Robust Exponential Worst Cases for Divide-et-Impera Algorithms for Parity Games

The McNaughton-Zielonka divide et impera algorithm is the simplest and most flexible approach available in the literature for determining the winner in a parity game. Despite its theoretical worst-case complexity and the negative reputation as a poorly effective algorithm in practice, it has been shown to rank among the best techniques for the solution of such games. Also, it proved to be resistant to a lower bound attack, even more than the strategy improvements approaches, and only recently a family of games on which the algorithm requires exponential time has been provided by Friedmann. An easy analysis of this family shows that a simple memoization technique can help the algorithm solve the family in polynomial time. The same result can also be achieved by exploiting an approach based on the dominion-decomposition techniques proposed in the literature. These observations raise the question whether a suitable combination of dynamic programming and game-decomposition techniques can improve on the exponential worst case of the original algorithm. In this paper we answer this question negatively, by providing a robustly exponential worst case, showing that no intertwining of the above mentioned techniques can help mitigating the exponential nature of the divide et impera approaches.Comment: In Proceedings GandALF 2017, arXiv:1709.0176

Quasipolynomial Set-Based Symbolic Algorithms for Parity Games

Solving parity games, which are equivalent to modal $\mu$-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 $O(d \cdot \log n)$ symbolic space. Moreover, for the important special case of $d \leq \log n$, our algorithm requires only polynomially many symbolic operations and poly-logarithmic symbolic space.Comment: Published at LPAR-22 in 201

Solving parity games: Explicit vs symbolic

In this paper we provide a broad investigation of the symbolic approach for solving Parity Games. Specifically, we implement in a fresh tool, called, four symbolic algorithms to solve Parity Games and compare their performances to the corresponding explicit versions for different classes of games. By means of benchmarks, we show that for random games, even for constrained random games, explicit algorithms actually perform better than symbolic algorithms. The situation changes, however, for structured games, where symbolic algorithms seem to have the advantage. This suggests that when evaluating algorithms for parity-game solving, it would be useful to have real benchmarks and not only random benchmarks, as the common practice has been

On the Succinctness of Alternating Parity Good-For-Games Automata

We study alternating parity good-for-games (GFG) automata, i.e., alternating parity automata where both conjunctive and disjunctive choices can be resolved in an online manner, without knowledge of the suffix of the input word still to be read. We show that they can be exponentially more succinct than both their nondeterministic and universal counterparts. Furthermore, we present a single exponential determinisation procedure and an Exptime upper bound to the problem of recognising whether an alternating automaton is GFG. We also study the complexity of deciding "half-GFGness", a property specific to alternating automata that only requires nondeterministic choices to be resolved in an online manner. We show that this problem is PSpace-hard already for alternating automata on finite words

A Recursive Approach to Solving Parity Games in Quasipolynomial Time

Zielonka's classic recursive algorithm for solving parity games is perhaps the simplest among the many existing parity game algorithms. However, its complexity is exponential, while currently the state-of-the-art algorithms have quasipolynomial complexity. Here, we present a modification of Zielonka's classic algorithm that brings its complexity down to $n^{O\left(\log\left(1+\frac{d}{\log n}\right)\right)}$, for parity games of size $n$ with $d$ priorities, in line with previous quasipolynomial-time solutions.</jats:p