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1. Implicitly Learning to Reason in First-Order Logic

   Belle, Vaishak; Juba, Brendan (December 2019, Advances in neural information processing systems)

   We consider the problem of answering queries about formulas of first-order logic based on background knowledge partially represented explicitly as other formulas, and partially represented as examples independently drawn from a fixed probability distribution. PAC semantics, introduced by Valiant, is one rigorous, general proposal for learning to reason in formal languages: although weaker than classical entailment, it allows for a powerful model theoretic framework for answering queries while requiring minimal assumptions about the form of the distribution in question. To date, however, the most significant limitation of that approach, and more generally most machine learning approaches with robustness guarantees, is that the logical language is ultimately essentially propositional, with finitely many atoms. Indeed, the theoretical findings on the learning of relational theories in such generality have been resoundingly negative. This is despite the fact that first-order logic is widely argued to be most appropriate for representing human knowledge. In this work, we present a new theoretical approach to robustly learning to reason in first-order logic, and consider universally quantified clauses over a countably infinite domain. Our results exploit symmetries exhibited by constants in the language, and generalize the notion of implicit learnability to show how queries can be computed against (implicitly) learned first-order background knowledge. 

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2. Cyclotomic nil-Brauer and singular Soergel bimodules of type D

   https://doi.org/10.1090/ert/710  

   Bodish, Elijah; Brundan, Jonathan; Elias, Ben (January 2026, Representation Theory)

   We study a new family of strict monoidal categories, which are cyclotomic quotients of the nil-Brauer category. We construct a monoidal functor from the cyclotomic nil-Brauer category of level l to another monoidal category constructed from singular Soergel bimodules of type D. We conjecture that our functor is an equivalence of categories. Although we can prove neither fullness nor faithfulness at this point, we are able to show that the functor induces an isomorphism at the level of Grothendieck rings. We compute these rings and their canonical bases, and give diagrammatic descriptions of the corresponding primitive idempotents. 

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3. Towards Conditional Independence Test for Relational Data

   Lee, Sanghack; Honavar, Vasant G. (January 2017, Uncertainty in artificial intelligence)

   Conditional independence (CI) tests play a central role in statistical inference, machine learning, and causal discovery. Most existing CI tests assume that the samples are indepen- dently and identically distributed (i.i.d.). How- ever, this assumption often does not hold in the case of relational data. We define Relational Conditional Independence (RCI), a generaliza- tion of CI to the relational setting. We show how, under a set of structural assumptions, we can test for RCI by reducing the task of test- ing for RCI on non-i.i.d. data to the problem of testing for CI on several data sets each of which consists of i.i.d. samples. We develop Kernel Relational CI test (KRCIT), a nonpara- metric test as a practical approach to testing for RCI by relaxing the structural assumptions used in our analysis of RCI. We describe re- sults of experiments with synthetic relational data that show the benefits of KRCIT relative to traditional CI tests that don’t account for the non-i.i.d. nature of relational data. 

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4. Affine and cyclotomic webs

   https://doi.org/10.1112/jlms.70278  

   Song, Linliang; Wang, Weiqiang (September 2025, Journal of the London Mathematical Society)

   Abstract Generalizing the polynomial web category, we introduce a diagrammatic ‐linear monoidal category,the affine web category, for any commutative ring . Integral bases consisting of elementary diagrams are obtained for the affine web category and its cyclotomic quotient categories. Connections between cyclotomic web categories and finite ‐algebras are established, leading to a diagrammatic presentation of idempotent subalgebras of ‐Schur algebras introduced by Brundan–Kleshchev. The affine web category will be used as a basic building block of another ‐linear monoidal category,the affine Schur category, formulated in a sequel. 

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5. Higher arity self-distributive operations in Cascades and their cohomology

   https://doi.org/10.1142/S0219498821501164  

   Elhamdadi, Mohamed; Saito, Masahico; Zappala, Emanuele (July 2021, Journal of Algebra and Its Applications)

   null (Ed.)

   We investigate constructions of higher arity self-distributive operations, and give relations between cohomology groups corresponding to operations of different arities. For this purpose we introduce the notion of mutually distributive [Formula: see text]-ary operations generalizing those for the binary case, and define a cohomology theory labeled by these operations. A geometric interpretation in terms of framed links is described, with the scope of providing algebraic background of constructing [Formula: see text]-cocycles for framed link invariants. This theory is also studied in the context of symmetric monoidal categories. Examples from Lie algebras, coalgebras and Hopf algebras are given. 

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