We present KReach, a tool for deciding reachability in general Petri nets. The tool is a full implementation of Kosaraju's original 1982 decision procedure for reachability in VASS. We believe this to be the first implementation of its kind. We include a comprehensive suite of libraries for development with Vector Addition Systems (with States) in the Haskell programming language. KReach serves as a practical tool, and acts as an effective teaching aid for the theory behind the algorithm. Preliminary tests suggest that there are some classes of Petri nets for which we can quickly show unreachability. In particular, using KReach for coverability problems, by reduction to reachability, is competitive even against state-of-the-art coverability checkers.

Finitary Idealized Concurrent Algol ($$\mathsf {FICA}$$ FICA ) is a prototypical programming language combining functional, imperative, and concurrent computation. There exists a fully abstract game model of $$\mathsf {FICA}$$ FICA , which in principle can be used to prove equivalence and safety of $$\mathsf {FICA}$$ FICA programs. Unfortunately, the problems are undecidable for the whole language, and only very rudimentary decidable sub-languages are known.We propose leafy automata as a dedicated automata-theoretic formalism for representing the game semantics of $$\mathsf {FICA}$$ FICA . The automata use an infinite alphabet with a tree structure. We show that the game semantics of any $$\mathsf {FICA}$$ FICA term can be represented by traces of a leafy automaton. Conversely, the traces of any leafy automaton can be represented by a $$\mathsf {FICA}$$ FICA term. Because of the close match with $$\mathsf {FICA}$$ FICA , we view leafy automata as a promising starting point for finding decidable subclasses of the language and, more generally, to provide a new perspective on models of higher-order concurrent computation.Moreover, we identify a fragment of $$\mathsf {FICA}$$ FICA that is amenable to verification by translation into a particular class of leafy automata. Using a locality property of the latter class, where communication between levels is restricted and every other level is bounded, we show that their emptiness problem is decidable by reduction to Petri net reachability.

A difficulty in treating such matters as the four-colour theorem is to specify any map considered, otherwise than by drawing it; the present note offers a system of notation for the purpose, founded on the division of a plane surface into regular hexagons, the honeycomb pattern.Each province on the map is represented by a hexagon of the honeycomb, perfect or imperfect. For a province of 6-p sides, p consecutive sides of the hexagon with the equilateral triangles standing on them are taken out by cuts along radii of the circumscribed circle; if p is negative, -p of the triangles are accordingly doubled. The bounding radii might be brought into coincidence so as to restore a polygon of 6-p sides, but it could not lie flat in a plane.The map is supposed to cover a sphere; all provinces are taken to be simply connected and only three meet at each vertex. [These restrictions may be removed afterwards by cancelling sides, but they are necessary if the map is to be represented on the honeycomb figure with one hexagon for each province.]Lay down the hexagons, perfect and imperfect, one by one, attaching each in succession along a single side to one of those that are already in position, so as to correspond with the arrangement on the given map.When all are in position they form a "net" covering a simply connected part of the plane. The perimeter of the net is a single closed line consisting of sides and radii of the hexagons, and these correspond in pairs so that, when corresponding edges are brought together, the original map is restored, except for deformation.The map is thus specified for topological purposes by the coordinates of the successive vertices of the perimeter of its net and by the correspondence. If two sides of one of the equilateral triangles are taken as axes, every coordinate is a multiple of the common side.If the p-th and q-th sides correspond and also the r-th and s-th, then r, s must both lie between p and q or else p and q between r and s. Otherwise a path within the net joining corresponding points of the p-th. and g-th and a path within the net joining corresponding points of the r-th and 5-th could cross each other in one point only, and the original map would not be on a simply connected surface.