derive analytical bounds

This study addresses the limitations of traditional network calculus, which assumes non-negative service curves and struggles to analyze complex systems with feedback control. By rigorously examining the properties of subadditive functions, the authors reveal that allowing negative service curves in feedback systems often leads to unstable analyses. To overcome this issue while preserving the non-negativity assumption, they develop a refined network calculus framework that integrates network calculus theory, subadditive function analysis, and system stability verification. Applying this approach to the complex feedback system proposed by Hamscher et al., the method achieves accurate modeling and tight performance bounds, effectively circumventing the instability inherent in prior techniques and significantly enhancing the applicability and reliability of network calculus in closed-loop systems.

Detecting design patterns and modeling code refactoring in large-scale, complex software systems remains challenging due to difficulties in representation learning and low computational efficiency. Method: This paper proposes Analytical Software Engineering (ASE), a novel design paradigm. Its core innovations include: (1) a language-agnostic, compact code abstraction—Behavior-Structure Sequence (BSS); (2) an Optimized Design Refactoring (ODR) framework integrating heuristic search with software metric encoding to eliminate iterative computation overhead inherent in conventional refactoring; and (3) a unified design balancing abstraction, tool accessibility, compatibility, and extensibility. Results: Experiments demonstrate that ASE significantly improves both design pattern detection accuracy and refactoring efficiency, validating its effectiveness for maintainability assessment and sustainable optimization. It establishes a new theoretical foundation and a scalable framework for automated analysis of complex software metrics.

Existing expression-template-based operator-overloading automatic differentiation (AD) tools (e.g., CoDiPack) lack native complex-number support, forcing complex operations to be decomposed into redundant real-valued arithmetic, inflating computational graph size and derivative tape memory overhead. Method: We propose a comprehensive complex-number integration scheme within the expression-template AD framework, encapsulating complex arithmetic and elementary functions—including optimized complex linear algebra—directly in the template logic, thereby avoiding explicit real/imaginary part splitting. Contribution/Results: This work presents the first end-to-end complex AD implementation in an expression-template AD system. Experiments on synthetic benchmarks show up to 42% reduction in memory footprint and up to 3.1× speedup in gradient computation, significantly improving differentiation efficiency and scalability for complex-valued applications.

Automated derivation of closed-form analytical solutions to complex differential equations remains a longstanding challenge due to the combinatorial explosion of symbolic search spaces and the difficulty of enforcing physical constraints. Method: This paper proposes a neural-symbolic collaborative framework that integrates formal grammar–driven symbolic expression generation with variational latent modeling and constraint-satisfaction optimization, enabling probabilistic search over continuous embedding manifolds for candidate solutions satisfying both governing equations and boundary conditions. Contribution/Results: It introduces the first unified, end-to-end differentiable architecture jointly modeling formal grammar generation and neural-symbolic embeddings, bridging numerical approximation and symbolic reasoning. Evaluated on diverse ordinary and partial differential equations, the method substantially outperforms commercial symbolic solvers (e.g., Mathematica) and purely neural approaches, producing high-accuracy, interpretable, and generalizable analytical solutions—thereby breaking a fundamental bottleneck in automated analytical discovery.

This work addresses the automatic verification of expected output bounds for probabilistic programs featuring general loops, continuous distributions, and conditional branching—where the integral semantics induced by continuous sampling impede conventional invariant-based reasoning. We propose a Riemann-sum-based approximation of the expected semantics, transforming integral bounds into quantitative invariants expressible in SMT logic. This constitutes the first systematic integration of Riemann integration into probabilistic program verification, accompanied by formal convergence guarantees for the approximation and a proof that the verification problem is coRE-complete. We implement a prototype within the Caesar verification framework, supporting intermediate-language encoding and SMT-driven inference; it successfully verifies multiple benchmarks involving continuous sampling and loops. Our approach bridges discrete program verifiers with continuous probabilistic analysis, enabling existing discrete verification tools to scale to programs with continuous distributions.

This work addresses the lack of machine-verifiable formalizations of line search methods in nonlinear optimization, which has hindered algorithmic reliability. Within the Lean 4 theorem prover, it presents the first systematic formalization of several classical line search criteria—including Armijo, Goldstein, Wolfe, and their nonmonotone variants—alongside rigorous definitions of gradient descent, descent directions, and backtracking step-size selection. The study fully verifies the Zoutendijk convergence theorem within this framework, thereby establishing a comprehensive formal foundation for line search theory. This contribution significantly enhances the verifiability and trustworthiness of nonlinear optimization algorithms through mechanized mathematical reasoning.

This work addresses the limitations of current automatic formalization research, which predominantly focuses on well-supported mathematical domains and relies solely on kernel acceptance rate as a quality metric, thereby neglecting the practical needs of underrepresented areas such as numerical analysis and lacking comprehensive evaluation. For the first time, we employ a Lean 4 coding agent to formalize an entire textbook—*Numerical Methods for Ordinary Differential Equations*—from scratch and introduce a three-dimensional evaluation framework that jointly assesses semantic correctness, Mathlib reusability, and cross-file reusability. Through LLM-as-judge, semantic validation, and dependency analysis, we uncover pervasive issues in existing systems, including incomplete statements and weakened assumptions, demonstrating that kernel acceptance rate substantially overestimates formalization quality. Our approach establishes a reproducible, multidimensional auditing paradigm for trustworthy automated formalization.

This work proposes a novel method for the automated discovery and verification of lower confidence bounds on the mean. By introducing a general relaxation framework parameterized by order statistics, the problem of finding optimal confidence bounds is formulated as a computationally tractable optimization problem, which unifies classical results such as Hoeffding’s inequality. The approach integrates mixed-integer linear programming with optimization relaxation theory to enable, for the first time, the automatic construction and formal verification of confidence bounds. In particular, when the order-statistic function is linear—as in the case of Hoeffding-type bounds—the method yields a mixed-integer linear program of linear size, allowing efficient approximation and rigorous validation of the target confidence bound.