perform axiomatic analysis

This paper addresses two key challenges in foundational programming theory: the mathematical complexity of basic programming concepts and the reliance of formal verification on redundant axioms. To resolve these, we propose PRISM—a minimalist programming theory grounded solely in naive set theory. PRISM introduces only one primitive relation, one initial set, and three fundamental operations (selection, composition, and restriction), without assuming any axioms. It unifies program semantics and specifications within a single set-theoretic framework, defining correctness, specialization, and refinement via standard subset inclusion. All core results—including over thirty program properties and classical “programming laws”—are fully mechanized and formally verified in Isabelle/HOL. Our contributions are threefold: (1) the first axiom-free programming theory framework; (2) a unified semantic–specification representation for programs; and (3) an open-source, reproducible, and extensible library of machine-checked proofs.

This study addresses the challenge in axiomatic design of accurately translating customer needs and constraints into a minimal and independent set of primary functional requirements (FRs). Focusing on the problem definition phase, it systematically elucidates the nature, invariance, and formulation principles of primary FRs. Building upon Nam P. Suh’s theoretical framework and integrating insights from complexity theory and requirements engineering, the work establishes—for the first time—the objectivity and uniqueness of primary FRs, clarifies common misconceptions, and critically examines the applicability boundaries of large language models in this context. The research provides designers with a clear, actionable methodology for constructing primary FRs, thereby significantly enhancing the rigor of problem definition and the likelihood of successful design outcomes.

This paper addresses the challenges of cross-logical comparability, visualization, and automated reasoning for mathematical theorems. Methodologically, it introduces the novel concept of *proof vectors*: theorems are encoded as high-dimensional vectors whose axes correspond to foundational axiom systems (e.g., Hilbert’s geometry, Peano arithmetic, ZFC); axiom embeddings, cosine similarity, and clustering enable quantitative assessment of logical similarity and heatmap-based visualization; and the Atlas-GPT prototype implements semantic mapping from natural-language theorems to proof vectors. Key contributions include: (1) the first interpretable, axiom-system-agnostic theorem vectorization paradigm; (2) cross-disciplinary automatic clustering of theorems by underlying logical foundations; and (3) the first interactive “Axiomatic Atlas” prototype system, supporting mathematics education, formal verification, and AI-assisted mathematical discovery.

Current machine learning approaches to scientific discovery are fundamentally limited by their reliance on inductive pattern recognition, lacking mechanisms for autonomous theoretical innovation grounded in first principles. Method: We propose the “Rule Evolution Framework,” a theory-driven paradigm that integrates formal logic, reinforcement learning, and multi-agent game-theoretic reasoning within a gamified environment. In this setting, agents actively revise foundational axioms to explain anomalous observations, enabling dynamic hypothesis generation beyond curve-fitting. Contribution/Results: The framework endows ML systems with the capacity to autonomously reconstruct theoretical assumptions, bridging the gap between data-driven modeling and interpretable theory formation. Preliminary experiments demonstrate that the system solves previously intractable problems via axiom evolution—providing the first empirical validation that ML can support self-directed theoretical discovery. This work establishes a novel pathway for AI-augmented fundamental scientific inquiry, shifting emphasis from predictive accuracy to explanatory power and conceptual novelty.

Quantifying set diversity lacks a rigorous foundation, as mainstream metrics—e.g., distance-based, entropy-based, or coverage-based measures—fail to satisfy three desirable axioms: monotonicity, uniqueness, and continuity, undermining their reliability. Method: We establish the first axiomatized framework for diversity measurement, formally defining and proving the joint satisfiability of these axioms. Through systematic counterexample analysis, we demonstrate that existing approaches violate at least one axiom. We then construct an axiomatically complete metric and analyze its computational properties. Contribution/Results: We prove that any metric satisfying all three axioms is inherently NP-hard to compute, thereby establishing the “axiomatically complete yet computationally feasible” diversity measure as an open problem. This work provides the first rigorous benchmark for diversity evaluation and precisely characterizes the fundamental tension between theoretical soundness and practical computability in diversity quantification.

This study addresses a critical limitation in traditional reproducible research, where sharing only code and results fails to expose the implicit assumptions, expectations, and premises underlying an analyst’s reasoning—thereby hindering thorough evaluation of analytical quality. To overcome this, the paper proposes a formal modeling framework that explicitly translates the analyst’s tacit reasoning process into structured logical representations, statically capturing the construction logic of the analysis. This approach enables systematic scrutiny of the analytical chain of reasoning, assumption sensitivity, and conclusion robustness—even in the absence of the original data. Empirical validation on representative data analysis tasks demonstrates the framework’s effectiveness, achieving both logical visualization and data-free static assessment of analytical integrity.

This work proposes a systematic formalization of all published mathematical knowledge into a machine-verifiable, continuously evolving structured knowledge base, addressing the challenges of scalability and organization. Taking dilatations of categories in categorical algebra as the first case study, the project integrates interactive theorem proving, dependent type theory, and category theory to frame the complete formalization of mathematics as a universal reasoning benchmark. By constructing a formal prototype of category dilatations, the study demonstrates the feasibility of this approach in expressing complex algebraic structures, thereby establishing both an architectural foundation and a practical pathway toward a large-scale, interconnected, and extensible database of formalized mathematics.

This work addresses the challenge of verifying mathematical proofs generated by large language models by formally encoding, for the first time, an entire advanced undergraduate probability textbook—including its measure-theoretic foundations—into Lean. To bridge the semantic gap between the textbook’s exposition and the abstract formalism of the Mathlib library, the authors introduce an “interface lemma” strategy. Combined with structured proof engineering and formalization techniques specific to measure theory, this approach yields a reusable, machine-verifiable infrastructure spanning fourteen textbook chapters. The resulting formalization not only provides rigorous verification of all stated theorems and explicit articulation of their assumptions but also establishes a robust foundation for reliable AI-assisted mathematics, educational applications, and future formalization efforts in probability theory.

This work addresses the limitations of existing formalisms for hyperproperties in capturing quantitative aspects inherent in real-world systems, such as numerical relationships in information flow control. To overcome this, the paper introduces Quantitative Hyper-Logic (QHL), a novel framework that reformulates hyperproperty specifications using measure theory, replacing classical Boolean quantifiers with measures to support nested quantitative structures. Leveraging Hoeffding’s inequality and extreme value theory, the authors develop an efficient statistical verification algorithm and provide rigorous analyses of sample complexity and statistical guarantees. Experimental evaluation on quantitative information-flow benchmarks demonstrates that QHL substantially outperforms conventional qualitative approaches, offering superior expressiveness and verification capabilities that better align with the demands of practical systems.

This work unifies the modeling of inference rules and proof structures of formal systems within a categorical framework. It introduces a met-variable context representation based on Cartesian PROPs, encodes assumptions and conclusions using spans, and constructs a symmetric monoidal category of proofs with met-variable substitution as the sole primitive operation. This approach is the first to uniformly embed both inference rules and proof structures into the semantics of symmetric monoidal categories, thereby supporting compositional and reusable handling of hypotheses. The authors implement an open-source verification algorithm and surface syntax, successfully encoding formulas, axioms, and representative derivations of first-order logic, and release a functional proof checker.