case analysis

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.

This work addresses long-standing open problems in extremal set theory, such as Chvátal’s conjecture, by introducing a novel paradigm for customized search space partitioning based on solution construction strategies, replacing conventional domain-agnostic lookahead-based methods. By integrating this approach with a proof-generating exact mixed-integer linear programming (MILP) solver, the proposed framework substantially enhances search efficiency. Empirical evaluation demonstrates successful verification of the largest finite instance of Chvátal’s conjecture to date, marking significant progress toward resolving this fundamental problem in combinatorics.

This paper addresses the lack of a unified metatheoretic characterization for program logics handling multi-branching effects—such as nondeterminism and probabilism. We propose a novel program logic framework centered on algebraic choice structures. Methodologically, we are the first to embed algebraic effects modeling directly into the core of Hoare logic, integrating modal semantics with a relatively complete proof system that supports general loops and uniform reasoning across effect types (e.g., nondeterministic and probabilistic). Our main contributions are: (1) the first relatively complete proof system for Hoare logic strictly extending it to cover multiple branching effects; (2) a unified metatheoretic account of multi-result programs; and (3) formal support for cross-model reuse of proof fragments—enabling verification transfer between distinct semantic models (e.g., relational, probabilistic, or game-based interpretations).

Termination analysis of programs has long relied on disparate, specialized logics, making it difficult to uniformly characterize termination, non-termination, and partial correctness—especially for nondeterministic and probabilistic programs. To address this, we propose the first unified program logic framework that simultaneously models and reasons about total correctness, partial correctness, and non-termination within a single formalism. Our logic extends Hoare logic by integrating incorrectness logic and semantics for nondeterminism and probability, and establishes a rigorous metatheory—including soundness and relative completeness. We validate its expressiveness and practicality through multiple case studies, demonstrating support for compositional verification across programming paradigms. The framework significantly enhances the generality, unification, and engineering applicability of termination analysis.

Automated verification in separation logic (SL) has long relied on ad hoc heuristics, lacking a systematic metatheory and suffering from poor scalability. Method: This paper establishes the first general SL metatheory grounded in category theory and algebraic structures—specifically functors, homomorphisms, and modules over rings—systematically integrating abstract algebra into SL automation. The framework supports compositional model instantiation and modular predicate synthesis for any data structure admitting an algebraic characterization. All results are formally verified in Isabelle/HOL, and an automatic algebraic instantiation algorithm is developed. Contribution/Results: Experiments demonstrate fully automated algebraic modeling of complex imperative program semantics—including lists, trees, and graphs—and yield inference engines whose performance matches state-of-the-art hand-crafted systems. This approach decisively overcomes the scalability limitations inherent in heuristic-based methods.

该研究通过消除几何学框架探讨局部最优对象能否由共享部署规则实现，分析信息、架构等因素对缺陷修复的影响。

This work addresses the high degree of manual effort and tediousness inherent in existing automated reasoning algorithms—such as those for hyper-exponential quantifier elimination—for complexity analysis. The paper proposes a higher-order abstract interpretation framework grounded in operator semantics, which automatically abstracts symbolic programs into numerical recurrence relations. By integrating termination analysis, fixed-point theory, and SMT solving techniques, the method enables fully automated derivation and verification of asymptotic upper bounds on computational complexity. This approach substantially reduces human intervention while significantly enhancing the automation, efficiency, and scalability of complexity analysis for intricate algorithms.

This study addresses the challenge in axiomatic design of accurately translating customer needs and constraints into a minimal and independent set of primary functional requirements (FRs). Focusing on the problem definition phase, it systematically elucidates the nature, invariance, and formulation principles of primary FRs. Building upon Nam P. Suh’s theoretical framework and integrating insights from complexity theory and requirements engineering, the work establishes—for the first time—the objectivity and uniqueness of primary FRs, clarifies common misconceptions, and critically examines the applicability boundaries of large language models in this context. The research provides designers with a clear, actionable methodology for constructing primary FRs, thereby significantly enhancing the rigor of problem definition and the likelihood of successful design outcomes.

This work addresses the problem of multi-objective expected reward optimization in infinite-state Markov decision processes, aiming to synthesize policies that approximate the Pareto front. To this end, it introduces the first deductive program-level reasoning framework that integrates multi-objective optimization with weak expectation semantics. The approach features a novel multi-objective expectation transformer and employs a convex hull power domain to symbolically represent post-expectation tuples. By combining hybrid determinization rules for policy synthesis with operational semantics modeling, the method enables symbolic policy synthesis over infinite state spaces. Experimental evaluation demonstrates its effectiveness in solving multi-objective optimization problems across several case studies.

This study addresses the equivalence verification problem between two fundamental representations of finite closure systems—implicational and intersectional canonical bases—specifically, whether an intersectional basis fully captures all closed sets generated by a given set of implications. By integrating techniques from computational complexity theory, formal concept analysis, and functional dependency theory, the work establishes for the first time that this problem is coNP-complete, even when restricted to acyclic implication sets with premises of size at most three. This result precisely characterizes the computational complexity of verifying completeness in closure system representations, rules out the existence of output-polynomial algorithms even in restricted settings such as acyclic convex geometries, and provides new lower bounds for related problems including characteristic model identification.