Bi-reachability in Petri nets with data

We investigate Petri nets with data, an extension of plain Petri nets where tokens carry values from an infinite data domain, and executability of transitions is conditioned by equalities between data values. We provide a decision procedure for the bi-reachability problem: given a Petri net and its two configurations, we ask if each of the configurations is reachable from the other. This pushes forward the decidability borderline, as the bi-reachability problem subsumes the coverability problem (which is known to be decidable) and is subsumed by the reachability problem (whose decidability status is unknown).

Solvability of orbit-finite systems of linear equations

We study orbit-finite systems of linear equations, in the setting of sets with atoms. Our principal contribution is a decision procedure for solvability of such systems. The procedure works for every field (and even commutative ring) under mild effectiveness assumptions, and reduces a given orbit-finite system to a number of finite ones: exponentially many in general, but polynomially many when atom dimension of input systems is fixed. Towards obtaining the procedure we push further the theory of vector spaces generated by orbit-finite sets, and show that each such vector space admits an orbit-finite basis. This fundamental property is a key tool in our development, but should be also of wider interest.

Improved Ackermannian lower bound for the VASS reachability problem

This draft is a follow-up of the Ackermannian lower bound for the reachability problem in vector addition systems with states (VASS), recently announced by Czerwi\'nski and Orlikowski. Independently, the same result has been announced by Leroux, but with a significantly different proof. We provide a simplification of the former construction, thus improving the lower bound for VASS in fixed dimension: while Czerwi\'nski and Orlikowski prove $F_k$-hardness in dimension $6k$, and Leroux in dimension $4k+9$, the simplified construction yields $F_k$-hardness already in dimension $3k+2$.