Toghrul Karimov

Valérie Berthé, Toghrul Karimov, Joris Nieuwveld et al.

We investigate the decidability of the monadic second-order (MSO) theory of the structure $\langle \mathbb{N};<,P_1, \ldots,P_d \rangle$, for various unary predicates $P_1,\ldots,P_d \subseteq \mathbb{N}$. We focus in particular on 'arithmetic' predicates arising in the study of linear recurrence sequences, such as fixed-base powers $k^{\mathbf{N}} = \{k^n : n \in \mathbb{N}\}$, $k$-th powers $\mathbf{N}^k = \{n^k : n \in \mathbb{N}\}$, and the set of terms of the Fibonacci sequence $\mathsf{Fib} = \{0,1,2,3,5,8,13,\ldots\}$ (and similarly for other linear recurrence sequences having a single, non-repeated, dominant characteristic root). We obtain several new unconditional and conditional decidability results, a select sample of which are the following: $\bullet$ The MSO theory of $\langle \mathbb{N};<, 2^{\mathbf{N}}, \mathsf{Fib} \rangle$ is decidable; $\bullet$ The MSO theory of $\langle \mathbb{N};<, 2^{\mathbf{N}}, 3^{\mathbf{N}}, 6^{\mathbf{N}} \rangle$ is decidable; $\bullet$ The MSO theory of $\langle \mathbb{N};<, 2^{\mathbf{N}}, 3^{\mathbf{N}}, 5^{\mathbf{N}} \rangle$ is decidable assuming Schanuel's conjecture; $\bullet$ The MSO theory of $\langle \mathbb{N};<, 4^{\mathbf{N}}, \mathbf{N}^2 \rangle$ is decidable; $\bullet$ The MSO theory of $\langle \mathbb{N};<, 2^{\mathbf{N}}, \mathbf{N}^2 \rangle$ is Turing-equivalent to the MSO theory of $\langle \mathbb{N};<,S \rangle$, where $S$ is the predicate corresponding to the binary expansion of $\sqrt{2}$. (As the binary expansion of $\sqrt{2}$ is widely believed to be normal, the corresponding MSO theory is in turn expected to be decidable.) These results are obtained by exploiting and combining techniques from dynamical systems, number theory, and automata theory.

Toghrul Karimov, Hassan Imani, Allan Kazakov

Post-training quantization (PTQ) is crucial for deploying efficient object detection models, like YOLO, on resource-constrained devices. However, the impact of reduced precision on model robustness to real-world input degradations such as noise, blur, and compression artifacts is a significant concern. This paper presents a comprehensive empirical study evaluating the robustness of YOLO models (nano to extra-large scales) across multiple precision formats: FP32, FP16 (TensorRT), Dynamic UINT8 (ONNX), and Static INT8 (TensorRT). We introduce and evaluate a degradation-aware calibration strategy for Static INT8 PTQ, where the TensorRT calibration process is exposed to a mix of clean and synthetically degraded images. Models were benchmarked on the COCO dataset under seven distinct degradation conditions (including various types and levels of noise, blur, low contrast, and JPEG compression) and a mixed-degradation scenario. Results indicate that while Static INT8 TensorRT engines offer substantial speedups (~1.5-3.3x) with a moderate accuracy drop (~3-7% mAP50-95) on clean data, the proposed degradation-aware calibration did not yield consistent, broad improvements in robustness over standard clean-data calibration across most models and degradations. A notable exception was observed for larger model scales under specific noise conditions, suggesting model capacity may influence the efficacy of this calibration approach. These findings highlight the challenges in enhancing PTQ robustness and provide insights for deploying quantized detectors in uncontrolled environments. All code and evaluation tables are available at https://github.com/AllanK24/QRID.