# Improved Set-based Symbolic Algorithms for Parity Games

## TL;DR

This paper introduces new set-based symbolic algorithms for solving parity games that significantly reduce symbolic operations and space requirements, improving scalability for analyzing reactive systems.

## Contribution

The paper presents two novel set-based symbolic algorithms for parity games, achieving lower symbolic operations and linear space, including the first linear space algorithm with sub-exponential symbolic operations.

## Key findings

- First algorithm: O(n^{c/2+1}) symbolic operations, linear space.
- Second algorithm: O(n^{c/3+1}) symbolic operations, linear space.
- First linear space algorithm with sub-exponential symbolic operations.

## Abstract

Graph games with {\omega}-regular winning conditions provide a mathematical framework to analyze a wide range of problems in the analysis of reactive systems and programs (such as the synthesis of reactive systems, program repair, and the verification of branching time properties). Parity conditions are canonical forms to specify {\omega}-regular winning conditions. Graph games with parity conditions are equivalent to {\mu}-calculus model checking, and thus a very important algorithmic problem. Symbolic algorithms are of great significance because they provide scalable algorithms for the analysis of large finite-state systems, as well as algorithms for the analysis of infinite-state systems with finite quotient. A set-based symbolic algorithm uses the basic set operations and the one-step predecessor operators. We consider graph games with $n$ vertices and parity conditions with $c$ priorities. While many explicit algorithms exist for graph games with parity conditions, for set-based symbolic algorithms there are only two algorithms (notice that we use space to refer to the number of sets stored by a symbolic algorithm): (a) the basic algorithm that requires $O(n^c)$ symbolic operations and linear space; and (b) an improved algorithm that requires $O(n^{c/2+1})$ symbolic operations but also $O(n^{c/2+1})$ space (i.e., exponential space). In this work we present two set-based symbolic algorithms for parity games: (a) our first algorithm requires $O(n^{c/2+1})$ symbolic operations and only requires linear space; and (b) developing on our first algorithm, we present an algorithm that requires $O(n^{c/3+1})$ symbolic operations and only linear space. We also present the first linear space set-based symbolic algorithm for parity games that requires at most a sub-exponential number of symbolic operations.