DynamicHS: Incremental Diagnosis via Stateful HS-Tree

• DynamicHS is a stateful variant of Reiter’s HS-Tree that incrementally updates a hitting-set search tree to compute minimal diagnoses efficiently.
• It employs a two-stage redundancy check—Quick and Complete—to prune invalid nodes and minimize computational cost during sequential diagnosis.
• DynamicHS preserves soundness, completeness, and best-first enumeration while significantly reducing reasoning calls and search tree size in complex diagnostic scenarios.

DynamicHS is a stateful variant of Reiter’s HS-Tree algorithm, specifically designed to streamline the computation of minimal diagnoses in sequential diagnosis (SD) tasks, where successive measurements incrementally refine system knowledge. DynamicHS maintains and incrementally updates a hitting-set search tree across iterations, reusing previous conflict and diagnosis computations, and thus minimizes redundant operations that characterize the stateless application of HS-Tree. The approach achieves significant reductions in reasoning calls and overall computational cost, particularly in domains with expensive logical inference, while rigorously preserving the theoretical properties of soundness, completeness, and best-first enumeration (Rodler, 2020, Rodler, 2019).

1. Diagnosis Problem Instance and Reiter’s HS-Tree

A diagnosis problem instance (DPI) is defined as $\langle K, B, P, N \rangle$, where $K$ is a set of component axioms, $B$ denotes background knowledge assumed to be correct, $P$ contains positive measurements (sentences that must be entailed), and $N$ contains negative measurements (sentences that must not be entailed). Diagnoses are minimal sets $D \subseteq K$ such that modifying the system by removing $D$ from $K$ restores the system to a state consistent with $P$ and not entailing any $n \in N$. Minimal conflicts are subsets $K$0 such that $K$1 is inconsistent or entails a negative measurement, with none of their proper subsets having this property. Minimal diagnoses correspond to the minimal hitting sets of the minimal conflicts collection:

$K$2

HS-Tree algorithmically constructs a search tree whose nodes represent candidate diagnoses, employing a priority queue ordered by a measure $K$3 (typically probability or cardinality), and repeatedly labels nodes by discovering and adding minimal conflicts through calls to a reasoning service. In classical sequential diagnosis, updating the DPI after each measurement results in discarding and reconstructing the entire HS-Tree, incurring high redundant computation costs (Rodler, 2020, Rodler, 2019).

2. Core Principles and Innovations of DynamicHS

DynamicHS addresses inefficiencies of stateless HS-Tree by employing statefulness—preserving and incrementally adapting the existing tree, conflicts, and diagnosis candidates with each new measurement. The key principles include:

• Reuse-and-Adapt Paradigm: The search tree, priority queues, computed conflicts, duplicates, and stored diagnoses are maintained across iterations. Upon DPI update, only the portions of the tree affected by the new measurement are examined for redundancy and pruned or relabeled as needed.
• Redundancy Checking: Nodes invalidated by the new measurement are identified using a two-stage redundancy check: (1) a Quick Redundancy Check (QRC) invokes a single call to findMinConflict on a union of the node’s conflict labels minus the node’s path; (2) if QRC is inconclusive, a Complete Redundancy Check (CRC) iteratively tests each stored conflict label.
• Conflict Reuse with Minimality Validation: When reusing a stored conflict $K$4 to relabel a node, DynamicHS first tests whether $K$5 remains minimal relative to the new DPI. If the true minimal conflict has strictly shrunk, the affected nodes are pruned and $K$6 is replaced by its subset.

3. Algorithmic Workflow and Data Structures

DynamicHS maintains a collection of data structures essential for stateful diagnosis:

• $K$7: Priority queue of open nodes yet to be labeled.
• $K$8: Pool of duplicate nodes (used for node replacement during pruning).
• $K$9: Supersets of known diagnoses (non-minimal nodes).
• $B$0: Set of computed minimal conflicts available for reuse.
• $B$1: Diagnoses consistent with the latest measurement.
• $B$2: Diagnoses invalidated by the most recent measurement.

The main workflow involves:

• Initializing State: The initial DPI and data structures are constructed, with the root node placed in the open queue.
• Tree Update: Upon receiving a new measurement, UPDATE-TREE operates only on $B$3, performing lazy and efficient pruning of invalidated branches and relabeling with new minimal conflicts if necessary.
• Diagnosis Candidate Expansion: For each candidate node, DLABEL determines its status via non-minimality checks, conflict reuse/minimality checks, or new conflict computation, adding valid diagnoses or expanding nodes using minimal conflicts.
• Best-First Enumeration: The priority queue ensures expansion and output of diagnoses in the preferred order according to the cost or probability function $B$4.

A high-level representation of the procedures, explicitly laid out in the literature, is reflected in LaTeX-style pseudocode provided in (Rodler, 2020, Rodler, 2019).

4. Measurement Selection Heuristics

DynamicHS can be combined with established measurement selection heuristics to guide the acquisition of system measurements that maximally reduce diagnostic ambiguity:

• ENT (Expected Entropy): Selects the measurement point minimizing expected entropy over the outcome predictions of leading diagnoses.
• SPL (Split-in-Half): Chooses the measurement that achieves the most balanced partition (by cardinality) of diagnoses between positive/negative predictions.
• MPS (Most-Probable-Singleton): Prioritizes measurements that maximize the probability of eliminating all but one leading diagnosis with a single query.

Let $B$5 be the current set of minimal diagnoses and $B$6 a possible measurement. The heuristic computes $B$7 and $B$8 as the sets of diagnoses predicting negative or positive, respectively, for $B$9, then applies the corresponding scoring function as specified in the original references (Rodler, 2020).

5. Empirical Performance and Complexity

Comparative analysis between DynamicHS and classical HS-Tree establishes that DynamicHS achieves substantial speedups and reductions in costly reasoner calls, especially in knowledge-base debugging scenarios:

• In evaluations on 20 real-world ontologies with sizes ranging from 42 to 33,000 axioms (OWL 2 DL or fragments), and across six diagnosis thresholds and three heuristics, DynamicHS outperformed HS-Tree in 96% of sessions.
• Median runtime reduction of $P$052% and up to $P$175% was observed, with hard instances showing median saving of $P$264% and maximum of $P$389%—i.e., HS-Tree at times taking 9× longer.
• Median reduction of $P$474% in hard reasoning calls, with the overhead of pruning operations negligible (less than 5% of time in 89% of scenarios).
• Memory overhead for stateful duplicate and node storage remained below a 2× factor in most cases.
• DynamicHS required a median of 50–100% fewer nodes per session, substantially reducing search tree size and redundant evaluations (Rodler, 2020).

6. Theoretical Guarantees

DynamicHS preserves all crucial theoretical properties of the original HS-Tree:

• Soundness: All outputs are minimal diagnoses for the current DPI.
• Completeness: All minimal diagnoses can be found, provided unbounded computation.
• Best-First Enumeration: The use of a priority queue ordered by any monotonic measure (e.g., probability or cardinality) ensures ordered enumeration.
• Logic and Reasoner Independence: DynamicHS operates as a meta-algorithm over any monotonic reasoning service, and does not rely on the specific logical language. Stateful incremental updates guarantee that addition of a single measurement to the DPI leads only to local and necessary revision of the tree, without risking the loss of any minimal diagnoses or violating best-first ordering (Rodler, 2020, Rodler, 2019).

7. Practical Recommendations and Prospects

DynamicHS is particularly effective in domains where expensive reasoning operations dominate the diagnosis bottleneck (e.g., OWL 2 DL ontology debugging, hardware and software diagnosis with complex constraints), and the transition between successive DPIs is characterized by small measurement-based refinements. For deployment, the literature recommends:

• Efficient duplicate detection via hash-based set equality on node paths.
• Use of lightweight list or array structures for node representations.
• Lazy validation strategies to defer reasoner calls until conflict reuse is unavoidable.
• Priority queue designs tailored to the chosen enumeration criterion.

Further extensions proposed in the literature include parallelized tree expansion, integration with alternate diagnosis generators (e.g., Inv-HS-Tree), and adaptively exploiting the stateful tree to refine measurement selection heuristics. The analysis of worst-case state growth and incremental conflict computation strategies remain open areas for advancement (Rodler, 2020, Rodler, 2019).