the Ackermanncompleteness of the (semilinear) homespace problem. For this we also show that, given a Petri net with an initial marking, the set of minimal reachable markings can be constructed in Ackermannian time.
BibTeX  Entry
@InProceedings{jancar_et_al:LIPIcs.CONCUR.2023.36,
author = {Jan\v{c}ar, Petr and Leroux, J\'{e}r\^{o}me},
title = {{The Semilinear HomeSpace Problem Is AckermannComplete for Petri Nets}},
booktitle = {34th International Conference on Concurrency Theory (CONCUR 2023)},
pages = {36:136:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959772990},
ISSN = {18688969},
year = {2023},
volume = {279},
editor = {P\'{e}rez, Guillermo A. and Raskin, JeanFran\c{c}ois},
publisher = {Schloss Dagstuhl  LeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2023/19030},
URN = {urn:nbn:de:0030drops190300},
doi = {10.4230/LIPIcs.CONCUR.2023.36},
annote = {Keywords: Petri nets, homespace property, semilinear sets, Ackermannian complexity}
}
Keywords: 

Petri nets, homespace property, semilinear sets, Ackermannian complexity 
Collection: 

34th International Conference on Concurrency Theory (CONCUR 2023) 
Issue Date: 

2023 
Date of publication: 

07.09.2023 