Abstract
We study the reachability problem for affine $\mathbb{Z}$-VASS, which are integer vector addition systems with states in which transitions perform affine transformations on the counters. This problem is easily seen to be undecidable in general, and we therefore restrict ourselves to affine $\mathbb{Z}$-VASS with the finite-monoid property (afmp-$\mathbb{Z}$-VASS). The latter have the property that the monoid generated by the matrices appearing in their affine transformations is finite. The class of afmp-$\mathbb{Z}$-VASS encompasses classical operations of counter machines such as resets, permutations, transfers and copies. We show that reachability in an afmp-$\mathbb{Z}$-VASS reduces to reachability in a $\mathbb{Z}$-VASS whose control-states grow linearly in the size of the matrix monoid. Our construction shows that reachability relations of afmp-$\mathbb{Z}$-VASS are semilinear, and in particular enables us to show that reachability in $\mathbb{Z}$-VASS with transfers and $\mathbb{Z}$-VASS with copies is PSPACE-complete. We then focus on the reachability problem for affine $\mathbb{Z}$-VASS with monogenic monoids: (possibly infinite) matrix monoids generated by a single matrix. We show that, in a particular case, the reachability problem is decidable for this class, disproving a conjecture about affine $\mathbb{Z}$-VASS with infinite matrix monoids we raised in a preliminary version of this paper. We complement this result by presenting an affine $\mathbb{Z}$-VASS with monogenic matrix monoid and undecidable reachability relation.
Highlights
Vector addition systems with states (VASS) are a fundamental model of computation comprising a finite-state controller with a finite number of counters ranging over the natural numbers
We focus on the reachability problem for affine Z-VASS with monogenic monoids: matrix monoids generated by a single matrix
In a particular case, the reachability problem is decidable for this class, disproving a conjecture about affine Z-VASS with infinite matrix monoids we raised in a preliminary version of this paper
Summary
Vector addition systems with states (VASS) are a fundamental model of computation comprising a finite-state controller with a finite number of counters ranging over the natural numbers. We show that a PSPACE lower bound of the reachability problem already holds for the extension of Z-VASS that only use permutation matrices in their transition updates This in turn gives PSPACE-completeness of interesting classes such as transfer Z-VASS and copy Z-VASS. We show that an affine Z-VASS that has both transfers and copies may not have the finite-monoid property, and that the reachability problem for this class becomes undecidable. We complement this result by investigating the case of monogenic classes, i.e. classes of monoids with a single generator.
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