Simulation as Coarsest Partition Problem

Abstract

The problem of determining the coarsest partition stable with respect to a given binary relation, is known to be equivalent to the problem of finding the maximal bisimulation on a given structure. Such an equivalence has suggested efficient algorithms for the computation of the maximal bisimulation relation. In this paper the simulation problem is rewritten in terms of coarsest stable partition problem allowing a more algebraic understanding of the simulation equivalence. On this ground, a new algorithm for deciding simulation is proposed. Such a procedure improves on either space or time complexity of previous simulation algorithms.

Saving Time in a Space-Efficient Simulation Algorithm

“…The importance of the simulation preorder and equivalence in compositional verification has been stated on more than one occasion [1,2,3,4,5,6]. It has been shown a natural preorder for matching implementations and specifications, when the preservation of the branching structure is important [7].…”

“…Unfortunately, this algorithm suffered from quadratic space complexity in the number of simulation classes, which was improved upon in [12,6] to O(|P||S| log(|S|)). The space complexity of O(|S| log(|P|) + |P| 2 ) for minimizing Kripke structures by simulation equivalence is achieved in [2], an algorithm later shown flawed and mended in [5]. This complexity is considered optimal when representing the simulation preorder as a partition-relation pair by keeping similar states in same partition classes, while representing the preorder as a relation between the classes.…”

“…This complexity is considered optimal when representing the simulation preorder as a partition-relation pair by keeping similar states in same partition classes, while representing the preorder as a relation between the classes. Unfortunately, this algorithm has an inferior time complexity of O(|P| 2 |→|) [2,5]. The space complexity of [12], has been iteratively improved to O(|P| 2 log(|S|) + |S| log(|P|)) [10,13], based on original algorithm of [4].…”

“…This allows us to take a different approach from others by improving the time complexity of the space-efficient algorithm of [2], instead of improving the space complexity of [4]. Unlike [2,5], we employ splitters for our refinement operation in the vein of [15,1].…”

“…Unlike [2,5], we employ splitters for our refinement operation in the vein of [15,1]. Moreover, we employ the "process the smaller half" method, that enables efficient refinement of the partitions and we also exploit properties of the topological order induced by the preorder.…”