Accurate and Efficient Expression Evaluation and Linear Algebra

We survey and unify recent results on the existence of accurate algorithms for evaluating multivariate polynomials, and more generally for accurate numerical linear algebra with structured matrices. By "accurate" we mean that the computed answer has relative error less than 1, i.e., has some correct leading digits. We also address efficiency, by which we mean algorithms that run in polynomial time in the size of the input. Our results will depend strongly on the model of arithmetic: Most of our results will use the so-called Traditional Model (TM). We give a set of necessary and sufficient conditions to decide whether a high accuracy algorithm exists in the TM, and describe progress toward a decision procedure that will take any problem and provide either a high accuracy algorithm or a proof that none exists. When no accurate algorithm exists in the TM, it is natural to extend the set of available accurate operations by a library of additional operations, such as $x+y+z$, dot products, or indeed any enumerable set which could then be used to build further accurate algorithms. We show how our accurate algorithms and decision procedure for finding them extend to this case. Finally, we address other models of arithmetic, and the relationship between (im)possibility in the TM and (in)efficient algorithms operating on numbers represented as bit strings.Comment: 49 pages, 6 figures, 1 tabl

Annotating patient clinical records with syntactic chunks and named entities: the Harvey corpus

The free text notes typed by physicians during patient consultations contain valuable information for the study of disease and treatment. These notes are difficult to process by existing natural language analysis tools since they are highly telegraphic (omitting many words), and contain many spelling mistakes, inconsistencies in punctuation, and non-standard word order. To support information extraction and classification tasks over such text, we describe a de-identified corpus of free text notes, a shallow syntactic and named entity annotation scheme for this kind of text, and an approach to training domain specialists with no linguistic background to annotate the text. Finally, we present a statistical chunking system for such clinical text with a stable learning rate and good accuracy, indicating that the manual annotation is consistent and that the annotation scheme is tractable for machine learning

Lower Bounds on the Depth of Monotone Arithmetic Computations

AbstractConsider an arithmetic expression of lengthninvolving only the operations {+,×} and non-negative constants. We prove lower bounds on the depth of any binary computation tree over the same sets of operations and constants that computes such an expression. We exhibit a family of arithmetic expressions that requires computation trees of depth at least 1.5log2n−O(1), thus proving a conjecture of S. R. Kosaraju (1986,in“Proc. of the 18th ACM Symp. on Theory Computing,” pp. 231–239). In contrast, Kosaraju showed how to compute such expressions with computation trees of depth 2log2n+O(1)

A formally verified compiler back-end

This article describes the development and formal verification (proof of semantic preservation) of a compiler back-end from Cminor (a simple imperative intermediate language) to PowerPC assembly code, using the Coq proof assistant both for programming the compiler and for proving its correctness. Such a verified compiler is useful in the context of formal methods applied to the certification of critical software: the verification of the compiler guarantees that the safety properties proved on the source code hold for the executable compiled code as well

Towards Revealing the Mystery behind Chain of Thought: a Theoretical Perspective

Recent studies have discovered that Chain-of-Thought prompting (CoT) can dramatically improve the performance of Large Language Models (LLMs), particularly when dealing with complex tasks involving mathematics or reasoning. Despite the enormous empirical success, the underlying mechanisms behind CoT and how it unlocks the potential of LLMs remain elusive. In this paper, we take a first step towards theoretically answering these questions. Specifically, we examine the capacity of LLMs with CoT in solving fundamental mathematical and decision-making problems. We start by giving an impossibility result showing that any bounded-depth Transformer cannot directly output correct answers for basic arithmetic/equation tasks unless the model size grows super-polynomially with respect to the input length. In contrast, we then prove by construction that autoregressive Transformers of a constant size suffice to solve both tasks by generating CoT derivations using a commonly-used math language format. Moreover, we show LLMs with CoT are capable of solving a general class of decision-making problems known as Dynamic Programming, thus justifying its power in tackling complex real-world tasks. Finally, extensive experiments on four tasks show that, while Transformers always fail to predict the answers directly, they can consistently learn to generate correct solutions step-by-step given sufficient CoT demonstrations.Comment: 33 page

Feasible arithmetic computations: Valiant's hypothesis

An account of Valiant's theory of p-computable versus p-definable polynomials, an arithmetic analogue of the Boolean theory of P versus NP, is presented, with detailed proofs of Valiant's central results

Physically Equivalent Intelligent Systems for Reasoning Under Uncertainty at Nanoscale

Machines today lack the inherent ability to reason and make decisions, or operate in the presence of uncertainty. Machine-learning methods such as Bayesian Networks (BNs) are widely acknowledged for their ability to uncover relationships and generate causal models for complex interactions. However, their massive computational requirement, when implemented on conventional computers, hinders their usefulness in many critical problem areas e.g., genetic basis of diseases, macro finance, text classification, environment monitoring, etc. We propose a new non-von Neumann technology framework purposefully architected across all layers for solving these problems efficiently through physical equivalence, enabled by emerging nanotechnology. The architecture builds on a probabilistic information representation and multi-domain mixed-signal circuit style, and is tightly coupled to a nanoscale physical layer that spans magnetic and electrical domains. Based on bottom-up device-circuit-architecture simulations, we show up to four orders of magnitude performance improvement (using computational resolution of 0.1) vs. best-of-breed multi-core machines with 100 processors, for BNs with about a million variables. Smaller problem sizes of ~100 variables can be realized at 20 mW power consumption and very low area around a few tenths of a mm2. Our vision is to enable solving complex Bayesian problems in real time, as well as enable intelligence capabilities at a small scale everywhere, ushering in a new era of machine intelligence