Tom van Dijk

Sylvan: Multi-Core Decision Diagrams

Decision diagrams such as binary decision diagrams and multi-valued decision diagrams play an important role in various fields, including symbolic model checking. An ongoing challenge is to develop datastructures and algorithms for modern multi-core architectures. The BDD package Sylvan provides one contribution by implementing parallelized BDD operations and thus allowing sequential algorithms to exploit the power of multi-core machines. We present several extensions to Sylvan. We implement parallel operations on list decision diagrams, a variant of multi-valued decision diagrams that is useful for symbolic model checking. We also substitute several core components of Sylvan by new designs, such as the work-stealing framework, the unique table and the operation cache. Furthermore, we combine parallel operations with parallelization on a higher level, by partitioning the transition relation. We show that this results in an improved speedup using the model checking toolset ltsmin. We also demonstrate that the parallelization of symbolic model checking for explicit-state modeling languages with an on-the-fly next-state function, as supported by ltsmin, scales well.

Lace: non-blocking split deque for work-stealing

Work-stealing is an efficient method to implement load balancing in fine-grained task parallelism. Typically, concurrent deques are used for this purpose. A disadvantage of many concurrent deques is that they require expensive memory fences for local deque operations.

Multi-core and/or Symbolic Model Checking

We review our progress in high-performance model checking. Our multi-core model checker is based on a scalable hash-table design and parallel random-walk traversal. Our symbolic model checker is based on Multiway Decision Diagrams and the saturation strategy. The LTSmin tool is based on the PINS architecture, decoupling model checking algorithms from the input specification language. Consequently, users can stay in their own specification language and postpone the choice between parallel or symbolic model checking. We support widely different specification languages including those of SPIN (Promela), mCRL2 and UPPAAL (timed automata). So far, multi-core and symbolic algorithms had very little in common, forcing the user in the end to make a wise trade-off between memory or speed. Recently, however, we designed a novel multi-core BDD package called Sylvan. This forms an excellent basis for scalable parallel symbolic model checking.

Multi-core Symbolic Bisimulation Minimisation

Bisimulation minimisation alleviates the exponential growth of transition systems in model checking by computing the smallest system that has the same behavior as the original system according to some notion of equivalence. One popular strategy to compute a bisimulation minimisation is signature-based partition refinement. This can be performed symbolically using binary decision diagrams to allow models with larger state spaces to be minimised. This paper studies strong and branching symbolic bisimulation for labeled transition systems, continuous-time markov chains, and interactive markov chains. We introduce the notion of partition refinement with partial signatures. We extend the parallel BDD library Sylvan to parallelize the signature refinement algorithm, and develop a new parallel BDD algorithm to refine a partition, which conserves previous block numbers and uses a parallel data structure to store block assignments. We also present a specialized BDD algorithm for the computation of inert transitions. The experimental evaluation, based on benchmarks from the literature, demonstrates a speedup of upi¾?to 95x sequentially. In addition, we find parallel speedups of upi¾?to 17x due to parallelisation with 48 cores. Finally, we present the implementation of these algorithms as a versatile tool that can be customized for bisimulation minimisation in various contexts.

Sylvan: multi-core framework for decision diagrams

Decision diagrams, such as binary decision diagrams, multi-terminal binary decision diagrams and multi-valued decision diagrams, play an important role in various fields. They are especially useful to represent the characteristic function of sets of states and transitions in symbolic model checking. Most implementations of decision diagrams do not parallelize the decision diagram operations. As performance gains in the current era now mostly come from parallel processing, an ongoing challenge is to develop datastructures and algorithms for modern multi-core architectures. The decision diagram package Sylvan provides a contribution by implementing parallelized decision diagram operations and thus allowing sequential algorithms that use decision diagrams to exploit the power of multi-core machines. This paper discusses the design and implementation of Sylvan, especially an improvement to the lock-free unique table that uses bit arrays, the concurrent operation cache and the implementation of parallel garbage collection. We extend Sylvan with multi-terminal binary decision diagrams for integers, real numbers and rational numbers. This extension also allows for custom MTBDD leaves and operations and we provide an example implementation of GMP rational numbers. Furthermore, we show how the provided framework can be integrated in existing tools to provide out-of-the-box parallel BDD algorithms, as well as support for the parallelization of higher-level algorithms. As a case study, we parallelize on-the-fly symbolic reachability in the model checking toolset LTSmin. We experimentally demonstrate that the parallelization of symbolic model checking for explicit-state modeling languages, as supported by LTSmin, scales well. We also show that improvements in the design of the unique table result in faster execution of on-the-fly symbolic reachability.

Oink: An Implementation and Evaluation of Modern Parity Game Solvers.

Parity games have important practical applications in formal verification and synthesis, especially to solve the model-checking problem of the modal mu-calculus. They are also interesting from the theory perspective, as they are widely believed to admit a polynomial solution, but so far no such algorithm is known. In recent years, a number of new algorithms and improvements to existing algorithms have been proposed. We implement a new and easy to extend tool Oink, which is a high-performance implementation of modern parity game algorithms. We further present a comprehensive empirical evaluation of modern parity game algorithms and solvers, both on real world benchmarks and randomly generated games. Our experiments show that our new tool Oink outperforms the current state-of-the art.

Multi-core Decision Diagrams

Decision diagrams are fundamental data structures that revolutionized fields such as model checking, automated reasoning and decision processes. As performance gains in the current era mostly come from parallel processing, an ongoing challenge is to develop data structures and algorithms for modern multicore architectures. This chapter describes the parallelization of decision diagram operations as implemented in the parallel decision diagram package Sylvan, which allows sequential algorithms that use decision diagrams to exploit the power of multi-core machines.

A Comparative Study of BDD Packages forźProbabilistic Symbolic Model Checking

Symbolic data structures using Binary Decision Diagrams BDDs have been successfully used in the last decades to analyse large systems. While various BDD and MTBDD packages have been developed in the community, the CUDD package remains the default choice of most of the symbolic probabilistic model checkers. In this paper, we provide the first comparative study of the performance of various BDD/MTBDD packages for this purpose. We provide experimental results for several well-known probabilistic benchmarks and study the effect of several optimisations. Our experiments show that no BDD package dominates on a single core, but that parallelisation yields significant speedups.

Distributed binary decision diagrams for symbolic reachability

Decision diagrams are used in symbolic verification to concisely represent state spaces. A crucial symbolic verification algorithm is reachability: systematically exploring all reachable system states. Although both parallel and distributed reachability algorithms exist, a combined solution is relatively unexplored. This paper contributes BDD-based reachability algorithms targeting compute clusters: high-performance networks of multi-core machines. The proposed algorithms may use the entire memory of every machine, allowing larger models to be processed while increasing performance by using all available computational power. To do this effectively, a distributed hash table, cluster-based work stealing algorithms, and several caching structures have been designed that all utilise the newest networking technology. The approach is evaluated extensively on a large collection of models, thereby demonstrating speedups up to 51,1x with 32 machines. The proposed algorithms not only benefit from the large amounts of available memory on compute clusters, but also from all available computational resources.

Multi-core symbolic bisimulation minimisation

We introduce parallel symbolic algorithms for bisimulation minimisation, to combat the combinatorial state space explosion along three different paths. Bisimulation minimisation reduces a transition system to the smallest system with equivalent behaviour. We consider strong and branching bisimilarity for interactive Markov chains, which combine labelled transition systems and continuous-time Markov chains. Large state spaces can be represented concisely by symbolic techniques, based on binary decision diagrams. We present specialised BDD operations to compute the maximal bisimulation using signature-based partition refinement. We also study the symbolic representation of the quotient system and suggest an encoding based on representative states, rather than block numbers. Our implementation extends the parallel, shared memory, BDD library Sylvan, to obtain a significant speedup on multi-core machines. We propose the usage of partial signatures and of disjunctively partitioned transition relations, to increase the parallelisation opportunities. Also our new parallel data structure for block assignments increases scalability. We provide SigrefMC, a versatile tool that can be customised for bisimulation minimisation in various contexts. In particular, it supports models generated by the high-performance model checker LTSmin, providing access to specifications in multiple formalisms, including process algebra. The extensive experimental evaluation is based on various benchmarks from the literature. We demonstrate a speedup up to 95