Cost-Optimal Decision Diagrams for Stochastic Boolean Function Evaluation

For a finite set $O$ of Boolean functions, we consider the class of propositional formulas built using the functions in $O$ as connectives. We determine, for each possible choice of $O$, the complexity of various fitting and learning problems. These include: finding a formula that fits a given labeled sample, finding a small one (an Occam algorithm), minimizing the number of misclassified examples when the sample is not realizable (empirical risk minimization), and several forms of PAC learning. Our results apply both to formulas (represented as trees) and to circuits. We also briefly discuss the status of the same questions for other kinds of propositional fragments.

Reducing the number of focal elements of a mass function is classically driven by an intrinsic distance, such as Jaccard or Jousselme, that keeps the approximation close to the original as a body of evidence. We consider instead the case where the mass function feeds a linear combinatorial optimisation problem with evidential costs. What should then be preserved is not the closeness of the two mass functions, but the quality of the decision they induce. We introduce a decision-aware approximation that targets the regret of the decision: one decides with the cheaper approximation and is evaluated under the true mass function. On a minimal shortest path, the distance-optimal approximation flips the decision while a decision-aware merge preserves it, and this occurs on a non-negligible fraction of random instances. We prove a one-point bound that localises the regret at the true optimum, turn it into an exact dynamic program for the scalar case, and extend it to an online version that prunes focal elements before the final cost is known. In experiments the decision-aware compressor flips the decision less often than representation-aware compression, for both the linear criterion and a non-linear proxy read-out.

Causal queries are often only partially identifiable from observational data, and experiments that could tighten the resulting bounds are typically costly. We study the problem of selecting, prior to observing experimental outcomes, a cost-constrained subset of experiments that maximally tightens bounds on a target query. We formalize this as the max-potency problem, where epistemic potency measures the worst-case reduction in bound width guaranteed by an experiment, and show that this problem is NP-hard via a reduction from 0-1 knapsack. Building on the polynomial-programming framework of Duarte et al. (2023), we give a general procedure for evaluating epistemic potency in discrete settings. To control the super-exponential search space, we introduce two graphical pruning criteria that depend only on the causal graph and the query: a novel path-interception rule that exploits district structure to certify zero potency in linear time, and an identifiability check based on the ID algorithm. On Erdos-Renyi random graphs and 11 bnlearn benchmark networks, the two criteria together prune 50-88% of candidate experiments on average without solving a single polynomial program. For the general subset search, we show that ID-pruned experiments are combinatorially inert, yielding a super-exponential reduction in the number of subsets evaluated. We close with an end-to-end demonstration on observational NHANES data, selecting optimal experiments for estimating the effect of physical activity on diabetes.