Faster Algorithms for Alternating Refinement Relations

TL;DR

This paper introduces faster algorithms for computing alternating and fair simulation relations in reactive systems, significantly improving efficiency over previous methods through new algorithmic approaches.

Contribution

The paper presents new algorithms for fair and alternating simulation that are faster and more space-efficient than existing solutions.

Findings

Fair simulation algorithm improved to O(n^3 * m) time

Alternating simulation algorithm improved to O(m^2) time

Iterative algorithm matches game-based efficiency with better space use

Abstract

One central issue in the formal design and analysis of reactive systems is the notion of refinement that asks whether all behaviors of the implementation is allowed by the specification. The local interpretation of behavior leads to the notion of simulation. Alternating transition systems (ATSs) provide a general model for composite reactive systems, and the simulation relation for ATSs is known as alternating simulation. The simulation relation for fair transition systems is called fair simulation. In this work our main contributions are as follows: (1) We present an improved algorithm for fair simulation with B\"uchi fairness constraints; our algorithm requires $O(n^3 \cdot m)$ time as compared to the previous known $O(n^6)$-time algorithm, where $n$ is the number of states and $m$ is the number of transitions. (2) We present a game based algorithm for alternating simulation that requires $O(m^2)$-time as compared to the previous known $O((n \cdot m)^2)$-time algorithm, where $n$ is the number of states and $m$ is the size of transition relation. (3) We present an iterative algorithm for alternating simulation that matches the time complexity of the game based algorithm, but is more space efficient than the game based algorithm.