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Oral exams are compelling because (1) students can be asked to provide their reasoning for an answer and (2) their interactive nature permits follow-up questions to resolve ambiguous answers. But oral exams scale poorly. In this paper, we present scalable conversational assessment, an LLM-based system that conducts interactive text-based conversations with students to assess their ability to make and support an argument. We use a two-agent architecture to separate concerns and reduce the likelihood that answers are leaked to the student: the interviewer interacts directly with the student, but only receives guidance about the next question to ask the student; the hidden evaluator knows the goals of the whole activity, tracks the student's progress using a rubric, and guides the interviewer on when to follow-up on the current question or proceed to the next question. The system was first tested using a collection of simulated students with LLM personas of a particular ability level or adversarial tactic (e.g., answer-extractors, prompt-injectors, confident fabricators). A novel element of this evaluation was that it required the use of a fictional assessment topic to prevent personas that shouldn't have information from using knowledge present in the LLM's pre-training. The system was also deployed in a small, elective AI literacy course in both formative and summative contexts. Through manual and post-facto AI review of the transcripts we identified opportunities to improve the follow-up question generation and ensure that grading only credited ideas that the student (and not the interviewer) contributed.

We investigate the minimal width of deep neural networks that can approximate all continuous functions from $\mathbb{R}^n$ to $\mathbb{R}^m$ in the topology of uniform convergence on compact subsets. We show that whenever the real projective space $\mathbb{RP}^{n-1}$ admits a smooth immersion into $\mathbb{R}^n$, the minimal width is at least $n+\min\{n,m\}$, assuming that the activation function is continuous and monotone. As a consequence, the minimal width of ReLU networks is $n+m$ whenever $n$ is equal to 2, 3, 4, 7 or 8 and $1\le m \le n$, extending results of Hanin-Sellke and Hwang.

In this article, we show that the problem of deciding whether a finite relational structure $\mathbf{A}$ admits a Maltsev polymorphism is a computational problem, which can be solved in polynomial time. We provide a proof of this result by reducing the problem to deciding whether a binary relational structure, associated with $\mathbf{A}$ admits such a polymorphism. In our construction of such an algorithm, we make an extensive use of a property of binary relations, which is implied by the existence of a Maltsev polymorphism, the so-called rectangularity property.

We study the query complexity of envy-free division for cake, chores, and resources with both positive and negative values. For arbitrary nonnegative nonatomic chore costs, we give a deterministic exact protocol using $2^{n\log_2 n+O(n)}$ queries and cuts. For both cake and chores, a Las Vegas protocol uses $O(n^{12}\log(2n))$ expected queries and $O(n^{11}\log(2n))$ expected cuts. Conversely, every exact protocol with fixed positive success probability requires $Ω(n^2)$ expected queries in general, even with arbitrarily many output cuts at no query cost. For mixed cake and chores, no uniform finite expected-query bound exists even when every Cut returns the leftmost root or every Cut returns the rightmost root. This holds for any fixed positive multiplicative approximation factor and success probability, already when all net totals are one. An additive impossibility holds on the same normalized domain.