Quasipolynomial Set-Based Symbolic Algorithms for Parity Games
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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.
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