Semilinear Home-space is Decidable for Petri Nets
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A set of configurations 𝐇 is an home-space for a set of configurations 𝐗 of a Petri net if every configuration reachable from 𝐗 can reach 𝐇. The semilinear home-space problem for Petri nets asks, given a Petri net A, and semilinear sets of configurations 𝐗,𝐇 if 𝐇 is an home-space for 𝐗. In 1989, Davide 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 using the same periods. In this paper, we show that the general problem is decidable. This result is obtained by proving a duality between the reachability problem and the non-home-space problem. More formally, we prove that for any Petri net A and for any linear set of configurations 𝐋, we can effectively compute a semilinear set 𝐖 of configurations such that for every set 𝐗, the set 𝐋 is not an home-space for 𝐗 if, and only if 𝐖 is reachable from 𝐗.
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