Reasoning about quantities in natural language

Quantitative reasoning involves understanding the use of quantities and numeric relations in text, and reasoning with respect to them. It forms an essential part of everyday interaction. However, little work from the Natural Language Processing community has focused on quantitative reasoning. In this thesis, we investigate the challenges in performing automated quantitative reasoning over natural language text. We formulate several tasks to tackle some of the fundamental problems of quantitative reasoning, and address the problem of developing robust statistical methods for these tasks. We show that standard NLP tools are not sufficient to obtain the abstraction needed for quantitative reasoning; the standard NLP pipeline needs to be extended in various ways. We propose several technical ideas for these extensions. We first look at the problem of detecting and normalizing quantities expressed in free form text, and show that correct detection and normalization can support several simple quantitative inferences. We then focus on numeric relation extraction from sentences, and show that several natural properties of language can be leveraged to effectively extract numeric relations from a sentence. We finally investigate the problem of quantitative reasoning over multiple quantities mentioned across several sentences. We develop a decomposition strategy which allows reasoning over pairs of numbers to be combined effectively to perform global reasoning. We also look at the problem of effectively using math domain knowledge in quantitative reasoning. On this front, we first propose graph representations called "unit dependency graphs'', and show that these graph representations can be used to effectively incorporate dimensional analysis knowledge in quantitative reasoning. Next, we develop a general framework to incorporate any declarative knowledge into quantitative reasoning. This framework is used to incorporate several mathematical concepts into textual quantitative reasoning, leading to robust reasoning systems

Programming with a Differentiable Forth Interpreter

Given that in practice training data is scarce for all but a small set of problems, a core question is how to incorporate prior knowledge into a model. In this paper, we consider the case of prior procedural knowledge for neural networks, such as knowing how a program should traverse a sequence, but not what local actions should be performed at each step. To this end, we present an end-to-end differentiable interpreter for the programming language Forth which enables programmers to write program sketches with slots that can be filled with behaviour trained from program input-output data. We can optimise this behaviour directly through gradient descent techniques on user-specified objectives, and also integrate the program into any larger neural computation graph. We show empirically that our interpreter is able to effectively leverage different levels of prior program structure and learn complex behaviours such as sequence sorting and addition. When connected to outputs of an LSTM and trained jointly, our interpreter achieves state-of-the-art accuracy for end-to-end reasoning about quantities expressed in natural language stories.Comment: 34th International Conference on Machine Learning (ICML 2017

Learning by Seeing by Doing: Arithmetic Word Problems

Learning by doing in pursuit of real-world goals has received much attention from education researchers but has been unevenly supported by mathematics education software at the elementary level, particularly as it involves arithmetic word problems. In this article, we give examples of doing-oriented tools that might promote children\u27s ability to see significant abstract structures in mathematical situations. The reflection necessary for such seeing is motivated by activities and contexts that emphasize affective and social aspects. Natural language, as a representation already familiar to children, is key in these activities, both as a means of mathematical expression and as a link between situations and various abstract representations. These tools support children\u27s ownership of a mathematical problem and its expression; remote sharing of problems and data; software interpretation of children\u27s own word problems; play with dynamically linked representations with attention to children\u27s prior connections; and systematic problem variation based on empirically determined level of difficulty

Developing algebraic and didactical knowledge in pre-service primary teacher education

This study analyzes the contribution of a teaching experiment for the development of prospective primary teachers regarding knowledge of algebra and of algebra teaching as well as their professional identity. The case study of a prospective teachersuggests that an exploratory approach combining content and pedagogy supports this development, especially in the need to propose challenging tasks, to provide opportunity for students’ autonomous work and collective discussions and to be attentive to children’s representations and strategies in order to promote algebraic thinking

Linear superposition as a core theorem of quantum empiricism

Clarifying the nature of the quantum state $|\Psi\rangle$ is at the root of the problems with insight into (counterintuitive) quantum postulates. We provide a direct-and math-axiom free-empirical derivation of this object as an element of a vector space. Establishing the linearity of this structure-quantum superposition-is based on a set-theoretic creation of ensemble formations and invokes the following three principia: $(\textsf{I})$ quantum statics, $(\textsf{II})$ doctrine of a number in the physical theory, and $(\textsf{III})$ mathematization of matching the two observations with each other; quantum invariance. All of the constructs rest upon a formalization of the minimal experimental entity: observed micro-event, detector click. This is sufficient for producing the $\mathbb C$-numbers, axioms of linear vector space (superposition principle), statistical mixtures of states, eigenstates and their spectra, and non-commutativity of observables. No use is required of the concept of time. As a result, the foundations of theory are liberated to a significant extent from the issues associated with physical interpretations, philosophical exegeses, and mathematical reconstruction of the entire quantum edifice.Comment: No figures. 64 pages; 68 pages(+4), overall substantial improvements; 70 pages(+2), further improvement