Atropos: Games, Complexity & Biomedical Informatics

• Atropos is a multi-faceted topic involving a combinatorial game based on Sperner’s lemma, where players alternate coloring a triangulated graph and risk immediate loss by forming a rainbow triangle.
• The Atropos‑k variant generalizes adjacency rules and demonstrates PSPACE-completeness via reductions from TQBF, using sophisticated gadgets like switches and crossovers.
• Atropos also refers to a biomedical evidence framework that integrates real-world data and retrieval-augmented generation, enhancing evidence-supported decision making in healthcare.

Atropos encompasses several significant concepts across computational complexity, combinatorial game theory, and biomedical informatics: (1) a family of combinatorial games defined on Sperner-labeled triangulations (notably, the original Atropos and its variants such as Atropos‑k), (2) frameworks and datasets in real-world evidence summarization and biomedical retrieval-augmented generation (RAG) systems (including proprietary resources originally named the Atropos Evidence Library), and (3) foundational implications for the interplay of logic, topology, complexity theory, and applied AI in healthcare.

1. Atropos Game: Definition and Core Mechanics

The original Atropos game is a turn-based, two-player, perfect-information combinatorial game played on a finite, triangular subgraph of a triangular lattice. Three distinct colors (commonly red, green, blue) are assigned to the vertices of the triangle, and boundary edges inherit corresponding colorings. All internal nodes are initially uncolored.

On each turn, a player colors an uncolored node with one of the available colors. The first move is unrestricted, but all subsequent moves must color an uncolored node that is adjacent to the most recently colored node. The pivotal rule is that creating a "rainbow triangle"—a triangle whose three vertices have pairwise distinct colors—results in an immediate loss for the acting player.

Sperner’s lemma ensures that any Sperner-labeled triangulation (under these boundary conditions) must contain at least one rainbow triangle, thereby guaranteeing a forced win for one player and eliminating the possibility of a draw.

2. Atropos‑k: Variant and Complexity Analysis

Atropos‑k generalizes adjacency by permitting moves within a distance k (shortest-path metric) of the most recently colored node. The original game coincides with k=1. For k=∞, the adjacency restriction is abolished, allowing play on any uncolored node.

The primary contribution concerning this variant is the demonstration that, for any fixed k ≥ 2, deciding whether the current player has a forced win in Atropos‑k is PSPACE-complete. The proof employs a reduction from True Quantified Boolean Formula (TQBF), employing sophisticated gadgets to simulate logical variable assignment and quantifier alternations. The reduction involves designing paths and gadgets (switches, merges, multi-switches, crossovers, and check gadgets), which, via parity and reachability constraints, enforce the alternation and winning conditions mimicking TQBF's evaluation structure (Yang et al., 4 Mar 2024).

This result positions Atropos‑k among canonical PSPACE-complete games—such as generalized Chess, Go, and Hex—demonstrating that local coloring constraints can encode the full expressive power of polynomial-space computation.

3. Topological Foundations: Sperner’s Lemma

Sperner's lemma forms the theoretical foundation of the Atropos game. In any Sperner-labeled triangulation of a triangle (with vertices assigned the three colors, and boundaries labeled accordingly), there must exist at least one "rainbow" simplex. This non-constructive guarantee underpins the losing condition in Atropos—once all internal nodes are colored, a rainbow triangle necessarily emerges.

Sperner’s lemma is both a topological result (used in proofs of Brouwer’s fixed point theorem) and a constructive tool for designing combinatorial games with forced-outcome structures. In the context of Atropos and its variants, it ensures that every path through the game space must end in a decisive outcome rather than a draw, and provides a topological guide to strategy analysis and complexity construction.

4. Computational Complexity and Reduction Techniques

Demonstrating PSPACE-completeness for Atropos‑k involves encoding arbitrary TQBF instances within the game's move graph. The reduction is carefully constructed using:

• Switch gadgets simulating ∀/∃ quantifiers by enforcing control of key moves to the adversary or hero, via carefully chosen distances within the path network.
• Multi-switch and merge gadgets facilitating clause selection and variable assignment, reflecting CNF clause checking.
• Parity adjustments via control of node distances ensure that the right player is faced with crucial existential or universal choice points at each subgame.

The result is that the game tree of Atropos‑k closely models the solution tree of the source TQBF formula—the existence of a forced win becomes isomorphic to the truth of the formula itself. A notable open question is the computational complexity of unrestricted Atropos (k=∞), which remains unresolved within this framework.

5. Biomedical Informatics: Atropos in Evidence Summarization and Retrieval

Separately, "Atropos" appears as an eponym—initially denoting a proprietary evidence library (the Atropos Evidence Library, later renamed "Alexandria") designed for real-world evidence (RWE) aggregation, synthesis, and retrieval-augmented generation for biomedical applications. This system aggregates observational studies, electronic health records, and claims data, enabling the generation of evidence-supported answers to physician and researcher queries (Baldwin et al., 30 Jun 2025).

Within recent frameworks such as Answered with Evidence, Alexandria (née Atropos) operates alongside PubMed-based retrieval systems to supply LLMs with both conventional and novel evidence sources. In studies, answers strictly grounded in Alexandria were evidence-supported for approximately 50% of biomedical questions, with a combined evidence-based coverage (when merged with PubMed-sourced RAG) exceeding 70% of queries (Baldwin et al., 30 Jun 2025).

In RWESummary—a MedHELM benchmarking task for evaluating LLM summarization of structured RWE studies—the test set and evaluation pipeline are based on proprietary Atropos Health data repositories. This supports assessment of direction of effect, numeric accuracy, and completeness, yielding robust LLM benchmarking and validation in clinically realistic settings. Gemini 2.5 models have demonstrated the highest accuracy across the rubric's weighted metrics when evaluated with this data (Mukerji et al., 23 Jun 2025).

6. Implications and Intersections

The Atropos game and its variants illustrate the deep connection between logic, topology, and computational complexity, with practical implications for the analysis of forced-move combinatorial games and reductions of logic-based decision problems. The encoding techniques based on Sperner’s lemma set a template for modeling alternation and quantification in other game-theoretic contexts.

In biomedical informatics, the Atropos Evidence Library (“Alexandria”) and its integration into LLM evaluation pipelines exemplify the operationalization of real-world evidence into automated reasoning systems. The synergy between structured observational data and published literature substantially increases the rate of high-confidence, evidence-supported answers for biomedical queries, providing a bridge between theoretical informatics and practical healthcare decision support.

7. Future Directions and Open Challenges

• Computational Complexity: Determining the exact complexity class of unrestricted Atropos (k=∞) remains open; novel reductions or proof techniques will be required given the breakdown of bounded-distance gadget constructions.
• Generalization: Extending gadget-based reductions to board variants, alternate coloring rules, or higher-dimensional analogues represents a direction for further complexity-theoretic investigation.
• Clinical RAG Frameworks: Further augmentation of biomedical evidence systems through integration of domain-specific evidence bases and the expansion of real-world data ingestion (beyond the Atropos/Alexandria dataset) could further enhance LLM reliability.
• Evaluation Rubrics: The RWESummary and Answered with Evidence frameworks establish methodological precedents for benchmarking LLM output in sensitive domains. Future work may emphasize interoperability and harmonization across proprietary and public datasets, as well as the transparent weighting of evaluation criteria to optimize clinical utility.

Atropos thus stands at the confluence of combinatorics, topology, computational complexity, and biomedical knowledge integration, manifesting both as a formal, highly expressive combinatorial game and as a suite of applied methodologies in evidence-based medicine.