Foundation for a series of efficient simulation algorithms

TL;DR

This paper introduces a foundational approach for developing efficient simulation algorithms that optimize both time and memory resources, solving an open problem with the best known complexity.

Contribution

It presents new notions of maximal transitions and stability, enabling the design of simulation algorithms with optimal time and space complexities.

Findings

First algorithm with optimal time complexity O(|Psim|.|→|)

Achieves low space complexity O(|Psim|^2 log |Psim| + |Q| log |Q|)

Provides a theoretical foundation for future efficient simulation algorithms

Abstract

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, $\rightarrow$) be a given transition system and Rinit be an initial preorder over Q. Until now, algorithms to compute Rsim , the coarsest simulation included in Rinit , 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(|Psim |.|$\rightarrow$|), and a bit space complexity in O(|Psim |^2. log(|Psim |) + |Q|. log(|Q|)), with Psim the partition induced by Rsim.