formal definition design

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.

Defining mathematical concepts formally remains a critical bottleneck in interactive theorem proving: steep learning curves hinder newcomers, and undergraduate-level formalization progresses slowly. This paper investigates the generality, readability, and type-system compatibility of definitions, using Lean’s mathlib as an empirical foundation. We systematically analyze hundreds of equivalent definitions across diverse mathematical domains, evaluating them via usability metrics—theorem verification success rate, proof conciseness, and interface orthogonality. We identify three key determinants of definition quality: abstraction level, constructive strength, and interface granularity; from these, we distill reusable design principles. Furthermore, we contrast definition strategies in computer algebra systems (CAS) and, for the first time, establish a cross-system formal definition design guide. Our framework significantly improves the efficiency of standardized knowledge construction and long-term collaborative sustainability in libraries such as mathlib.

This work addresses the prevailing lack of systematic understanding of foundational formal theories in current AI compiler design, which hinders rigorous evaluation of the completeness and desirability of intermediate representations and compilation abstractions. For the first time, it systematically establishes precise correspondences between core mechanisms of MLIR—such as term rewriting systems, refinement calculi, and abstract interpretation—and classical formal theories. By grounding compiler abstractions in formal semantics, the paper clarifies the theoretical underpinnings of these constructs, articulates a precise notion of “design completeness,” and provides assessable criteria and guiding principles to navigate trade-offs between engineering pragmatism and theoretical ideals.

This study addresses the limited semantic transparency and poor comprehensibility of existing conceptual models, which stem from their reliance on low-level syntactic constructs to represent domain abstractions, thereby hindering effective system design and stakeholder communication. To overcome this, the paper proposes a language-agnostic abstract symbol engineering approach that identifies, formalizes, visualizes, and validates recurring syntactic configuration patterns, replacing them with high-level, semantically transparent abstract symbols. The method is instantiated as the DeCleaR extension to Dynamic Condition Response (DCR) graphs. Empirical evaluation demonstrates that DeCleaR significantly enhances perceived model quality, pragmatic quality, and user preference compared to standard DCR graphs.

This paper establishes an algebraic semantics foundation for programming languages with computational effects (e.g., state, I/O, exceptions). To address this, it introduces the *effect multicategory* as the central semantic model and—crucially—first defines and studies multicategories enriched over a *duoidal category*, rigorously proving their equivalence. This equivalence naturally yields definitions of effect algebras and 2-morphisms. Methodologically, the work integrates duoidal category theory, pre-multicategories, tensor and Cartesian product structures on functor categories valued in sets, and enriched category theory. The main contributions are: (i) a unified algebraic semantic framework subsuming both pure and effectful computation; (ii) the first general algebraic model for effectful programming languages based on higher-order categorical structures—specifically, 2-morphisms; and (iii) a rigorous theoretical foundation enabling axiomatization, logical reasoning, and principled language design for effect systems.

This study addresses the problem of systematically constructing logical systems corresponding to program abstractions to support formal reasoning. By associating logical systems with finite abstract domains, the work proposes a general method: for a given abstract lattice, it constructs a logic whose Lindenbaum–Tarski algebra is isomorphic to the abstraction, and derives corresponding axioms and inference rules. This approach establishes, for the first time, a systematic connection between abstract interpretation and proof theory as well as algebraic logic, enabling logical modeling of non-Cartesian abstractions such as octagons. The resulting logical connectives and inference systems preserve the concretization map and, under suitable conditions, satisfy soundness and completeness. The framework naturally extends to Cartesian products, multi-variable settings, and non-Cartesian abstract domains.

Traditional algebraic rewriting is unreliable for expressions involving measurements due to domain inconsistencies arising from repeated observations and division operations. This work proposes a unified semantic framework that simultaneously tracks both the provenance and definedness of expressions, enabling sound one-way rewriting and interchangeability judgments. By introducing label-sensitive bracketing semantics, admissible domain refinement, and a relative variant of support sets, the authors develop domain-safe rewriting rules and formally prove restoration and strictness theorems. All results are fully formalized in Lean 4 without any use of `sorry`, revealing fundamental limitations: simplifications are generally irreversible, equivalence over a common domain is insufficient, and label erasure inherently causes information loss.

This work addresses the limitations of traditional stratified definitions in logic, where the prohibition of negatively occurring defined predicates restricts expressive power for advanced applications such as relational reasoning. By relaxing the stratification condition, the paper integrates generic (nabla) quantification and general induction into an extended logical framework called G, achieving—for the first time—compatibility between weakly stratified definitions and these two mechanisms. Relying on monotonic fixed-point semantics and structural induction, the authors establish that this extension preserves logical consistency while substantially enhancing the capacity for formal reasoning about properties of programming languages. This result provides a theoretical foundation for extending the Abella proof assistant to support a broader class of inductive definitions.

This study rigorously formalizes Russell’s theory of definite descriptions within the framework of modern structural proof theory, preserving both its semantic content and syntactic representation. By employing sequent calculus, the work constructs three cut-free formal systems that, for the first time, unify multiple existing formalizations of the theory within a single proof-theoretic setting. A lambda operator is introduced as a scope marker to enhance expressive power. The appendix further translates these systems into natural deduction, thereby improving their practical usability. This contribution not only systematically compares and extends current approaches but also successfully integrates Russell’s theory into contemporary proof-theoretic methodology, achieving both theoretical rigor and practical applicability.

This work proposes a parameterized, unified realizability framework that overcomes the limitations of traditional realizability interpretations, which require explicit witnesses for existential quantifiers and struggle to uniformly handle atomic formulas alongside quantified statements. By abstracting and formally characterizing the semantic treatment of atomic formulas, the framework encompasses a wide spectrum of classical and modern realizability interpretations. It is shown to be compatible with various logical systems, including Heyting arithmetic, where its expressive power and consistency are rigorously verified. This approach enables a systematic integration and comparative analysis of diverse realizability methods within a single coherent setting.