It is shown that every polyhedron in R3 can be guarded by at most 56 of its edges. This result holds even if the boundary disconnected (i.e., has “holes”), and regardless genus each connected component boundary. When a homeomorphic to ball all faces are triangles, bound improved slightly 2936

This paper deals with conservative reachability analysis of a class of hybrid systems with continuous dynamics described by linear differential inclusions, convex invariants and guards, and linear reset maps. We present an approach for computing over-approximations of the set of reachable states. It is based on the notion of support function and thus it allows us to consider invariants, guards ...

In his seminal paper [5], Granger presents an analysis which infers linear congruence relations between integer variables. For affine programs without guards, his analysis is complete, i.e., infers all such congruences. No upper complexity bound, though, has been found for Granger’s algorithm. Here, we present a variation of this analysis which runs in polynomial time. Moreover, we provide an i...

We present a tool for the robustness analysis of timed automata, that can check whether a given time-abstract behaviour of a timed automaton is still present when the guards are perturbed. The perturbation model we consider is shrinking, which corresponds to increasing lower bounds and decreasing upper bounds in the clock guards by parameters. The tool synthesizes these parameters for which the...

We provide a spectrum of results for the Universal Guard Problem, in which one is to obtain a small set of points (“guards”) that are “universal” in their ability to guard any of a set of possible polygonal domains in the plane. We give upper and lower bounds on the number of universal guards that are always sufficient to guard all polygons having a given set of n vertices, or to guard all poly...

We provide a spectrum of results for the Universal Guard Problem, in which one is to obtain a small set of points (“guards”) that are “universal” in their ability to guard any of a set of possible polygonal domains in the plane. We give upper and lower bounds on the number of universal guards that are always sufficient to guard all polygons having a given set of n vertices, or to guard all poly...

We show that model-checking flat counter systems over CTL* (with arithmetical constraints on counter values) has the same complexity as the satisfiability problem for Presburger arithmetic. The lower bound already holds with the temporal operator EF only, no arithmetical constraints in the logical language and with guards on transitions made of simple linear constraints. This complements our un...

We address the problem of stationing guards in vertices of a simple polygon in such a way that the whole polygon is guarded and the number of guards is minimum. It is known that this is an NP-hard Art Gallery Problem with relevant practical applications. In this paper we present an approximation method that solves the problem by successive approximations, which we introduced in [21]. We report ...

We prove that bn+h 4 c vertex guards are always sufficient to see the entire interior of an n-vertex orthogonal polygon P with an arbitrary number h of holes provided that there exists a quadrilateralization whose dual graph is a cactus. Our proof is based upon 4-coloring of a quadrilateralization graph, and it is similar to that of Kahn and others for orthogonal polygons without holes. Consequ...