The significance of learning constraints from data is underscored by its potential applications in real-world problem-solving. While constraints are popular for modeling and solving, the approaches to learning constraints from data remain relatively scarce. Furthermore, the intricate task of modeling demands expertise and is prone to errors, thus constraint acquisition methods offer a solution by automating this process through learnt constraints from examples or behaviours of solutions and non-solutions. This work introduces a novel approach grounded in Deep Neural Network (DNN) based on Symbolic Regression that, by setting suitable loss functions, constraints can be extracted directly from datasets. Using the present approach, direct formulation of constraints was achieved. Furthermore, given the broad pre-developed architectures and functionalities of DNN, connections and extensions with other frameworks could be foreseen.
{"title":"Deep Neural Network for Constraint Acquisition through Tailored Loss Function","authors":"Eduardo Vyhmeister, Rocio Paez, Gabriel Gonzalez","doi":"arxiv-2403.02042","DOIUrl":"https://doi.org/arxiv-2403.02042","url":null,"abstract":"The significance of learning constraints from data is underscored by its\u0000potential applications in real-world problem-solving. While constraints are\u0000popular for modeling and solving, the approaches to learning constraints from\u0000data remain relatively scarce. Furthermore, the intricate task of modeling\u0000demands expertise and is prone to errors, thus constraint acquisition methods\u0000offer a solution by automating this process through learnt constraints from\u0000examples or behaviours of solutions and non-solutions. This work introduces a\u0000novel approach grounded in Deep Neural Network (DNN) based on Symbolic\u0000Regression that, by setting suitable loss functions, constraints can be\u0000extracted directly from datasets. Using the present approach, direct\u0000formulation of constraints was achieved. Furthermore, given the broad\u0000pre-developed architectures and functionalities of DNN, connections and\u0000extensions with other frameworks could be foreseen.","PeriodicalId":501033,"journal":{"name":"arXiv - CS - Symbolic Computation","volume":"54 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140033239","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}
Jean-Guillaume DumasUGA, LJK, CASC, Alexis GalanCASC, Bruno GrenetCASC, Aude MaignanCASC, Daniel S. Roche
We consider the private set union (PSU) problem, where two parties each hold a private set of elements, and they want one of the parties (the receiver) to learn the union of the two sets and nothing else. Our protocols are targeted for the unbalanced case where the receiver's set size is larger than the sender's set size, with the goal of minimizing the costs for the sender both in terms of communication volume and local computation time. This setting is motivated by applications where the receiver has significantly more data (input set size) and computational resources than the sender which might be realized on a small, low-power device. Asymptotically, we achieve communication cost linear in the sender's (smaller) set size, and computation costs for sender and receiver which are nearly-linear in their respective set sizes. To our knowledge, ours is the first algorithm to achieve nearly-linear communication and computation for PSU in this unbalanced setting. Our protocols utilize fully homomorphic encryption (FHE) and, optionally, linearly homomorphic encryption (LHE) to perform the necessary computations while preserving privacy. The underlying computations are based on univariate polynomial arithmetic realized within homomorphic encryption, namely fast multiplication, modular reduction, and multi-point evaluation. These asymptotically fast HE polynomial arithmetic algorithms may be of independent interest.
我们考虑的是私人集合联合（PSU）问题，即双方各自持有一个私人元素集合，他们希望其中一方（接收方）只学习两个集合的联合，而不学习其他内容。我们的协议针对的是接收方集合大小大于发送方集合大小的不平衡情况，目标是最大限度地降低发送方在通信量和本地计算时间方面的成本。在这种情况下，接收方的数据量（输入集大小）和计算资源都明显多于发送方，而这种应用可能是在小型、低功耗设备上实现的。渐进地，我们实现了通信成本与发送方（较小）数据集大小的线性关系，以及发送方和接收方计算成本与各自数据集大小的近似线性关系。据我们所知，我们的算法是第一种在这种不平衡设置下实现 PSU 的近线性通信和计算的算法。我们的协议利用全同态加密（FHE）和可选的线性同态加密（LHE）来执行必要的计算，同时保护隐私。基本计算基于在同态加密中实现的单变量多项式运算，即快速乘法、模块化还原和多点评估。这些近似快速的 HE 多项式算术算法可能会引起独立的兴趣。
{"title":"Communication Optimal Unbalanced Private Set Union","authors":"Jean-Guillaume DumasUGA, LJK, CASC, Alexis GalanCASC, Bruno GrenetCASC, Aude MaignanCASC, Daniel S. Roche","doi":"arxiv-2402.16393","DOIUrl":"https://doi.org/arxiv-2402.16393","url":null,"abstract":"We consider the private set union (PSU) problem, where two parties each hold\u0000a private set of elements, and they want one of the parties (the receiver) to\u0000learn the union of the two sets and nothing else. Our protocols are targeted\u0000for the unbalanced case where the receiver's set size is larger than the\u0000sender's set size, with the goal of minimizing the costs for the sender both in\u0000terms of communication volume and local computation time. This setting is\u0000motivated by applications where the receiver has significantly more data (input\u0000set size) and computational resources than the sender which might be realized\u0000on a small, low-power device. Asymptotically, we achieve communication cost\u0000linear in the sender's (smaller) set size, and computation costs for sender and\u0000receiver which are nearly-linear in their respective set sizes. To our\u0000knowledge, ours is the first algorithm to achieve nearly-linear communication\u0000and computation for PSU in this unbalanced setting. Our protocols utilize fully\u0000homomorphic encryption (FHE) and, optionally, linearly homomorphic encryption\u0000(LHE) to perform the necessary computations while preserving privacy. The\u0000underlying computations are based on univariate polynomial arithmetic realized\u0000within homomorphic encryption, namely fast multiplication, modular reduction,\u0000and multi-point evaluation. These asymptotically fast HE polynomial arithmetic\u0000algorithms may be of independent interest.","PeriodicalId":501033,"journal":{"name":"arXiv - CS - Symbolic Computation","volume":"15 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139977247","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}
We construct explicit pseudorandom generators that fool $n$-variate polynomials of degree at most $d$ over a finite field $mathbb{F}_q$. The seed length of our generators is $O(d log n + log q)$, over fields of size exponential in $d$ and characteristic at least $d(d-1)+1$. Previous constructions such as Bogdanov's (STOC 2005) and Derksen and Viola's (FOCS 2022) had either suboptimal seed length or required the field size to depend on $n$. Our approach follows Bogdanov's paradigm while incorporating techniques from Lecerf's factorization algorithm (J. Symb. Comput. 2007) and insights from the construction of Derksen and Viola regarding the role of indecomposability of polynomials.
{"title":"Optimal Pseudorandom Generators for Low-Degree Polynomials Over Moderately Large Fields","authors":"Ashish Dwivedi, Zeyu Guo, Ben Lee Volk","doi":"arxiv-2402.11915","DOIUrl":"https://doi.org/arxiv-2402.11915","url":null,"abstract":"We construct explicit pseudorandom generators that fool $n$-variate\u0000polynomials of degree at most $d$ over a finite field $mathbb{F}_q$. The seed\u0000length of our generators is $O(d log n + log q)$, over fields of size\u0000exponential in $d$ and characteristic at least $d(d-1)+1$. Previous\u0000constructions such as Bogdanov's (STOC 2005) and Derksen and Viola's (FOCS\u00002022) had either suboptimal seed length or required the field size to depend on\u0000$n$. Our approach follows Bogdanov's paradigm while incorporating techniques from\u0000Lecerf's factorization algorithm (J. Symb. Comput. 2007) and insights from the\u0000construction of Derksen and Viola regarding the role of indecomposability of\u0000polynomials.","PeriodicalId":501033,"journal":{"name":"arXiv - CS - Symbolic Computation","volume":"65 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139909598","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}
Pascal Giorgi, Bruno Grenet, Armelle Perret du Cray, Daniel S. Roche
We consider the classical problems of interpolating a polynomial given a black box for evaluation, and of multiplying two polynomials, in the setting where the bit-lengths of the coefficients may vary widely, so-called unbalanced polynomials. Writing s for the total bit-length and D for the degree, our new algorithms have expected running time $tilde{O}(s log D)$, whereas previous methods for (resp.) dense or sparse arithmetic have at least $tilde{O}(sD)$ or $tilde{O}(s^2)$ bit complexity.
我们考虑了给定黑盒求值的多项式插值和两个多项式相乘的经典问题，在这种情况下，系数的比特长度可能变化很大，即所谓的不平衡多项式。用 s 表示总位长，用 D 表示度数，我们的新算法的预期运行时间为 $tilde{O}(s log D)$，而以前的密集或稀疏算术方法至少有 $tilde{O}(sD)$ 或 $tilde{O}(s^2)$ 的位复杂度。
{"title":"Fast interpolation and multiplication of unbalanced polynomials","authors":"Pascal Giorgi, Bruno Grenet, Armelle Perret du Cray, Daniel S. Roche","doi":"arxiv-2402.10139","DOIUrl":"https://doi.org/arxiv-2402.10139","url":null,"abstract":"We consider the classical problems of interpolating a polynomial given a\u0000black box for evaluation, and of multiplying two polynomials, in the setting\u0000where the bit-lengths of the coefficients may vary widely, so-called unbalanced\u0000polynomials. Writing s for the total bit-length and D for the degree, our new\u0000algorithms have expected running time $tilde{O}(s log D)$, whereas previous\u0000methods for (resp.) dense or sparse arithmetic have at least $tilde{O}(sD)$ or\u0000$tilde{O}(s^2)$ bit complexity.","PeriodicalId":501033,"journal":{"name":"arXiv - CS - Symbolic Computation","volume":"313 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139767434","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}
Krylov methods rely on iterated matrix-vector products $A^k u_j$ for an $ntimes n$ matrix $A$ and vectors $u_1,ldots,u_m$. The space spanned by all iterates $A^k u_j$ admits a particular basis -- the emph{maximal Krylov basis} -- which consists of iterates of the first vector $u_1, Au_1, A^2u_1,ldots$, until reaching linear dependency, then iterating similarly the subsequent vectors until a basis is obtained. Finding minimal polynomials and Frobenius normal forms is closely related to computing maximal Krylov bases. The fastest way to produce these bases was, until this paper, Keller-Gehrig's 1985 algorithm whose complexity bound $O(n^omega log(n))$ comes from repeated squarings of $A$ and logarithmically many Gaussian eliminations. Here $omega>2$ is a feasible exponent for matrix multiplication over the base field. We present an algorithm computing the maximal Krylov basis in $O(n^omegaloglog(n))$ field operations when $m in O(n)$, and even $O(n^omega)$ as soon as $min O(n/log(n)^c)$ for some fixed real $c>0$. As a consequence, we show that the Frobenius normal form together with a transformation matrix can be computed deterministically in $O(n^omega loglog(n)^2)$, and therefore matrix exponentiation~$A^k$ can be performed in the latter complexity if $log(k) in O(n^{omega-1-varepsilon})$, for $varepsilon>0$. A key idea for these improvements is to rely on fast algorithms for $mtimes m$ polynomial matrices of average degree $n/m$, involving high-order lifting and minimal kernel bases.
{"title":"Computing Krylov iterates in the time of matrix multiplication","authors":"Vincent Neiger, Clément Pernet, Gilles Villard","doi":"arxiv-2402.07345","DOIUrl":"https://doi.org/arxiv-2402.07345","url":null,"abstract":"Krylov methods rely on iterated matrix-vector products $A^k u_j$ for an\u0000$ntimes n$ matrix $A$ and vectors $u_1,ldots,u_m$. The space spanned by all\u0000iterates $A^k u_j$ admits a particular basis -- the emph{maximal Krylov basis}\u0000-- which consists of iterates of the first vector $u_1, Au_1, A^2u_1,ldots$,\u0000until reaching linear dependency, then iterating similarly the subsequent\u0000vectors until a basis is obtained. Finding minimal polynomials and Frobenius\u0000normal forms is closely related to computing maximal Krylov bases. The fastest\u0000way to produce these bases was, until this paper, Keller-Gehrig's 1985\u0000algorithm whose complexity bound $O(n^omega log(n))$ comes from repeated\u0000squarings of $A$ and logarithmically many Gaussian eliminations. Here\u0000$omega>2$ is a feasible exponent for matrix multiplication over the base\u0000field. We present an algorithm computing the maximal Krylov basis in\u0000$O(n^omegaloglog(n))$ field operations when $m in O(n)$, and even\u0000$O(n^omega)$ as soon as $min O(n/log(n)^c)$ for some fixed real $c>0$. As a\u0000consequence, we show that the Frobenius normal form together with a\u0000transformation matrix can be computed deterministically in $O(n^omega\u0000loglog(n)^2)$, and therefore matrix exponentiation~$A^k$ can be performed in\u0000the latter complexity if $log(k) in O(n^{omega-1-varepsilon})$, for\u0000$varepsilon>0$. A key idea for these improvements is to rely on fast\u0000algorithms for $mtimes m$ polynomial matrices of average degree $n/m$,\u0000involving high-order lifting and minimal kernel bases.","PeriodicalId":501033,"journal":{"name":"arXiv - CS - Symbolic Computation","volume":"25 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139767521","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}
Autonomous Intelligent Agents are employed in many applications upon which the life and welfare of living beings and vital social functions may depend. Therefore, agents should be trustworthy. A priori certification techniques (i.e., techniques applied prior to system's deployment) can be useful, but are not sufficient for agents that evolve, and thus modify their epistemic and belief state, and for open Multi-Agent Systems, where heterogeneous agents can join or leave the system at any stage of its operation. In this paper, we propose/refine/extend dynamic (runtime) logic-based self-checking techniques, devised in order to be able to ensure agents' trustworthy and ethical behaviour.
{"title":"Ensuring trustworthy and ethical behaviour in intelligent logical agents","authors":"Stefania Costantini","doi":"arxiv-2402.07547","DOIUrl":"https://doi.org/arxiv-2402.07547","url":null,"abstract":"Autonomous Intelligent Agents are employed in many applications upon which\u0000the life and welfare of living beings and vital social functions may depend.\u0000Therefore, agents should be trustworthy. A priori certification techniques\u0000(i.e., techniques applied prior to system's deployment) can be useful, but are\u0000not sufficient for agents that evolve, and thus modify their epistemic and\u0000belief state, and for open Multi-Agent Systems, where heterogeneous agents can\u0000join or leave the system at any stage of its operation. In this paper, we\u0000propose/refine/extend dynamic (runtime) logic-based self-checking techniques,\u0000devised in order to be able to ensure agents' trustworthy and ethical\u0000behaviour.","PeriodicalId":501033,"journal":{"name":"arXiv - CS - Symbolic Computation","volume":"98 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139767368","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}
Shaoshi Chen, Ruyong Feng, Manuel Kauers, Xiuyun Li
We propose a summation analog of the paradigm of parallel integration. Using this paradigm, we make some first steps towards an indefinite summation algorithm applicable to summands that rationally depend on the summation index and a P-recursive sequence and its shifts. Under the assumption that the corresponding difference field has no unnatural constants, we are able to compute a bound on the normal part of the denominator of a potential closed form. We can also handle the numerator. Our algorithm is incomplete so far as we cannot predict the special part of the denominator. However, we do have some structural results about special polynomials for the setting under consideration.
我们提出了一种类似于并行积分范式的求和方法。利用这一范式，我们迈出了第一步，建立了一种不定求和算法，适用于合理地依赖于求和指数和 P 递推序列及其移位的求和。假定相应的差分场没有非自然常数，我们就能计算出潜在闭合形式分母法向部分的约束。我们还可以处理分子。由于我们无法预测分母的特殊部分，所以我们的算法并不完整。不过，我们确实有一些关于所考虑的特殊多项式的结构性结果。
{"title":"Towards a Parallel Summation Algorithm","authors":"Shaoshi Chen, Ruyong Feng, Manuel Kauers, Xiuyun Li","doi":"arxiv-2402.04684","DOIUrl":"https://doi.org/arxiv-2402.04684","url":null,"abstract":"We propose a summation analog of the paradigm of parallel integration. Using\u0000this paradigm, we make some first steps towards an indefinite summation\u0000algorithm applicable to summands that rationally depend on the summation index\u0000and a P-recursive sequence and its shifts. Under the assumption that the\u0000corresponding difference field has no unnatural constants, we are able to\u0000compute a bound on the normal part of the denominator of a potential closed\u0000form. We can also handle the numerator. Our algorithm is incomplete so far as\u0000we cannot predict the special part of the denominator. However, we do have some\u0000structural results about special polynomials for the setting under\u0000consideration.","PeriodicalId":501033,"journal":{"name":"arXiv - CS - Symbolic Computation","volume":"17 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139909656","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}
Marius-Constantin Dinu, Claudiu Leoveanu-Condrei, Markus Holzleitner, Werner Zellinger, Sepp Hochreiter
We introduce SymbolicAI, a versatile and modular framework employing a logic-based approach to concept learning and flow management in generative processes. SymbolicAI enables the seamless integration of generative models with a diverse range of solvers by treating large language models (LLMs) as semantic parsers that execute tasks based on both natural and formal language instructions, thus bridging the gap between symbolic reasoning and generative AI. We leverage probabilistic programming principles to tackle complex tasks, and utilize differentiable and classical programming paradigms with their respective strengths. The framework introduces a set of polymorphic, compositional, and self-referential operations for data stream manipulation, aligning LLM outputs with user objectives. As a result, we can transition between the capabilities of various foundation models endowed with zero- and few-shot learning capabilities and specialized, fine-tuned models or solvers proficient in addressing specific problems. In turn, the framework facilitates the creation and evaluation of explainable computational graphs. We conclude by introducing a quality measure and its empirical score for evaluating these computational graphs, and propose a benchmark that compares various state-of-the-art LLMs across a set of complex workflows. We refer to the empirical score as the "Vector Embedding for Relational Trajectory Evaluation through Cross-similarity", or VERTEX score for short. The framework codebase and benchmark are linked below.
{"title":"SymbolicAI: A framework for logic-based approaches combining generative models and solvers","authors":"Marius-Constantin Dinu, Claudiu Leoveanu-Condrei, Markus Holzleitner, Werner Zellinger, Sepp Hochreiter","doi":"arxiv-2402.00854","DOIUrl":"https://doi.org/arxiv-2402.00854","url":null,"abstract":"We introduce SymbolicAI, a versatile and modular framework employing a\u0000logic-based approach to concept learning and flow management in generative\u0000processes. SymbolicAI enables the seamless integration of generative models\u0000with a diverse range of solvers by treating large language models (LLMs) as\u0000semantic parsers that execute tasks based on both natural and formal language\u0000instructions, thus bridging the gap between symbolic reasoning and generative\u0000AI. We leverage probabilistic programming principles to tackle complex tasks,\u0000and utilize differentiable and classical programming paradigms with their\u0000respective strengths. The framework introduces a set of polymorphic,\u0000compositional, and self-referential operations for data stream manipulation,\u0000aligning LLM outputs with user objectives. As a result, we can transition\u0000between the capabilities of various foundation models endowed with zero- and\u0000few-shot learning capabilities and specialized, fine-tuned models or solvers\u0000proficient in addressing specific problems. In turn, the framework facilitates\u0000the creation and evaluation of explainable computational graphs. We conclude by\u0000introducing a quality measure and its empirical score for evaluating these\u0000computational graphs, and propose a benchmark that compares various\u0000state-of-the-art LLMs across a set of complex workflows. We refer to the\u0000empirical score as the \"Vector Embedding for Relational Trajectory Evaluation\u0000through Cross-similarity\", or VERTEX score for short. The framework codebase\u0000and benchmark are linked below.","PeriodicalId":501033,"journal":{"name":"arXiv - CS - Symbolic Computation","volume":"22 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139666345","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}
We present efficient methods for calculating linear recurrences of hypergeometric double sums and, more generally, of multiple sums. In particular, we supplement this approach with the algorithmic theory of contiguous relations, which guarantees the applicability of our method for many input sums. In addition, we elaborate new techniques to optimize the underlying key task of our method to compute rational solutions of parameterized linear recurrences.
{"title":"Creative Telescoping for Hypergeometric Double Sums","authors":"Peter Paule, Carsten Schneider","doi":"arxiv-2401.16314","DOIUrl":"https://doi.org/arxiv-2401.16314","url":null,"abstract":"We present efficient methods for calculating linear recurrences of\u0000hypergeometric double sums and, more generally, of multiple sums. In\u0000particular, we supplement this approach with the algorithmic theory of\u0000contiguous relations, which guarantees the applicability of our method for many\u0000input sums. In addition, we elaborate new techniques to optimize the underlying\u0000key task of our method to compute rational solutions of parameterized linear\u0000recurrences.","PeriodicalId":501033,"journal":{"name":"arXiv - CS - Symbolic Computation","volume":"323 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139588164","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}
Machine-learning methods are gradually being adopted in a great variety of social, economic, and scientific contexts, yet they are notorious for struggling with exact mathematics. A typical example is computer algebra, which includes tasks like simplifying mathematical terms, calculating formal derivatives, or finding exact solutions of algebraic equations. Traditional software packages for these purposes are commonly based on a huge database of rules for how a specific operation (e.g., differentiation) transforms a certain term (e.g., sine function) into another one (e.g., cosine function). Thus far, these rules have usually needed to be discovered and subsequently programmed by humans. Focusing on the paradigmatic example of solving linear equations in symbolic form, we demonstrate how the process of finding elementary transformation rules and step-by-step solutions can be automated using reinforcement learning with deep neural networks.
{"title":"Symbolic Equation Solving via Reinforcement Learning","authors":"Lennart Dabelow, Masahito Ueda","doi":"arxiv-2401.13447","DOIUrl":"https://doi.org/arxiv-2401.13447","url":null,"abstract":"Machine-learning methods are gradually being adopted in a great variety of\u0000social, economic, and scientific contexts, yet they are notorious for\u0000struggling with exact mathematics. A typical example is computer algebra, which\u0000includes tasks like simplifying mathematical terms, calculating formal\u0000derivatives, or finding exact solutions of algebraic equations. Traditional\u0000software packages for these purposes are commonly based on a huge database of\u0000rules for how a specific operation (e.g., differentiation) transforms a certain\u0000term (e.g., sine function) into another one (e.g., cosine function). Thus far,\u0000these rules have usually needed to be discovered and subsequently programmed by\u0000humans. Focusing on the paradigmatic example of solving linear equations in\u0000symbolic form, we demonstrate how the process of finding elementary\u0000transformation rules and step-by-step solutions can be automated using\u0000reinforcement learning with deep neural networks.","PeriodicalId":501033,"journal":{"name":"arXiv - CS - Symbolic Computation","volume":"8 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139554610","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}
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