inductive construction

This work investigates the goal-directed generation of mathematically meaningful theorems—or lemmas suitable for automated proof—from a given set of axioms. To this end, it introduces a novel approach grounded in the propositions-as-types paradigm, which systematically partitions the space of proof terms according to inductive levels and integrates proof-term enumeration with compression techniques, including separation-based reduction, DAG compression, and combinatory logic. This framework enables the efficient construction and compact representation of proof structures. Experimental evaluation on a fragment of Metamath’s set.mm library demonstrates that the method successfully produces nontrivial and semantically relevant theorems, thereby confirming its feasibility and advantages in the context of automated theorem discovery.

This study systematically evaluates the rigorous proof reasoning and explicit construction capabilities of large language models on Olympiad-level combinatorics problems. To this end, we introduce a benchmark comprising 100 expert-annotated competition problems, categorizing tasks into analytical (proof-oriented) and constructive (implementation-oriented) types. We propose a unified evaluation protocol that integrates rubric-guided proof assessment with deterministic verification of constructions, enhanced by a Best@4 multi-solution sampling strategy. Experimental results show that the strongest model achieves an average score of 65.4% overall (75.3% under Best@4), with markedly divergent performance across the two task types, revealing current limitations in creative mathematical reasoning—particularly on existence and construction problems. This work presents the first fine-grained distinction and joint evaluation of these capabilities, offering a new benchmark and diagnostic framework for mathematical reasoning research.

This paper addresses efficiency and performance bottlenecks in learning from combinatorial data—such as web pages, social networks, and molecular structures—by proposing the first unified framework for connectivity-aware modeling, integrating topological data analysis (TDA) with graph representation learning. Methodologically, it systematically employs persistent homology to capture higher-order topological features, synergizes hypergraph modeling with graph neural networks (GNNs), and incorporates combinatorial optimization to ensure algorithmic scalability. Compared to conventional approaches, the framework achieves an average 12.7% improvement in prediction accuracy across molecular property prediction, community detection, and web page ranking tasks, while reducing time complexity to near-linear. This advancement significantly enhances structural connectivity modeling capability and cross-domain generalizability.

This paper addresses the disconnect between theorem verification and pedagogical frameworks in formal mathematics education. Focusing on the Lean formalization of Tutte’s Matching Theorem from graph theory, it introduces a lightweight, modular, teacher-free educational formalization framework. Methodologically, it integrates a progressive verification strategy with task decomposition, fully formalizing the theorem in Lean using mathlib and successfully merging it into the mathlib main library. Concurrently, it designs a reusable, low-threshold pedagogical pathway adaptable to both university curricula and open-source community settings. Key contributions include: (1) the first deep integration of core theorem formalization into educational framework design; (2) a novel formalization learning paradigm enabling autonomous, collaborative, and iterative engagement by learners across diverse backgrounds; and (3) empirical validation through multiple successful implementations, demonstrating substantial reduction in the barrier to entry for formal methods.

This paper addresses the fragmentation and lack of interoperability among structural complexity measures across graph theory, geometric group theory, and dynamical systems. Methodologically, it introduces a unified “structured decomposition” framework grounded in category theory: (i) it is the first to formalize diverse domain-specific decomposition paradigms categorically; (ii) it establishes a general duality theory linking decompositions to object completions; and (iii) it defines composable width functors enabling cross-model quantification, comparison, and translation of structural complexity. Key contributions include: a unified categorical characterization of over ten complexity parameters—including treewidth, layered treewidth, and hypergraph treewidth—revealing their intrinsic structural relationships; and a novel parameterized tractability paradigm for NP-hard problems, grounded in decomposition width. The framework achieves both theoretical unification and algorithmic realizability.

This work addresses the absence of rigorous formalizations of abstract simplicial complexes and their stellar subdivisions in existing proof systems. It presents the first purely combinatorial formal framework for abstract simplicial complexes grounded in combinatorial topology, implemented in the Lean theorem prover. The framework encompasses fundamental operations such as morphisms, links, and joins, and systematically investigates their interaction with stellar subdivision. Key contributions include the first formalization of stellar subdivision in any proof assistant, the verification of several crucial identities—some previously undocumented in the literature—for the study of triangulated manifolds, and the proof of significant theorems such as the invariance of links under subdivision. This development establishes a reliable formal foundation for computational topology.

Traditional graph representations face significant challenges in graph isomorphism testing and symmetry-aware visualization due to high computational complexity and low efficiency. This work proposes “graph linear notation”—a complete graph invariant derived from canonical form algorithms—and establishes it, for the first time, as an equivalent definition for finite graphs. This representation not only substantially simplifies graph isomorphism comparison and symmetry-aware visualization but also naturally accommodates the extension and application of classical graph-theoretic concepts, such as coloring and paths, within its framework. By unifying these capabilities, the proposed notation offers a highly efficient and coherent new paradigm for structural graph analysis.

This study addresses the problem of efficiently extracting sequent calculus derivations from linear logic proof nets without altering their underlying graph structure. To this end, we propose a local coloring graph model based on half-edge coloring, which generalizes Yeo’s theorem to uniformly identify splitting vertices. Our approach integrates the Danos–Regnier correctness criterion, a cusp minimization lemma, and path analysis techniques, enabling modular reconstruction of sequent derivations directly from the original proof net. The method applies to linear logic systems featuring the mix rule and lacking multiplicative units. Beyond unifying and simplifying several existing graph-theoretic results, our work establishes that, in the absence of cusp cycles, a splitting vertex always exists, thereby guaranteeing an efficient sequentialization procedure.

This work addresses a longstanding limitation in Multiplicative Exponential Linear Logic (MELL), where existing correctness criteria are merely necessary and thus insufficient for effectively reconstructing sequent calculus proofs. We introduce a geometric constraint that strengthens connectivity-based correctness conditions to become both sufficient and efficiently decidable, leading to the definition of a new fragment, VMELL, which unifies classical and intuitionistic polarizations. By integrating proof nets, Danos–Regnier correctness graphs, and geometric analysis, we provide an explicit translation from bang-calculus terms into VMELL and precisely characterize their structure at the proof-net level. This framework successfully simulates the cut-elimination dynamics of bang reduction and uniformly accommodates the standard translations of both call-by-name and call-by-value λ-calculi.

This work proposes a formalization of algorithms within an intensional computability framework and clarifies their relationship to implementations in computational models. Treating computational models as monoid actions on configuration spaces, programs are modeled as dynamical systems constrained by such actions. Algorithms are defined as finite directed graphs of partial maps over edge-labeled abstract data structures, explicitly separating control flow from data operations. By leveraging tools from category theory, dynamical systems theory, and graph theory, the approach constructs a rigorous semantic framework that, for the first time, treats algorithms as abstract specifications of computational behavior and precisely characterizes the structure-preserving implementation relation between programs and algorithms, thereby deepening our understanding of the nature of computation.