Reachability in One-Dimensional Pushdown Vector Addition Systems is Decidable

TL;DR

It is shown that for every one-dimensional GVAS (1-GVAS) one can compute another 1-GVAS, which has the same reachability relation as the original one and additionally has the so-called thin property, which implies decidability of the problem for all 1-GVAS.

Abstract

We consider the model of one-dimensional Pushdown Vector Addition Systems (1-PVAS), a fundamental computational model simulating both recursive and concurrent behaviours. Our main result is decidability of the reachability problem for 1-PVAS, an important open problem investigated for at least a decade. In the algorithm we actually consider an equivalent model of Grammar Vector Addition Systems (GVAS). We prove the main result by showing that for every one-dimensional GVAS (1-GVAS) one can compute another 1-GVAS, which has the same reachability relation as the original one and additionally has the so-called thin property. Due to the work of Atig and Ganty from 2011, thin 1-GVAS have decidable reachability problem, therefore our construction implies decidability of the problem for all 1-GVAS. Moreover, we also show that if reachability in thin 1-GVAS can be decided in elementary time then also reachability in all 1-GVAS can be decided in elementary time.

Figures (6)

• Figure 1: Direction of the flow.
• Figure 2: Derivation with counters.
• Figure 3: Partial derivation
• Figure 4: This represents a partial derivation of current node $N_0$ (and an $(a_0, X_0)$-cycle if $(a_0,X_0) = (a_4,X_4)$). The notations represented there are the same as in the definitions. The green triangles represent subtrees where input and output are specified everywhere. What only matters about those trees is their yields.
• Figure 5: Correspondence between derivations in $G$ and $H$

Theorems & Definitions (47)

• theorem 1
• theorem 2
• corollary 3
• claim 4
• proof
• claim 5
• claim 6
• lemma 7
• lemma 8
• remark 9