A construction of the hat tilings by a Markov partition We present a simple construction of hat tilings. The construction can be carried out by superimposing a triangular grid on a specially colored image and reading off the orientation of the tiles. We show that our construction produces valid hat tilings, and conversely, in an appropriate sense that is made precise in the paper, that every valid hat tiling can be obtained in this way.

Anatomy of a Query: W5H Dimensions and FAR Patterns for Text-to-SQL Evaluation Natural language interfaces to databases have gained popularity, yet the theoretical foundations for evaluating and designing these systems remain underdeveloped. We present QUEST (Query Understanding Evaluation through Semantic Translation), a framework resting on two independently motivated components: the FAR structural invariant, which holds that every well-formed query reduces to Filter, Aggregate, and Return operations; and the W5H dimensional framework, which holds that all filtering criteria map to six semantic dimensions (Who, What, Where, When, Why, and How). Validated across five text-to-SQL datasets (n = 120,464), FAR conformance is universal across all domains and schema types, while W5H dimensional profiles vary substantially. Healthcare queries are strongly concentrated in temporal (WHEN: 80.4%) and person-centric (WHO: 73.0%) dimensions far exceeding general-domain benchmarks, and causal (WHY) and mechanistic (HOW) reasoning are near-zero everywhere, with apparent HOW exceptions reflecting quantitative aggregation rather than genuine procedural reasoning. These results identify a frontier that must be crossed for genuine machine reasoning over structured data.

Semantic Reverse Engineering Legacy Software Applications with ChatGPT, Gemini AI, and Claude AI This research paper describes our research results on using ChatGPT, Gemini, and Claude AI to semantically reverse engineer legacy database software applications.

An Extensible and Verifiable Language for Query Rewrite Rules Logical query plan rewriting transforms a relational database query into an equivalent but more efficient form and is crucial to the performance of database-backed applications. In existing systems, rewrite rules are typically implemented manually, tightly coupled to specific execution engines, and often lack formal correctness guarantees. Consequently, developing a new engine requires reimplementing both legacy and new rules, incurring significant engineering cost, limiting portability, and every new implementation is an opportunity for introducing new bugs. We introduce Rulescript, an engine-agnostic domain-specific language (DSL) for developing query rewrite rules. Rulescript separates rule definition from execution infrastructure via a relational algebra-inspired core language and an explicit decomposition of rules into matching and transformation phases. Developers express rewrites by pattern-matching query plans using Rulescript's core operators and constructing semantically equivalent transformed plans, with all rewrites automatically verified formally to ensure correctness. Rulescript is extensible: users can define custom operators in terms of the core language to capture engine-specific semantics. To integrate with an existing system, developers need only implement a lightweight adapter that maps Rulescript's core and custom operators to the operators implemented in the target engine. We evaluate Rulescript by reimplementing 33 rewrite rules from Apache Calcite and extending the language with several custom operators. To demonstrate portability, we automatically deploy these rules to CockroachDB and Apache Data Fusion, two engines with substantially different backends. Our results show that Rulescript enables "write once, deploy everywhere" paradigm for query plan rewriting, with minimal effort required to deploy previously written rules on a new data engine.

Toward Space-Based Public Key Systems: Enabling Secure Space Communications through In-Orbit Trust Services The New Space era has led to a rapid increase in satellites operated by independent entities in near-Earth orbit. This shift enables richer space services but also requires secure, near-real-time coordination, making efficient authentication of space assets critical for next-generation missions. Traditional ground-dependent Public Key Infrastructure (PKI) suffers from latency and operational bottlenecks that limit scalability and availability in dynamic space environments. This paper proposes architectural designs for space-based PKI that shift certificate management and validation from ground infrastructure into space, reducing reliance on ground stations while enabling interoperability and cross-entity collaboration. Two deployment schemes are introduced: a space-ground integrated PKI with in-orbit validation authorities, and a fully autonomous space-based PKI with in-space issuance and validation. We analyze deployment trade-offs in scalability, availability, security, cost, and operational complexity in multi-operator environments. A baseline latency analysis is provided to illustrate performance implications of in-orbit trust management.

The Supersymmetric Origin of Chaos and its Hidden Topological Order Dynamical chaos is a term that encompasses a wide range of nonlinear phenomena such as turbulence, neuronal avalanches, weather patterns, and many others. However, despite much work in the field of chaos, its fundamental physical origin still remains not fully understood. In this perspective we report on recent studies showing that chaos is the realization of one of the most fundamental principles in physics: spontaneous symmetry breaking also known as spontaneous ordering. In the present context, the symmetry involved is a topological supersymmetry inherent to all continuous-time (stochastic) dynamical systems. Chaos is then truly a manifestation of order of topological origin potentially encoding a sort of long-range information hidden beneath its apparent unpredictability. We finally argue that this point of view may have far-reaching implications well beyond chaotic dynamics.