probabilistic method

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.

Many existence problems in combinatorial design remain open, lacking constructive solutions or effective search heuristics. Method: We propose a novel framework that integrates reasoning-oriented large language models (LLMs) into the constructive solving protocol CPro1, enabling end-to-end generation of executable search heuristics directly from problem specifications. Our approach unifies LLM-driven code generation, automated correctness verification, hyperparameter optimization, and execution feedback in a closed loop. Contribution/Results: Applied to 16 long-standing open instances from the *Handbook of Combinatorial Designs* (2006), our method successfully constructs solutions for 7 cases—including three problem classes resolved for the first time. Moreover, it discovers several new combinatorial structures recently reported in 2025 literature. By automating heuristic discovery and validation, this work substantially advances the frontier of constructive combinatorial design automation.

Existing program logics cannot fully characterize the output distribution of probabilistic concurrent programs, and no distribution-level formal verification methodology exists for such programs. Method: We propose the first distributional verification logic supporting programs combining probabilistic and concurrent features, systematically integrating independence, conditional distributions, and invariants into Outcome Logic, and introducing the first probabilistic concurrent separation principle. By unifying probabilistic separation logic, concurrent separation logic, and Outcome Logic, we design distribution-aware assertions, randomized resource models, and context-compositional proof rules to enable modular, compositional distributional verification. Results: Our logic is the first to precisely model independent execution, conditional dependencies, and concurrency invariants at the distribution level. It supports fully automated formal verification of representative probabilistic concurrent programs, thereby filling a fundamental theoretical gap in distributional verification of concurrent probabilistic programs.

Randomized algorithms instruction often suffers from a disconnect between theoretical foundations and practical implementation, alongside insufficient coverage of modern developments. Method: This project develops a systematic lecture-note framework for advanced undergraduate and graduate students, integrating core probabilistic tools—including expectation, Chernoff and Hoeffding bounds, martingales, Markov chains, and the Lovász Local Lemma—with classical and cutting-edge algorithmic analyses (e.g., randomized quicksort, hashing, MCMC, approximate counting, and derandomization). Notably, it is the first to incorporate foundational quantum computing concepts and distributed randomized algorithms into a course at this level. Contribution/Results: The resulting resource is logically coherent, self-contained, and immediately deployable in teaching. It has served as the primary textbook for Yale University’s CPSC 4690/5690 course for multiple years and is widely adopted as a key reference in randomized algorithms courses across numerous global institutions.

Traditional probabilistic laws face persistent challenges—including indeterminate boundaries of physical possibility and acute empirical underdetermination. Method: This paper proposes a novel metaphysical framework grounded in algorithmic randomness, introducing a “probability-constrained law” paradigm that replaces generative chance laws. It jointly employs Kolmogorov complexity and relative frequency as dual constraints, and integrates nonstandard probability models with possible-worlds semantics to rigorously delimit the set of physically possible histories. Contribution/Results: The work achieves the first systematic synthesis of algorithmic information theory with a neo-Humean (i.e., non-Humean) conception of laws, thereby resolving one class of empirical underdetermination while uncovering and characterizing a previously overlooked type. It substantially enhances both the empirical testability and metaphysical constraint strength of probabilistic laws, providing a critical pathway toward a unified account of non-Humean laws.

This work addresses fundamental challenges in combinatorics and algorithm design by developing efficient random and exhaustive generation methods for combinatorial structures. It proposes a unified generation framework that integrates bijective, enumerative, algebraic, and analytic combinatorics with probabilistic algorithms and rigorous algorithmic analysis. The resulting methodology not only advances theoretical computer science but also yields significant applications in bioinformatics, combinatorics on words, and number theory. Through this interdisciplinary approach, the study demonstrates the effectiveness and broad applicability of novel generation algorithms, highlighting their potential to bridge theoretical insights with practical computational problems across diverse scientific domains.

This work addresses the challenges in combinatorial counting arising from intricate structural and arithmetic constraints, which hinder manual derivation and cause existing methods to break problem symmetries. To overcome these limitations, the paper introduces Cofola, a typed declarative language that unifies combinatorial counting as a weighted first-order model counting (WFOMC) problem with coefficient extraction constraints—the first such formulation. Cofola naturally expresses common combinatorial structures including sets, multisets, permutations, and partitions. Its three-stage compilation pipeline integrates preprocessing, symmetry-preserving decomposition, and ordering axiom encoding—such as lexicographic symmetry breaking and sequence/cycle axioms—to enable efficient solving while preserving inherent symmetries. Experimental results demonstrate that Cofola substantially outperforms existing frameworks in both expressiveness and computational efficiency across a diverse benchmark suite, ranging from textbook examples to complex multi-object scenarios.

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.