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With geometrical techniques we hope to bring new insight into the reachability problem for vector-addition systems, which is pertaining in many areas in computer science theory but unsolved ever since it was proposed by Karp and Miller. We show that the reachability-problem is decidable in dim. ≤ 3 and give a partial solution for the general problem covering a wide instance of it. In both cases a semi-linear characterization is obtained for the complete set of solutions. Contrary to common belief the complete solution in dim. 3 does not immediately generalize, and our analysis gives evidence why. A few generalizations and application to long-standing classificational problems in language theory are discussed.

A partial solution to the reachability-problem for vector-addition systems