interval arithmetic certification

To address rounding errors in floating-point arithmetic that compromise the correctness of MDP model checking, this paper presents the first end-to-end formal verification of interval iteration under IEEE 754 floating-point semantics. Leveraging Isabelle/HOL and the Refinement Framework, we extend the refinement calculus to support directed rounding modes, enabling rigorous correctness proofs spanning abstract semantics, algorithmic implementation, and LLVM-level code. Our approach bridges theoretical analysis with actual hardware floating-point behavior, ensuring mathematical precision guarantees while achieving competitive performance on standard MDP benchmarks—matching state-of-the-art tools without false positives or negatives induced by rounding errors. The core contributions are: (i) the first complete formal verification of interval iteration in a realistic floating-point setting; and (ii) a novel extension of the refinement framework to reason about directed rounding, thereby enabling sound and scalable quantitative verification of probabilistic systems under IEEE 754 constraints.

This study addresses the computational challenges in solving high-dimensional uncertain nonlinear systems via interval methods, which are often hindered by prohibitive costs. The authors establish an algorithmic worst-case analysis framework to systematically quantify how initial search volume, target tolerance, and fundamental verification operations influence computational complexity. They derive upper bounds on time and space complexity for several interval-based approaches, integrating techniques such as interval bisection, subdivision-filtering, constraint propagation, interval Newton, and Krawczyk methods, alongside interval linear algebra and inclusion functions. Their analysis reveals that naive interval computations of matrix determinants and inverses suffer from factorial-time bottlenecks. Notably, while the Krawczyk and interval Newton methods exhibit comparable dominant-order complexity, the former requires only real-matrix inversion at the midpoint, yielding significantly higher efficiency—providing crucial theoretical support for practical applications such as biochemical reaction networks.

This work addresses the problem of constructing certified approximations of high-dimensional surfaces defined by non-square systems, with rigorous guarantees of existence and uniqueness. To this end, we generalize the classical Krawczyk test—originally formulated for square systems—to handle non-square systems and higher-dimensional algebraic varieties. By integrating interval arithmetic with techniques from analytic system solving, we develop a general-purpose algorithm for certified surface approximation. A prototype implementation demonstrates the efficacy of our approach on several complex surface instances, significantly broadening the class of geometric objects amenable to formal certification.

Machine learning and cyber-physical systems increasingly demand numerically sound, formally verified computational methods. Method: This paper introduces the first end-to-end verifiable numerical methods framework built on Isabelle/HOL and ITrees. It enables users to declaratively specify numerical programs—including variants and invariants—using a high-level, user-friendly specification language; formal verification is automated via ITrees-based operational semantics and the HOL-Analysis library. Contribution/Results: We extend the formalization of Taylor’s theorem with higher-order derivatives and the Peano remainder form, strengthening the underlying mathematical foundation. Leveraging code generation, we fully verify the bisection and fixed-point iteration methods and produce certified, executable code. Our framework significantly improves both the efficiency and practical applicability of formal verification for numerical algorithms.

This work addresses the problem of deriving provably tight floating-point rounding error bounds for numerical programs featuring conditional branches, no loops, and mixed-precision arithmetic. Methodologically, it unifies the modeling of conditional control flow and precision heterogeneity via two novel quantitative metrics—“instability jumps” and “window width”—and integrates interval arithmetic, abstract interpretation, and precision-aware semantic modeling, augmented with abstraction-guided global optimization. Its key contribution is the first formal framework enabling joint, compositional analysis of conditional branching and mixed precision, achieving both high bound tightness and practical analysis efficiency. Experimental evaluation on standard benchmarks demonstrates significantly tighter error bounds compared to prior approaches. Furthermore, the framework successfully guides precision configuration—e.g., step size and search direction—in the conjugate gradient method, empirically validating its utility in supporting design-time trade-offs among accuracy, error bounds, and computational efficiency.

This work addresses the challenge of propagating interval uncertainty in engineering systems, which traditionally relies on computationally expensive optimization to determine output bounds. The authors reformulate this task as an interval-valued regression problem and propose using neural network surrogates to directly predict upper and lower output bounds, thereby avoiding repeated calls to costly simulations or optimization routines. For the first time, the study systematically investigates the application of both multilayer perceptrons (MLPs) and Deep Operator Networks (DeepONets) for interval prediction, comparing three strategies: standard propagation, IBP/CROWN bound propagation, and interval neural networks (INNs). Experimental results demonstrate that the proposed approach achieves comparable estimation accuracy while significantly improving computational efficiency, outperforming conventional optimization-driven methods.

This work explores partial progress toward the formal verification of the Riemann Hypothesis without claiming its proof, leveraging verifiable AI-assisted reasoning to precisely identify outstanding mathematical obstacles. We introduce the first integration of AI-driven automated reasoning with formal verification through a recursively self-improving Verifiable Generative Physical Transformer (VGPT-RSI), producing Coq-verified finite certificates for two related problems: a boundary certificate for parameterized safety lower bounds and a finite formal certificate for Lagarias’ criterion. Our approach combines outward-rounding interval arithmetic, Arb/FLINT ball arithmetic, and Rocq/CoqInterval formal checking. The results explicitly expose three unresolved bottlenecks: the formalization of Lagarias’ equivalence, the infinite generalization of the global tail theorem, and the reduction of potential counterexamples to colossally abundant numbers.

Floating-point round-off errors are ubiquitous in numerically intensive programs arising in fields such as scientific computing and optimization. As floating-point errors potentially lead to unexpected and catastrophic program failures, one must derive guaranteed round-off thresholds to ensure the correctness of these programs. However, deterministic round-off thresholds tend to be too conservative to be usable in practice, since they often involve large round-off errors that occur with small probability. Probabilistic thresholds relax deterministic ones by specifying that the probability of the round-off error exceeding a threshold is below a given confidence. In this work, we propose a novel approach to probabilistic round-off analysis, by applying concentration inequalities over the Taylor expansion from FPTaylor (TOPLAS 2018). A major obstacle in applying concentration inequalities is that the Taylor expansion involves absolute value operators that make the calculation of the expected values of the first order partial differential terms difficult. Our first step to overcome this obstacle is a sound over-approximation that removes the absolute value operators in polynomial expressions. Then, we show how to handle fractional expressions by a transformation into polynomial case. Finally, we show how to improve our approach with range partitioning. Our approach is scalable since the key computational part is the calculation of expected values of polynomial expressions with independent variables, for which the linear and independence properties of expectation boost the computation. Experimental results show that our approach is orders of magnitude more time efficient, while producing thresholds with comparable precision against the state of the art.

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.

Existing quantum neural networks lack effective certified training methods, making it difficult to guarantee prediction correctness under adversarial perturbations. This work introduces interval bound propagation (IBP) into quantum machine learning for the first time, proposing a Quantum Interval Bound Propagation (QIBP) method that tracks upper and lower bounds of layer-wise outputs during training to construct a certified training framework supporting adversarial robustness guarantees. The study systematically evaluates the trade-offs between interval arithmetic and affine arithmetic within this framework. Experimental results demonstrate that models trained with QIBP provide rigorous guarantees of classification correctness within predefined perturbation bounds, significantly enhancing the robustness of decision boundaries.