Foundation for a series of efficient simulation algorithms

Compute the coarsest simulation preorder included in an initial preorder is used to reduce the resources needed to analyze a given transition system. This technique is applied on many models like Kripke structures, labeled graphs, labeled transition systems or even word and tree automata. Let (Q,→) be a given transition system and ℛ
_init
be an initial preorder over Q. Until now, algorithms to compute ℛ
_sim
, the coarsest simulation included in ℛ
_init
, are either memory efficient or time efficient but not both. In this paper we propose the foundation for a series of efficient simulation algorithms with the introduction of the notion of maximal transitions and the notion of stability of a preorder with respect to a coarser one. As an illustration we solve an open problem by providing the first algorithm with the best published time complexity, O(|P
_sim
|.|→|), and a bit space complexity in O(|P
_sim
|
²
.log(|P
_sim
|)+|Q|.log(|Q|)), with P
_sim
the partition induced by ℛ
_sim
.