Latticed Simulation Relations and Games

Multi-valued Kripke structures are Kripke structures in which the atomic propositions and the transitions are not Boolean and can take values from some set. In particular, latticed Kripke structures, in which the elements in the set are partially ordered, are useful in abstraction, query checking, and reasoning about multiple view-points. The challenges that formal methods involve in the Boolean setting are carried over, and in fact increase, in the presence of multi-valued systems and logics. We lift to the latticed setting two basic notions that have been proven useful in the Boolean setting. We first define latticed simulation between latticed Kripke structures. The relation maps two structures M1 and M2 to a lattice element that essentially denotes the truth value of the statement “every behavior of M1 is also a behavior of M2”. We show that latticedsimulation is logically characterized by the universal fragment of latticed μ-calculus, and can be calculated in polynomial time. We then proceed to defining latticed two-player games. Such games are played along graphs in which each transition have a value in the lattice. The value of the game essentially denotes the truth value of the statement “the ∨-player can force the game to computations that satisfy the winning condition”. An earlier definition of such games involved a zig-zagged traversal of paths generated during the game. Our definition involves a forward traversal of the paths, and it leads to better understanding of multi-valued games. In particular, we prove a min-max property for such games, and relate latticed simulation with latticed games.