Ackermann Encoding, Bisimulations, and OBDDs

In this paper we consider the problem of computing and representing the bisimulation on a given Kripke structure. Such a problem, central in Model Checking, has been tackled by many authors and various solutions have been given. In particular, the algorithm proposed by Kanellakis and Smolka [7] is used in many model checkers with explicit-state representation, while the algorithm proposed by Bouali and de Simone [3] is used in the case of symbolic representation. The algorithms presented in [2] and [8] are studied to obtain better performances in the case of the so-called on-the-fly Model Checking. The algorithm proposed by Paige and Tarjan in [11] is still the best in terms of worst case time complexity (O(|E| log |N |)). In [5] has been proposed an algorithm which works on explicit representations and which in the worst case has the same time complexity as the one of Paige and Tarjan, but which in many cases has a linear time complexity O(|E|) and requires less space during the computation. A particularly interesting line of research is witnessed by the work presented in [3] which gives particular relevance to the design of algorithms whose execution can be easily coordinated with efficient (symbolic) representation techniques for the underlying Kripke structures. The OBDD data structures (see [4]) employed in symbolic model checking (see [10]), allow the representation in memory of much larger structures and set new standards for the size of Kripke structures to which Model Checking can be applied. However, it is not always immediate to map the further space-saving method of bisimulation reduction on OBDD-representation of Kripke structures: especially designed algorithms must be proposed for this purpose and different complexity parameters must be considered [6]. The method proposed in this paper aims at carrying out the activity of bisimulation computation and OBDD-representation in one single step. The technique is based on the fact that a Kripke structure is nothing but a (in general redundant) representation of an hereditarily finite set. Starting from Ackermann’s encoding [9, 1] of hereditarily finite sets as natural numbers, we propose an extension of such an encoding to non-well-founded hereditarily finite sets. Such an encoding turns out to naturally suggest an OBDD-representation which, moreover, simplifies the calculation. A key notion behind our encoding is the notion of rank, already used in [5]. But here the encoding give us a sort of global positive strategy, while in [5] we combine positive and negative strategy. The paper is organized as follows: in Section 2 we present the problem; in Section 3 we recall the Ackermann encoding in the case of acyclic graphs, while in Section 4