We study the problem of formally explaining why a candidate was not selected by a given tournament rule, by identifying sub-tournaments in which the candidate loses independently of how the rest of the tournament is completed. We define destructive minimal supports as any minimal sub-tournaments satisfying this property, which in formal explainable artificial intelligence correspond to abductive explanations for the question "Why does the loser lose the tournament?". For six common tournament solutions (maximin, uncovered set and its weighted variant, top-cycle, Copeland, and Borda) we provide characterizations of when a candidate is either a necessary loser or a possible winner, we determine the size of the smallest destructive minimal supports, complemented by polynomial-time algorithms for their computation except for the case of the Borda rule which is suspected to be NP-complete.
Andreas Maier, Siming Bayer, Patrick Krausscs.AI cs.LG
Explanation requires ground truth: to verify an account of a system we must know its inner functioning-just what is missing where explainable AI (XAI) is most needed. Systems we can study fall into two camps. Simple, procedural one-decision trees, rule lists, sparse linear models-have a known but trivial mechanism, so explaining them tests nothing; genuinely complex ones-deep networks, real-world tasks-need XAI but have no ground-truth inner functioning, so an explanation can be plausible, confident, and wrong with no way to tell. We remove this dichotomy with a study object both genuinely complex and fully specified-inspectable by construction-and, so gradient methods apply, fully differentiable. We reimplement the Atari 2600 Video Computer System (VCS)-a real computer architecture, and the cradle of deep reinforcement learning-as two independent end-to-end differentiable emulators in Julia (jutari) and JAX (jaxtari), each validated bit-for-bit against xitari. Both reproduce xitari on all 64 supported Arcade Learning Environment (ALE) games: 64/64 byte-identical RAM and 64/64 pixel-identical screens. Treating the cartridge ROM as a weight tensor, RAM as a soft tape, and control flow as gates, we prove the differentiable (soft) execution equals the original (hard) one bit-for-bit in the forward pass at any finite temperature, while exposing surrogate gradients where the bit logic has none. The JAX port also opens a GPU path: batched differentiable rollouts reach millions of environment-steps/s on one commodity GPU. The system was built in roughly 137 active hours over 29 calendar days, much of it written autonomously by coding agents. This paper builds and validates the foundation, showing-theoretically and in a qualitative gradient study-that gradient-based XAI on it is feasible. Both ports' full code is available under the MIT license at https://github.com/akmaier/UnderstandingVCS.
Estimating the economic contribution of a single patent inside a product that embodies tens of thousands of patents is a long-standing unsolved problem in intellectual property economics. We propose PatentXAI, a framework that treats patent valuation as a problem of explainable AI: given a characteristic function v(S) encoding the revenue achievable by patent subset S, a patent's Shapley value measures its fair share of product profit in a way that satisfies efficiency, symmetry, dummy, and additivity. To make computation tractable we restrict each patent's coalition to its Markov Blanket inside a knowledge graph, grounded in the C-SVE conditional independence theorem (Li et al., 2020). Scaling experiments from n=12 to n=100 patents using Pareto-distributed coverage graphs report median Markov Blanket size of 32.9 percent of n at n=100, with 90th-percentile blanket size of 55.2 percent of n, and runtime of 10 milliseconds per patent. Difference against exact ground truth at n=12 is 0.088; difference against a high-sample Monte Carlo reference at n=100 is 0.062 plus or minus 0.003. A dense-component experiment shows that when 80 percent of patents share one component, the blanket correctly expands to cover that dense cluster, and the difference versus reference falls to 0.039 because the pooled computation becomes more accurate on homogeneous portfolios. Profit allocation proceeds hierarchically: exact Shapley distributes total profit among macro-components, then centrality-weighted Shapley distributes each component budget among covering patents. Estimating v(S) from real data is the primary open problem; we distinguish this from the computational contribution and outline a concrete roadmap for empirical validation using public ETSI, USPTO, and Lens.org datasets.