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Reduction systems and degree bounds for integration 积分的还原系统和程度界限
arXiv - CS - Symbolic Computation Pub Date : 2024-04-19 DOI: arxiv-2404.13042
Hao Du, Clemens G. Raab
{"title":"Reduction systems and degree bounds for integration","authors":"Hao Du, Clemens G. Raab","doi":"arxiv-2404.13042","DOIUrl":"https://doi.org/arxiv-2404.13042","url":null,"abstract":"In symbolic integration, the Risch--Norman algorithm aims to find closed\u0000forms of elementary integrals over differential fields by an ansatz for the\u0000integral, which usually is based on heuristic degree bounds. Norman presented\u0000an approach that avoids degree bounds and only relies on the completion of\u0000reduction systems. We give a formalization of his approach and we develop a\u0000refined completion process, which terminates in more instances. In some\u0000situations when the algorithm does not terminate, one can detect patterns\u0000allowing to still describe infinite reduction systems that are complete. We\u0000present such infinite systems for the fields generated by Airy functions and\u0000complete elliptic integrals, respectively. Moreover, we show how complete\u0000reduction systems can be used to find rigorous degree bounds. In particular, we\u0000give a general formula for weighted degree bounds and we apply it to find tight\u0000bounds for above examples.","PeriodicalId":501033,"journal":{"name":"arXiv - CS - Symbolic Computation","volume":"69 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140635124","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Constant-Depth Arithmetic Circuits for Linear Algebra Problems 线性代数问题的定深算术电路
arXiv - CS - Symbolic Computation Pub Date : 2024-04-16 DOI: arxiv-2404.10839
Robert Andrews, Avi Wigderson
{"title":"Constant-Depth Arithmetic Circuits for Linear Algebra Problems","authors":"Robert Andrews, Avi Wigderson","doi":"arxiv-2404.10839","DOIUrl":"https://doi.org/arxiv-2404.10839","url":null,"abstract":"We design polynomial size, constant depth (namely, $mathsf{AC}^0$)\u0000arithmetic formulae for the greatest common divisor (GCD) of two polynomials,\u0000as well as the related problems of the discriminant, resultant, B'ezout\u0000coefficients, squarefree decomposition, and the inversion of structured\u0000matrices like Sylvester and B'ezout matrices. Our GCD algorithm extends to any\u0000number of polynomials. Previously, the best known arithmetic formulae for these\u0000problems required super-polynomial size, regardless of depth. These results are based on new algorithmic techniques to compute various\u0000symmetric functions in the roots of polynomials, as well as manipulate the\u0000multiplicities of these roots, without having access to them. These techniques\u0000allow $mathsf{AC}^0$ computation of a large class of linear and polynomial\u0000algebra problems, which include the above as special cases. We extend these techniques to problems whose inputs are multivariate\u0000polynomials, which are represented by $mathsf{AC}^0$ arithmetic circuits. Here\u0000too we solve problems such as computing the GCD and squarefree decomposition in\u0000$mathsf{AC}^0$.","PeriodicalId":501033,"journal":{"name":"arXiv - CS - Symbolic Computation","volume":"49 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140614861","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Towards a Research Community in Interpretable Reinforcement Learning: the InterpPol Workshop 建立可解释强化学习研究社区：InterpPol 研讨会
arXiv - CS - Symbolic Computation Pub Date : 2024-04-16 DOI: arxiv-2404.10906
Hector Kohler, Quentin Delfosse, Paul Festor, Philippe Preux
{"title":"Towards a Research Community in Interpretable Reinforcement Learning: the InterpPol Workshop","authors":"Hector Kohler, Quentin Delfosse, Paul Festor, Philippe Preux","doi":"arxiv-2404.10906","DOIUrl":"https://doi.org/arxiv-2404.10906","url":null,"abstract":"Embracing the pursuit of intrinsically explainable reinforcement learning\u0000raises crucial questions: what distinguishes explainability from\u0000interpretability? Should explainable and interpretable agents be developed\u0000outside of domains where transparency is imperative? What advantages do\u0000interpretable policies offer over neural networks? How can we rigorously define\u0000and measure interpretability in policies, without user studies? What\u0000reinforcement learning paradigms,are the most suited to develop interpretable\u0000agents? Can Markov Decision Processes integrate interpretable state\u0000representations? In addition to motivate an Interpretable RL community centered\u0000around the aforementioned questions, we propose the first venue dedicated to\u0000Interpretable RL: the InterpPol Workshop.","PeriodicalId":501033,"journal":{"name":"arXiv - CS - Symbolic Computation","volume":"183 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140614865","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Connectivity in Symmetric Semi-Algebraic Sets 对称半代数集合中的连通性
arXiv - CS - Symbolic Computation Pub Date : 2024-04-15 DOI: arxiv-2404.09749
Cordian Riener, Robin Schabert, Thi Xuan Vu
{"title":"Connectivity in Symmetric Semi-Algebraic Sets","authors":"Cordian Riener, Robin Schabert, Thi Xuan Vu","doi":"arxiv-2404.09749","DOIUrl":"https://doi.org/arxiv-2404.09749","url":null,"abstract":"Semi-algebraic set is a subset of the real space defined by polynomial\u0000equations and inequalities. In this paper, we consider the problem of deciding\u0000whether two given points in a semi-algebraic set are connected. We restrict to\u0000the case when all equations and inequalities are invariant under the action of\u0000the symmetric group and their degrees at most $d<n$, where $n$ is the number of\u0000variables. Additionally, we assume that the two points are in the same\u0000fundamental domain of the action of the symmetric group, by assuming that the\u0000coordinates of two given points are sorted in non-decreasing order. We\u0000construct and analyze an algorithm that solves this problem, by taking\u0000advantage of the group action, and has a complexity being polynomial in $n$.","PeriodicalId":501033,"journal":{"name":"arXiv - CS - Symbolic Computation","volume":"38 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140582682","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Computing with Hypergeometric-Type Terms 使用超几何类型术语进行计算
arXiv - CS - Symbolic Computation Pub Date : 2024-04-15 DOI: arxiv-2404.10143
Bertrand Teguia Tabuguia
{"title":"Computing with Hypergeometric-Type Terms","authors":"Bertrand Teguia Tabuguia","doi":"arxiv-2404.10143","DOIUrl":"https://doi.org/arxiv-2404.10143","url":null,"abstract":"Take a multiplicative monoid of sequences in which the multiplication is\u0000given by Hadamard product. The set of linear combinations of interleaving\u0000monoid elements then yields a ring. We consider such a construction for the\u0000monoid of hypergeometric sequences, yielding what we call the ring of\u0000hypergeometric-type sequences -- a subring of the ring of holonomic sequences.\u0000We present two algorithms in this setting: one for computing holonomic\u0000recurrence equations from hypergeometric-type normal forms and the other for\u0000finding products of hypergeometric-type terms. These are newly implemented\u0000commands in our Maple package $texttt{HyperTypeSeq}$, which we also describe.","PeriodicalId":501033,"journal":{"name":"arXiv - CS - Symbolic Computation","volume":"37 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140614829","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Study of Emotion Concept Formation by Integrating Vision, Physiology, and Word Information using Multilayered Multimodal Latent Dirichlet Allocation 利用多层多模态潜在德里希勒分配整合视觉、生理和文字信息的情绪概念形成研究
arXiv - CS - Symbolic Computation Pub Date : 2024-04-12 DOI: arxiv-2404.08295
Kazuki Tsurumaki, Chie Hieida, Kazuki Miyazawa
{"title":"Study of Emotion Concept Formation by Integrating Vision, Physiology, and Word Information using Multilayered Multimodal Latent Dirichlet Allocation","authors":"Kazuki Tsurumaki, Chie Hieida, Kazuki Miyazawa","doi":"arxiv-2404.08295","DOIUrl":"https://doi.org/arxiv-2404.08295","url":null,"abstract":"How are emotions formed? Through extensive debate and the promulgation of\u0000diverse theories , the theory of constructed emotion has become prevalent in\u0000recent research on emotions. According to this theory, an emotion concept\u0000refers to a category formed by interoceptive and exteroceptive information\u0000associated with a specific emotion. An emotion concept stores past experiences\u0000as knowledge and can predict unobserved information from acquired information.\u0000Therefore, in this study, we attempted to model the formation of emotion\u0000concepts using a constructionist approach from the perspective of the\u0000constructed emotion theory. Particularly, we constructed a model using\u0000multilayered multimodal latent Dirichlet allocation , which is a probabilistic\u0000generative model. We then trained the model for each subject using vision,\u0000physiology, and word information obtained from multiple people who experienced\u0000different visual emotion-evoking stimuli. To evaluate the model, we verified\u0000whether the formed categories matched human subjectivity and determined whether\u0000unobserved information could be predicted via categories. The verification\u0000results exceeded chance level, suggesting that emotion concept formation can be\u0000explained by the proposed model.","PeriodicalId":501033,"journal":{"name":"arXiv - CS - Symbolic Computation","volume":"46 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140582790","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Performant Dynamically Typed E-Graphs in Pure Julia 纯 Julia 中的高性能动态类型电子图
arXiv - CS - Symbolic Computation Pub Date : 2024-04-12 DOI: arxiv-2404.08751
Alessandro Cheli, Niklas Heim
{"title":"Performant Dynamically Typed E-Graphs in Pure Julia","authors":"Alessandro Cheli, Niklas Heim","doi":"arxiv-2404.08751","DOIUrl":"https://doi.org/arxiv-2404.08751","url":null,"abstract":"We introduce the third major version of Metatheory.jl, a Julia library for\u0000general-purpose metaprogramming and symbolic computation. Metatheory.jl\u0000provides a flexible and performant implementation of e-graphs and Equality\u0000Saturation (EqSat) that addresses the two-language problem in high-level\u0000compiler optimizations, symbolics and metaprogramming. We present results from\u0000our ongoing optimization efforts, comparing the state-of-the-art egg Rust\u0000library's performance against our system and show that performant EqSat\u0000implementations are possible without sacrificing the comfort of a direct 1-1\u0000integration with a dynamic, high-level and an interactive host programming\u0000language.","PeriodicalId":501033,"journal":{"name":"arXiv - CS - Symbolic Computation","volume":"63 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140583129","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Complexity of Probabilistic Reasoning for Neurosymbolic Classification Techniques 神经符号分类技术的概率推理复杂性
arXiv - CS - Symbolic Computation Pub Date : 2024-04-12 DOI: arxiv-2404.08404
Arthur Ledaguenel, Céline Hudelot, Mostepha Khouadjia
{"title":"Complexity of Probabilistic Reasoning for Neurosymbolic Classification Techniques","authors":"Arthur Ledaguenel, Céline Hudelot, Mostepha Khouadjia","doi":"arxiv-2404.08404","DOIUrl":"https://doi.org/arxiv-2404.08404","url":null,"abstract":"Neurosymbolic artificial intelligence is a growing field of research aiming\u0000to combine neural network learning capabilities with the reasoning abilities of\u0000symbolic systems. Informed multi-label classification is a sub-field of\u0000neurosymbolic AI which studies how to leverage prior knowledge to improve\u0000neural classification systems. A well known family of neurosymbolic techniques\u0000for informed classification use probabilistic reasoning to integrate this\u0000knowledge during learning, inference or both. Therefore, the asymptotic\u0000complexity of probabilistic reasoning is of cardinal importance to assess the\u0000scalability of such techniques. However, this topic is rarely tackled in the\u0000neurosymbolic literature, which can lead to a poor understanding of the limits\u0000of probabilistic neurosymbolic techniques. In this paper, we introduce a\u0000formalism for informed supervised classification tasks and techniques. We then\u0000build upon this formalism to define three abstract neurosymbolic techniques\u0000based on probabilistic reasoning. Finally, we show computational complexity\u0000results on several representation languages for prior knowledge commonly found\u0000in the neurosymbolic literature.","PeriodicalId":501033,"journal":{"name":"arXiv - CS - Symbolic Computation","volume":"51 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140582663","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
WebPie: A Tiny Slice of Dependent Typing WebPie：依赖输入法的小小切片
arXiv - CS - Symbolic Computation Pub Date : 2024-04-08 DOI: arxiv-2404.05457
Christophe ScholliersGhent University
{"title":"WebPie: A Tiny Slice of Dependent Typing","authors":"Christophe ScholliersGhent University","doi":"arxiv-2404.05457","DOIUrl":"https://doi.org/arxiv-2404.05457","url":null,"abstract":"Dependently typed programming languages have become increasingly relevant in\u0000recent years. They have been adopted in industrial strength programming\u0000languages and have been extremely successful as the basis for theorem provers.\u0000There are however, very few entry level introductions to the theory of language\u0000constructs for dependently typed languages, and even less sources on didactical\u0000implementations. In this paper, we present a small dependently typed\u0000programming language called WebPie. The main features of the language are\u0000inductive types, recursion and case matching. While none of these features are\u0000new, we believe this article can provide a step forward towards the\u0000understanding and systematic construction of dependently typed languages for\u0000researchers new to dependent types.","PeriodicalId":501033,"journal":{"name":"arXiv - CS - Symbolic Computation","volume":"31 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140582872","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Power Series Composition in Near-Linear Time 近线性时间中的功率序列构成
arXiv - CS - Symbolic Computation Pub Date : 2024-04-08 DOI: arxiv-2404.05177
Yasunori Kinoshita, Baitian Li
{"title":"Power Series Composition in Near-Linear Time","authors":"Yasunori Kinoshita, Baitian Li","doi":"arxiv-2404.05177","DOIUrl":"https://doi.org/arxiv-2404.05177","url":null,"abstract":"We present an algebraic algorithm that computes the composition of two power\u0000series in $mathop{tilde{mathrm O}}(n)$ time complexity. The previous best\u0000algorithms are $mathop{mathrm O}(n^{1+o(1)})$ by Kedlaya and Umans (FOCS\u00002008) and an $mathop{mathrm O}(n^{1.43})$ algebraic algorithm by Neiger,\u0000Salvy, Schost and Villard (JACM 2023). Our algorithm builds upon the recent Graeffe iteration approach to manipulate\u0000rational power series introduced by Bostan and Mori (SOSA 2021).","PeriodicalId":501033,"journal":{"name":"arXiv - CS - Symbolic Computation","volume":"117 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140582664","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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