Sławomir Lasota

Arka Ghosh, Sławomir Lasota

We study existence and computability of finite bases for ideals of polynomials over infinitely many variables. In our setting, variables come from a countable logical structure A, and embeddings from A to A act on polynomials by renaming variables. First, we give a sufficient and necessary condition for A to guarantee the following generalisation of Hilbert's Basis Theorem: every polynomial ideal which is equivariant, i.e. invariant under renaming of variables, is finitely generated. Second, we develop an extension of classical Buchberger's algorithm to compute a Gröbner basis of a given equivariant ideal. This implies decidability of the membership problem for equivariant ideals. Finally, we sketch upon various applications of these results to register automata, Petri nets with data, orbit-finitely generated vector spaces, and orbit-finite systems of linear equations.

Łukasz Kamiński, Sławomir Lasota

We investigate the reachability problem in symmetric vector addition systems with states (VASS), where transitions are invariant under a group of permutations of coordinates. One extremal case, the trivial groups, yields general VASS. In another extremal case, the symmetric groups, we show that the reachability problem can be solved in PSPACE, regardless of the dimension of input VASS (to be contrasted with Ackermannian complexity in general VASS). We also consider other groups, in particular alternating and cyclic ones. Furthermore, motivated by the open status of the reachability problem in data VASS, we estimate the gain in complexity when the group arises as a combination of the trivial and symmetric groups.

Arka Ghosh, Piotr Hofman, Sławomir Lasota

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.

Sławomir Lasota

Petri nets, equivalently presentable as vector addition systems with states, are an established model of concurrency with widespread applications. The reachability problem, where we ask whether from a given initial configuration there exists a sequence of valid execution steps reaching a given final configuration, is the central algorithmic problem for this model. The complexity of the problem has remained, until recently, one of the hardest open questions in verification of concurrent systems. A first upper bound has been provided only in 2015 by Leroux and Schmitz, then refined by the same authors to non-primitive recursive Ackermannian upper bound in 2019. The exponential space lower bound, shown by Lipton already in 1976, remained the only known for over 40 years until a breakthrough non-elementary lower bound by Czerwi{ń}ski, Lasota, Lazic, Leroux and Mazowiecki in 2019. Finally, a matching Ackermannian lower bound announced this year by Czerwi{ń}ski and Orlikowski, and independently by Leroux, established the complexity of the problem. Our primary contribution is an improvement of the former construction, making it conceptually simpler and more direct. On the way we improve the lower bound for vector addition systems with states in fixed dimension (or, equivalently, Petri nets with fixed number of places): while Czerwi{ń}ski and Orlikowski prove $F_k$-hardness (hardness for $k$th level in Grzegorczyk Hierarchy) in dimension $6k$, our simplified construction yields $F_k$-hardness already in dimension $3k+2$.