An Efficient Simplex Coverability Algorithm in E2 with Application to Stochastic Sequential Machines

The problem of determining the existence of a simplex which covers a given convex polytope inside another given convex polytope in two-dimensional Euclidean space is shown to be efficiently solvable, and an effective procedure is derived to find a suitable covering simplex or show that none exists. This solution provides a finite algorithm for satisfying a necessary condition in the search for a simplicial (fewest states) stochastic sequential machine (SSM) which either covers or is covered by a given SSM of rank three. Theorems are proved which establish necessary and sufficient conditions restricting the class of corresponding simplexes through which a search must proceed to those whose vertices lie in the boundary of the bounding polytope and whose facet-supporting flats contain certain specified vertices of the polytope to be covered. The problem is then reduced to testing roots in the finite solution tree of a set of second degree algebraic equations against a finite table of linear constraints. An algorithm to generate the constraint sets for the equations to be solved is also presented along with examples of its application.