# Quasipolynomial Set-Based Symbolic Algorithms for Parity Games

Solving parity games, which are equivalent to modal μ-calculus model checking, is a central algorithmic problem in formal methods. Besides the standard computation model with the explicit representation of games, another important theoretical model of computation is that of set-based symbolic algorithms. Set-based symbolic algorithms use basic set operations and one-step predecessor operations on the implicit description of games, rather than the explicit representation. The significance of symbolic algorithms is that they provide scalable algorithms for large finite-state systems, as well as for infinite-state systems with finite quotient. Consider parity games on graphs with n vertices and parity conditions with d priorities. While there is a rich literature of explicit algorithms for parity games, the main results for set-based symbolic algorithms are as follows: (a) an algorithm that requires O(n^d) symbolic operations and O(d) symbolic space; and (b) an improved algorithm that requires O(n^d/3+1) symbolic operations and O(n) symbolic space. Our contributions are as follows: (1) We present a black-box set-based symbolic algorithm based on the explicit progress measure algorithm. Two important consequences of our algorithm are as follows: (a) a set-based symbolic algorithm for parity games that requires quasi-polynomially many symbolic operations and O(n) symbolic space; and (b) any future improvement in progress measure based explicit algorithms imply an efficiency improvement in our set-based symbolic algorithm for parity games. (2) We present a set-based symbolic algorithm that requires quasi-polynomially many symbolic operations and O(d ·log n) symbolic space. Moreover, for the important special case of d ≤log n, our algorithm requires only polynomially many symbolic operations and poly-logarithmic symbolic space.

• 70 publications
• 14 publications
• 71 publications
• 6 publications
research
04/15/2021

### Symbolic Time and Space Tradeoffs for Probabilistic Verification

We present a faster symbolic algorithm for the following central problem...
research
09/23/2020

### Symbolic Parity Game Solvers that Yield Winning Strategies

Parity games play an important role for LTL synthesis as evidenced by re...
research
09/17/2019

### Simple Fixpoint Iteration To Solve Parity Games

A naive way to solve the model-checking problem of the mu-calculus uses ...
research
02/15/2022

### Fast Symbolic Algorithms for Omega-Regular Games under Strong Transition Fairness

We consider fixpoint algorithms for two-player games on graphs with ω-re...
research
03/19/2020

### The Strahler number of a parity game

The Strahler number of a rooted tree is the largest height of a perfect ...
research
01/13/2020

### A Universal Attractor Decomposition Algorithm for Parity Games

An attractor decomposition meta-algorithm for solving parity games is gi...
research
06/27/2017

### Symbolic Versus Numerical Computation and Visualization of Parameter Regions for Multistationarity of Biological Networks

We investigate models of the mitogenactivated protein kinases (MAPK) net...