equivalence proof

Existing resources for computational complexity reductions lack systematic organization, visualization, and extensibility. Method: This paper introduces an interactive, graph-database–based online platform that unifies diverse reduction paradigms—including classical complexity classes (e.g., NP, #P), parameterized classes (W[1], W[2]), gap-preserving reductions, and PCP theorems—within a single directed graph model linking problems and reductions. The platform supports semantic search, multi-dimensional filtering, and community-driven curation, and employs a modular software architecture enabling seamless integration of new complexity classes and reduction types. Contribution/Results: The platform publicly releases an interconnected knowledge graph covering core classes such as NP, #P, SSP-NP, W[1], W[2], and PCP. It establishes an open, verifiable, evolvable, and reusable knowledge infrastructure for complexity theory research, advancing both pedagogy and collaborative investigation of reduction hierarchies.

This paper establishes a rigorous categorical semantics for e-graphs (equivalence graphs) within the framework of monadic categories, supporting double-pushout (DPO) rewriting. Method: The authors generalize e-graphs to monadic categories by introducing *equivalence hypergraphs* (e-hypergraphs)—a compositional structure whose vertices are algebras over a monad and whose hyperedges encode algebraic operations, thereby internalizing structural equations up to isomorphism. The approach integrates category theory, semilattice-enriched categories, and hypergraph-based combinatorial modeling to yield a sound and complete semantic framework. Contribution/Results: The resulting framework provides an algebraic and monadic foundation for equivalence reasoning in e-graph–based program optimization, and extends the formal applicability of e-graphs to SMT solving and algebraic optimization—enabling principled, categorical treatment of equational rewriting beyond traditional graph-based methods.

This work addresses the high formalization complexity of verifying map equivalences in homotopy type theory. We propose two lightweight techniques: (1) decomposing maps into composites of elementary equivalences, and (2) systematically leveraging the 3-for-2 property of equivalences to refine logical reasoning. To our knowledge, this is the first formal framework that jointly exploits both techniques, significantly reducing proof complexity and redundancy. Implemented within the Coq/Agda toolchain, we formally verify a foundational theorem in synthetic homotopy theory. The resulting proof is concise, highly modular, and strongly reusable—demonstrating the practical efficacy of our approach. Our methodology provides a transferable, principled foundation for equivalence reasoning in homotopical settings, advancing the automation and scalability of formal proofs in univalent mathematics.

This paper addresses the challenges of fragility analysis and poor constructivity in observational equivalence proofs. We propose a stepwise reasoning method based on hypergraph rewriting. Our key contributions are threefold: (1) We formally characterize robustness as a critical sufficient condition for observational equivalence—a novel formulation; (2) We establish a neighborhood-based local reasoning framework that supports generalized observational equivalence definitions and verification under syntactically restricted contexts and quantitative step bounds; (3) Leveraging a hypergraph rewriting abstract machine inspired by geometric interaction, we structurally model function abstraction and application in higher-order stateful lambda calculus. Experimental evaluation on call-by-value lambda calculus demonstrates that our approach significantly improves modularity, constructivity, and analyzability of fragility in observational equivalence proofs.

This paper addresses the lack of formal semantic foundations for reduction logic—i.e., goal-directed backward reasoning—by establishing, for the first time, a systematic formal semantic framework for reduction operators. Methodologically, it integrates categorical semantics with operational semantics to develop the logical metatheory of reduction calculus and proposes mathematically precise criteria for determining the validity of reduction operators. The core contribution lies in overcoming the unidirectionality inherent in traditional deductive logic, thereby enabling an ontological and verifiable formalization of the fundamental human- and machine-shared inference pattern: “deriving sufficient premises from a given conclusion.” This work provides a rigorous semantic foundation and soundness guarantees for goal-driven AI tasks, including automated planning, theorem discovery, and commonsense reasoning.

This study addresses the problem of characterizing graphic sequences and generating all their realizations. By introducing a “2-reduction” operation—subtracting one from each of two elements in the sequence—that preserves graphic equivalence, the authors develop a unified inductive reduction framework. This approach not only streamlines the reproducibility of classical characterization theorems such as Erdős–Gallai but also enables efficient construction of all graph realizations corresponding to a given sequence. Furthermore, it yields a novel equivalent characterization of graphic sequences, uncovering a deep connection between their intrinsic mathematical structure and underlying algorithmic mechanisms.

This work investigates how to characterize the strength of propositional proof systems via provable reductions to TFNP search problems. To this end, it introduces a novel class of implicit proof systems ⟨EF, R⟩, where R is a TFNP problem such that the task of finding falsifying assignments for unsatisfiable formulas reduces polynomially to R, and the correctness of this reduction is verifiable in Extended Frege (EF). The main contributions include establishing, for the first time, a polynomial equivalence between ⟨EF, Resolution⟩ and the classical sequent calculus G₁; proving that ⟨EF, Iter⟩ is likewise equivalent to both G₁ and ⟨EF, Resolution⟩; and demonstrating that EF-provably correct reductions are strictly stronger than FP-computability. Moreover, for any sufficiently strong proof system P, there exists a search problem Rₚ in FP such that ⟨EF, Rₚ⟩ is polynomially equivalent to P.

Traditional logical relations struggle to accommodate directed reductions in type theory, impeding normalization proofs in dependent type systems. This work presents the first logical relation model within simplicial homotopy type theory that integrates contravariant computability predicates with directed quotient inductive types. Reduction is internalized as an inequality type, and a comonadic flat modality cleanly separates vertical reductions from horizontal parametricity. Leveraging built-in functoriality and universal properties, the approach supports computability reasoning under directed reduction, enabling a successful proof of directed Boolean normalization. The method extends to systems featuring dependent types and universes, yielding the first formalization of representation independence with proof relevance.

This work presents a systematic investigation of the interpolation property across classical, intuitionistic, modal, and substructural logics, proposing a unified, constructive, modular, and syntax-driven methodology that integrates Maehara’s and Pitts’ classical techniques into a general proof-theoretic framework. By uncovering a structural correspondence between interpolation and well-behaved proof systems, the study not only establishes the existence of interpolation theorems for a wide range of logics but also delivers a reusable blueprint for constructing interpolation proofs within modern formal proof systems. This approach emphasizes the role of syntactic structure in enabling modular and scalable interpolation results, thereby advancing the theoretical understanding and practical applicability of interpolation in diverse logical settings.

This work addresses the poor readability, modularity, and maintainability of formal proofs generated by large language models, which often fall short of high-quality mathematical library standards. Inspired by human proof-refactoring practices, the authors propose a four-stage agent framework that systematically decomposes proof refactoring into candidate fragment extraction, auxiliary lemma design, component verification, and original proof repair. Departing from length-based or other single-metric optimizations, the approach prioritizes structural quality. Experiments on Lean-generated proofs from PutnamBench and Putnam2025 demonstrate that the method significantly outperforms the Claude Code baseline in human readability and signature quality, establishing the first automated pipeline for structure-oriented proof refactoring.