Model-constructing satisfiability calculus (MCSAT) framework has been applied to SMT problems on different arithmetic theories. NLSAT, an implementation using cylindrical algebraic decomposition for explanation, is especially competitive among nonlinear real arithmetic constraints. However, current Conflict-Driven Clause Learning (CDCL)-style algorithms only consider literal information for decision, and thus ignore clause-level influence on arithmetic variables. As a consequence, NLSAT encounters unnecessary conflicts caused by improper literal decisions. In this work, we analyze the literal decision caused conflicts, and introduce clause-level information with a direct effect on arithmetic variables. Two main algorithm improvements are presented: clause-level feasible-set based look-ahead mechanism and arithmetic propagation based branching heuristic. We implement our solver named clauseSMT on our dynamic variable ordering framework. Experiments show that clauseSMT is competitive on nonlinear real arithmetic theory against existing SMT solvers (cvc5, Z3, Yices2), and outperforms all these solvers on satisfiable instances of SMT(QF_NRA) in SMT-LIB. The effectiveness of our proposed methods are also studied.
{"title":"clauseSMT: A NLSAT-Based Clause-Level Framework for Satisfiability Modulo Nonlinear Real Arithmetic Theory","authors":"Zhonghan Wang","doi":"arxiv-2406.02122","DOIUrl":"https://doi.org/arxiv-2406.02122","url":null,"abstract":"Model-constructing satisfiability calculus (MCSAT) framework has been applied\u0000to SMT problems on different arithmetic theories. NLSAT, an implementation\u0000using cylindrical algebraic decomposition for explanation, is especially\u0000competitive among nonlinear real arithmetic constraints. However, current\u0000Conflict-Driven Clause Learning (CDCL)-style algorithms only consider literal\u0000information for decision, and thus ignore clause-level influence on arithmetic\u0000variables. As a consequence, NLSAT encounters unnecessary conflicts caused by\u0000improper literal decisions. In this work, we analyze the literal decision caused conflicts, and introduce\u0000clause-level information with a direct effect on arithmetic variables. Two main\u0000algorithm improvements are presented: clause-level feasible-set based\u0000look-ahead mechanism and arithmetic propagation based branching heuristic. We\u0000implement our solver named clauseSMT on our dynamic variable ordering\u0000framework. Experiments show that clauseSMT is competitive on nonlinear real\u0000arithmetic theory against existing SMT solvers (cvc5, Z3, Yices2), and\u0000outperforms all these solvers on satisfiable instances of SMT(QF_NRA) in\u0000SMT-LIB. The effectiveness of our proposed methods are also studied.","PeriodicalId":501033,"journal":{"name":"arXiv - CS - Symbolic Computation","volume":"58 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141252324","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}
Christoph Koutschan, Anton Ponomarchuk, Josef Schicho
Any continuous piecewise-linear function $Fcolon mathbb{R}^{n}to mathbb{R}$ can be represented as a linear combination of $max$ functions of at most $n+1$ affine-linear functions. In our previous paper [``Representing piecewise linear functions by functions with small arity'', AAECC, 2023], we showed that this upper bound of $n+1$ arguments is tight. In the present paper, we extend this result by establishing a correspondence between the function $F$ and the minimal number of arguments that are needed in any such decomposition. We show that the tessellation of the input space $mathbb{R}^{n}$ induced by the function $F$ has a direct connection to the number of arguments in the $max$ functions.
任何连续的片断线性函数 $Fcolon mathbb{R}^{n}tomathbb{R}$ 都可以表示为最多 $n+1$ 仿真线性函数的 $max$ 函数的线性组合。在我们之前的论文["Representingpiecewise linear functions by functions with small arity'', AAECC, 2023]中，我们证明了这个 $n+1$ 参数的上限是很紧的。在本文中，我们通过建立函数 $F$ 与任何此类分解所需的最小参数数之间的对应关系来扩展这一结果。我们证明，函数 $F$ 所诱导的输入空间 $mathbb{R}^{n}$ 的细分与 $max$ 函数中的参数数有直接联系。
{"title":"Representing Piecewise-Linear Functions by Functions with Minimal Arity","authors":"Christoph Koutschan, Anton Ponomarchuk, Josef Schicho","doi":"arxiv-2406.02421","DOIUrl":"https://doi.org/arxiv-2406.02421","url":null,"abstract":"Any continuous piecewise-linear function $Fcolon mathbb{R}^{n}to\u0000mathbb{R}$ can be represented as a linear combination of $max$ functions of\u0000at most $n+1$ affine-linear functions. In our previous paper [``Representing\u0000piecewise linear functions by functions with small arity'', AAECC, 2023], we\u0000showed that this upper bound of $n+1$ arguments is tight. In the present paper,\u0000we extend this result by establishing a correspondence between the function $F$\u0000and the minimal number of arguments that are needed in any such decomposition.\u0000We show that the tessellation of the input space $mathbb{R}^{n}$ induced by\u0000the function $F$ has a direct connection to the number of arguments in the\u0000$max$ functions.","PeriodicalId":501033,"journal":{"name":"arXiv - CS - Symbolic Computation","volume":"66 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141252320","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}
Xusheng ZhiUniversity of Wisconsin-Madison and Peking University, Thomas RepsUniversity of Wisconsin-Madison
Binary Decision Diagrams (BDDs) are widely used for the representation of Boolean functions. Context-Free-Language Ordered Decision Diagrams (CFLOBDDs) are a plug-compatible replacement for BDDs -- roughly, they are BDDs augmented with a certain form of procedure call. A natural question to ask is, ``For a given Boolean function $f$, what is the relationship between the size of a BDD for $f$ and the size of a CFLOBDD for $f$?'' Sistla et al. established that, in the best case, the CFLOBDD for a function $f$ can be exponentially smaller than any BDD for $f$ (regardless of what variable ordering is used in the BDD); however, they did not give a worst-case bound -- i.e., they left open the question, ``Is there a family of functions ${ f_i }$ for which the size of a CFLOBDD for $f_i$ must be substantially larger than a BDD for $f_i$?'' For instance, it could be that there is a family of functions for which the BDDs are exponentially more succinct than any corresponding CFLOBDDs. This paper studies such questions, and answers the second question posed above in the negative. In particular, we show that by using the same variable ordering in the CFLOBDD that is used in the BDD, the size of a CFLOBDD for any function $f$ cannot be far worse than the size of the BDD for $f$. The bound that relates their sizes is polynomial: If BDD $B$ for function $f$ is of size $|B|$ and uses variable ordering $textit{Ord}$, then the size of the CFLOBDD $C$ for $f$ that also uses $textit{Ord}$ is bounded by $O(|B|^3)$. The paper also shows that the bound is tight: there is a family of functions for which $|C|$ grows as $Omega(|B|^3)$.
{"title":"Polynomial Bounds of CFLOBDDs against BDDs","authors":"Xusheng ZhiUniversity of Wisconsin-Madison and Peking University, Thomas RepsUniversity of Wisconsin-Madison","doi":"arxiv-2406.01525","DOIUrl":"https://doi.org/arxiv-2406.01525","url":null,"abstract":"Binary Decision Diagrams (BDDs) are widely used for the representation of\u0000Boolean functions. Context-Free-Language Ordered Decision Diagrams (CFLOBDDs)\u0000are a plug-compatible replacement for BDDs -- roughly, they are BDDs augmented\u0000with a certain form of procedure call. A natural question to ask is, ``For a\u0000given Boolean function $f$, what is the relationship between the size of a BDD\u0000for $f$ and the size of a CFLOBDD for $f$?'' Sistla et al. established that, in\u0000the best case, the CFLOBDD for a function $f$ can be exponentially smaller than\u0000any BDD for $f$ (regardless of what variable ordering is used in the BDD);\u0000however, they did not give a worst-case bound -- i.e., they left open the\u0000question, ``Is there a family of functions ${ f_i }$ for which the size of a\u0000CFLOBDD for $f_i$ must be substantially larger than a BDD for $f_i$?'' For\u0000instance, it could be that there is a family of functions for which the BDDs\u0000are exponentially more succinct than any corresponding CFLOBDDs. This paper studies such questions, and answers the second question posed\u0000above in the negative. In particular, we show that by using the same variable\u0000ordering in the CFLOBDD that is used in the BDD, the size of a CFLOBDD for any\u0000function $f$ cannot be far worse than the size of the BDD for $f$. The bound\u0000that relates their sizes is polynomial: If BDD $B$ for function $f$ is of size\u0000$|B|$ and uses variable ordering $textit{Ord}$, then the size of the CFLOBDD\u0000$C$ for $f$ that also uses $textit{Ord}$ is bounded by $O(|B|^3)$. The paper also shows that the bound is tight: there is a family of functions\u0000for which $|C|$ grows as $Omega(|B|^3)$.","PeriodicalId":501033,"journal":{"name":"arXiv - CS - Symbolic Computation","volume":"17 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141252119","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}
Alice Bizzarri, Chung-En Yu, Brian Jalaian, Fabrizio Riguzzi, Nathaniel D. Bastian
The prevailing approaches in Network Intrusion Detection Systems (NIDS) are often hampered by issues such as high resource consumption, significant computational demands, and poor interpretability. Furthermore, these systems generally struggle to identify novel, rapidly changing cyber threats. This paper delves into the potential of incorporating Neurosymbolic Artificial Intelligence (NSAI) into NIDS, combining deep learning's data-driven strengths with symbolic AI's logical reasoning to tackle the dynamic challenges in cybersecurity, which also includes detailed NSAI techniques introduction for cyber professionals to explore the potential strengths of NSAI in NIDS. The inclusion of NSAI in NIDS marks potential advancements in both the detection and interpretation of intricate network threats, benefiting from the robust pattern recognition of neural networks and the interpretive prowess of symbolic reasoning. By analyzing network traffic data types and machine learning architectures, we illustrate NSAI's distinctive capability to offer more profound insights into network behavior, thereby improving both detection performance and the adaptability of the system. This merging of technologies not only enhances the functionality of traditional NIDS but also sets the stage for future developments in building more resilient, interpretable, and dynamic defense mechanisms against advanced cyber threats. The continued progress in this area is poised to transform NIDS into a system that is both responsive to known threats and anticipatory of emerging, unseen ones.
{"title":"A Synergistic Approach In Network Intrusion Detection By Neurosymbolic AI","authors":"Alice Bizzarri, Chung-En Yu, Brian Jalaian, Fabrizio Riguzzi, Nathaniel D. Bastian","doi":"arxiv-2406.00938","DOIUrl":"https://doi.org/arxiv-2406.00938","url":null,"abstract":"The prevailing approaches in Network Intrusion Detection Systems (NIDS) are\u0000often hampered by issues such as high resource consumption, significant\u0000computational demands, and poor interpretability. Furthermore, these systems\u0000generally struggle to identify novel, rapidly changing cyber threats. This\u0000paper delves into the potential of incorporating Neurosymbolic Artificial\u0000Intelligence (NSAI) into NIDS, combining deep learning's data-driven strengths\u0000with symbolic AI's logical reasoning to tackle the dynamic challenges in\u0000cybersecurity, which also includes detailed NSAI techniques introduction for\u0000cyber professionals to explore the potential strengths of NSAI in NIDS. The\u0000inclusion of NSAI in NIDS marks potential advancements in both the detection\u0000and interpretation of intricate network threats, benefiting from the robust\u0000pattern recognition of neural networks and the interpretive prowess of symbolic\u0000reasoning. By analyzing network traffic data types and machine learning\u0000architectures, we illustrate NSAI's distinctive capability to offer more\u0000profound insights into network behavior, thereby improving both detection\u0000performance and the adaptability of the system. This merging of technologies\u0000not only enhances the functionality of traditional NIDS but also sets the stage\u0000for future developments in building more resilient, interpretable, and dynamic\u0000defense mechanisms against advanced cyber threats. The continued progress in\u0000this area is poised to transform NIDS into a system that is both responsive to\u0000known threats and anticipatory of emerging, unseen ones.","PeriodicalId":501033,"journal":{"name":"arXiv - CS - Symbolic Computation","volume":"43 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141252118","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}
Viktor Martinek, Julia Reuter, Ophelia Frotscher, Sanaz Mostaghim, Markus Richter, Roland Herzog
We study the addition of shape constraints and their consideration during the parameter estimation step of symbolic regression (SR). Shape constraints serve as a means to introduce prior knowledge about the shape of the otherwise unknown model function into SR. Unlike previous works that have explored shape constraints in SR, we propose minimizing shape constraint violations during parameter estimation using gradient-based numerical optimization. We test three algorithm variants to evaluate their performance in identifying three symbolic expressions from a synthetically generated data set. This paper examines two benchmark scenarios: one with varying noise levels and another with reduced amounts of training data. The results indicate that incorporating shape constraints into the expression search is particularly beneficial when data is scarce. Compared to using shape constraints only in the selection process, our approach of minimizing violations during parameter estimation shows a statistically significant benefit in some of our test cases, without being significantly worse in any instance.
我们研究了在符号回归（SR）的参数估计步骤中增加形状约束及其考虑因素。形状约束是一种在 SR 中引入关于未知模型函数形状的先验知识的方法。与之前在 SR 中探讨形状约束的工作不同，我们建议在参数估计过程中使用基于梯度的数值优化来最小化违反形状约束的情况。我们测试了三种算法变体，以评估它们在从合成生成的数据集中识别三种符号表达式时的性能。本论文对两种基准情景进行了测试：一种是噪声水平不同的情景，另一种是训练数据量减少的情景。结果表明，在数据稀缺的情况下，将形状约束纳入表达式搜索尤其有益。与仅在选择过程中使用形状约束相比，我们在参数估计过程中最小化违规的方法在一些测试案例中显示出了统计学上的显著优势，而在任何情况下都没有明显的劣势。
{"title":"Shape Constraints in Symbolic Regression using Penalized Least Squares","authors":"Viktor Martinek, Julia Reuter, Ophelia Frotscher, Sanaz Mostaghim, Markus Richter, Roland Herzog","doi":"arxiv-2405.20800","DOIUrl":"https://doi.org/arxiv-2405.20800","url":null,"abstract":"We study the addition of shape constraints and their consideration during the\u0000parameter estimation step of symbolic regression (SR). Shape constraints serve\u0000as a means to introduce prior knowledge about the shape of the otherwise\u0000unknown model function into SR. Unlike previous works that have explored shape\u0000constraints in SR, we propose minimizing shape constraint violations during\u0000parameter estimation using gradient-based numerical optimization. We test three algorithm variants to evaluate their performance in identifying\u0000three symbolic expressions from a synthetically generated data set. This paper\u0000examines two benchmark scenarios: one with varying noise levels and another\u0000with reduced amounts of training data. The results indicate that incorporating\u0000shape constraints into the expression search is particularly beneficial when\u0000data is scarce. Compared to using shape constraints only in the selection\u0000process, our approach of minimizing violations during parameter estimation\u0000shows a statistically significant benefit in some of our test cases, without\u0000being significantly worse in any instance.","PeriodicalId":501033,"journal":{"name":"arXiv - CS - Symbolic Computation","volume":"13 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141252120","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}
Bigraphs are a versatile modelling formalism that allows easy expression of placement and connectivity relations in a graphical format. System evolution is user defined as a set of rewrite rules. This paper presents a practical, yet detailed guide to developing, executing, and reasoning about bigraph models, including recent extensions such as parameterised, instantaneous, prioritised and conditional rules, and probabilistic and stochastic rewriting.
{"title":"Practical Modelling with Bigraphs","authors":"Blair Archibald, Muffy Calder, Michele Sevegnani","doi":"arxiv-2405.20745","DOIUrl":"https://doi.org/arxiv-2405.20745","url":null,"abstract":"Bigraphs are a versatile modelling formalism that allows easy expression of\u0000placement and connectivity relations in a graphical format. System evolution is\u0000user defined as a set of rewrite rules. This paper presents a practical, yet\u0000detailed guide to developing, executing, and reasoning about bigraph models,\u0000including recent extensions such as parameterised, instantaneous, prioritised\u0000and conditional rules, and probabilistic and stochastic rewriting.","PeriodicalId":501033,"journal":{"name":"arXiv - CS - Symbolic Computation","volume":"3 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141252124","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}
Aditya Ganeshan, Ryan Y. Huang, Xianghao Xu, R. Kenny Jones, Daniel Ritchie
The ability to edit 3D assets from natural language presents a compelling paradigm to aid in the democratization of 3D content creation. However, while natural language is often effective at communicating general intent, it is poorly suited for specifying precise manipulation. To address this gap, we introduce ParSEL, a system that enables controllable editing of high-quality 3D assets from natural language. Given a segmented 3D mesh and an editing request, ParSEL produces a parameterized editing program. Adjusting the program parameters allows users to explore shape variations with a precise control over the magnitudes of edits. To infer editing programs which align with an input edit request, we leverage the abilities of large-language models (LLMs). However, while we find that LLMs excel at identifying initial edit operations, they often fail to infer complete editing programs, and produce outputs that violate shape semantics. To overcome this issue, we introduce Analytical Edit Propagation (AEP), an algorithm which extends a seed edit with additional operations until a complete editing program has been formed. Unlike prior methods, AEP searches for analytical editing operations compatible with a range of possible user edits through the integration of computer algebra systems for geometric analysis. Experimentally we demonstrate ParSEL's effectiveness in enabling controllable editing of 3D objects through natural language requests over alternative system designs.
用自然语言编辑 3D 资产的能力为 3D 内容创作的民主化提供了一个引人注目的范式。然而，虽然自然语言通常能有效传达一般意图，但却不太适合指定精确操作。为了弥补这一不足，我们推出了 ParSEL 系统，它可以通过自然语言对高质量的 3D 资产进行可控编辑。给定一个分割的三维网格和一个编辑请求，ParSEL 会生成一个参数化的编辑程序。通过调整程序参数，用户可以探索形状的变化，并精确控制编辑的幅度。为了推断出与输入编辑请求相一致的编辑程序，我们利用了大型语言模型（LLM）的能力。然而，尽管我们发现 LLM 擅长识别初始编辑操作，但它们往往无法推断出完整的编辑程序，并产生违反形状语义的输出。为了克服这个问题，我们引入了分析编辑推广算法（AEP），这种算法通过附加操作来扩展种子编辑，直到形成完整的编辑程序。与传统方法不同的是，AEP 通过整合计算机代数系统的计量分析，寻找与一系列可能的用户编辑相兼容的分析编辑操作。通过实验，我们证明了 ParSEL 在通过自然语言请求对 3D 物体进行可控编辑方面的有效性。
{"title":"ParSEL: Parameterized Shape Editing with Language","authors":"Aditya Ganeshan, Ryan Y. Huang, Xianghao Xu, R. Kenny Jones, Daniel Ritchie","doi":"arxiv-2405.20319","DOIUrl":"https://doi.org/arxiv-2405.20319","url":null,"abstract":"The ability to edit 3D assets from natural language presents a compelling\u0000paradigm to aid in the democratization of 3D content creation. However, while\u0000natural language is often effective at communicating general intent, it is\u0000poorly suited for specifying precise manipulation. To address this gap, we\u0000introduce ParSEL, a system that enables controllable editing of high-quality 3D\u0000assets from natural language. Given a segmented 3D mesh and an editing request,\u0000ParSEL produces a parameterized editing program. Adjusting the program\u0000parameters allows users to explore shape variations with a precise control over\u0000the magnitudes of edits. To infer editing programs which align with an input\u0000edit request, we leverage the abilities of large-language models (LLMs).\u0000However, while we find that LLMs excel at identifying initial edit operations,\u0000they often fail to infer complete editing programs, and produce outputs that\u0000violate shape semantics. To overcome this issue, we introduce Analytical Edit\u0000Propagation (AEP), an algorithm which extends a seed edit with additional\u0000operations until a complete editing program has been formed. Unlike prior\u0000methods, AEP searches for analytical editing operations compatible with a range\u0000of possible user edits through the integration of computer algebra systems for\u0000geometric analysis. Experimentally we demonstrate ParSEL's effectiveness in\u0000enabling controllable editing of 3D objects through natural language requests\u0000over alternative system designs.","PeriodicalId":501033,"journal":{"name":"arXiv - CS - Symbolic Computation","volume":"149 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141192393","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}
For a given ideal I in K[x_1,...,x_n,y_1,...,y_m] in a polynomial ring with n+m variables, we want to find all elements that can be written as f-g for some f in K[x_1,...,x_n] and some g in K[y_1,...,y_m], i.e., all elements of I that contain no term involving at the same time one of the x_1,...,x_n and one of the y_1,...,y_m. For principal ideals and for ideals of dimension zero, we give a algorithms that compute all these polynomials in a finite number of steps.
对于具有 n+m 个变量的多项式环 K[x_1,...,x_n,y_1,...,y_m] 中的给定理想 I，我们希望找到所有元素，对于 K[x_1,...,x_n] 中的某个 f 和 K[y_1,...,y_m] 中的某个 g，都可以写成 f-g，也就是说、即 I 中的所有元素都不包含同时涉及 x_1,...,x_n 中的一个项和 y_1,...,y_m 中的一个项。对于主理想和维数为零的理想，我们给出了用有限步数计算所有这些多项式的算法。
{"title":"On the Problem of Separating Variables in Multivariate Polynomial Ideals","authors":"Manfred Buchacher, Manuel Kauers","doi":"arxiv-2405.19223","DOIUrl":"https://doi.org/arxiv-2405.19223","url":null,"abstract":"For a given ideal I in K[x_1,...,x_n,y_1,...,y_m] in a polynomial ring with\u0000n+m variables, we want to find all elements that can be written as f-g for some\u0000f in K[x_1,...,x_n] and some g in K[y_1,...,y_m], i.e., all elements of I that\u0000contain no term involving at the same time one of the x_1,...,x_n and one of\u0000the y_1,...,y_m. For principal ideals and for ideals of dimension zero, we give\u0000a algorithms that compute all these polynomials in a finite number of steps.","PeriodicalId":501033,"journal":{"name":"arXiv - CS - Symbolic Computation","volume":"56 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141192463","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}
Julia Reuter, Viktor Martinek, Roland Herzog, Sanaz Mostaghim
When developing empirical equations, domain experts require these to be accurate and adhere to physical laws. Often, constants with unknown units need to be discovered alongside the equations. Traditional unit-aware genetic programming (GP) approaches cannot be used when unknown constants with undetermined units are included. This paper presents a method for dimensional analysis that propagates unknown units as ''jokers'' and returns the magnitude of unit violations. We propose three methods, namely evolutive culling, a repair mechanism, and a multi-objective approach, to integrate the dimensional analysis in the GP algorithm. Experiments on datasets with ground truth demonstrate comparable performance of evolutive culling and the multi-objective approach to a baseline without dimensional analysis. Extensive analysis of the results on datasets without ground truth reveals that the unit-aware algorithms make only low sacrifices in accuracy, while producing unit-adherent solutions. Overall, we presented a promising novel approach for developing unit-adherent empirical equations.
在建立经验方程时，领域专家要求这些方程必须准确并符合物理规律。通常情况下，需要与方程一起发现未知单位的常数。当包含有确定单位的未知常数时，传统的单位感知遗传编程（GP）方法就无法使用了。本文提出了一种维度分析方法，将未知单位作为 "小丑 "进行传播，并返回违反单位的大小。我们提出了三种方法，即进化剔除、配对机制和多目标方法，将维度分析集成到 GP 算法中。在具有地面实况的数据集上进行的实验表明，进化剔除和多目标方法的性能与不进行维度分析的基线相当。对无地面实况数据集的结果进行的广泛分析表明，单元感知算法只牺牲了较低的准确性，同时产生了单元相干解。
{"title":"Unit-Aware Genetic Programming for the Development of Empirical Equations","authors":"Julia Reuter, Viktor Martinek, Roland Herzog, Sanaz Mostaghim","doi":"arxiv-2405.18896","DOIUrl":"https://doi.org/arxiv-2405.18896","url":null,"abstract":"When developing empirical equations, domain experts require these to be\u0000accurate and adhere to physical laws. Often, constants with unknown units need\u0000to be discovered alongside the equations. Traditional unit-aware genetic\u0000programming (GP) approaches cannot be used when unknown constants with\u0000undetermined units are included. This paper presents a method for dimensional\u0000analysis that propagates unknown units as ''jokers'' and returns the magnitude\u0000of unit violations. We propose three methods, namely evolutive culling, a\u0000repair mechanism, and a multi-objective approach, to integrate the dimensional\u0000analysis in the GP algorithm. Experiments on datasets with ground truth\u0000demonstrate comparable performance of evolutive culling and the multi-objective\u0000approach to a baseline without dimensional analysis. Extensive analysis of the\u0000results on datasets without ground truth reveals that the unit-aware algorithms\u0000make only low sacrifices in accuracy, while producing unit-adherent solutions.\u0000Overall, we presented a promising novel approach for developing unit-adherent\u0000empirical equations.","PeriodicalId":501033,"journal":{"name":"arXiv - CS - Symbolic Computation","volume":"100 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141192454","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 introduce an efficient method for learning linear models from uncertain data, where uncertainty is represented as a set of possible variations in the data, leading to predictive multiplicity. Our approach leverages abstract interpretation and zonotopes, a type of convex polytope, to compactly represent these dataset variations, enabling the symbolic execution of gradient descent on all possible worlds simultaneously. We develop techniques to ensure that this process converges to a fixed point and derive closed-form solutions for this fixed point. Our method provides sound over-approximations of all possible optimal models and viable prediction ranges. We demonstrate the effectiveness of our approach through theoretical and empirical analysis, highlighting its potential to reason about model and prediction uncertainty due to data quality issues in training data.
{"title":"Learning from Uncertain Data: From Possible Worlds to Possible Models","authors":"Jiongli Zhu, Su Feng, Boris Glavic, Babak Salimi","doi":"arxiv-2405.18549","DOIUrl":"https://doi.org/arxiv-2405.18549","url":null,"abstract":"We introduce an efficient method for learning linear models from uncertain\u0000data, where uncertainty is represented as a set of possible variations in the\u0000data, leading to predictive multiplicity. Our approach leverages abstract\u0000interpretation and zonotopes, a type of convex polytope, to compactly represent\u0000these dataset variations, enabling the symbolic execution of gradient descent\u0000on all possible worlds simultaneously. We develop techniques to ensure that\u0000this process converges to a fixed point and derive closed-form solutions for\u0000this fixed point. Our method provides sound over-approximations of all possible\u0000optimal models and viable prediction ranges. We demonstrate the effectiveness\u0000of our approach through theoretical and empirical analysis, highlighting its\u0000potential to reason about model and prediction uncertainty due to data quality\u0000issues in training data.","PeriodicalId":501033,"journal":{"name":"arXiv - CS - Symbolic Computation","volume":"43 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141192464","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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