Mathematically, the summation satisfies Commutative Property. Note semirings differ from natural arithmetic operators in two aspects. We prove Lemma 1 by induction. In practice, for link prediction we find it only takes a very small number of iterations (e.g., Here we prove the time complexity for NBFNet and other GNN frameworks.

We show, contrary to the optimism about LLM's problem-solving abilities, fueled by the recent gold medals that were attained, that a problem exists -- Yu Tsumura's 554th problem -- that a) is within the scope of an IMO problem in terms of proof sophistication, b) is not a combinatorics problem which has caused issues for LLMs, c) requires fewer proof techniques than typical hard IMO problems, d) has a publicly available solution (likely in the training data of LLMs), and e) that cannot be readily solved by any existing off-the-shelf LLM (commercial or open-source).

Math is constructed by people for people: just as natural language corpora reflect not just propositions but the communicative goals of language users, the math data that models are trained on reflects not just idealized mathematical entities but rich communicative intentions. While there are important advantages to treating math in a purely symbolic manner, we here hypothesize that there are complementary benefits to treating math as situated linguistic communication and that language models are well suited for this goal, in ways that are not fully appreciated. We illustrate these points with two case studies. First, we ran an experiment in which we found that language models interpret the equals sign in a humanlike way--generating systematically different word problems for the same underlying equation arranged in different ways. Second, we found that language models prefer proofs to be ordered in naturalistic ways, even though other orders would be logically equivalent. We advocate for AI systems that learn from and represent the communicative intentions latent in human-generated math. Mathematical propositions are first of all English sentences; not only English sentences, but each mathematical proposition has a resemblance to certain non-mathematical propositions.

This study explores the effectiveness of Large Language Models (LLMs) in creating personalized "mirror stories" that reflect and resonate with individual readers' identities, addressing the significant lack of diversity in literature. We present MirrorStories, a corpus of 1,500 personalized short stories generated by integrating elements such as name, gender, age, ethnicity, reader interest, and story moral. We demonstrate that LLMs can effectively incorporate diverse identity elements into narratives, with human evaluators identifying personalized elements in the stories with high accuracy. Through a comprehensive evaluation involving 26 diverse human judges, we compare the effectiveness of MirrorStories against generic narratives. We find that personalized LLM-generated stories not only outscore generic human-written and LLM-generated ones across all metrics of engagement (with average ratings of 4.22 versus 3.37 on a 5-point scale), but also achieve higher textual diversity while preserving the intended moral. We also provide analyses that include bias assessments and a study on the potential for integrating images into personalized stories.

Recurrent Neural Cascades (RNC) are the class of recurrent neural networks with no cyclic dependencies among recurrent neurons. Their subclass RNC+ with positive recurrent weights has been shown to be closely connected to the star-free regular languages, which are the expressivity of many well-established temporal logics. The existing expressivity results show that the regular languages captured by RNC+ are the star-free ones, and they leave open the possibility that RNC+ may capture languages beyond regular. We exclude this possibility for languages that include an identity element, i.e., an input that can occur an arbitrary number of times without affecting the output. Namely, in the presence of an identity element, we show that the languages captured by RNC+ are exactly the star-free regular languages. Identity elements are ubiquitous in temporal patterns, and hence our results apply to a large number of applications. The implications of our results go beyond expressivity. At their core, we establish a close structural correspondence between RNC+ and semiautomata cascades, showing that every neuron can be equivalently captured by a three-state semiautomaton. A notable consequence of this result is that RNC+ are no more succinct than cascades of three-state semiautomata.