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753篇论文匹配“Neuroscience, Cognitive Science”
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Neuroscience, Cognitive Science

Yule Wang, Chengrui Li, Weihan Li, Anqi Wu

Understanding the neural basis of behavior is a fundamental goal in neuroscience. Current research in large-scale neuro-behavioral data analysis often relies on decoding models, which quantify behavioral information in neural data but lack details on behavior encoding. This raises an intriguing scientific question: "how can we enable in-depth exploration of neural representations in behavioral tasks, revealing interpretable neural dynamics associated with behaviors". However, addressing this issue is challenging due to the varied behavioral encoding across different brain regions and mixed selectivity at the population level. To tackle this limitation, our approach, named ("BeNeDiff"), first identifies a fine-grained and disentangled neural subspace using a behavior-informed latent variable model. It then employs state-of-the-art generative diffusion models to synthesize behavior videos that interpret the neural dynamics of each latent factor. We validate the method on multi-session datasets containing widefield calcium imaging recordings across the dorsal cortex. Through guiding the diffusion model to activate individual latent factors, we verify that the neural dynamics of latent factors in the disentangled neural subspace provide interpretable quantifications of the behaviors of interest. At the same time, the neural subspace in BeNeDiff demonstrates high disentanglement and neural reconstruction quality.

Neuroscience, Cognitive Science

Eli Sennesh, Hao Wu, Tommaso Salvatori

Unexpected stimuli induce "error" or "surprise" signals in the brain. The theory of predictive coding promises to explain these observations in terms of Bayesian inference by suggesting that the cortex implements variational inference in a probabilistic graphical model. However, when applied to machine learning tasks, this family of algorithms has yet to perform on par with other variational approaches in high-dimensional, structured inference problems. To address this, we introduce a novel predictive coding algorithm for structured generative models, that we call divide-and-conquer predictive coding (DCPC); it differs from other formulations of predictive coding, as it respects the correlation structure of the generative model and provably performs maximum-likelihood updates of model parameters, all without sacrificing biological plausibility. Empirically, DCPC achieves better numerical performance than competing algorithms and provides accurate inference in a number of problems not previously addressed with predictive coding. We provide an open implementation of DCPC in Pyro on Github.

Neuroscience, Cognitive Science

Sangwoo Hwang, Seunghyun Lee, Dahoon Park, Donghun Lee, Jaeha Kung

Event-driven spiking neural networks(SNNs) are promising neural networks that reduce the energy consumption of continuously growing AI models. Recently, keeping pace with the development of transformers, transformer-based SNNs were presented. Due to the incompatibility of self-attention with spikes, however, existing transformer-based SNNs limit themselves by either restructuring self-attention architecture or conforming to non-spike computations. In this work, we propose a novel transformer-to-SNN conversion method that outputs an end-to-end spike-based transformer, named SpikedAttention. Our method directly converts the well-trained transformer without modifying its attention architecture. For the vision task, the proposed method converts Swin Transformer into an SNN without post-training or conversion-aware training, achieving state-of-the-art SNN accuracy on ImageNet dataset, i.e., 80.0\% with 28.7M parameters. Considering weight accumulation, neuron potential update, and on-chip data movement, SpikedAttention reduces energy consumption by 42\% compared to the baseline ANN, i.e., Swin-T. Furthermore, for the first time, we demonstrate that SpikedAttention successfully converts a BERT model to an SNN with only 0.3\% accuracy loss on average consuming 58\% less energy on GLUE benchmark. Our code is available at Github ( https://github.com/sangwoohwang/SpikedAttention ).

Neuroscience, Cognitive Science

Paul Soulos, Henry Conklin, Mattia Opper, Paul Smolensky, Jianfeng Gao, Roland Fernandez

Neural networks continue to struggle with compositional generalization, and this issue is exacerbated by a lack of massive pre-training. One successful approach for developing neural systems which exhibit human-like compositional generalization is $\textit{hybrid}$ neurosymbolic techniques. However, these techniques run into the core issues that plague symbolic approaches to AI: scalability and flexibility. The reason for this failure is that at their core, hybrid neurosymbolic models perform symbolic computation and relegate the scalable and flexible neural computation to parameterizing a symbolic system. We investigate a $\textit{unified}$ neurosymbolic system where transformations in the network can be interpreted simultaneously as both symbolic and neural computation. We extend a unified neurosymbolic architecture called the Differentiable Tree Machine in two central ways. First, we significantly increase the model’s efficiency through the use of sparse vector representations of symbolic structures. Second, we enable its application beyond the restricted set of tree2tree problems to the more general class of seq2seq problems. The improved model retains its prior generalization capabilities and, since there is a fully neural path through the network, avoids the pitfalls of other neurosymbolic techniques that elevate symbolic computation over neural computation.

Neuroscience, Cognitive Science

Spandan Madan, Will Xiao, Mingran Cao, Hanspeter Pfister, Margaret Livingstone, Gabriel Kreiman

We characterized the generalization capabilities of deep neural network encoding models when predicting neuronal responses from the visual cortex to flashed images. We collected MacaqueITBench, a large-scale dataset of neuronal population responses from the macaque inferior temporal (IT) cortex to over $300,000$ images, comprising $8,233$ unique natural images presented to seven monkeys over $109$ sessions. Using MacaqueITBench, we investigated the impact of distribution shifts on models predicting neuronal activity by dividing the images into Out-Of-Distribution (OOD) train and test splits. The OOD splits included variations in image contrast, hue, intensity, temperature, and saturation. Compared to the performance on in-distribution test images---the conventional way in which these models have been evaluated---models performed worse at predicting neuronal responses to out-of-distribution images, retaining as little as $20\\%$ of the performance on in-distribution test images. Additionally, the relative ranking of different models in terms of their ability to predict neuronal responses changed drastically across OOD shifts. The generalization performance under OOD shifts can be well accounted by a simple image similarity metric---the cosine distance between image representations extracted from a pre-trained object recognition model is a strong predictor of neuronal predictivity under different distribution shifts. The dataset of images, neuronal firing rate recordings, and computational benchmarks are hosted publicly at: https://github.com/Spandan-Madan/benchmarking_ood_generalization_visual_cortex.

Neuroscience, Cognitive Science

Christopher Wang, Adam Yaari, Aaditya Singh, Vighnesh Subramaniam, Dana Rosenfarb, Jan DeWitt, Pranav Misra, Joseph Madsen, Scellig Stone, Gabriel Kreiman 等

We present the Brain Treebank, a large-scale dataset of electrophysiological neural responses, recorded from intracranial probes while 10 subjects watched one or more Hollywood movies. Subjects watched on average 2.6 Hollywood movies, for an average viewing time of 4.3 hours, and a total of 43 hours. The audio track for each movie was transcribed with manual corrections. Word onsets were manually annotated on spectrograms of the audio track for each movie. Each transcript was automatically parsed and manually corrected into the universal dependencies (UD) formalism, assigning a part of speech to every word and a dependency parse to every sentence. In total, subjects heard over 38,000 sentences (223,000 words), while they had on average 168 electrodes implanted. This is the largest dataset of intracranial recordings featuring grounded naturalistic language, one of the largest English UD treebanks in general, and one of only a few UD treebanks aligned to multimodal features. We hope that this dataset serves as a bridge between linguistic concepts, perception, and their neural representations. To that end, we present an analysis of which electrodes are sensitive to language features while also mapping out a rough time course of language processing across these electrodes. The Brain Treebank is available at https://BrainTreebank.dev/

Neuroscience, Cognitive Science

Francesco Damiani, Akiyuki Anzai, Jan Drugowitsch, Gregory DeAngelis, Ruben Moreno Bote

A pivotal brain computation relies on the ability to sustain perception-action loops. Stochastic optimal control theory offers a mathematical framework to explain these processes at the algorithmic level through optimality principles. However, incorporating a realistic noise model of the sensorimotor system — accounting for multiplicative noise in feedback and motor output, as well as internal noise in estimation — makes the problem challenging. Currently, the algorithm that is commonly used is the one proposed in the seminal study in (Todorov, 2005). After discovering some pitfalls in the original derivation, i.e., unbiased estimation does not hold, we improve the algorithm by proposing an efficient gradient descent-based optimization that minimizes the cost-to-go while only imposing linearity of the control law. The optimal solution is obtained by iteratively propagating in closed form the sufficient statistics to compute the expected cost and then minimizing this cost with respect to the filter and control gains. We demonstrate that this approach results in a significantly lower overall cost than current state-of-the-art solutions, particularly in the presence of internal noise, though the improvement is present in other circumstances as well, with theoretical explanations for this enhanced performance. Providing the optimal control law is key for inverse control inference, especially in explaining behavioral data under rationality assumptions.

Neuroscience, Cognitive Science

Hui Zheng, Haiteng Wang, Weibang Jiang, Zhongtao Chen, Li He, Peiyang Lin, Penghu Wei, Guoguang Zhao, Yunzhe Liu

Invasive brain-computer interfaces with Electrocorticography (ECoG) have shown promise for high-performance speech decoding in medical applications, but less damaging methods like intracranial stereo-electroencephalography (sEEG) remain underexplored. With rapid advances in representation learning, leveraging abundant recordings to enhance speech decoding is increasingly attractive. However, popular methods often pre-train temporal models based on brain-level tokens, overlooking that brain activities in different regions are highly desynchronized during tasks. Alternatively, they pre-train spatial-temporal models based on channel-level tokens but fail to evaluate them on challenging tasks like speech decoding, which requires intricate processing in specific language-related areas. To address this issue, we collected a well-annotated Chinese word-reading sEEG dataset targeting language-related brain networks from 12 subjects. Using this benchmark, we developed the Du-IN model, which extracts contextual embeddings based on region-level tokens through discrete codex-guided mask modeling. Our model achieves state-of-the-art performance on the 61-word classification task, surpassing all baselines. Model comparisons and ablation studies reveal that our design choices, including (\romannumeral1) temporal modeling based on region-level tokens by utilizing 1D depthwise convolution to fuse channels in the ventral sensorimotor cortex (vSMC) and superior temporal gyrus (STG) and (\romannumeral2) self-supervision through discrete codex-guided mask modeling, significantly contribute to this performance. Overall, our approach -- inspired by neuroscience findings and capitalizing on region-level representations from specific brain regions -- is suitable for invasive brain modeling and represents a promising neuro-inspired AI approach in brain-computer interfaces. Code and dataset are available at https://github.com/liulab-repository/Du-IN.

Neuroscience, Cognitive Science

chenlin zhou, Han Zhang, Zhaokun Zhou, Liutao Yu, Liwei Huang, Xiaopeng Fan, Li Yuan, Zhengyu Ma, Huihui Zhou, Yonghong Tian

Spiking Transformers, which integrate Spiking Neural Networks (SNNs) with Transformer architectures, have attracted significant attention due to their potential for low energy consumption and high performance. However, there remains a substantial gap in performance between SNNs and Artificial Neural Networks (ANNs). To narrow this gap, we have developed QKFormer, a direct training spiking transformer with the following features: i) _Linear complexity and high energy efficiency_, the novel spike-form Q-K attention module efficiently models the token or channel attention through binary vectors and enables the construction of larger models. ii) _Multi-scale spiking representation_, achieved by a hierarchical structure with the different numbers of tokens across blocks. iii) _Spiking Patch Embedding with Deformed Shortcut (SPEDS)_, enhances spiking information transmission and integration, thus improving overall performance. It is shown that QKFormer achieves significantly superior performance over existing state-of-the-art SNN models on various mainstream datasets. Notably, with comparable size to Spikformer (66.34 M, 74.81\%), QKFormer (64.96 M) achieves a groundbreaking top-1 accuracy of **85.65\%** on ImageNet-1k, substantially outperforming Spikformer by **10.84\%**. To our best knowledge, this is the first time that directly training SNNs have exceeded 85\% accuracy on ImageNet-1K.

Neuroscience, Cognitive Science

William Redman, Francisco Acosta, Santiago Acosta-Mendoza, Nina Miolane

Success in collaborative and competitive environments, where agents must work with or against each other, requires individuals to encode the position and trajectory of themselves and others. Decades of neurophysiological experiments have shed light on how brain regions [e.g., medial entorhinal cortex (MEC), hippocampus] encode the self's position and trajectory. However, it has only recently been discovered that MEC and hippocampus are modulated by the positions and trajectories of others. To understand how encoding spatial information of multiple agents shapes neural representations, we train a recurrent neural network (RNN) model that captures properties of MEC to path integrate trajectories of two agents simultaneously navigating the same environment. We find significant differences between these RNNs and those trained to path integrate only a single agent. At the individual unit level, RNNs trained to path integrate more than one agent develop weaker grid responses, stronger border responses, and tuning for the relative position of the two agents. At the population level, they develop more distributed and robust representations, with changes in network dynamics and manifold topology. Our results provide testable predictions and open new directions with which to study the neural computations supporting spatial navigation.

Neuroscience, Cognitive Science

Tassilo Wald, Constantin Ulrich, Priyank Jaini, Gregor Koehler, David Zimmerer, Stefan Denner, Fabian Isensee, Michael Baumgartner, Klaus Maier-Hein

What representation do deep neural networks learn? How similar are images to each other for neural networks? Despite the overwhelming success of deep learning methods key questions about their internal workings still remain largely unanswered, due to their internal high dimensionality and complexity. To address this, one approach is to measure the similarity of activation responses to various inputs.Representational Similarity Matrices (RSMs) distill this similarity into scalar values for each input pair.These matrices encapsulate the entire similarity structure of a system, indicating which input lead to similar responses.While the similarity between images is ambiguous, we argue that the spatial location of semantic objects does neither influence human perception nor deep learning classifiers. Thus this should be reflected in the definition of similarity between image responses for computer vision systems. Revisiting the established similarity calculations for RSMs we expose their sensitivity to spatial alignment. In this paper we propose to solve this through _semantic RSMs_, which are invariant to spatial permutation. We measure semantic similarity between input responses by formulating it as a set-matching problem. Further, we quantify the superiority of _semantic_ RSMs over _spatio-semantic_ RSMs through image retrieval and by comparing the similarity between representations to the similarity between predicted class probabilities.

Neuroscience, Cognitive Science

Ziquan Wei, Tingting Dan, Jiaqi Ding, Guorong Wu

Although modern imaging technologies allow us to study connectivity between two distinct brain regions $\textit{in-vivo}$, an in-depth understanding of how anatomical structure supports brain function and how spontaneous functional fluctuations emerge remarkable cognition is still elusive. Meanwhile, tremendous efforts have been made in the realm of machine learning to establish the nonlinear mapping between neuroimaging data and phenotypic traits. However, the absence of neuroscience insight in the current approaches poses significant challenges in understanding cognitive behavior from transient neural activities. To address this challenge, we put the spotlight on the coupling mechanism of structural connectivity (SC) and functional connectivity (FC) by formulating such network neuroscience question into an expressive graph representation learning problem for high-order topology. Specifically, we introduce the concept of $\textit{topological detour}$ to characterize how a ubiquitous instance of FC (direct link) is supported by neural pathways (detour) physically wired by SC, which forms a cyclic loop interacted by brain structure and function. In the clich\'e of machine learning, the multi-hop detour pathway underlying SC-FC coupling allows us to devise a novel multi-head self-attention mechanism within Transformer to capture multi-modal feature representation from paired graphs of SC and FC. Taken together, we propose a biological-inspired deep model, coined as $\textit{NeuroPath}$, to find putative connectomic feature representations from the unprecedented amount of neuroimages, which can be plugged into various downstream applications such as task recognition and disease diagnosis. We have evaluated $\textit{NeuroPath}$ on large-scale public datasets including Human Connectome Project (HCP) and UK Biobank (UKB) under different experiment settings of supervised and zero-shot learning, where the state-of-the-art performance by our $\textit{NeuroPath}$ indicates great potential in network neuroscience.

Neuroscience, Cognitive Science

Shihan Ma, Bo Hu, Tianyu Jia, Alexander Clarke, Blanka Zicher, Arnault Caillet, Dario Farina, Jose C Principe

The cortico-spinal neural pathway is fundamental for motor control and movement execution, and in humans it is typically studied using concurrent electroencephalography (EEG) and electromyography (EMG) recordings. However, current approaches for capturing high-level and contextual connectivity between these recordings have important limitations. Here, we present a novel application of statistical dependence estimators based on orthonormal decomposition of density ratios to model the relationship between cortical and muscle oscillations. Our method extends from traditional scalar-valued measures by learning eigenvalues, eigenfunctions, and projection spaces of density ratios from realizations of the signal, addressing the interpretability, scalability, and local temporal dependence of cortico-muscular connectivity. We experimentally demonstrate that eigenfunctions learned from cortico-muscular connectivity can accurately classify movements and subjects. Moreover, they reveal channel and temporal dependencies that confirm the activation of specific EEG channels during movement.

Neuroscience, Cognitive Science

David Lipshutz, Eero Simoncelli

Efficient coding theory posits that sensory circuits transform natural signals into neural representations that maximize information transmission subject to resource constraints. Local interneurons are thought to play an important role in these transformations, shaping patterns of circuit activity to facilitate and direct information flow. However, the relationship between these coordinated, nonlinear, circuit-level transformations and the properties of interneurons (e.g., connectivity, activation functions) remains unknown. Here, we propose a normative computational model that establishes such a relationship. Our model is derived from an optimal transport objective that conceptualizes the circuit's input-response function as transforming the inputs to achieve a target response distribution. The circuit, which is comprised of primary neurons that are recurrently connected to a set of local interneurons, continuously optimizes this objective by dynamically adjusting both the synaptic connections between neurons as well as the interneuron activation functions. In an application motivated by redundancy reduction theory, we demonstrate that when the inputs are natural image statistics and the target distribution is a spherical Gaussian, the circuit learns a nonlinear transformation that significantly reduces statistical dependencies in neural responses. Overall, our results provide a framework in which the distribution of circuit responses is systematically and nonlinearly controlled by adjustment of interneuron connectivity and activation functions.

Neuroscience, Cognitive Science

Guobin Shen, Dongcheng Zhao, Xiang He, Linghao Feng, Yiting Dong, Jihang Wang, Qian Zhang, Yi Zeng

Decoding non-invasive brain recordings is pivotal for advancing our understanding of human cognition but faces challenges due to individual differences and complex neural signal representations. Traditional methods often require customized models and extensive trials, lacking interpretability in visual reconstruction tasks. Our framework integrates 3D brain structures with visual semantics using a *Vision Transformer 3D*. This unified feature extractor efficiently aligns fMRI features with multiple levels of visual embeddings, eliminating the need for subject-specific models and allowing extraction from single-trial data. The extractor consolidates multi-level visual features into one network, simplifying integration with Large Language Models (LLMs). Additionally, we have enhanced the fMRI dataset with diverse fMRI-image-related textual data to support multimodal large model development. Integrating with LLMs enhances decoding capabilities, enabling tasks such as brain captioning, complex reasoning, concept localization, and visual reconstruction. Our approach demonstrates superior performance across these tasks, precisely identifying language-based concepts within brain signals, enhancing interpretability, and providing deeper insights into neural processes. These advances significantly broaden the applicability of non-invasive brain decoding in neuroscience and human-computer interaction, setting the stage for advanced brain-computer interfaces and cognitive models.

Neuroscience, Cognitive Science

Mengting Xu, De Ma, Huajin Tang, Qian Zheng, Gang Pan

Currently, researchers think that the inherent robustness of spiking neural networks (SNNs) stems from their biologically plausible spiking neurons, and are dedicated to developing more bio-inspired models to defend attacks. However, most work relies solely on experimental analysis and lacks theoretical support, and the direct-encoding method and fixed membrane potential leak factor they used in spiking neurons are simplified simulations of those in the biological nervous system, which makes it difficult to ensure generalizability across all datasets and networks. Contrarily, the biological nervous system can stay reliable even in a highly complex noise environment, one of the reasons is selective visual attention and non-fixed membrane potential leaks in biological neurons. This biological finding has inspired us to design a highly robust SNN model that closely mimics the biological nervous system. In our study, we first present a unified theoretical framework for SNN robustness constraint, which suggests that improving the encoding method and evolution of the membrane potential leak factor in spiking neurons can improve SNN robustness. Subsequently, we propose a robust SNN (FEEL-SNN) with Frequency Encoding (FE) and Evolutionary Leak factor (EL) to defend against different noises, mimicking the selective visual attention mechanism and non-fixed leak observed in biological systems. Experimental results confirm the efficacy of both our FE, EL, and FEEL methods, either in isolation or in conjunction with established robust enhancement algorithms, for enhancing the robustness of SNNs.

Neuroscience, Cognitive Science

Laurent Bonnasse-Gahot, Christophe Pallier

Over the past decade, studies of naturalistic language processing where participants are scanned while listening to continuous text have flourished. Using word embeddings at first, then large language models, researchers have created encoding models to analyze the brain signals. Presenting these models with the same text as the participants allows to identify brain areas where there is a significant correlation between the functional magnetic resonance imaging (fMRI) time series and the ones predicted by the models' artificial neurons. One intriguing finding from these studies is that they have revealed highly symmetric bilateral activation patterns, somewhat at odds with the well-known left lateralization of language processing. Here, we report analyses of an fMRI dataset where we manipulate the complexity of large language models, testing 28 pretrained models from 8 different families, ranging from 124M to 14.2B parameters. First, we observe that the performance of models in predicting brain responses follows a scaling law, where the fit with brain activity increases linearly with the logarithm of the number of parameters of the model (and its performance on natural language processing tasks). Second, although this effect is present in both hemispheres, it is stronger in the left than in the right hemisphere. Specifically, the left-right difference in brain correlation follows a scaling law with the number of parameters. This finding reconciles computational analyses of brain activity using large language models with the classic observation from aphasic patients showing left hemisphere dominance for language.

Neuroscience, Cognitive Science

Ábel Ságodi, Guillermo Martín-Sánchez, Piotr Sokol, Memming Park

Continuous attractors offer a unique class of solutions for storing continuous-valued variables in recurrent system states for indefinitely long time intervals.Unfortunately, continuous attractors suffer from severe structural instability in general---they are destroyed by most infinitesimal changes of the dynamical law that defines them.This fragility limits their utility especially in biological systems as their recurrent dynamics are subject to constant perturbations.We observe that the bifurcations from continuous attractors in theoretical neuroscience models display various structurally stable forms.Although their asymptotic behaviors to maintain memory are categorically distinct, their finite-time behaviors are similar.We build on the persistent manifold theory to explain the commonalities between bifurcations from and approximations of continuous attractors.Fast-slow decomposition analysis uncovers the existence of a persistent slow manifold that survives the seemingly destructive bifurcation, relating the flow within the manifold to the size of the perturbation. Moreover, this allows the bounding of the memory error of these approximations of continuous attractors.Finally, we train recurrent neural networks on analog memory tasks to support the appearance of these systems as solutions and their generalization capabilities.Therefore, we conclude that continuous attractors are functionally robust and remain useful as a universal analogy for understanding analog memory.

Applications · Neuroscience, Cognitive Science

Hyunseok Oh, Youngki Lee

Spiking neural network (SNN) is studied in multidisciplinary domains to (i) enable order-of-magnitudes energy-efficient AI inference, and (ii) computationally simulate neuroscientific mechanisms. The lack of discrete theory obstructs the practical application of SNN by limiting its performance and nonlinearity support. We present a new optimization-theoretic perspective of the discrete dynamics of spiking neuron. We prove that a discrete dynamical system of simple integrate-and-fire models approximates the subgradient method over unconstrained optimization problems. We practically extend our theory to introduce a novel sign gradient descent (signGD)-based neuronal dynamics that can (i) approximate diverse nonlinearities beyond ReLU, and (ii) advance ANN-to-SNN conversion performance in low time-steps. Experiments on large-scale datasets show that our technique achieve (i) state-of-the-art performance in ANN-to-SNN conversion, and (ii) is first to convert new DNN architectures, e.g., ConvNext, MLP-Mixer, and ResMLP. We publicly share our source code at www.github.com/snuhcs/snn_signgd .

Applications · Neuroscience, Cognitive Science

Yewen Pu, Saujas Vaduguru, Priyan Vaithilingam, Elena Glassman, Daniel Fried

The usage of Rational Speech Acts (RSA) framework has been successful in building *pragmatic* program synthesizers that return programs which, in addition to being logically consistent with user-generated examples, account for the fact that a user chooses their examples informatively. We present a general method of amortizing the slow, exact RSA synthesizer. Our method first compiles a communication dataset of partially ranked programs by querying the exact RSA synthesizer. It then distills a *global ranking* -- a single, total ordering of all programs, to approximate the partial rankings from this dataset. This global ranking is then used at inference time to rank multiple logically consistent candidate programs generated from a fast, non-pragmatic synthesizer. Experiments on two program synthesis domains using our ranking method resulted in orders of magnitudes of speed ups compared to the exact RSA synthesizer, while being more accurate than a non-pragmatic synthesizer. Finally, we prove that in the special case of synthesis from a single example, this approximation is exact.