Annals of Mathematics and Artificial Intelligence最新文献

{"title":"Towards a structural analysis of interesting geometric theorems: analysis of a survey","authors":"Pedro Quaresma,&nbsp;Pierluigi Graziani","doi":"10.1007/s10472-026-10020-6","DOIUrl":"10.1007/s10472-026-10020-6","url":null,"abstract":"<div><p>A longstanding challenge in automated theorem generation is not merely to derive new statements, but to identify which of them are mathematically interesting. Motivated by Wos’ Problem 31, this paper reports the results of a first survey, within a two-stage expert-survey study on theorem interestingness in geometry, and explores whether some of the dimensions elicited from human judgements admit formally computable structural counterparts. The descriptive analysis identifies six recurring, non-exclusive and non-exhaustive dimensions of interestingness: simplicity and clarity, surprise, conceptual depth, usefulness and applicability, aesthetic value, and historical, pedagogical, or emotional relevance. We then examine these dimensions in relation to a modern geometrographic analysis of area-method proofs, considering the overall complexity of a proof, the number and complexity of its steps, the occurrence and diversity of difficult lemmas, and the <i>Geometrographic Readability Coefficient of Proofs</i> (GRCP). The comparison suggests that simplicity, clarity, and certain aspects of conceptual depth may have plausible structural counterparts in proof features. By contrast, surprise, usefulness, aesthetic value, and historical or pedagogical relevance remain strongly dependent on mathematical and human context. The analysis is exploratory and does not establish statistical correlations or validate GRCP as a measure of theorem interestingness. Rather, it provides an empirical and formal basis for hypotheses to be tested in the second survey and in subsequent comparisons with human judgements of proof readability.</p></div>","PeriodicalId":7971,"journal":{"name":"Annals of Mathematics and Artificial Intelligence","volume":"94 5","pages":"843 - 866"},"PeriodicalIF":1.6,"publicationDate":"2026-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10472-026-10020-6.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149331303","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On integrated information under Krohn-Rhodes decomposition","authors":"Arnaud Mayeux","doi":"10.1007/s10472-026-10021-5","DOIUrl":"10.1007/s10472-026-10021-5","url":null,"abstract":"<div><p>We provide explicit examples of finite systems that exhibit non-zero integrated information while already in Krohn-Rhodes form. These constructions serve as counterexamples to claims in the literature that Krohn-Rhodes decomposition systematically removes intrinsic integration. This indicates that the relationship between algebraic decomposition and integrated information is more nuanced than commonly assumed.</p></div>","PeriodicalId":7971,"journal":{"name":"Annals of Mathematics and Artificial Intelligence","volume":"94 5","pages":"739 - 749"},"PeriodicalIF":1.6,"publicationDate":"2026-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10472-026-10021-5.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149331299","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Locating semialgebraic hardness in matrix-factorization relaxations of clustering","authors":"Angshul Majumdar","doi":"10.1007/s10472-026-10022-4","DOIUrl":"10.1007/s10472-026-10022-4","url":null,"abstract":"<div><p>Clustering is usually analysed as a discrete partitioning problem, yet many of its practically important formulations are continuous structured factorizations over the reals. This paper asks where the difficulty of those continuous formulations actually resides. We isolate three recurring families—prototype factorizations, orthogonality-constrained spectral factorizations, and symmetric similarity factorizations—and show that they behave in genuinely different ways. For the spectral branch we prove a collapse theorem: nonnegativity together with column orthonormality forces the columns of the embedding to have pairwise disjoint supports, so the relaxation degenerates into a partition search and its difficulty is combinatorial rather than semialgebraic; without nonnegativity the same decision problem is settled by Ky Fan’s maximum principle and carries no complexity content at all. For the similarity branch the picture is the opposite. Building on a rank-two nonnegative Gram encoding established in a companion preprint by the present author, we prove that similarity-factorization feasibility is <span>(exists mathbb {R})</span>-complete at rank two when the only side information consists of <i>tie constraints</i>, that is, assertions that two prescribed pairs of objects are exactly equally similar. Tie constraints are ordinary comparison-based side information. We then settle the rank parameter as well, and the two constraint languages separate. Under arbitrary affine relations the problem is already existential-real complete at rank <i>one</i>, where no embedding geometry survives at all, so that language measures nothing about the factorization; under tie constraints rank one is decidable in polynomial time, so rank two is the exact threshold and the hardness is a genuine property of nonnegative Gram geometry. The rank-two template with prescribed similarity values but no relational side information is also polynomial-time decidable. Together these results give the first evidence that hardness in continuous clustering relaxations is branch-dependent: algebraic on the similarity side, combinatorial on the spectral side.</p></div>","PeriodicalId":7971,"journal":{"name":"Annals of Mathematics and Artificial Intelligence","volume":"94 5","pages":"773 - 795"},"PeriodicalIF":1.6,"publicationDate":"2026-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149331304","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Preface for Special issue on Data Science meets Optimization III","authors":"Yaoxin Wu,&nbsp;Jayanta Mandi,&nbsp;Hoong Chuin Lau,&nbsp;Michele Lombardi,&nbsp;Neil Yorke-Smith,&nbsp;Yingqian Zhang","doi":"10.1007/s10472-026-10015-3","DOIUrl":"10.1007/s10472-026-10015-3","url":null,"abstract":"","PeriodicalId":7971,"journal":{"name":"Annals of Mathematics and Artificial Intelligence","volume":"94 3","pages":"327 - 328"},"PeriodicalIF":1.6,"publicationDate":"2026-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148613197","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the complexity of least general generalization with commutative functions","authors":"Tatsuya Akutsu,&nbsp;Atsuhiro Takasu","doi":"10.1007/s10472-026-10014-4","DOIUrl":"10.1007/s10472-026-10014-4","url":null,"abstract":"<div><p>Least general generalization is the problem of finding the most specific term that can generate all given terms via substitutions. It is also known as anti-unification and is considered a dual of the most general unification. It is known that both the least general generalization problem and the unification problem can be solved in polynomial time; however the latter becomes NP-hard when commutative function symbols are allowed. In this paper, we study the complexity of the least general generalization problem with commutative functions, formulated as a minimization problem. In particular, we show that this problem is NP-hard even if both input and output terms are restricted so that each variable occurs only once. We also show that this restricted variant can be solved in polynomial time if both the maximum arity and the number of input terms are bounded by constants. For the general terms, we present exponential-time algorithms for finding the most specific term in some restricted cases and in the general case.</p></div>","PeriodicalId":7971,"journal":{"name":"Annals of Mathematics and Artificial Intelligence","volume":"94 5","pages":"817 - 841"},"PeriodicalIF":1.6,"publicationDate":"2026-06-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10472-026-10014-4.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149331302","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Reducing the number of trees in a bag of binary decision trees governed by the majority vote","authors":"Tatsuya Akutsu,&nbsp;Avraham A. Melkman,&nbsp;Atsuhiro Takasu","doi":"10.1007/s10472-026-10013-5","DOIUrl":"10.1007/s10472-026-10013-5","url":null,"abstract":"<div><p>In this paper, we focus on the prediction phase of a random forest and study the problem of representing a bag of decision trees using a smaller bag of decision trees, where we only consider binary decision problems on the binary domain and simple decision trees in which an internal node is limited to querying the Boolean value of a single variable. As a main result, we show that given a constant integer <i>c</i> the majority function of an odd number <i>n</i> of variables can be represented by a bag of <span>(n-2c)</span> decision trees each of which has size polynomial in <i>n</i>. We also show that a general bag of <i>n</i> decision trees can be represented by another bag containing only <span>(n-2c)</span> polynomial-size decision trees, provided a small classification error is allowed. A related result on the <i>k</i>-out-of-<i>n</i> functions is presented as well.</p></div>","PeriodicalId":7971,"journal":{"name":"Annals of Mathematics and Artificial Intelligence","volume":"94 5","pages":"797 - 815"},"PeriodicalIF":1.6,"publicationDate":"2026-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10472-026-10013-5.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149331305","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A comparison of the Myerson value and the position value","authors":"Ayşe Mutlu Derya","doi":"10.1007/s10472-026-10011-7","DOIUrl":"10.1007/s10472-026-10011-7","url":null,"abstract":"<div><p>In the realm of graph-restricted games, the underlying network structure plays a pivotal role, enforcing a key constraint: communication between agents is possible only if a valid path connecting them exists within the network. This constraint strongly shapes the dynamics and strategic considerations, particularly in value allocation settings. The Myerson value and the position value are two prominent allocation rules in such contexts. While classical axiomatic characterizations of these rules often rely on structural assumptions, such as component additivity of the value function or restrictions on the network, these assumptions limit their applicability in general, non-additive environments. We provide new axiomatic characterizations of both values that apply to arbitrary (including non-additive) value functions. Our key axiom specifies how an allocation rule should respond when a source network <i>g</i> and all its supergraphs experience a uniform perturbation <span>(epsilon )</span>, which may represent either an increase or a decrease in value. For the Myerson value, this perturbation is shared equally among the agents with links in <i>g</i> (EDBA). For the position value, it is shared equally among the links in <i>g</i> and then split between their endpoints (EDBL). Combined with a null-game property, each axiom uniquely identifies its respective value. This parallel axiomatization highlights the fundamental contrast between agent-oriented and link-oriented allocation and extends both values beyond previously assumed structural restrictions.</p></div>","PeriodicalId":7971,"journal":{"name":"Annals of Mathematics and Artificial Intelligence","volume":"94 4","pages":"419 - 451"},"PeriodicalIF":1.6,"publicationDate":"2026-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148927610","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Automated conjecturing with TxGraffiti","authors":"Randy Davila","doi":"10.1007/s10472-026-10005-5","DOIUrl":"10.1007/s10472-026-10005-5","url":null,"abstract":"<div><p><i>Texas Graffiti</i> (<i>TxGraffiti</i>) is an automated conjecturing system that operates on a finite, versioned <i>snapshot table</i> of mathematical objects equipped with precomputed numerical invariants and Boolean predicates. For a chosen target invariant and hypothesis predicate, <i>TxGraffiti</i> searches over predicate–predictor combinations and generates <i>table-true</i> conditional inequalities by fitting coefficients within simple, human-readable templates (e.g., univariate affine upper and lower bounds) via a sequence of small linear programs. The resulting candidates are ranked and compressed using heuristics that measure informativeness (e.g., touch/sharp behavior) and remove redundancy or transitive implication, producing a compact set of interpretable conjectures suitable for further testing and proof. We describe the system architecture, data curation workflow, optimization models, and filtering procedures, and we provide an interactive web interface together with an open-source Python implementation. Although our examples focus on graph invariants, the snapshot-table paradigm and optimization-and-filtering pipeline apply broadly whenever objects admit computable features and predicates.</p></div>","PeriodicalId":7971,"journal":{"name":"Annals of Mathematics and Artificial Intelligence","volume":"94 4","pages":"453 - 476"},"PeriodicalIF":1.6,"publicationDate":"2026-04-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148927611","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A prudent logic of partial functions","authors":"Đorđe Marković,&nbsp;Robbe Van den Eede,&nbsp;Marc Denecker","doi":"10.1007/s10472-025-09999-1","DOIUrl":"10.1007/s10472-025-09999-1","url":null,"abstract":"<div><p>Partial functions are ubiquitous in Knowledge Representation applications, ranging from practical, e.g., business applications, to more abstract, e.g., mathematical and programming applications. Expressing propositions about partial functions may lead to non/denoting terms. These result in undefinedness errors and ambiguity, causing subtle modeling and reasoning problems. In our approach, formulae are well-defined (<i>true</i> or <i>false</i>) and non/ambiguous in all structures. We develop a base extension of three-valued predicate logic, in which partial function terms are <i>guarded</i> by domain expressions, ensuring the well/definedness property despite the three-valued nature of the underlying logic. This property allows us to define the satisfaction relation of the new logic in terms of the standard two-valued logic of total functions. To tackle the verbosity of this core language, we propose different ways to increase convenience by means of disambiguating annotations and non-commutative connectives. As a practically relevant result, we prove that many different <i>unnesting</i> methods, which eliminate (partial) functions by replacing them with their graph predicates, are equivalence-preserving in the proposed language. Furthermore, we present an extension of the logic with <i>definitions</i> of partial functions and give their semantics in terms of the existing semantics for inductive definitions of sets. Finally, we explore the connection to functional programming.</p></div>","PeriodicalId":7971,"journal":{"name":"Annals of Mathematics and Artificial Intelligence","volume":"94 4","pages":"477 - 519"},"PeriodicalIF":1.6,"publicationDate":"2026-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148927561","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Non-parametric binary regression in metric spaces with logarithmic loss","authors":"Ariel Avital,&nbsp;Klim Efremenko,&nbsp;Aryeh Kontorovich,&nbsp;David Tolpin,&nbsp;Bo Waggoner","doi":"10.1007/s10472-025-09995-5","DOIUrl":"10.1007/s10472-025-09995-5","url":null,"abstract":"<div><p>In Bernoulli discriminative models, log-likelihood is a natural — and, in a well-defined sense, universal — choice of risk score. In this general setting, we propose a non-parametric variant of binary regression, where the model is regularized to be a Lipschitz function taking a metric space to [0, 1]. Our choice of logarithmic loss corresponds to the log-likelihood risk score. This setting presents novel computational and statistical challenges. On the computational front, we derive an efficient optimization algorithm based on interior point methods (IPM); an attractive feature is that it is parameter-free (that is, does not require tuning an update step size). On the statistical front, the unbounded loss function presents a problem for classic generalization bounds, based on covering-number and Rademacher techniques. Additionally, an impossibility result we prove shows that the unboundedness presents an inherent obstruction to learnability. We get around this challenge via an adaptive truncation approach, and also derive a lower bound indicating that the truncation is, in some sense, necessary. To our knowledge, our approach provides the first rigorous computational and theoretical results in this area. Finally, we present encouraging empirical results.</p></div>","PeriodicalId":7971,"journal":{"name":"Annals of Mathematics and Artificial Intelligence","volume":"94 4","pages":"521 - 547"},"PeriodicalIF":1.6,"publicationDate":"2026-02-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10472-025-09995-5.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148927262","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}