Journal of Symbolic Computation最新文献

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D-algebraic functions D 代函数
IF 0.6 4区 数学
Journal of Symbolic Computation Pub Date : 2024-08-28 DOI: 10.1016/j.jsc.2024.102377
Rida Ait El Manssour , Anna-Laura Sattelberger , Bertrand Teguia Tabuguia
{"title":"D-algebraic functions","authors":"Rida Ait El Manssour ,&nbsp;Anna-Laura Sattelberger ,&nbsp;Bertrand Teguia Tabuguia","doi":"10.1016/j.jsc.2024.102377","DOIUrl":"10.1016/j.jsc.2024.102377","url":null,"abstract":"<div><p>Differentially-algebraic (D-algebraic) functions are solutions of polynomial equations in the function, its derivatives, and the independent variables. We revisit closure properties of these functions by providing constructive proofs. We present algorithms to compute algebraic differential equations for compositions and arithmetic manipulations of univariate D-algebraic functions and derive bounds for the order of the resulting differential equations. We apply our methods to examples in the sciences.</p></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"128 ","pages":"Article 102377"},"PeriodicalIF":0.6,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142150874","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Invariant neural architecture for learning term synthesis in instantiation proving 在实例化证明中学习术语合成的不变神经架构
IF 0.6 4区 数学
Journal of Symbolic Computation Pub Date : 2024-08-28 DOI: 10.1016/j.jsc.2024.102375
Jelle Piepenbrock , Josef Urban , Konstantin Korovin , Miroslav Olšák , Tom Heskes , Mikoláš Janota
{"title":"Invariant neural architecture for learning term synthesis in instantiation proving","authors":"Jelle Piepenbrock ,&nbsp;Josef Urban ,&nbsp;Konstantin Korovin ,&nbsp;Miroslav Olšák ,&nbsp;Tom Heskes ,&nbsp;Mikoláš Janota","doi":"10.1016/j.jsc.2024.102375","DOIUrl":"10.1016/j.jsc.2024.102375","url":null,"abstract":"<div><p>The development of strong CDCL-based propositional (SAT) solvers has greatly advanced several areas of automated reasoning (AR). One of the directions in AR is therefore to make use of SAT solvers in expressive formalisms such as first-order logic, for which large corpora of general mathematical problems exist today. This is possible due to Herbrand's theorem, which allows reduction of first-order problems to propositional problems by instantiation. The core challenge is synthesizing the appropriate instances from the typically infinite Herbrand universe.</p><p>In this work, we develop a machine learning system targeting this task, addressing its combinatorial and invariance properties. In particular, we develop a GNN2RNN architecture based on a graph neural network (GNN) that learns from problems and their solutions independently of many symmetries and symbol names (addressing the abundance of Skolems), combined with a recurrent neural network (RNN) that proposes for each clause its instantiations. The architecture is then combined with an efficient ground solver and, starting with zero knowledge, iteratively trained on a large corpus of mathematical problems. We show that the system is capable of solving many problems by such educated guessing, finding proofs for 32.12% of the training set. The final trained system solves 19.74% of the unseen test data on its own. We also observe that the trained system finds solutions that the iProver and CVC5 systems did not find.</p></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"128 ","pages":"Article 102375"},"PeriodicalIF":0.6,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0747717124000798/pdfft?md5=03f2c9a993930436ebd44dced50d3406&pid=1-s2.0-S0747717124000798-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142150873","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Graph sequence learning for premise selection 用于前提选择的图序列学习
IF 0.6 4区 数学
Journal of Symbolic Computation Pub Date : 2024-08-27 DOI: 10.1016/j.jsc.2024.102376
Edvard K. Holden, Konstantin Korovin
{"title":"Graph sequence learning for premise selection","authors":"Edvard K. Holden,&nbsp;Konstantin Korovin","doi":"10.1016/j.jsc.2024.102376","DOIUrl":"10.1016/j.jsc.2024.102376","url":null,"abstract":"<div><p>Premise selection is crucial for large theory reasoning with automated theorem provers as the sheer size of the problems quickly leads to resource exhaustion. This paper proposes a premise selection method inspired by the machine learning domain of image captioning, where language models automatically generate a suitable caption for a given image. Likewise, we attempt to generate the sequence of axioms required to construct the proof of a given conjecture. In our <em>axiom captioning</em> approach, a pre-trained graph neural network is combined with a language model via transfer learning to encapsulate both the inter-axiom and conjecture-axiom relationships. We evaluate different configurations of our method and experience a 14% improvement in the number of solved problems over a baseline.</p></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"128 ","pages":"Article 102376"},"PeriodicalIF":0.6,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0747717124000804/pdfft?md5=f758e854b5cedd39b04e5e1431d3d6d8&pid=1-s2.0-S0747717124000804-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142162490","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Symmetric SAGE and SONC forms, exactness and quantitative gaps 对称 SAGE 和 SONC 形式、精确性和数量差距
IF 0.6 4区 数学
Journal of Symbolic Computation Pub Date : 2024-08-13 DOI: 10.1016/j.jsc.2024.102374
Philippe Moustrou , Cordian Riener , Thorsten Theobald , Hugues Verdure
{"title":"Symmetric SAGE and SONC forms, exactness and quantitative gaps","authors":"Philippe Moustrou ,&nbsp;Cordian Riener ,&nbsp;Thorsten Theobald ,&nbsp;Hugues Verdure","doi":"10.1016/j.jsc.2024.102374","DOIUrl":"10.1016/j.jsc.2024.102374","url":null,"abstract":"<div><p>The classes of sums of arithmetic-geometric exponentials (SAGE) and of sums of nonnegative circuit polynomials (SONC) provide nonnegativity certificates which are based on the inequality of the arithmetic and geometric means. We study the cones of symmetric SAGE and SONC forms and their relations to the underlying symmetric nonnegative cone.</p><p>As main results, we provide several symmetric cases where the SAGE or SONC property coincides with nonnegativity and we present quantitative results on the differences in various situations. The results rely on characterizations of the zeroes and the minimizers for symmetric SAGE and SONC forms, which we develop. Finally, we also study symmetric monomial mean inequalities and apply SONC certificates to establish a generalized version of Muirhead's inequality.</p></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"127 ","pages":"Article 102374"},"PeriodicalIF":0.6,"publicationDate":"2024-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0747717124000786/pdfft?md5=ff647fd07ebbb483b3bff891ef5b3479&pid=1-s2.0-S0747717124000786-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142039673","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A short proof for the parameter continuation theorem 参数延续定理的简短证明
IF 0.6 4区 数学
Journal of Symbolic Computation Pub Date : 2024-08-03 DOI: 10.1016/j.jsc.2024.102373
Viktoriia Borovik, Paul Breiding
{"title":"A short proof for the parameter continuation theorem","authors":"Viktoriia Borovik,&nbsp;Paul Breiding","doi":"10.1016/j.jsc.2024.102373","DOIUrl":"10.1016/j.jsc.2024.102373","url":null,"abstract":"<div><p>The Parameter Continuation Theorem is the theoretical foundation for polynomial homotopy continuation, which is one of the main tools in computational algebraic geometry. In this note, we give a short proof using Gröbner bases. Our approach gives a method for computing discriminants.</p></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"127 ","pages":"Article 102373"},"PeriodicalIF":0.6,"publicationDate":"2024-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0747717124000774/pdfft?md5=f847eab8eb976e0f1998987fb2d287a9&pid=1-s2.0-S0747717124000774-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141947539","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Fast evaluation and root finding for polynomials with floating-point coefficients 浮点系数多项式的快速求值和求根
IF 0.6 4区 数学
Journal of Symbolic Computation Pub Date : 2024-07-20 DOI: 10.1016/j.jsc.2024.102372
Rémi Imbach , Guillaume Moroz
{"title":"Fast evaluation and root finding for polynomials with floating-point coefficients","authors":"Rémi Imbach ,&nbsp;Guillaume Moroz","doi":"10.1016/j.jsc.2024.102372","DOIUrl":"10.1016/j.jsc.2024.102372","url":null,"abstract":"<div><p>Evaluating or finding the roots of a polynomial <span><math><mi>f</mi><mo>(</mo><mi>z</mi><mo>)</mo><mo>=</mo><msub><mrow><mi>f</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mrow><mi>f</mi></mrow><mrow><mi>d</mi></mrow></msub><msup><mrow><mi>z</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> with floating-point number coefficients is a ubiquitous problem. By using a piecewise approximation of <em>f</em> obtained with a careful use of the Newton polygon of <em>f</em>, we improve state-of-the-art upper bounds on the number of operations to evaluate and find the roots of a polynomial. In particular, if the coefficients of <em>f</em> are given with <em>m</em> significant bits, we provide for the first time an algorithm that finds all the roots of <em>f</em> with a relative condition number lower than <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi></mrow></msup></math></span>, using a number of bit operations quasi-linear in the bit-size of the floating-point representation of <em>f</em>. Notably, our new approach handles efficiently polynomials with coefficients ranging from <span><math><msup><mrow><mn>2</mn></mrow><mrow><mo>−</mo><mi>d</mi></mrow></msup></math></span> to <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>d</mi></mrow></msup></math></span>, both in theory and in practice.</p></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"127 ","pages":"Article 102372"},"PeriodicalIF":0.6,"publicationDate":"2024-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141947541","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Computing character tables and Cartan matrices of finite monoids with fixed point counting 用定点计数计算有限单体的字符表和卡坦矩阵
IF 0.6 4区 数学
Journal of Symbolic Computation Pub Date : 2024-07-20 DOI: 10.1016/j.jsc.2024.102371
Balthazar Charles
{"title":"Computing character tables and Cartan matrices of finite monoids with fixed point counting","authors":"Balthazar Charles","doi":"10.1016/j.jsc.2024.102371","DOIUrl":"10.1016/j.jsc.2024.102371","url":null,"abstract":"<div><p>In this paper we present an algorithm for efficiently counting fixed points in a finite monoid <em>M</em> under a conjugacy-like action. We then prove a formula for the character table of <em>M</em> in terms of fixed points, which allows for the effective computation of both the character table of <em>M</em> other a field of null characteristic, as well as its Cartan matrix, using a formula from [Thiéry '12], again in terms of fixed points. We discuss the implementation details of the resulting algorithms and provide benchmarks of their performance.</p></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"127 ","pages":"Article 102371"},"PeriodicalIF":0.6,"publicationDate":"2024-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141947566","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Signature-based standard basis algorithm under the framework of GVW algorithm GVW 算法框架下基于签名的标准基础算法
IF 0.6 4区 数学
Journal of Symbolic Computation Pub Date : 2024-07-19 DOI: 10.1016/j.jsc.2024.102370
Dong Lu , Dingkang Wang , Fanghui Xiao , Xiaopeng Zheng
{"title":"Signature-based standard basis algorithm under the framework of GVW algorithm","authors":"Dong Lu ,&nbsp;Dingkang Wang ,&nbsp;Fanghui Xiao ,&nbsp;Xiaopeng Zheng","doi":"10.1016/j.jsc.2024.102370","DOIUrl":"10.1016/j.jsc.2024.102370","url":null,"abstract":"<div><p>The GVW algorithm, one of the most important so-called signature-based algorithms, is designed to eliminate a large number of useless polynomial reductions from Buchberger's algorithm. The cover theorem serves as the theoretical foundation of the GVW algorithm, and up to now, it applies only to a certain class of monomial orders, namely global orders and a special class of local orders. In this paper we extend this theorem to any semigroup order, which can be either global, local or even mixed. Building upon the pioneering idea of the Mora normal form algorithm, we propose a more comprehensive and general proof for the cover theorem while bypassing the need to choose a minimal element from an infinite set of monomials in all the existing proofs. Therefore, the algorithm for signature-based standard bases is presented for any semigroup order under the framework of the GVW algorithm, and an example is given to provide an illustration of the algorithm.</p></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"127 ","pages":"Article 102370"},"PeriodicalIF":0.6,"publicationDate":"2024-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141840958","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On symmetry adapted bases in trigonometric optimization 关于三角优化中的对称适配基
IF 0.6 4区 数学
Journal of Symbolic Computation Pub Date : 2024-07-17 DOI: 10.1016/j.jsc.2024.102369
Tobias Metzlaff
{"title":"On symmetry adapted bases in trigonometric optimization","authors":"Tobias Metzlaff","doi":"10.1016/j.jsc.2024.102369","DOIUrl":"10.1016/j.jsc.2024.102369","url":null,"abstract":"<div><p>The problem of computing the global minimum of a trigonometric polynomial is computationally hard. We address this problem for the case, where the polynomial is invariant under the exponential action of a finite group. The strategy is to follow an established relaxation strategy in order to obtain a converging hierarchy of lower bounds. Those bounds are obtained by numerically solving semi-definite programs (SDPs) on the cone of positive semi-definite Hermitian Toeplitz matrices, which is outlined in the book of Dumitrescu <span><span>Dumitrescu (2007)</span></span>. To exploit the invariance, we show that the group has an induced action on the Toeplitz matrices and prove that the feasible region of the SDP can be restricted to the invariant matrices, whilst retaining the same solution. Then we construct a symmetry adapted basis tailored to this group action, which allows us to block-diagonalize invariant matrices and thus reduce the computational complexity to solve the SDP.</p><p>The approach is in its generality novel for trigonometric optimization and complements the one that was proposed as a poster at the ISSAC 2022 conference <span><span>Hubert et al. (2022)</span></span> and later extended to <span><span>Hubert et al. (2024)</span></span>. In the previous work, we first used the invariance of the trigonometric polynomial to obtain a classical polynomial optimization problem on the orbit space and subsequently relaxed the problem to an SDP. Now, we first make the relaxation and then exploit invariance.</p><p>Partial results of this article have been presented as a poster at the ISSAC 2023 conference <span><span>Metzlaff (2023)</span></span>.</p></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"127 ","pages":"Article 102369"},"PeriodicalIF":0.6,"publicationDate":"2024-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0747717124000737/pdfft?md5=5897477ca2a3bd75103332cdce197f3a&pid=1-s2.0-S0747717124000737-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141779956","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Approximate GCD of several multivariate sparse polynomials based on SLRA interpolation 基于 SLRA 插值的多个多元稀疏多项式的近似 GCD
IF 0.6 4区 数学
Journal of Symbolic Computation Pub Date : 2024-07-15 DOI: 10.1016/j.jsc.2024.102368
Kosaku Nagasaka
{"title":"Approximate GCD of several multivariate sparse polynomials based on SLRA interpolation","authors":"Kosaku Nagasaka","doi":"10.1016/j.jsc.2024.102368","DOIUrl":"10.1016/j.jsc.2024.102368","url":null,"abstract":"<div><p>To compute the greatest common divisor (GCD) of a set of multivariate polynomials, modular algorithms are typically employed to prevent any growth in the coefficient polynomials in the intermediate expressions. However, when dealing with multivariate polynomials with a priori errors on their coefficients, using such modular algorithms becomes challenging. This is because any resulting approximate GCD computed in one variable may have perturbations depending on the evaluation point and may not be an image of the same desired multivariate approximate GCD. This necessitates computing it as given multivariate polynomials, and operating with large matrices whose size is exponential in the number of variables. In this paper, we present a new modular algorithm, suitable for dense cases and effective for sparse ones, called “SLRA interpolation”. This algorithm uses the multidimensional fast Fourier transform (FFT) and the structured low-rank approximation (SLRA) of non-square block diagonal matrices. The SLRA interpolation technique may reduce the time-complexity for one iteration in the computation of approximate GCD of several multivariate polynomials, especially for the sparse case.</p></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"127 ","pages":"Article 102368"},"PeriodicalIF":0.6,"publicationDate":"2024-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141712328","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","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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