perform log-domain arithmetic

This paper investigates the structural complexity of multi-output functions computable in logarithmic space by single-pass, read-write, in-place Turing machines. It formally defines the new complexity class *inplaceFL* (and its extension *inplaceFCL*) and systematically characterizes their computational power. Using relativization, circuit complexity analysis, and catalytic computation models, the paper establishes several unconditional and conditional separations under standard complexity assumptions: (1) *FL* ⊈ *inplaceFL*; (2) integer multiplication and NC⁰₄ circuit evaluation are not in *inplaceFL*, assuming cryptographic hardness; (3) NC⁰₂ circuit evaluation admits an *inplaceFL* algorithm; (4) matrix multiplication and inversion over finite fields are in *inplaceFCL*. This work provides the first structural complexity-theoretic characterization of inherent limitations of in-place computation, reveals a novel barrier to proving *CL* ⊆ *P*, and delivers foundational insights into computability under strict space constraints.

To address the high computational cost and low hardware efficiency of floating-point operations in deep learning training, this paper proposes a hardware-aware low-precision logarithmic fixed-point training method tailored for accelerators. The approach innovatively incorporates a bit-width–aware mechanism into logarithmic addition approximation, jointly optimizing piecewise linear approximation and simulated annealing to achieve Pareto-optimal trade-offs between accuracy and hardware overhead. Leveraging the logarithmic number system (LNS) and bit-accurate C++ simulation, end-to-end training is realized using 12-bit integer arithmetic. Experimental results on VGG-11 and VGG-16 demonstrate accuracy comparable to 32-bit floating-point training, while reducing multiply-accumulate (MAC) unit area by 32.5% and energy consumption by 53.5%. This work establishes a novel hardware–software co-design paradigm for low-precision deep learning training.

This study addresses the challenge of numerical instability in scientific software caused by floating-point precision errors, particularly in safety-critical contexts where traditional methods struggle with complex expressions. It presents the first systematic evaluation of large language models (LLMs) for improving numerical stability by detecting and rewriting unstable arithmetic expressions. The experiments encompass 2,470 expressions featuring nested conditionals, high-precision literals, and multi-variable arithmetic, evaluated across six prominent LLMs. Results demonstrate that LLMs outperform baseline methods in 65.4% of cases and successfully stabilize 97.9% of the 431 instances where baselines completely fail. Nevertheless, limitations persist in handling control flow constructs and high-precision literals, highlighting areas for future improvement.

This study investigates the algebraicity and arithmetic properties of hypergeometric functions over the rational numbers, finite fields, and p-adic fields. Leveraging the SageMath computer algebra system, the work integrates techniques from algebraic number theory, finite field theory, and p-adic analysis to systematically implement, for the first time in an open-source framework, algorithms capable of determining algebraicity, computing valuations, and solving for minimal polynomials in positive characteristic. This implementation fills a critical gap in existing computational toolchains by enabling uniform arithmetic analysis of hypergeometric functions across multiple number-theoretic domains, thereby substantially enhancing SageMath’s capacity for algebraic manipulation of such functions.

Repetitive computation in change-sensitive programs—such as database queries, compilers, and real-time analytics—incurs substantial overhead and undermines complexity control. Method: We propose the “incrementalization” paradigm, formalizing incremental computation as a discrete analogue of differentiation and establishing its theoretical foundation in discrete computation. Our approach introduces an “iterate–incrementalize–implement” design framework, featuring a novel meta-level abstraction-driven model for algorithmic complexity refinement, integrating higher-order abstractions over data, control flow, and modules with formal incrementalization transformations. Contribution/Results: We deliver a reusable, formally verifiable incrementalization methodology that guarantees correctness while significantly improving computational efficiency and enhancing controllability of algorithmic complexity. The framework enables systematic, principled application of incremental computation across diverse domains, bridging theory and practice in program optimization and reactive systems.

This work investigates which combinatorial and number-theoretic counting functions are computable in logarithmic space, i.e., belong to the complexity class #L. By developing a framework that counts accepting paths of nondeterministic logspace Turing machines and integrating tools from combinatorial encoding, discrete geometry, and representation theory, the study systematically establishes the #L-computability of numerous classical functions. Key contributions include the first unified inclusion of Catalan numbers, Stirling numbers, and the number of standard Young tableaux within #L; proofs that multinomial coefficients, linear extensions of trees, and GL₂-plethysm coefficients under bounded outer partitions lie in #L or are verifiable in log² space; and a novel conditional approach to refuting their #P-completeness, thereby substantially expanding the theoretical frontier of low-complexity counting problems.

This work investigates the puzzling dense spectral patterns exhibited by small Transformers on modular multiplication tasks. The authors propose replacing the conventional additive discrete Fourier transform (DFT) with a multiplicative feature transformation that aligns with the underlying multiplicative group structure. By integrating discrete logarithm-based reordering, Gini coefficient–based sparsity measurements, and MLP neuron tuning analysis, they demonstrate for the first time that the model implicitly converts multiplication into addition in the discrete logarithm domain. This approach increases the embedding spectrum’s Gini coefficient from 0.07 to 0.58, with 96.9% of MLP neurons precisely tuned to a single multiplicative frequency, thereby confirming the existence of a “discrete logarithm clock” mechanism. These findings establish a novel paradigm for neural network interpretability grounded in algebraic structure alignment.

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 presents the first complete formal verification in Lean 4 of the informal Euclidean domain algorithms originally described in the 1986 ICON language. By separating concerns into mathematical definitions, computable implementations, and output formatting, the project constructs a computable mirror atop Mathlib’s `EuclideanDomain` hierarchy and integrates a regression testing infrastructure to reproduce the original outputs. All 14 algorithms are formally specified, with core procedures such as integer GCD and the extended Euclidean algorithm accompanied by machine-checked correctness proofs. The formalization precisely delineates the boundaries between computability and mathematical correctness while fully replicating the benchmark results reported in Ericson’s technical report.

该研究通过仅使用幂和主对数，解决了标签系统单步计算的表达问题，将离散更新转换为连续运算。