The Topology of Ill-Posed Questions: Persistent Homology for Detection and Steering in LLMs

Ill-posed questions, including ambiguous, underspecified, or contradictory queries, may admit no valid answer or multiple plausible answers, posing a challenge for large language models (LLMs). Existing approaches largely analyze ill-posedness through model outputs and often focus on specific subclasses. We investigate whether diverse sources of ill-posedness can be represented within a unified topology of LLM internal states and whether this structure can be used to steer response behavior. We model the contextual hidden states of prompt tokens at each transformer layer as a point cloud and characterize its geometry using finite zero-dimensional persistent homology. Each layer is summarized by three compact descriptors: mean finite lifetime, normalized lifetime entropy, and largest-lifetime concentration. Concatenating these descriptors across layers yields a topology representation of the question. We further introduce topology-conditioned activation steering, which retrieves topologically similar examples and constructs query-specific activation interventions that encourage source-aware clarification or abstention. Across three open-weight LLMs, topology features consistently outperform prompt-based and pooled-hidden-state baselines for ill-posedness classification, improving average accuracy from \(67.4\%\) to \(78.9\%\) on AmbigQA, from \(79.9\%\) to \(88.5\%\) on SituatedQA, and from \(57.6\%\) to \(69.6\%\) on CLAMBER 9-way classification. Topology-conditioned steering increases the average total acceptable response rate from \(61.4\%\) to \(70.6\%\) and grounded acceptable responses from \(11.9\%\) to \(16.4\%\). These results show that persistent homology provides both an interpretable representation of ill-posedness and an effective mechanism for targeted response steering.

ACDZero: Graph-Embedding-Based Tree Search for Mastering Automated Cyber Defense

Automated cyber defense (ACD) seeks to protect computer networks with minimal or no human intervention, reacting to intrusions by taking corrective actions such as isolating hosts, resetting services, deploying decoys, or updating access controls. However, existing approaches for ACD, such as deep reinforcement learning (RL), often face difficult exploration in complex networks with large decision/state spaces and thus require an expensive amount of samples. Inspired by the need to learn sample-efficient defense policies, we frame ACD in CAGE Challenge 4 (CAGE-4 / CC4) as a context-based partially observable Markov decision problem and propose a planning-centric defense policy based on Monte Carlo Tree Search (MCTS). It explicitly models the exploration-exploitation tradeoff in ACD and uses statistical sampling to guide exploration and decision making. We make novel use of graph neural networks (GNNs) to embed observations from the network as attributed graphs, to enable permutation-invariant reasoning over hosts and their relationships. To make our solution practical in complex search spaces, we guide MCTS with learned graph embeddings and priors over graph-edit actions, combining model-free generalization and policy distillation with look-ahead planning. We evaluate the resulting agent on CC4 scenarios involving diverse network structures and adversary behaviors, and show that our search-guided, graph-embedding-based planning improves defense reward and robustness relative to state-of-the-art RL baselines.