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States are increasingly adopting changes to K-12 funding systems in order to promote and encourage student engagement in secondary-level career and technical education (CTE). Two of the most prevalent reforms include: a) establishing tiered weights for CTE in school funding formulas based on the connection between a program of study and workforce needs and b) incentive grant programs that provide funds based on student attainment of industry-recognized credentials. However, it is unclear whether and how these changes induce higher levels of meaningful and useful CTE engagement. This study evaluates the impact of these policy changes on state funding for CTE and high school level CTE enrollment.

Catholic schools have seen more than a 30% decline in enrollment over the past 20 years. While some of the decline in enrollment may have been spurred by secular trends or the Church abuse scandal, the increase in schools of choice, principally public charter schools, may explain at least some of this decline. In this paper we estimate the effect of the opening of charter schools in proximity to Catholic schools across the entire U.S. We find that the opening of a nearby charter school has a negative impact on Catholic school enrollment and increases the likelihood that the school will close. We also find that charter openings induce greater racial isolation. Findings are especially pronounced in K8 schools, rather than high schools.

Equational Unification is a critical problem in many areas such as automated theorem proving and security protocol analysis. In this paper, we focus on XOR-Unification, that is, unification modulo the theory of exclusive-or. This theory contains an operator with the properties Associativity, Commutativity, Nilpotency, and the presence of an identity. In the proof assistant Coq, we implement an algorithm that solves XOR unification problems, whose design was inspired by Liu and Lynch, and prove it sound, complete, and terminating. Using Coq's code extraction capability we obtain an implementation in the programming language OCaml., Comment: In Proceedings ICLP 2024, arXiv:2502.08453

We introduce the Formally Verified Automated Programming Progress Standards, or FVAPPS, a benchmark of 4715 samples for writing programs and proving their correctness, the largest formal verification benchmark, including 1083 curated and quality controlled samples. Previously, APPS provided a benchmark and dataset for programming puzzles to be completed in Python and checked against unit tests, of the kind seen in technical assessments in the software engineering industry. Building upon recent approaches for benchmarks in interactive theorem proving, we generalize the unit tests to Lean 4 theorems given without proof (i.e., using Lean's "sorry" keyword). On the 406 theorems of 100 randomly selected samples, Sonnet correctly proves 30% and Gemini correctly proves 18%. We challenge the machine learning and program synthesis communities to solve both each general purpose programming problem and its associated correctness specifications. The benchmark is available at https://huggingface.co/datasets/quinn-dougherty/fvapps., Comment: 8 pages, to appear at the 2025LLM4Code Workshop at ICSE 2025

Self-supervised learning (SSL) models have become crucial in speech processing, with recent advancements concentrating on developing architectures that capture representations across multiple timescales. The primary goal of these multi-scale architectures is to exploit the hierarchical nature of speech, where lower-resolution components aim to capture representations that align with increasingly abstract concepts (e.g., from phones to words to sentences). Although multi-scale approaches have demonstrated some improvements over single-scale models, the precise reasons for these enhancements have poor empirical support. In this study, we present an initial analysis of layer-wise representations in multi-scale architectures, with a focus on Canonical Correlation Analysis (CCA) and Mutual Information (MI). We apply this analysis to Multi-Resolution HuBERT (MR-HuBERT) and find that (1) the improved performance on SUPERB tasks is primarily due to the auxiliary low-resolution loss rather than the downsampling itself, and (2) downsampling to lower resolutions neither improves downstream performance nor correlates with higher-level information (e.g., words), though it does improve computational efficiency. These findings challenge assumptions about the multi-scale nature of MR-HuBERT and motivate the importance of disentangling computational efficiency from learning better representations.

A chessboard has the property that every row and every column has as many white squares as black squares. In this mostly methodological note, we address the problem of counting such rectangular arrays with a fixed (numeric) number of rows, but an arbitrary (symbolic) number of columns. We first address the ``vanilla" problem where there are no restrictions, and then go on to discuss the still-more-challenging problem of counting such binary arrays that are not permitted to contain a specified (finite) set of horizontal patterns, and a specified set of vertical patterns. While we can rigorously prove that each such sequence satisfies some linear recurrence equation with polynomial coefficients, actually finding these recurrences poses major {\it symbolic}-computational challenges, that we can only meet in some small cases. In fact, just generating as many as possible terms of these sequences is a big {\it numeric}-computational challenge. This was tackled by computer whiz Ron H. Hardin, who contributed several such sequences, and computed quite a few terms of each. We extend Hardin's sequences quite considerably. We also talk about the much easier problem of counting such restricted arrays without balance conditions., Comment: Expanded computations and added references

This article introduces a finite piecewise Euclidean cell complex homeomorphic to the space of monic centered complex polynomials of degree $d$ whose critical values lie in a fixed closed rectangular region. We call this the branched rectangle complex since its points are indexed by marked $d$-sheeted planar branched covers of the fixed rectangle. The vertices of the cell structure are indexed by the combinatorial "basketballs" studied by Martin, Savitt and Singer. Structurally, the branched rectangle complex is a full subcomplex of a direct product of two copies of the order complex of the noncrossing partition lattice. Topologically, it is homeomorphic to the closed $2n$-dimensional ball where $n=d-1$. Metrically, the simplices in each factor are orthoschemes. It can also be viewed as a compactification of the space of all monic centered complex polynomials of degree $d$. We also introduce a finite piecewise Euclidean cell complex homeomorphic to the space of monic centered complex polynomials of degree $d$ whose critical values lie in a fixed closed annular region. We call this the branched annulus complex since its points are indexed by marked $d$-sheeted planar branched covers of the fixed annulus.It can be constructed from the branched rectangle complex as a cellular quotient by isometric face identifications. And it can be viewed as a compactification of the space of all monic centered complex polynomials of degree $d$ with distinct roots. Finally, the branched annulus complex deformation retracts to the branched circle complex, which we identify with the dual braid complex. Our explicit embedding of the dual braid complex as a spine for the space of polynomials with distinct roots provides a direct proof that these two classifying spaces for the braid group are homotopy equivalent., Comment: 82 pages, 36 figures

Theory of Computing (ToC) is an important course in CS curricula because of its connections to other CS courses as a foundation for them. Traditional ToC course grading schemes are mostly exam-based, and sometimes a small weight for traditional proof-type assignments. Recent work experimented with a new type of assignment, namely a ``mock conference'' project wherein students approach and present ToC problems as if they were submitting to a ``real'' CS conference. In this paper we massively scaffold this existing project and provide our experiences in running such a conference in our own ToC course.

Finite State Machines are a concept widely taught in undergraduate theory of computing courses. Educators typically use tools with static representations of FSMs to help students visualize these objects and processes; however, all existing tools require manual editing by the instructor. In this poster, we created an automatic visualization tool for FSMs that generates videos of FSM simulation, named Finite State Machine with Input and Process Render (FSMIPR). Educators can input any formal definition of an FSM and an input string, and FSMIPR generates an accompanying video of its simulation. We believe that FSMIPR will be beneficial to students who learn difficult computer theory concepts. We conclude with future work currently in-progress with FSMIPR., Comment: Accepted to SIGCSE Virtual 2024

The comma sequence (1, 12, 35, 94, ...) is the lexicographically earliest sequence such that the difference of consecutive terms equals the concatenation of the digits on either side of the comma separating them. The behavior of a "generalized comma sequence" depends on the base the numbers are written in, as well as the sequence's initial values. We provide a computational proof that all comma sequences in bases 3 through 633 are finite. Relying on a combinatorial conjecture, Angelini et al. estimated that the final element of a comma sequence in base b should be roughly exp(O(b)). We prove their conjecture, but provide evidence that the correct estimate is actually exp(O(b log b))., Comment: This version handles much larger bases by using a superior cycle-detection algorithm suggested by Luke Pebody

Large Language Models (LLMs) have had considerable difficulty when prompted with mathematical questions, especially those within theory of computing (ToC) courses. In this paper, we detail two experiments regarding our own ToC course and the ChatGPT LLM. For the first, we evaluated ChatGPT's ability to pass our own ToC course's exams. For the second, we created a database of sample ToC questions and responses to accommodate other ToC offerings' choices for topics and structure. We scored each of ChatGPT's outputs on these questions. Overall, we determined that ChatGPT can pass our ToC course, and is adequate at understanding common formal definitions and answering "simple"-style questions, e.g., true/false and multiple choice. However, ChatGPT often makes nonsensical claims in open-ended responses, such as proofs., Comment: Accepted to SIGCSE Virtual 2024

Bulk heterojunction (BHJ) organic solar cells have made remarkable inroads towards 20% efficiency, yet nonradiative recombination losses ({\Delta}Vnr) remain high compared to silicon and perovskite photovoltaics. Interfaces buried within BHJ blends hold the key to recombination losses but access to their energetic landscape underpinning charge transfer (CT) states and their disorder remain elusive. Here, we reveal the energetic landscape and CT state manifold of modern BHJs with both spatial and energetic resolutions and link the offset between singlet (ES1) and CT energy (ES1-CT) and interfacial energetic disorder with {\Delta}Vnr. We do so by locally mapping the energy distributions of modern PM6-based BHJs with IT4F, Y6 and PC71BM acceptors and combine it, for the first time, with sensitive EQE measurements, to visualize and quantify donor (D) and acceptor (A) energetics at interfaces and associated them with CT states within a modified Marcus framework. A key new ability is the identification of the specific BHJ interfaces associated with the CT manifold, including where the lowest energy CT states reside. Moreover, we quantify energy levels and electronic disorders directly at these and other interfaces and connect these contributions to the energy losses. We delineate the influences of S1-CT offset and interfacial energetic disorder on {\Delta}Vnr across morphologically varied BHJs. Our results show both factors influencing energy losses in different ways. We demonstrate that PM6:Y6 can achieve low {\Delta}Vnr by forming a nominally sharp D/A interface with exceptionally low interfacial disorder via judicious processing combined with a low S1 to CT offset. This provides a design rule to minimize {\Delta}Vnr for modern NFAs: sharp D/A interfaces with low S1 to CT offset exhibiting minimal interfacial disorder., Comment: Manuscript: 30 pages, 4 figures, 2 tables Supplementary information: 41 pages, 24 figures, 7 tables

We introduce a comprehensive quantum solver for binary combinatorial optimization problems on gate-model quantum computers that outperforms any published alternative and consistently delivers correct solutions for problems with up to 127 qubits. We provide an overview of the internal workflow, describing the integration of a customized ansatz and variational parameter update strategy, efficient error suppression in hardware execution, and QPU-overhead-free post-processing to correct for bit-flip errors. We benchmark this solver on IBM quantum computers for several classically nontrivial unconstrained binary optimization problems -- the entire optimization is conducted on hardware with no use of classical simulation or prior knowledge of the solution. First, we demonstrate the ability to correctly solve Max-Cut instances for random regular graphs with a variety of densities using up to 120 qubits, where the graph topologies are not matched to device connectivity. Next, we apply the solver to higher-order binary optimization and successfully search for the ground state energy of a 127-qubit spin-glass model with linear, quadratic, and cubic interaction terms. Use of this new quantum solver increases the likelihood of finding the minimum energy by up to $\sim1,500\times$ relative to published results using a DWave annealer, and it can find the correct solution when the annealer fails. Furthermore, for both problem types, the Q-CTRL solver outperforms a heuristic local solver used to indicate the relative difficulty of the problems pursued. Overall, these results represent the largest quantum optimizations successfully solved on hardware to date, and demonstrate the first time a gate-model quantum computer has been able to outperform an annealer for a class of binary optimization problems., Comment: 19 pages, 8 figures

Prior research has clearly established the substantial expected payoffs to investments in early childhood education. However, the ability to deliver early childhood programs differs across communities with access to high quality programing especially hard to establish in rural communities. We study one program, Early Steps to School Success, to understand whether the provision of home visiting and book exchange programs in rural Kentucky can influence kindergarten readiness. Linking program data with the state longitudinal data system in Kentucky we create multiple comparison groups by matching children on known program qualification indicators to estimate whether Early Steps program participation was related to school readiness. Our estimates suggest that program participation resulted in small improvements to children's kindergarten readiness, as measured by the Brigance kindergarten readiness assessment overall score and sub-scores in language, cognitive, and physical development. Results are not sensitive to our choice of comparison group, though they appear driven by the experiences of children who participate from birth through age five or from ages three-to-five only. Our findings suggest that the Early Steps home visiting intervention may be a worthwhile intervention for improving kindergarten preparedness for children living in rural contexts.

The layered van der Waals material CrCl3 exhibits very strongly bound ligand field excitons that control optoelectronic applications and are connected with magnetic ordering by virtue of their d-orbital origin. Time-resolved photoluminescence of these exciton populations at room temperature shows that their relaxation is dominated by exciton-exciton annihilation and that the spontaneous decay lifetime is very long. These observations allow the rough quantification of the exciton annihilation rate constant and contextualization in light of a recent theory of universal scaling behavior of the annihilation process.

Modern tools for class-agnostic image segmentation (e.g., SegmentAnything) and open-set semantic understanding (e.g., CLIP) provide unprecedented opportunities for robot perception and mapping. While traditional closed-set metric-semantic maps were restricted to tens or hundreds of semantic classes, we can now build maps with a plethora of objects and countless semantic variations. This leaves us with a fundamental question: what is the right granularity for the objects (and, more generally, for the semantic concepts) the robot has to include in its map representation? While related work implicitly chooses a level of granularity by tuning thresholds for object detection, we argue that such a choice is intrinsically task-dependent. The first contribution of this paper is to propose a task-driven 3D scene understanding problem, where the robot is given a list of tasks in natural language and has to select the granularity and the subset of objects and scene structure to retain in its map that is sufficient to complete the tasks. We show that this problem can be naturally formulated using the Information Bottleneck (IB), an established information-theoretic framework. The second contribution is an algorithm for task-driven 3D scene understanding based on an Agglomerative IB approach, that is able to cluster 3D primitives in the environment into task-relevant objects and regions and executes incrementally. The third contribution is to integrate our task-driven clustering algorithm into a real-time pipeline, named Clio, that constructs a hierarchical 3D scene graph of the environment online using only onboard compute, as the robot explores it. Our final contribution is an extensive experimental campaign showing that Clio not only allows real-time construction of compact open-set 3D scene graphs, but also improves the accuracy of task execution by limiting the map to relevant semantic concepts.

Optimal transport (OT) and the related Wasserstein metric (W) are powerful and ubiquitous tools for comparing distributions. However, computing pairwise Wasserstein distances rapidly becomes intractable as cohort size grows. An attractive alternative would be to find an embedding space in which pairwise Euclidean distances map to OT distances, akin to standard multidimensional scaling (MDS). We present Wasserstein Wormhole, a transformer-based autoencoder that embeds empirical distributions into a latent space wherein Euclidean distances approximate OT distances. Extending MDS theory, we show that our objective function implies a bound on the error incurred when embedding non-Euclidean distances. Empirically, distances between Wormhole embeddings closely match Wasserstein distances, enabling linear time computation of OT distances. Along with an encoder that maps distributions to embeddings, Wasserstein Wormhole includes a decoder that maps embeddings back to distributions, allowing for operations in the embedding space to generalize to OT spaces, such as Wasserstein barycenter estimation and OT interpolation. By lending scalability and interpretability to OT approaches, Wasserstein Wormhole unlocks new avenues for data analysis in the fields of computational geometry and single-cell biology., Comment: Published at the Forty-first International Conference on Machine Learning (ICML2024)

Generalizing an argument of Matiyasevich, we illustrate a method to generate infinitely many diophantine equations whose solutions can be completely described by linear recurrences. In particular, we provide an integer-coefficient polynomial $p(x, y, z)$ whose only integer roots are consecutive triples of Tribonacci numbers., Comment: This version corrects a misstated date

This paper examines how the process of making higher education choices in the United States--whether to enter higher education, attend a particular college, or follow a particular path through college--produces and legitimates social inequality. The paper's central thesis is that a societal regime of many choices--while serving individual freedom and producing social well-being--"produces" societal inequality in a way that obscures that process of social reproduction for virtually all who participate in that choice regime. Students often make choices that do not serve their interests as well as they might wish, particularly if students are faced with many choices and do not have adequate information. The incidence of those suboptimal choices is not random but is socially stratified. It is higher for less advantaged people, and unequal provision of good information plays a crucial role in producing those socially stratified suboptimal choices. Secondly, the provision of many choices "legitimates" social inequality. Seemingly offered many choices in life, both the fortunate and unfortunate in society come to feel that much of the inequality they experience is due to their own actions and therefore is legitimate. The paper concludes by offering various prescriptions for reducing the socially stratifying impacts and ideological consequences of a high-choice regime. It lays out how we could more equally distribute high-quality information, nudge students toward better choice making, reduce the costs to students of suboptimal choices, and mitigate blaming self and others by demystifying the nature of choice. In making these arguments, this paper draws on the research literature in sociology of education, behavioral economics, cognitive psychology, and social psychology of inequality.