Model-Checking the Higher-Dimensional Modal mu-Calculus

The higher-dimensional modal mu-calculus is an extension of the mu-calculus in which formulas are interpreted in tuples of states of a labeled transition system. Every property that can be expressed in this logic can be checked in polynomial time, and conversely every polynomial-time decidable problem that has a bisimulation-invariant encoding into labeled transition systems can also be defined in the higher-dimensional modal mu-calculus. We exemplify the latter connection by giving several examples of decision problems which reduce to model checking of the higher-dimensional modal mu-calculus for some fixed formulas. This way generic model checking algorithms for the logic can then be used via partial evaluation in order to obtain algorithms for theses problems which may benefit from improvements that are well-established in the field of program verification, namely on-the-fly and symbolic techniques. The aim of this work is to extend such techniques to other fields as well, here exemplarily done for process equivalences, automata theory, parsing, string problems, and games.Comment: In Proceedings FICS 2012, arXiv:1202.317

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

Benchmarks for Parity Games (extended version)

We propose a benchmark suite for parity games that includes all benchmarks that have been used in the literature, and make it available online. We give an overview of the parity games, including a description of how they have been generated. We also describe structural properties of parity games, and using these properties we show that our benchmarks are representative. With this work we provide a starting point for further experimentation with parity games.Comment: The corresponding tool and benchmarks are available from https://github.com/jkeiren/paritygame-generator. This is an extended version of the paper that has been accepted for FSEN 201

The Fixpoint-Iteration Algorithm for Parity Games

It is known that the model checking problem for the modal mu-calculus reduces to the problem of solving a parity game and vice-versa. The latter is realised by the Walukiewicz formulas which are satisfied by a node in a parity game iff player 0 wins the game from this node. Thus, they define her winning region, and any model checking algorithm for the modal mu-calculus, suitably specialised to the Walukiewicz formulas, yields an algorithm for solving parity games. In this paper we study the effect of employing the most straight-forward mu-calculus model checking algorithm: fixpoint iteration. This is also one of the few algorithms, if not the only one, that were not originally devised for parity game solving already. While an empirical study quickly shows that this does not yield an algorithm that works well in practice, it is interesting from a theoretical point for two reasons: first, it is exponential on virtually all families of games that were designed as lower bounds for very particular algorithms suggesting that fixpoint iteration is connected to all those. Second, fixpoint iteration does not compute positional winning strategies. Note that the Walukiewicz formulas only define winning regions; some additional work is needed in order to make this algorithm compute winning strategies. We show that these are particular exponential-space strategies which we call eventually-positional, and we show how positional ones can be extracted from them.Comment: In Proceedings GandALF 2014, arXiv:1408.556

Model Checking Games for the Quantitative mu-Calculus

We investigate quantitative extensions of modal logic and the modal mu-calculus, and study the question whether the tight connection between logic and games can be lifted from the qualitative logics to their quantitative counterparts. It turns out that, if the quantitative mu-calculus is defined in an appropriate way respecting the duality properties between the logical operators, then its model checking problem can indeed be characterised by a quantitative variant of parity games. However, these quantitative games have quite different properties than their classical counterparts, in particular they are, in general, not positionally determined. The correspondence between the logic and the games goes both ways: the value of a formula on a quantitative transition system coincides with the value of the associated quantitative game, and conversely, the values of quantitative parity games are definable in the quantitative mu-calculus

Local Strategy Improvement for Parity Game Solving

The problem of solving a parity game is at the core of many problems in model checking, satisfiability checking and program synthesis. Some of the best algorithms for solving parity game are strategy improvement algorithms. These are global in nature since they require the entire parity game to be present at the beginning. This is a distinct disadvantage because in many applications one only needs to know which winning region a particular node belongs to, and a witnessing winning strategy may cover only a fractional part of the entire game graph. We present a local strategy improvement algorithm which explores the game graph on-the-fly whilst performing the improvement steps. We also compare it empirically with existing global strategy improvement algorithms and the currently only other local algorithm for solving parity games. It turns out that local strategy improvement can outperform these others by several orders of magnitude