The formal language for mathematical expressions to which our certification algorithm is applied is specified by the grammar depicted in Figure 1. The language is rich enough to cover all the examples in the main paper and this supplement. In this grammar, number is a placeholder for an arbitrary floating point number, variable is a placeholder for variable names starting with a Latin character and function is a placeholder for the supported elementary differentiable functions like exp,log and sum. Here, is used for transposition and a preceding . Here are some examples from the language (the fist example uses a transposition and the fifth and seventh example use elementwise operations): 2-norm Xw y 2: (X*w-y)'*(X*w-y) logistic log(1+exp(x)): log(1+exp(x)) 1 quadratic x2: x^2 relative entropy xlog(x/y): x*log(x/y), x>0, y>0 logistic regression Our implementation of the Hessian approach works on vectorized and normalized expression DAGs (directed acyclic graphs) for Hessians that contain every subexpression exactly once.

Neural networks with a latency requirement on the order of microseconds, like the ones used at the CERN Large Hadron Collider, are typically deployed on FPGAs fully unrolled and pipelined. A bottleneck for the deployment of such neural networks is area utilization, which is directly related to the required constant matrix-vector multiplication (CMVM) operations. In this work, we propose an efficient algorithm for implementing CMVM operations with distributed arithmetic (DA) on FPGAs that simultaneously optimizes for area consumption and latency. The algorithm achieves resource reduction similar to state-of-the-art algorithms while being significantly faster to compute. The proposed algorithm is open-sourced and integrated into the \texttt{hls4ml} library, a free and open-source library for running real-time neural network inference on FPGAs. We show that the proposed algorithm can reduce on-chip resources by up to a third for realistic, highly quantized neural networks while simultaneously reducing latency, enabling the implementation of previously infeasible networks.

We present the first correct-by-construction learning-based system for step-by-step mathematical integration. The key idea is to learn a policy, represented by a GPT transformer model, which guides the search for the right mathematical integration rule, to be carried out by a symbolic solver. Concretely, we introduce a symbolic engine with axiomatically correct actions on mathematical expressions, as well as the first dataset for step-by-step integration. Our GPT-style transformer model, trained on this synthetic data, demonstrates strong generalization by surpassing its own data generator in accuracy and efficiency, using 50% fewer search steps. Our experimental results with SoTA LLMs also demonstrate that the standard approach of fine-tuning LLMs on a set of question-answer pairs is insufficient for solving this mathematical task. This motivates the importance of discovering creative methods for combining LLMs with symbolic reasoning engines, of which our work is an instance. Large language models (LLMs) based on the transformer architecture (Vaswani et al., 2023) have demonstrated remarkable abilities across diverse tasks, such as language translation, code generation, and engaging human-like conversations (OpenAI, 2024). However, applying these models to mathematics presents significant challenges. Their autoregressive nature makes them prone to hallucinations and errors during inference.

Many structured prediction and reasoning tasks can be framed as program synthesis problems, where the goal is to generate a program in a domain-specific language (DSL) that transforms input data into the desired output. Unfortunately, purely neural approaches, such as large language models (LLMs), often fail to produce fully correct programs in unfamiliar DSLs, while purely symbolic methods based on combinatorial search scale poorly to complex problems. Motivated by these limitations, we introduce a hybrid approach, where LLM completions for a given task are used to learn a task-specific, context-free surrogate model, which is then used to guide program synthesis. We evaluate this hybrid approach on three domains, and show that it outperforms both unguided search and direct sampling from LLMs, as well as existing program synthesizers.

Large scale analytics engines have become a core dependency for modern data-driven enterprises to derive business insights and drive actions. These engines support a large number of analytic jobs processing huge volumes of data on a daily basis, and workloads are often inundated with overlapping computations across multiple jobs. Reusing common computation is crucial for efficient cluster resource utilization and reducing job execution time. Detecting common computation is the first and key step for reducing this computational redundancy. However, detecting equivalence on large-scale analytics engines requires efficient and scalable solutions that are fully automated. In addition, to maximize computation reuse, equivalence needs to be detected at the semantic level instead of just the syntactic level (i.e., the ability to detect semantic equivalence of seemingly different-looking queries). Unfortunately, existing solutions fall short of satisfying these requirements. In this paper, we take a major step towards filling this gap by proposing GEqO, a portable and lightweight machine-learning-based framework for efficiently identifying semantically equivalent computations at scale. GEqO introduces two machine-learning-based filters that quickly prune out nonequivalent subexpressions and employs a semi-supervised learning feedback loop to iteratively improve its model with an intelligent sampling mechanism. Further, with its novel database-agnostic featurization method, GEqO can transfer the learning from one workload and database to another. Our extensive empirical evaluation shows that, on TPC-DS-like queries, GEqO yields significant performance gains-up to 200x faster than automated verifiers-and finds up to 2x more equivalences than optimizer and signature-based equivalence detection approaches.

In deep learning inference, model parameters are pruned and quantized to reduce the model size. Compression methods and common subexpression (CSE) elimination algorithms are applied on sparse constant matrices to deploy the models on low-cost embedded devices. However, the state-of-the-art CSE elimination methods do not scale well for handling large matrices. They reach hours for extracting CSEs in a $200 \times 200$ matrix while their matrix multiplication algorithms execute longer than the conventional matrix multiplication methods. Besides, there exist no compression methods for matrices utilizing CSEs. As a remedy to this problem, a random search-based algorithm is proposed in this paper to extract CSEs in the column pairs of a constant matrix. It produces an adder tree for a $1000 \times 1000$ matrix in a minute. To compress the adder tree, this paper presents a compression format by extending the Compressed Sparse Row (CSR) to include CSEs. While compression rates of more than $50\%$ can be achieved compared to the original CSR format, simulations for a single-core embedded system show that the matrix multiplication execution time can be reduced by $20\%$.

We show that reverse mode automatic differentiation and symbolic differentiation are equivalent in the sense that they both perform the same operations when computing derivatives. This is in stark contrast to the common claim that they are substantially different. The difference is often illustrated by claiming that symbolic differentiation suffers from "expression swell" whereas automatic differentiation does not. Here, we show that this statement is not true. "Expression swell" refers to the phenomenon of a much larger representation of the derivative as opposed to the representation of the original function.

We examine whether self-supervised language modeling applied to mathematical formulas enables logical reasoning. We suggest several logical reasoning tasks that can be used to evaluate language models trained on formal mathematical statements, such as type inference, suggesting missing assumptions and completing equalities. To train language models for formal mathematics, we propose a novel skip-tree task. We find that models trained on the skip-tree task show surprisingly strong mathematical reasoning abilities, and outperform models trained on standard skip-sequence tasks. We also analyze the models' ability to formulate new conjectures by measuring how often the predictions are provable and useful in other proofs.

Writing programs that can create programs have been an elusive goal of artificial intelligence(AI) research for many years. As a matter of fact, the idea that AI agents can create their own programs if often seem as one of the differentiators of general AI vs. narrow AI. So important is this goal, that AI researchers have created a specific area of research known as Program Synthesis that focuses on addressing those challenges. The idea behind program synthesis is to create AI agents that can generate programs that match a given specification. We often use primitive versions of this technique when we take advantage of, for instance, the Flash Fill feature in Microsoft Excel.

Combining abstract, symbolic reasoning with continuous neural reasoning is a grand challenge of representation learning. As a step in this direction, we propose a new architecture, called neural equivalence networks, for the problem of learning continuous semantic representations of algebraic and logical expressions. These networks are trained to represent semantic equivalence, even of expressions that are syntactically very different. The challenge is that semantic representations must be computed in a syntax-directed manner, because semantics is compositional, but at the same time, small changes in syntax can lead to very large changes in semantics, which can be difficult for continuous neural architectures. We perform an exhaustive evaluation on the task of checking equivalence on a highly diverse class of symbolic algebraic and boolean expression types, showing that our model significantly outperforms existing architectures.