On Convexity of Right-Closed Integral Sets

Let N denote the set of non-negative integers. A set of non-negative, n-dimensional integral vectors, M ⊂ N, is said to be right-closed, if ((x ∈ M) ∧ (y ≥ x) ∧ (y ∈ N)) ⇒ (y ∈ M). In this paper, we present a polynomial time algorithm for testing the convexity of a right-closed set of integral vectors, when the dimension n is fixed. Right-closed set of integral vectors are infinitely large, by definition. We compute the convex-hull of an appropriately-defined finite subset of this infinite-set of vectors. We then check if a stylized Linear Program has a non-zero optimal value for a special collection of facets of this convex-hull. This result is to be viewed against the backdrop of the fact that checking the convexity of a real-valued, geometric set can only be accomplished in an approximate sense; and, the fact that most algorithms involving sets of real-valued vectors do not apply directly to their integral counterparts. This observation plays an important role in the efficient synthesis of Supervisory Policies that avoid Livelocks in Discrete-Event/Discrete-State Systems. *Corresponding author: E-mail: [email protected]; Salimi and Sreenivas; BJMCS, 20(1), 1-11, 2017; Article no.BJMCS.30348