Model checking with the standard Kripke models used in (Dynamic) Epistemic Logic leads to scalability issues. Hence alternative representations have been developed, in particular symbolic structures based on Binary Decision Diagrams (BDDs) and succinct models based on mental programs. While symbolic structures have been shown to perform well in practice, their theoretical complexity was not known so far. On the other hand, for succinct models model checking is known to be PSPACE-complete, but no implementations are available. We close this gap and directly relate the two representations. We show that model checking DEL on symbolic structures encoded with BDDs is also PSPACE-complete. In fact, already model checking Epistemic Logic without dynamics is PSPACE-complete on symbolic structures. We also provide direct translations between BDDs and mental programs. Both translations yield exponential outputs. For the translation from mental programs to BDDs we show that no small translation exists. For the other direction we conjecture the same.

This information would be provided by experts in the chosen domain, i.e. epidemiologists working on epidemic

While the game Connect-Four has been solved mathematically and the best move can be effectively computed with search based methods, a strong solution in the form of a look-up table was believed to be infeasible. In this paper, we revisit a symbolic search method based on binary decision diagrams to produce strong solutions. With our efficient implementation we were able to produce a 89.6 GB large look-up table in 47 hours on a single CPU core with 128 GB main memory for the standard $7 \times 6$ board size. In addition to this win-draw-loss evaluation, we include an alpha-beta search in our open source artifact to find the move which achieves the fastest win or slowest loss.

This work introduces VISY-REVE: a novel pipeline to validate image processing algorithms for Vision-Based Navigation. Traditional validation methods such as synthetic rendering or robotic testbed acquisition suffer from difficult setup and slow runtime. Instead, we propose augmenting image datasets in real-time with synthesized views at novel poses. This approach creates continuous trajectories from sparse, pre-existing datasets in open or closed-loop. In addition, we introduce a new distance metric between camera poses, the Boresight Deviation Distance, which is better suited for view synthesis than existing metrics. Using it, a method for increasing the density of image datasets is developed.

Safety-critical controllers of complex systems are hard to construct manually. Automated approaches such as controller synthesis or learning provide a tempting alternative but usually lack explainability. To this end, learning decision trees (DTs) have been prevalently used towards an interpretable model of the generated controllers. However, DTs do not exploit shared decision-making, a key concept exploited in binary decision diagrams (BDDs) to reduce their size and thus improve explainability. In this work, we introduce predicate decision diagrams (PDDs) that extend BDDs with predicates and thus unite the advantages of DTs and BDDs for controller representation. We establish a synthesis pipeline for efficient construction of PDDs from DTs representing controllers, exploiting reduction techniques for BDDs also for PDDs.

Among different interesting problems in XAI, explaining classifiers' decisions has attracted significant attention[Bae+10; MI22; Mar24]. Work in this area has proposed notions to explain classifiers' behavior on a specific feature vector (instance), providing so-called local explanations, as well as notions to explain the overall classifier's behavior regardless of any particular instance, providing so-called global explanations. For both notions, a key issue is to analyze the computational complexity of the problems at hand, since this is crucial to understand how to approach the development of algorithmic solutions and their inherent limits. In the field of XAI, there has been an extensive body of work addressing complexity issues[BAK24; OPS23; Are+23; CM23; CA23; CCM23; Hua+22; Aud+22; CM22; Bar+20a; Mar+20] This paper falls within this ongoing research stream. Specifically, we deal with global explanations, and focus on so-called global necessary reasons, defined as conditions that instances must satisfy in order to be classified with a class of interest. This notion has been considered by Ignatiev, Narodytska, and Marques-Silva[INM19], where an interesting relationship with another kind of global explanation is shown.

We propose a scalable preconditioned primal-dual hybrid gradient algorithm for solving partial differential equations (PDEs). We multiply the PDE with a dual test function to obtain an inf-sup problem whose loss functional involves lower-order differential operators. The Primal-Dual Hybrid Gradient (PDHG) algorithm is then leveraged for this saddle point problem. By introducing suitable precondition operators to the proximal steps in the PDHG algorithm, we obtain an alternative natural gradient ascent-descent optimization scheme for updating the neural network parameters. We apply the Krylov subspace method (MINRES) to evaluate the natural gradients efficiently. Such treatment readily handles the inversion of precondition matrices via matrix-vector multiplication. A posterior convergence analysis is established for the time-continuous version of the proposed method. The algorithm is tested on various types of PDEs with dimensions ranging from $1$ to $50$, including linear and nonlinear elliptic equations, reaction-diffusion equations, and Monge-Amp\`ere equations stemming from the $L^2$ optimal transport problems. We compare the performance of the proposed method with several commonly used deep learning algorithms such as physics-informed neural networks (PINNs), the DeepRitz method, weak adversarial networks (WANs), etc, for solving PDEs using the Adam and L-BFGS optimizers. The numerical results suggest that the proposed method performs efficiently and robustly and converges more stably.

In multicriteria decision-making, a user seeks a set of non-dominated solutions to a (constrained) multiobjective optimization problem, the so-called Pareto frontier. In this work, we seek to bring a state-of-the-art method for exact multiobjective integer linear programming into the heuristic realm. We focus on binary decision diagrams (BDDs) which first construct a graph that represents all feasible solutions to the problem and then traverse the graph to extract the Pareto frontier. Because the Pareto frontier may be exponentially large, enumerating it over the BDD can be time-consuming. We explore how restricted BDDs, which have already been shown to be effective as heuristics for single-objective problems, can be adapted to multiobjective optimization through the use of machine learning (ML). MORBDD, our ML-based BDD sparsifier, first trains a binary classifier to eliminate BDD nodes that are unlikely to contribute to Pareto solutions, then post-processes the sparse BDD to ensure its connectivity via optimization. Experimental results on multiobjective knapsack problems show that MORBDD is highly effective at producing very small restricted BDDs with excellent approximation quality, outperforming width-limited restricted BDDs and the well-known evolutionary algorithm NSGA-II.

Sparse linear regression (SLR) is a well-studied problem in statistics where one is given a design matrix $X\in\mathbb{R}^{m\times n}$ and a response vector $y=X\theta^*+w$ for a $k$-sparse vector $\theta^*$ (that is, $\|\theta^*\|_0\leq k$) and small, arbitrary noise $w$, and the goal is to find a $k$-sparse $\widehat{\theta} \in \mathbb{R}^n$ that minimizes the mean squared prediction error $\frac{1}{m}\|X\widehat{\theta}-X\theta^*\|^2_2$. While $\ell_1$-relaxation methods such as basis pursuit, Lasso, and the Dantzig selector solve SLR when the design matrix is well-conditioned, no general algorithm is known, nor is there any formal evidence of hardness in an average-case setting with respect to all efficient algorithms. We give evidence of average-case hardness of SLR w.r.t. all efficient algorithms assuming the worst-case hardness of lattice problems. Specifically, we give an instance-by-instance reduction from a variant of the bounded distance decoding (BDD) problem on lattices to SLR, where the condition number of the lattice basis that defines the BDD instance is directly related to the restricted eigenvalue condition of the design matrix, which characterizes some of the classical statistical-computational gaps for sparse linear regression. Also, by appealing to worst-case to average-case reductions from the world of lattices, this shows hardness for a distribution of SLR instances; while the design matrices are ill-conditioned, the resulting SLR instances are in the identifiable regime. Furthermore, for well-conditioned (essentially) isotropic Gaussian design matrices, where Lasso is known to behave well in the identifiable regime, we show hardness of outputting any good solution in the unidentifiable regime where there are many solutions, assuming the worst-case hardness of standard and well-studied lattice problems.