The BDD package Adiar manipulates Binary Decision Diagrams (BDDs) in external memory. This enables handling big BDDs, but the performance suffers when dealing with moderate-sized BDDs. This is mostly due to initializing expensive external memory data structures, even if their contents can fit entirely inside internal memory. The contents of these auxiliary data structures always correspond to a graph cut in an input or output BDD. Specifically, these cuts respect the levels of the BDD. We formalise the shape of these cuts and prove sound upper bounds on their maximum size for each BDD operation. We have implemented these upper bounds within Adiar. With these bounds, it can predict whether a faster internal memory variant of the auxiliary data structures can be used. In practice, this improves Adiar’s running time across the board. Specifically for the moderate-sized BDDs, this results in an average reduction of the computation time by $$86.1\%$$ (median of $$89.7\%$$ ). In some cases, the difference is even $$99.9\%$$ . When checking equivalence of hardware circuits from the EPFL Benchmark Suite, for one of the instances the time was decreased by 52 h.

Abstract We follow up on the idea of Lars Arge to rephrase the Reduce and Apply operations of Binary Decision Diagrams (BDDs) as iterative I/O-efficient algorithms. We identify multiple avenues to simplify and improve the performance of his proposed algorithms. Furthermore, we extend the technique to other common BDD operations, many of which are not derivable using Apply operations alone. We provide asymptotic improvements to the few procedures that can be derived using Apply. Our work has culminated in a BDD package named Adiar that is able to efficiently manipulate BDDs that outgrow main memory. This makes Adiar surpass the limits of conventional BDD packages that use recursive depth-first algorithms. It is able to do so while still achieving a satisfactory performance compared to other BDD packages: Adiar, in parts using the disk, is on instances larger than 9.5 GiB only 1.47 to 3.69 times slower compared to CUDD and Sylvan, exclusively using main memory. Yet, Adiar is able to obtain this performance at a fraction of the main memory needed by conventional BDD packages to function.

We show that the problem of deciding whether in a multi-player perfect information recursive game (i.e. a stochastic game with terminal rewards) there exists a stationary Nash equilibrium ensuring each player a certain payoff is ∃ℝ-complete. Our result holds for acyclic games, where a Nash equilibrium may be computed efficiently by backward induction, and even for deterministic acyclic games with non-negative terminal rewards. We further extend our results to the existence of Nash equilibria where a single player is surely winning. Combining our result with known gadget games without any stationary Nash equilibrium, we obtain that for cyclic games, just deciding existence of any stationary Nash equilibrium is ∃ℝ-complete. This holds for reach-a-set games, stay-in-a-set games, and for deterministic recursive games.