A Comparison of BDD-Based Parity Game Solvers

Parity games are two player games with omega-winning conditions, played on finite graphs. Such games play an important role in verification, satisfiability and synthesis. It is therefore important to identify algorithms that can efficiently deal with large games that arise from such applications. In this paper, we describe our experiments with BDD-based implementations of four parity game solving algorithms, viz. Zielonka's recursive algorithm, the more recent Priority Promotion algorithm, the Fixpoint-Iteration algorithm and the automata based APT algorithm. We compare their performance on several types of random games and on a number of cases taken from the Keiren benchmark set.Comment: In Proceedings GandALF 2018, arXiv:1809.0241

Improved Algorithms for Parity and Streett objectives

The computation of the winning set for parity objectives and for Streett objectives in graphs as well as in game graphs are central problems in computer-aided verification, with application to the verification of closed systems with strong fairness conditions, the verification of open systems, checking interface compatibility, well-formedness of specifications, and the synthesis of reactive systems. We show how to compute the winning set on $n$ vertices for (1) parity-3 (aka one-pair Streett) objectives in game graphs in time $O(n^{5/2})$ and for (2) k-pair Streett objectives in graphs in time $O(n^2 + nk \log n)$. For both problems this gives faster algorithms for dense graphs and represents the first improvement in asymptotic running time in 15 years

Zielonka's Recursive Algorithm: dull, weak and solitaire games and tighter bounds

Dull, weak and nested solitaire games are important classes of parity games, capturing, among others, alternation-free mu-calculus and ECTL* model checking problems. These classes can be solved in polynomial time using dedicated algorithms. We investigate the complexity of Zielonka's Recursive algorithm for solving these special games, showing that the algorithm runs in O(d (n + m)) on weak games, and, somewhat surprisingly, that it requires exponential time to solve dull games and (nested) solitaire games. For the latter classes, we provide a family of games G, allowing us to establish a lower bound of 2^(n/3). We show that an optimisation of Zielonka's algorithm permits solving games from all three classes in polynomial time. Moreover, we show that there is a family of (non-special) games M that permits us to establish a lower bound of 2^(n/3), improving on the previous lower bound for the algorithm.Comment: In Proceedings GandALF 2013, arXiv:1307.416

New Deterministic Algorithms for Solving Parity Games

We study parity games in which one of the two players controls only a small number $k$ of nodes and the other player controls the $n-k$ other nodes of the game. Our main result is a fixed-parameter algorithm that solves bipartite parity games in time $k^{O(\sqrt{k})}\cdot O(n^3)$, and general parity games in time $(p+k)^{O(\sqrt{k})} \cdot O(pnm)$, where $p$ is the number of distinct priorities and $m$ is the number of edges. For all games with $k = o(n)$ this improves the previously fastest algorithm by Jurdzi{\'n}ski, Paterson, and Zwick (SICOMP 2008). We also obtain novel kernelization results and an improved deterministic algorithm for graphs with small average degree

The tropical shadow-vertex algorithm solves mean payoff games in polynomial time on average

We introduce an algorithm which solves mean payoff games in polynomial time on average, assuming the distribution of the games satisfies a flip invariance property on the set of actions associated with every state. The algorithm is a tropical analogue of the shadow-vertex simplex algorithm, which solves mean payoff games via linear feasibility problems over the tropical semiring $(\mathbb{R} \cup \{-\infty\}, \max, +)$. The key ingredient in our approach is that the shadow-vertex pivoting rule can be transferred to tropical polyhedra, and that its computation reduces to optimal assignment problems through Pl\"ucker relations.Comment: 17 pages, 7 figures, appears in 41st International Colloquium, ICALP 2014, Copenhagen, Denmark, July 8-11, 2014, Proceedings, Part

A Multi-Core Solver for Parity Games

We describe a parallel algorithm for solving parity games,\ud with applications in, e.g., modal mu-calculus model\ud checking with arbitrary alternations, and (branching) bisimulation\ud checking. The algorithm is based on Jurdzinski's Small Progress\ud Measures. Actually, this is a class of algorithms, depending on\ud a selection heuristics.\ud \ud Our algorithm operates lock-free, and mostly wait-free (except for\ud infrequent termination detection), and thus allows maximum\ud parallelism. Additionally, we conserve memory by avoiding storage\ud of predecessor edges for the parity graph through strictly\ud forward-looking heuristics.\ud \ud We evaluate our multi-core implementation's behaviour on parity games\ud obtained from mu-calculus model checking problems for a set of\ud communication protocols, randomly generated problem instances, and\ud parametric problem instances from the literature.\ud \u

Qualitative Analysis of Partially-observable Markov Decision Processes

We study observation-based strategies for partially-observable Markov decision processes (POMDPs) with omega-regular objectives. An observation-based strategy relies on partial information about the history of a play, namely, on the past sequence of observations. We consider the qualitative analysis problem: given a POMDP with an omega-regular objective, whether there is an observation-based strategy to achieve the objective with probability~1 (almost-sure winning), or with positive probability (positive winning). Our main results are twofold. First, we present a complete picture of the computational complexity of the qualitative analysis of POMDP s with parity objectives (a canonical form to express omega-regular objectives) and its subclasses. Our contribution consists in establishing several upper and lower bounds that were not known in literature. Second, we present optimal bounds (matching upper and lower bounds) on the memory required by pure and randomized observation-based strategies for the qualitative analysis of POMDP s with parity objectives and its subclasses