Samuele Pollaci, Babis Kostopoulos, Marc Denecker, Bart Bogaerts
Approximation Fixpoint Theory (AFT) is an algebraic framework designed to study the semantics of non-monotonic logics. Despite its success, AFT is not readily applicable to higher-order definitions. To solve such an issue, we devise a formal mathematical framework employing concepts drawn from Category Theory. In particular, we make use of the notion of Cartesian closed category to inductively construct higher-order approximation spaces while preserving the structures necessary for the correct application of AFT. We show that this novel theoretical approach extends standard AFT to a higher-order environment, and generalizes the AFT setting of arXiv:1804.08335 .
{"title":"A Category-Theoretic Perspective on Higher-Order Approximation Fixpoint Theory (Extended Version)","authors":"Samuele Pollaci, Babis Kostopoulos, Marc Denecker, Bart Bogaerts","doi":"arxiv-2408.11712","DOIUrl":"https://doi.org/arxiv-2408.11712","url":null,"abstract":"Approximation Fixpoint Theory (AFT) is an algebraic framework designed to\u0000study the semantics of non-monotonic logics. Despite its success, AFT is not\u0000readily applicable to higher-order definitions. To solve such an issue, we\u0000devise a formal mathematical framework employing concepts drawn from Category\u0000Theory. In particular, we make use of the notion of Cartesian closed category\u0000to inductively construct higher-order approximation spaces while preserving the\u0000structures necessary for the correct application of AFT. We show that this\u0000novel theoretical approach extends standard AFT to a higher-order environment,\u0000and generalizes the AFT setting of arXiv:1804.08335 .","PeriodicalId":501208,"journal":{"name":"arXiv - CS - Logic in Computer Science","volume":"7 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192506","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}
Minghao Liu, David M. Cerna, Filipe Gouveia, Andrew Cropper
Knowledge refactoring compresses a logic program by introducing new rules. Current approaches struggle to scale to large programs. To overcome this limitation, we introduce a constrained optimisation refactoring approach. Our first key idea is to encode the problem with decision variables based on literals rather than rules. Our second key idea is to focus on linear invented rules. Our empirical results on multiple domains show that our approach can refactor programs quicker and with more compression than the previous state-of-the-art approach, sometimes by 60%.
{"title":"Scalable Knowledge Refactoring using Constrained Optimisation","authors":"Minghao Liu, David M. Cerna, Filipe Gouveia, Andrew Cropper","doi":"arxiv-2408.11530","DOIUrl":"https://doi.org/arxiv-2408.11530","url":null,"abstract":"Knowledge refactoring compresses a logic program by introducing new rules.\u0000Current approaches struggle to scale to large programs. To overcome this\u0000limitation, we introduce a constrained optimisation refactoring approach. Our\u0000first key idea is to encode the problem with decision variables based on\u0000literals rather than rules. Our second key idea is to focus on linear invented\u0000rules. Our empirical results on multiple domains show that our approach can\u0000refactor programs quicker and with more compression than the previous\u0000state-of-the-art approach, sometimes by 60%.","PeriodicalId":501208,"journal":{"name":"arXiv - CS - Logic in Computer Science","volume":"395 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192458","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}
Damiano Azzolini, Elena Bellodi, Rafael Kiesel, Fabrizio Riguzzi
Solving a decision theory problem usually involves finding the actions, among a set of possible ones, which optimize the expected reward, possibly accounting for the uncertainty of the environment. In this paper, we introduce the possibility to encode decision theory problems with Probabilistic Answer Set Programming under the credal semantics via decision atoms and utility attributes. To solve the task we propose an algorithm based on three layers of Algebraic Model Counting, that we test on several synthetic datasets against an algorithm that adopts answer set enumeration. Empirical results show that our algorithm can manage non trivial instances of programs in a reasonable amount of time. Under consideration in Theory and Practice of Logic Programming (TPLP).
{"title":"Solving Decision Theory Problems with Probabilistic Answer Set Programming","authors":"Damiano Azzolini, Elena Bellodi, Rafael Kiesel, Fabrizio Riguzzi","doi":"arxiv-2408.11371","DOIUrl":"https://doi.org/arxiv-2408.11371","url":null,"abstract":"Solving a decision theory problem usually involves finding the actions, among\u0000a set of possible ones, which optimize the expected reward, possibly accounting\u0000for the uncertainty of the environment. In this paper, we introduce the\u0000possibility to encode decision theory problems with Probabilistic Answer Set\u0000Programming under the credal semantics via decision atoms and utility\u0000attributes. To solve the task we propose an algorithm based on three layers of\u0000Algebraic Model Counting, that we test on several synthetic datasets against an\u0000algorithm that adopts answer set enumeration. Empirical results show that our\u0000algorithm can manage non trivial instances of programs in a reasonable amount\u0000of time. Under consideration in Theory and Practice of Logic Programming\u0000(TPLP).","PeriodicalId":501208,"journal":{"name":"arXiv - CS - Logic in Computer Science","volume":"74 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192461","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}
This expository note describes two convenient techniques in the context of homotopy type theory for proving and formalizing that a given map is an equivalence. The first technique decomposes the map as a series of basic equivalences, while the second refines this approach using the 3-for-2 property of equivalences. The techniques are illustrated by proving a basic result in synthetic homotopy theory.
{"title":"Formalizing equivalences without tears","authors":"Tom de Jong","doi":"arxiv-2408.11501","DOIUrl":"https://doi.org/arxiv-2408.11501","url":null,"abstract":"This expository note describes two convenient techniques in the context of\u0000homotopy type theory for proving and formalizing that a given map is an\u0000equivalence. The first technique decomposes the map as a series of basic\u0000equivalences, while the second refines this approach using the 3-for-2 property\u0000of equivalences. The techniques are illustrated by proving a basic result in\u0000synthetic homotopy theory.","PeriodicalId":501208,"journal":{"name":"arXiv - CS - Logic in Computer Science","volume":"7 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192460","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}
This work proposes a novel approach for automatic verification and synthesis of infinite-state reactive programs with respect to ${CTL}^*$ specifications, based on translation to Existential Horn Clauses (EHCs). $CTL^*$ is a powerful temporal logic, which subsumes the temporal logics LTL and CTL, both widely used in specification, verification, and synthesis of complex systems. EHCs with its solver E-HSF, is an extension of Constrained Horn Clauses, which includes existential quantification as well as the power of handling well-foundedness. We develop the translation system textit{Trans}, which given a verification problem consisting of a program $P$ and a specification $phi$, builds a set of EHCs which is satisfiable iff $P$ satisfies $phi$. We also develop a synthesis algorithm that given a program with holes in conditions and assignments, fills the holes so that the synthesized program satisfies the given $CTL^*$ specification. We prove that our verification and synthesis algorithms are both sound and relative complete. Finally, we present case studies to demonstrate the applicability of our algorithms for $CTL^*$ verification and synthesis.
{"title":"CTL* Verification and Synthesis using Existential Horn Clauses","authors":"Mishel Carelli, Orna Grumberg","doi":"arxiv-2408.11502","DOIUrl":"https://doi.org/arxiv-2408.11502","url":null,"abstract":"This work proposes a novel approach for automatic verification and synthesis\u0000of infinite-state reactive programs with respect to ${CTL}^*$ specifications,\u0000based on translation to Existential Horn Clauses (EHCs). $CTL^*$ is a powerful temporal logic, which subsumes the temporal logics LTL\u0000and CTL, both widely used in specification, verification, and synthesis of\u0000complex systems. EHCs with its solver E-HSF, is an extension of Constrained Horn Clauses,\u0000which includes existential quantification as well as the power of handling\u0000well-foundedness. We develop the translation system textit{Trans}, which given a verification\u0000problem consisting of a program $P$ and a specification $phi$, builds a set of\u0000EHCs which is satisfiable iff $P$ satisfies $phi$. We also develop a synthesis\u0000algorithm that given a program with holes in conditions and assignments, fills\u0000the holes so that the synthesized program satisfies the given $CTL^*$\u0000specification. We prove that our verification and synthesis algorithms are both sound and\u0000relative complete. Finally, we present case studies to demonstrate the\u0000applicability of our algorithms for $CTL^*$ verification and synthesis.","PeriodicalId":501208,"journal":{"name":"arXiv - CS - Logic in Computer Science","volume":"15 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192459","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}
Litian Huang, Xinguo Yu, Feng Xiong, Bin He, Shengbing Tang, Jiawen Fu
Solving Algebra Problems with Geometry Diagrams (APGDs) is still a challenging problem because diagram processing is not studied as intensively as language processing. To work against this challenge, this paper proposes a hologram reasoning scheme and develops a high-performance method for solving APGDs by using this scheme. To reach this goal, it first defines a hologram, being a kind of graph, and proposes a hologram generator to convert a given APGD into a hologram, which represents the entire information of APGD and the relations for solving the problem can be acquired from it by a uniform way. Then HGR, a hologram reasoning method employs a pool of prepared graph models to derive algebraic equations, which is consistent with the geometric theorems. This method is able to be updated by adding new graph models into the pool. Lastly, it employs deep reinforcement learning to enhance the efficiency of model selection from the pool. The entire HGR not only ensures high solution accuracy with fewer reasoning steps but also significantly enhances the interpretability of the solution process by providing descriptions of all reasoning steps. Experimental results demonstrate the effectiveness of HGR in improving both accuracy and interpretability in solving APGDs.
{"title":"Hologram Reasoning for Solving Algebra Problems with Geometry Diagrams","authors":"Litian Huang, Xinguo Yu, Feng Xiong, Bin He, Shengbing Tang, Jiawen Fu","doi":"arxiv-2408.10592","DOIUrl":"https://doi.org/arxiv-2408.10592","url":null,"abstract":"Solving Algebra Problems with Geometry Diagrams (APGDs) is still a\u0000challenging problem because diagram processing is not studied as intensively as\u0000language processing. To work against this challenge, this paper proposes a\u0000hologram reasoning scheme and develops a high-performance method for solving\u0000APGDs by using this scheme. To reach this goal, it first defines a hologram,\u0000being a kind of graph, and proposes a hologram generator to convert a given\u0000APGD into a hologram, which represents the entire information of APGD and the\u0000relations for solving the problem can be acquired from it by a uniform way.\u0000Then HGR, a hologram reasoning method employs a pool of prepared graph models\u0000to derive algebraic equations, which is consistent with the geometric theorems.\u0000This method is able to be updated by adding new graph models into the pool.\u0000Lastly, it employs deep reinforcement learning to enhance the efficiency of\u0000model selection from the pool. The entire HGR not only ensures high solution\u0000accuracy with fewer reasoning steps but also significantly enhances the\u0000interpretability of the solution process by providing descriptions of all\u0000reasoning steps. Experimental results demonstrate the effectiveness of HGR in\u0000improving both accuracy and interpretability in solving APGDs.","PeriodicalId":501208,"journal":{"name":"arXiv - CS - Logic in Computer Science","volume":"75 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192464","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}
Johannes K. Fichte, Markus Hecher, Yasir Mahmood, Arne Meier
Abstract argumentation is a popular toolkit for modeling, evaluating, and comparing arguments. Relationships between arguments are specified in argumentation frameworks (AFs), and conditions are placed on sets (extensions) of arguments that allow AFs to be evaluated. For more expressiveness, AFs are augmented with emph{acceptance conditions} on directly interacting arguments or a constraint on the admissible sets of arguments, resulting in dialectic frameworks or constrained argumentation frameworks. In this paper, we consider flexible conditions for emph{rejecting} an argument from an extension, which we call rejection conditions (RCs). On the technical level, we associate each argument with a specific logic program. We analyze the resulting complexity, including the structural parameter treewidth. Rejection AFs are highly expressive, giving rise to natural problems on higher levels of the polynomial hierarchy.
抽象论证是一种流行的工具包，用于对论据进行建模、评估和比较。论证框架（AFs）中规定了论证之间的关系，并在论证集合（扩展）上设置了条件，以便对 AFs 进行评估。为了提高表达能力，论证框架会在直接交互的论据上附加 "接受条件"，或在可接受的论据集上附加约束，从而形成辩证框架或约束论证框架。在本文中，我们考虑的是将一个论证从一个扩展中 "拒绝 "出去的灵活条件，我们称之为拒绝条件（RCs）。在技术层面，我们将每个论证与特定的逻辑程序关联起来。我们分析了由此产生的复杂性，包括结构参数树宽。拒绝 AF 具有很强的表达能力，因此会产生多项式层次结构中更高层次的自然问题。
{"title":"Rejection in Abstract Argumentation: Harder Than Acceptance?","authors":"Johannes K. Fichte, Markus Hecher, Yasir Mahmood, Arne Meier","doi":"arxiv-2408.10683","DOIUrl":"https://doi.org/arxiv-2408.10683","url":null,"abstract":"Abstract argumentation is a popular toolkit for modeling, evaluating, and\u0000comparing arguments. Relationships between arguments are specified in\u0000argumentation frameworks (AFs), and conditions are placed on sets (extensions)\u0000of arguments that allow AFs to be evaluated. For more expressiveness, AFs are\u0000augmented with emph{acceptance conditions} on directly interacting arguments\u0000or a constraint on the admissible sets of arguments, resulting in dialectic\u0000frameworks or constrained argumentation frameworks. In this paper, we consider\u0000flexible conditions for emph{rejecting} an argument from an extension, which\u0000we call rejection conditions (RCs). On the technical level, we associate each\u0000argument with a specific logic program. We analyze the resulting complexity,\u0000including the structural parameter treewidth. Rejection AFs are highly\u0000expressive, giving rise to natural problems on higher levels of the polynomial\u0000hierarchy.","PeriodicalId":501208,"journal":{"name":"arXiv - CS - Logic in Computer Science","volume":"24 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192463","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}
John Lindqvist, Fernando R. Velázquez-Quesada, Thomas Ågotnes
Motivated by the search for forms of distributed belief that do not collapse in the face of conflicting information, this paper introduces the notions of cautious and bold distributed belief. Both notions rely on maximally consistent subgroups of agents, with cautious quantifying universally and bold quantifying existentially. As a result, while the cautious distributed belief of a group is inconsistent only when all group members are individually inconsistent, the bold distributed belief of a group is never inconsistent. The paper discusses these two notions, presenting their respective modalities and semantic interpretations, discussing some of their basic properties, studying whether they preserve doxastic properties from the members of the group, and comparing them not only with standard distributed belief but also with one another, both at the level of modalities and at the level of language expressivity.
{"title":"Variations on distributed belief","authors":"John Lindqvist, Fernando R. Velázquez-Quesada, Thomas Ågotnes","doi":"arxiv-2408.10637","DOIUrl":"https://doi.org/arxiv-2408.10637","url":null,"abstract":"Motivated by the search for forms of distributed belief that do not collapse\u0000in the face of conflicting information, this paper introduces the notions of\u0000cautious and bold distributed belief. Both notions rely on maximally consistent\u0000subgroups of agents, with cautious quantifying universally and bold quantifying\u0000existentially. As a result, while the cautious distributed belief of a group is\u0000inconsistent only when all group members are individually inconsistent, the\u0000bold distributed belief of a group is never inconsistent. The paper discusses\u0000these two notions, presenting their respective modalities and semantic\u0000interpretations, discussing some of their basic properties, studying whether\u0000they preserve doxastic properties from the members of the group, and comparing\u0000them not only with standard distributed belief but also with one another, both\u0000at the level of modalities and at the level of language expressivity.","PeriodicalId":501208,"journal":{"name":"arXiv - CS - Logic in Computer Science","volume":"69 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192462","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}
Emanuele De AngelisCNR-IASI, Rome, Italy, Maurizio ProiettiCNR-IASI, Rome, Italy, Francesca ToniImperial, London, UK
Assumption-based Argumentation (ABA) is advocated as a unifying formalism for various forms of non-monotonic reasoning, including logic programming. It allows capturing defeasible knowledge, subject to argumentative debate. While, in much existing work, ABA frameworks are given up-front, in this paper we focus on the problem of automating their learning from background knowledge and positive/negative examples. Unlike prior work, we newly frame the problem in terms of brave reasoning under stable extensions for ABA. We present a novel algorithm based on transformation rules (such as Rote Learning, Folding, Assumption Introduction and Fact Subsumption) and an implementation thereof that makes use of Answer Set Programming. Finally, we compare our technique to state-of-the-art ILP systems that learn defeasible knowledge.
基于假设的论证（ABA）被认为是包括逻辑编程在内的各种非单调推理形式的统一形式主义。它允许捕捉可失败的知识，并进行论证辩论。在许多现有工作中，ABA 框架都是预先给出的，而在本文中，我们将重点放在从背景知识和正/负示例中自动学习 ABA 框架的问题上。与之前的工作不同，我们新提出了在 ABA 稳定扩展下的勇敢推理问题。我们提出了一种基于转换规则（如记诵学习、折叠、假设引入和事实归纳）的新算法，并利用答案集编程实现了该算法。最后，我们将我们的技术与最先进的学习可败知识的 ILP 系统进行了比较。
{"title":"Learning Brave Assumption-Based Argumentation Frameworks via ASP","authors":"Emanuele De AngelisCNR-IASI, Rome, Italy, Maurizio ProiettiCNR-IASI, Rome, Italy, Francesca ToniImperial, London, UK","doi":"arxiv-2408.10126","DOIUrl":"https://doi.org/arxiv-2408.10126","url":null,"abstract":"Assumption-based Argumentation (ABA) is advocated as a unifying formalism for\u0000various forms of non-monotonic reasoning, including logic programming. It\u0000allows capturing defeasible knowledge, subject to argumentative debate. While,\u0000in much existing work, ABA frameworks are given up-front, in this paper we\u0000focus on the problem of automating their learning from background knowledge and\u0000positive/negative examples. Unlike prior work, we newly frame the problem in\u0000terms of brave reasoning under stable extensions for ABA. We present a novel\u0000algorithm based on transformation rules (such as Rote Learning, Folding,\u0000Assumption Introduction and Fact Subsumption) and an implementation thereof\u0000that makes use of Answer Set Programming. Finally, we compare our technique to\u0000state-of-the-art ILP systems that learn defeasible knowledge.","PeriodicalId":501208,"journal":{"name":"arXiv - CS - Logic in Computer Science","volume":"69 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192470","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}
Martin Grohe, Christoph Standke, Juno Steegmans, Jan Van den Bussche
We lay the foundations for a database-inspired approach to interpreting and understanding neural network models by querying them using declarative languages. Towards this end we study different query languages, based on first-order logic, that mainly differ in their access to the neural network model. First-order logic over the reals naturally yields a language which views the network as a black box; only the input--output function defined by the network can be queried. This is essentially the approach of constraint query languages. On the other hand, a white-box language can be obtained by viewing the network as a weighted graph, and extending first-order logic with summation over weight terms. The latter approach is essentially an abstraction of SQL. In general, the two approaches are incomparable in expressive power, as we will show. Under natural circumstances, however, the white-box approach can subsume the black-box approach; this is our main result. We prove the result concretely for linear constraint queries over real functions definable by feedforward neural networks with a fixed number of hidden layers and piecewise linear activation functions.
{"title":"Query languages for neural networks","authors":"Martin Grohe, Christoph Standke, Juno Steegmans, Jan Van den Bussche","doi":"arxiv-2408.10362","DOIUrl":"https://doi.org/arxiv-2408.10362","url":null,"abstract":"We lay the foundations for a database-inspired approach to interpreting and\u0000understanding neural network models by querying them using declarative\u0000languages. Towards this end we study different query languages, based on\u0000first-order logic, that mainly differ in their access to the neural network\u0000model. First-order logic over the reals naturally yields a language which views\u0000the network as a black box; only the input--output function defined by the\u0000network can be queried. This is essentially the approach of constraint query\u0000languages. On the other hand, a white-box language can be obtained by viewing\u0000the network as a weighted graph, and extending first-order logic with summation\u0000over weight terms. The latter approach is essentially an abstraction of SQL. In\u0000general, the two approaches are incomparable in expressive power, as we will\u0000show. Under natural circumstances, however, the white-box approach can subsume\u0000the black-box approach; this is our main result. We prove the result concretely\u0000for linear constraint queries over real functions definable by feedforward\u0000neural networks with a fixed number of hidden layers and piecewise linear\u0000activation functions.","PeriodicalId":501208,"journal":{"name":"arXiv - CS - Logic in Computer Science","volume":"131 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192465","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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