monotonicity arguments

This work conducts ordinal analysis of a specific fragment of the μ-calculus whose proof-theoretic strength equals that of parameter-free Π¹₂-comprehension (Π¹₂-CA₀⁻). Method: We develop a staged analytical framework: proofs in the original system are systematically reconstructed in an extended system via “proof-modification,” progressively lifting the comprehension strength—first to Π¹₁-comprehension and ultimately characterizing the ordinal upper bound of Π¹₂-CA₀⁻. This constitutes the first systematic application of the proof-modification paradigm to ordinal analysis of subsystems of second-order arithmetic, unifying treatments of parameter-free and full Π¹₁-generalization. Contribution/Results: By integrating μ-calculus semantics, recursion theory, and techniques from Gödel’s constructible hierarchy, we determine the exact ordinal upper bound of Π¹₂-CA₀⁻ and demonstrate the scalability and consistency of our method across systems of varying Π¹₁-strength.

This work proposes a novel method to overcome the limitations of the classical absolute positivity criterion, which fails to handle nonlinear polynomial constraints involving universal quantifiers. Specifically, the approach addresses ∃∀ inequalities over the natural numbers by integrating monotonic algebra with well-founded order theory, thereby dispensing with the absolute positivity assumption. This advancement substantially broadens the class of constructible nonlinear polynomial interpretations. Experimental results demonstrate that the technique successfully solves constraint instances previously intractable to existing methods, thus extending the applicability of polynomial interpretations in termination and complexity analysis of term rewriting systems.

Formal proofs in proof assistants like Coq often break under semantic changes—e.g., modifications to equivalence relations in quotient types—posing challenges for existing automated repair techniques, which lack support for behavioral-level alterations. This work introduces the first proof repair algorithm tailored to quotient-type equivalence changes. We propose a setoid-based external modeling of quotient types, enabling joint handling of representation and behavioral changes. Furthermore, we construct the first internal correctness proof of the algorithm in Cubical Agda, grounded in univalent semantics to guarantee semantic consistency. Empirical evaluation demonstrates successful repair of multiple previously intractable quotient-type equivalence changes, substantially reducing manual intervention. The approach is validated across both Coq and Cubical Agda, confirming its feasibility, cross-platform applicability, and formal correctness.

This paper investigates the quantifier-free definability of specific connectives—such as those of Taranovsky and Kreisel–Połacik type—in intuitionistic second-order logic. Addressing limitations of conventional model-theoretic approaches in characterizing quantifier-free definable mappings between structures, it establishes, for the first time, a deep correspondence between such definability and cut elimination as well as local provable equivalence. The analysis employs proof-theoretic methods—including natural deduction and sequent calculus, local conservativity arguments, counterexample construction, and type-elimination techniques. Main contributions include: (i) complete criteria for quantifier-free definability in classical theories such as linear and discrete orders; (ii) correction and generalization of Tarski’s seminal result on real closed fields; and (iii) the first systematic proof-theoretic framework for definability theory in intuitionistic logic.

Horn logic lacks a unified proof-theoretic framework. Method: We introduce an abstract, logic-agnostic proof-theoretic model based on *g-sequents* (generalized binary graph-like sequents), formalizing them via abstract algebraic semantics, graph representation, and lattice-theoretic modeling; we develop an algebraic classification and interaction analysis theory for inference rules, and design a generic proof transformation algorithm that characterizes upper and lower bound correspondences of nested/marked sequent systems within abstract lattices. Contribution/Results: We achieve the first uniform characterization of diverse Horn-logic sequent-style systems; establish that cut-free nested/marked systems precisely correspond to extremal points (i.e., top and bottom elements) of the associated lattice; derive quantitative relationships between proof size and sequent complexity; and determine the computational complexity of lattice operations. This work bridges structural proof theory, algebraic logic, and lattice theory to provide a foundational, modular framework for Horn-logic reasoning.

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.

Existing logical frameworks, such as λProlog and LF, struggle to support automated proof search when syntactic equality is treated as a logical connective, particularly in the presence of quantifier alternations and polarity-sensitive occurrences of equality. This work proposes a proof-search method that extends unification mechanisms to directly reason with equality as defined by its introduction rules within a first-order sequent calculus. The approach constitutes the first automated proof-search procedure for such an equality treatment, yielding a lightweight, unification-aware logical framework compatible with various first-order systems. It successfully derives core mathematical principles—including the Peano axioms without induction—thereby addressing a significant gap in equational reasoning capabilities left by current frameworks.

Current evaluations of mathematical proofs generated by large language models predominantly focus on correctness, often overlooking critical dimensions such as clarity, conciseness, insightfulness, and transferability. This work introduces ProofRank, a novel benchmark that systematically defines and quantifies five scalable quality metrics: conciseness, computational simplicity, cognitive simplicity, diversity, and adaptability. Leveraging a dataset derived from mathematical competition problems, the study conducts a comprehensive evaluation of mainstream models using automated proxy metrics. The results reveal substantial variation in proof quality across models and demonstrate that the most correct proofs are not necessarily the highest-quality ones, highlighting a trade-off between correctness and holistic proof quality. These findings underscore the need for future evaluation frameworks to balance practical utility with formal correctness.

This work addresses the susceptibility of large language models to “context contamination” when verifying complex mathematical proofs, which often masks logical errors. The authors propose a step-level verification framework that preserves the full context of each inference step and strictly restricts the set of admissible theorems for reference, enabling fine-grained validation of research-level proofs. By integrating LLM-guided reasoning constraints, an adversarial benchmark (FirstProof Challenge), and systematic ablation studies, the method substantially outperforms conventional global evaluation approaches in precisely identifying subtle logical flaws. Remaining misjudgments predominantly stem not from severe hallucinations but from “over-rigor”—arising when domain-specific conventions are left implicit—thereby exposing latent ambiguities in existing expert benchmarks and advancing a more human-like, cautious paradigm for mathematical verification.

This study addresses the problem of ensuring safety, liveness, and fairness guarantees for argument strength in dynamic weighted argumentation dialogues. By integrating classical temporal logic properties into quantitative (bipolar) argumentation frameworks, the work constructs a dynamic evolution model based on weighted argumentation graphs to formally characterize how argument strengths vary over time. It introduces, for the first time, systematic definitions of strong and weak safety, cross-threshold liveness, and sequential fairness, clarifies their logical interrelationships, and identifies the core analytical challenges inherent in achieving general-purpose guarantees. Combining formal verification, temporal reasoning, and dynamic graph analysis, the approach provides a theoretical foundation for interpretable and trustworthy argumentative interactions.