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Chess endgame tables encode unapproximated game- theoretic values of endgame positions. The speed at which information is retrieved from these tables and their representation size are major limiting factors in their effective use. We explore and make novel extensions to three alternatives (decision trees, decision diagrams, and logic minimization) to the currently preferred implementation (Syzygy) for representing such tables. Syzygy is most compact, but also slowest at handling queries. Two-level logic minimization works well when the full compression algorithm can be run. Decision DAGs and multiterminal binary decision diagrams are comparable and offer the best querying times, with decision diagrams providing better compression. 

We introduce RexBDDs, binary decision diagrams (BDDs) that exploit reduction opportunities well beyond those of reduced ordered BDDs, zero-suppressed BDDs, and recent proposals integrating multiple reduction rules. RexBDDs also leverage (output) complement flags and (input) swap flags to potentially decrease the number of nodes by a factor of four. We define a reduced form of RexBDDs that ensures canonicity, and use a set of benchmarks to demonstrate their superior storage and runtime requirements compared to previous alternatives. 

Binary decision diagrams (BDDs) have been a huge success story in hardware and software verification and are increasingly applied to a wide range of combinatorial problems. While BDDs can encode boolean-valued functions of boolean-valued variables, many BDD variants have been proposed, not just to improve their efficiency, but to manage multivalued domains (a straightforward extension), multivalued ranges (using several competitive alternatives), and two-dimensional data (relations and matrices instead of sets or vectors). Orthogonally to these extensions, much effort has been spent on variable order heuristics, an essential aspect that can affect memory and time requirements by up to an exponential factor. We survey some of these exciting results and discuss some fruitful research directions for further work. Index Terms—Binary decision diagrams, canonicity, discrete function encoding, variable order heuristics, Markov chains 

Generating the state space of any finite discrete-state system using symbolic algorithms like saturation requires the use of decision diagrams or compatible structures for encoding its reachability set and transition relations. For systems that can be formally expressed using ordinary Petri Nets(PN), implicit relations, a static alternative to decision diagram-based representation of transition relations, can significantly improve the performance of saturation. However, in practice, some systems require more general models, such as self-modifying Petri nets, which cannot currently utilize implicit relations and thus use decision diagrams that are repeatedly rebuilt to accommodate the changing bounds of the system variables, potentially leading to overhead in saturation algorithm. This work introduces a hybrid representation for transition relations, that combines decision diagrams and implicit relations, to reduce the rebuilding overheads of the saturation algorithm for a general class of models. Experiments on several benchmark models across different tools demonstrate the efficiency of this representation. 

Decision diagrams are a well-established data structure for reachability set generation and model checking of high-level models such as Petri nets, due to their versatility and the availability of efficient algorithms for their construction. Using a decision diagram to represent the transition relation of each event of the high-level model, the saturation algorithm can be used to construct a decision diagram representing all states reachable from an initial set of states, via the occurrence of zero or more events. A difficulty arises in practice for models whose state variable bounds are unknown, as the transition relations cannot be constructed before the bounds are known. Previously, on-the-fly approaches have constructed the transition relations along with the reachability set during the saturation procedure. This can affect performance, as the transition relation decision diagrams must be rebuilt, and compute-table entries may need to be discarded, as the size of each state variable increases. In this paper, we introduce a different approach based on an implicit and unchanging representation for the transition relations, thereby avoiding the need to reconstruct the transition relations and discard compute-table entries. We modify the saturation algorithm to use this new representation, and demonstrate its effectiveness with experiments on several benchmark models. 

Various versions of binary decision diagrams (BDDs) have been proposed in the past, differing in the reduction rule needed to give meaning to edges skipping levels. The most widely adopted, fully-reduced BDDs and zero-suppressed BDDs, excel at encoding different types of boolean functions (if the function contains subfunctions independent of one or more underlying variables, or it tends to have value zero when one of its arguments is nonzero, respectively). Recently, new classes of BDDs have been proposed that, at the cost of some additional complexity and larger memory requirements per node, exploit both cases. We introduce a new type of BDD that we believe is conceptually simpler, has small memory requirements in terms of node size, tends to result in fewer nodes, and can easily be further extended with additional reduction rules. We present a formal definition, prove canonicity, and provide experimental results to support our efficiency claims 

The size of a BDD heavily depends on its variable order. Significant efforts have been made to find good variable orders, statically or dynamically. This paper concentrates on a related issue, transforming a BDD from one variable order to another, as needed when an application must cope with BDDs having different variable orders. Instead of rebuilding BDDs as done in previous work, we accomplish such transformation through a sequence of adjacent variable swaps. Since there are many ways to schedule these swaps, we propose and compare several heuristics to determine good schedules.