Mason DiCicco

Duncan Soiffer, Nathaniel Itty, Christopher D. Rosin et al.

A collection of hyperplanes $\mathcal{H}$ slices all edges of the $n$-dimensional hypercube $Q_n$ with vertex set $\{-1,1\}^n$ if, for every edge $e$ in the hypercube, there exists a hyperplane in $\mathcal{H}$ intersecting $e$ in its interior. Let $S(n)$ be the minimum number of hyperplanes needed to slice $Q_n$. We prove that $S(n) \leq \lceil \frac{4n}{5} \rceil$, except when $n$ is an odd multiple of $5$, in which case $S(n) \leq \frac{4n}{5} +1$. This improves upon the previously known upper bound of $S(n) \leq \lceil\frac{5n}{6} \rceil$ due to Paterson reported in 1971. We also obtain new lower bounds on the maximum number of edges in $Q_n$ that can be sliced using $k<n$ hyperplanes. We prove the improved upper bound on $S(n)$ by constructing $8$ hyperplanes slicing $Q_{10}$ aided by the recently introduced CPro1: an automatic tool that uses reasoning LLMs coupled with automated hyperparameter tuning to create search algorithms for the discovery of mathematical constructions.

Mason DiCicco, Eamon Worden, Conner Olsen et al.

Understanding the mathematical reasoning capabilities of Large Language Models (LLMs) is a central topic in the study of artificial intelligence. This new domain necessitates the creation of datasets of reasoning tasks for both training and benchmarking the performance of LLMs. To this end, we introduce the Karp dataset: The first dataset composed of detailed proofs of NP-completeness reductions. The reductions vary in difficulty, ranging from simple exercises of undergraduate courses to more challenging reductions from academic papers. We compare the performance of state-of-the-art models on this task and demonstrate the effect of fine-tuning with the Karp dataset on reasoning capacity.