Concurrency Theory: Lecture 4, 11 January 2017 ---------------------------------------------- Coverability tree Theorem: M is coverable iff we can find a marking M' >= M in the coverability tree. This follows from the following claim. Claim: Let M be an omega marking in the coverability tree with omega-marked places p1,p2,...,pk. Then given a simultaneous set of bounds n1,n2,...,nk, we can reach a marking M' from Min such that M' agrees with M on all places with a finite marking and for each pj with a omega marking, M'(pj) >= nj, where nj is the lower bound specified for pj. Proof: There is a path from Min to M in the coverability tree. Along this path, mark the positions where places in M are first promoted to omega. Segments between marked positions can be "pumped". Pumping a later segment could remove tokens from a place that was promoted earlier. Hence, pump the segments from last to first by appropriate amounts to ensure that all places are boosted sufficiently high to account for loss of tokens in the later part of the run. Corollary: Net is bounded iff no reachable marking has omega in coverability tree. Place invariants: - Weighted sum is the same for all reachable markings - A marking that does not satisfy the invariant is unreachable - Non-negative invariants and bounded places. Transition invariants: - Parikh vector of a loop is a transition invariant - Transition invariants of live and bounded nets Extensions to the model: - Arc weights: consider vector addition system model where transitions are arbitrary change vectors. - Place capacities: - Can eliminate using complementary places - Contact-free elementary net systems are a special case - Inhibitor arcs - Change the model, reachability becomes undecidable