Fatal attractors in parity games

We study a new form of attractor in parity games and use it to define solvers that run in PTIME and are partial in that they do not solve all games completely. Technically, for color c this new attractor determines whether player c%2 can reach a set of nodes X of color c whilst avoiding any nodes of color less than c. Such an attractor is fatal if player c%2 can attract all nodes in X back to X in this manner. Our partial solvers detect fixed-points of nodes based on fatal attractors and correctly classify such nodes as won by player c%2. Experimental results show that our partial solvers completely solve benchmarks that were constructed to challenge existing full solvers. Our partial solvers also have encouraging run times. For one partial solver we prove that its runtime is in O(|V |3), that its output game is independent of the order in which attractors are computed, and that it solves all B¨uchi games

Parity games : descriptive complexity and algorithms for new solvers

Parity games are 2-person, 0-sum, graph-based, and determined games that form an important foundational concept in formal methods (see e.g., [Zie98]), and their exact computational complexity has been an open problem for over twenty years now. In this thesis, we study algorithms that solve parity games in that they determine which nodes are won by which player, and where such decisions are supported with winning strategies. We modify and so improve a known algorithm but also propose new algorithmic approaches to solving parity games and to understanding their descriptive complexity. For all of our contributions, we write our own custom frameworks, in the Scala programming language, to perform tailored experiments and empirical studies to demonstrate and support our theoretical findings. First, we improve on one of the solver algorithms, based on small progress measures [Jur00], by use of concurrency. We show that, for many parity games, it is possible to deliver extra performance using this technique in a multi-core environment. Second, we design algorithms to reduce the computational complexity of parity games, and create implementations to observe and evaluate the behaviours of these reductions in our experimental settings. The measure Rabin index, arising from the design of the said algorithm, is shown to be a new descriptive complexity for parity games. Finally, we define a new family of attractors and derive new parity game solvers from them. Although these new solvers are “partial”, in that they do not solve all parity games completely, our experiments show that they do solve a set of benchmark games (i.e., games with known structures) designed to stress test solvers from PGSolver toolkit [FL10] completely, and some of these partial solvers deliver favourable performance against a known high performance solver in many circumstances

On The Growth Of Permutation Classes

We study aspects of the enumeration of permutation classes, sets of permutations closed downwards under the subpermutation order. First, we consider monotone grid classes of permutations. We present procedures for calculating the generating function of any class whose matrix has dimensions m × 1 for some m, and of acyclic and unicyclic classes of gridded permutations. We show that almost all large permutations in a grid class have the same shape, and determine this limit shape. We prove that the growth rate of a grid class is given by the square of the spectral radius of an associated graph and deduce some facts relating to the set of grid class growth rates. In the process, we establish a new result concerning tours on graphs. We also prove a similar result relating the growth rate of a geometric grid class to the matching polynomial of a graph, and determine the effect of edge subdivision on the matching polynomial. We characterise the growth rates of geometric grid classes in terms of the spectral radii of trees. We then investigate the set of growth rates of permutation classes and establish a new upper bound on the value above which every real number is the growth rate of some permutation class. In the process, we prove new results concerning expansions of real numbers in non-integer bases in which the digits are drawn from sets of allowed values. Finally, we introduce a new enumeration technique, based on associating a graph with each permutation, and determine the generating functions for some previously unenumerated classes. We conclude by using this approach to provide an improved lower bound on the growth rate of the class of permutations avoiding the pattern 1324. In the process, we prove that, asymptotically, patterns in Łukasiewicz paths exhibit a concentrated Gaussian distribution