Proof-checking mathematical texts in controlled natural language

The research conducted for this thesis has been guided by the vision of a computer program that could check the correctness of mathematical proofs written in the language found in mathematical textbooks. Given that reliable processing of unrestricted natural language input is out of the reach of current technology, we focused on the attainable goal of using a controlled natural language (a subset of a natural language defined through a formal grammar) as input language to such a program. We have developed a prototype of such a computer program, the Naproche system. This thesis is centered around the novel logical and linguistic theory needed for defining and motivating the controlled natural language and the proof checking algorithm of the Naproche system. This theory provides means for bridging the wide gap between natural and formal mathematical proofs. We explain how our system makes use of and extends existing linguistic formalisms in order to analyse the peculiarities of the language of mathematics. In this regard, we describe a phenomenon of this language previously not described by other logicians or linguists, the implicit dynamic function introduction, exemplified by constructs of the form "for every x there is an f(x) such that ...". We show how this function introduction can lead to a paradox analogous to Russell's paradox. To tackle this problem, we developed a novel foundational theory of functions called Ackermann-like Function Theory, which is equiconsistent to ZFC (Zermelo-Fraenkel set theory with the Axiom of Choice) and can be used for imposing limitations to implicit dynamic function introduction in order to avoid this paradox. We give a formal account of implicit dynamic function introduction by extending Dynamic Predicate Logic, a formalism developed by linguists to account for the dynamic nature of natural language quantification, to a novel formalism called Higher-Order Dynamic Predicate Logic, whose semantics is based on Ackermann-like Function Theory. Higher-Order Dynamic Predicate Logic also includes a formal account of the linguistic theory of presuppositions, which we use for clarifying and formally modelling the usage of potentially undefined terms (e.g. 1/x, which is undefined for x=0) and of definite descriptions (e.g. "the even prime number") in the language of mathematics. The semantics of the controlled natural language is defined through a translation from the controlled natural language into an extension of Higher-Order Dynamic Predicate Logic called Proof Text Logic. Proof Text Logic extends Higher-Order Dynamic Predicate Logic in two respects, which make it suitable for representing the content of mathematical texts: It contains features for representing complete texts rather than single assertions, and instead of being based on Ackermann-like Function Theory, it is based on a richer foundational theory called Class-Map-Tuple-Number Theory, which does not only have maps/functions, but also classes/sets, tuples, numbers and Booleans as primitives. The proof checking algorithm checks the deductive correctness of proof texts written in the controlled natural language of the Naproche system. Since the semantics of the controlled natural language is defined through a translation into the Proof Text Logic formalism, the proof checking algorithm is defined on Proof Text Logic input. The algorithm makes use of automated theorem provers for checking the correctness of single proof steps. In this way, the proof steps in the input text do not need to be as fine-grained as in formal proof calculi, but may contain several reasoning steps at once, just as is usual in natural mathematical texts. The proof checking algorithm has to recognize implicit dynamic function introductions in the input text and has to take care of presuppositions of mathematical statements according to the principles of the formal account of presuppositions mentioned above. We prove two soundness and two completeness theorems for the proof checking algorithm: In each case one theorem compares the algorithm to the semantics of Proof Text Logic and one theorem compares it to the semantics of standard first-order predicate logic. As a case study for the theory developed in the thesis, we illustrate the working of the Naproche system on a controlled natural language adaptation of the beginning of Edmund Landau's Grundlagen der Analysis.Beweisprüfung mathematischer Texte in kontrollierter natürlicher Sprache Die Forschung, die für diese Dissertation durchgeführt wurde, basiert auf der Vision eines Computerprogramms, das die Korrektheit von mathematischen Beweisen, die in der gewöhnlichen mathematischen Fachsprache verfasst sind, überprüfen kann. Da die zuverlässige automatische Bearbeitung von uneingeschränktem natürlich-sprachlichen Input außer Reichweite der gegenwärtigen Technologie ist, haben wir uns auf das erreichbare Ziel fokussiert, eine kontrollierte natürliche Sprache (eine Teilmenge der natürlichen Sprache, die durch eine formale Grammatik definiert ist) als Eingabesprache für ein solches Programm zu verwenden. Wir haben einen Prototypen eines solchen Computerprogramms, das Naproche-System, entwickelt. Die vorliegende Dissertation beschreibt die neuartigen logischen und linguistischen Theorien, die benötigt werden, um die kontrollierte natürliche Sprache und den Beweisprüfungs-Algorithmus des Naproche-Systems zu definieren und zu motivieren. Diese Theorien stellen Methoden zu Verfügung, die dazu verwendet werden können, die weite Kluft zwischen natürlichen und formalen mathematischen Beweisen zu überbrücken. Wir erklären, wie unser System existierende linguistische Formalismen verwendet und erweitert, um die Besonderheiten der mathematischen Fachsprache zu analysieren. In diesem Zusammenhang beschreiben wir ein Phänomen dieser Fachsprache, das bisher von Logikern und Linguisten nicht beschrieben wurde – die implizite dynamische Funktionseinführung, die durch Konstruktionen der vorm "für jedes x gibt es ein f(x), so dass ..." veranschaulicht werden kann. Wir zeigen, wie diese Funktionseinführung zu einer der Russellschen analogen Antinomie führt. Um dieses Problem zu lösen, haben wir eine neuartige Grundlagentheorie für Funktionen entwickelt, die Ackermann-artige Funktionstheorie, die äquikonsistent zu ZFC (Zermelo-Fraenkel-Mengenlehre mit Auswahlaxiom) ist und verwendet werden kann, um der impliziten dynamischen Funktionseinführung Grenzen zu setzen, die zur Vermeidung dieser Antinomie führen. Wir beschreiben die implizite dynamische Funktionseinführung formal, indem wir die Dynamische Prädikatenlogik – ein Formalismus, der von Linguisten entwickelt wurde, um die dynamischen Eigenschaften der natürlich-sprachlichen Quantifizierung zu erfassen – zur Dynamischen Prädikatenlogik Höherer Stufe erweitern, deren Semantik auf der Ackermann-artigen Funktionstheorie basiert. Die Dynamische Prädikatenlogik Höherer Stufe formalisiert auch die linguistische Theorie der Präsuppositionen, die wir verwenden, um den Gebrauch potentiell undefinierter Terme (z.B. der Term 1/x, der für x=0 undefiniert ist) und bestimmter Kennzeichnungen (z.B. "die gerade Primzahl") in der mathematischen Fachsprache zu modellieren. Die Semantik der kontrollierten natürlichen Sprache wird definiert durch eine Übersetzung dieser in eine Erweiterung der Dynamischen Prädikatenlogik Höherer Stufe mit der Bezeichnung Beweistext-Logik. Die Beweistext-Logik erweitert die Dynamische Prädikatenlogik Höherer Stufe in zwei Hinsichten: Sie stellt Funktionalitäten für die Repräsentation von vollständigen Texten, und nicht nur von Einzelaussagen, zur Verfügung, und anstatt auf der Ackermann-artigen Funktionstheorie zu basieren, basiert sie auf einer reichhaltigeren Grundlagentheorie – der Klassen-Abbildungs-Tupel-Zahlen-Theorie, die neben Abbildungen/Funktionen auch noch Klassen/Mengen, Tupel, Zahlen und boolesche Werte als Grundobjekte zur Verfügung stellt. Der Beweisprüfungs-Algorithmus prüft die deduktive Korrektheit von Beweistexten, die in der kontrollierten natürlichen Sprache des Naproche-Systems verfasst sind. Da die Semantik dieser kontrollierten natürlichen Sprache durch eine Übersetzung in die Beweistext-Logik definiert ist, ist der Beweisprüfungs-Algorithmus für Beweistext-Logik-Input definiert. Der Algorithmus verwendet automatische Beweiser für die Überprüfung einzelner Beweisschritte. Dadurch müssen die Beweisschritte in dem Eingabetext nicht so kleinschrittig sein wie in formalen Beweiskalkülen, sondern können mehrere Deduktionsschritte zu einem Schritt vereinen, so wie dies auch in natürlichen mathematischen Texten üblich ist. Der Beweisprüfungs-Algorithmus muss die impliziten Funktionseinführungen im Eingabetext erkennen und Präsuppositionen von mathematischen Aussagen auf Grundlage der oben erwähnten Präsuppositionstheorie behandeln. Wir beweisen zwei Korrektheits- und zwei Vollständigkeitssätze für den Beweisprüfungs-Algorithmus: Jeweils einer dieser Sätze vergleicht den Algorithmus mit der Semantik der Beweistext-Logik und jeweils einer mit der Semantik der üblichen Prädikatenlogik erster Stufe. Als Fallstudie für die in dieser Dissertation entwickelte Theorie veranschaulichen wir die Funktionsweise des Naproche-Systems an einem an die kontrollierte natürliche Sprache angepassten Anfangsabschnitt von Edmund Landaus Grundlagen der Analysis

The Naproche system: Proof-checking mathematical texts in controlled natural language

The Naproche system is a system for linguistically analysing and proof-checking mathematical texts written in a controlled natural language, i.e. a subset of the usual natural language of mathematical texts defined through a formal grammar. This paper gives an overview over the linguistic and logical techniques developed for the Naproche system. Special attention is given to the dynamic nature of quantification in natural language, to the phenomenon of implicit function introduction in mathematical texts, and to the usage of definitions for dynamically extending the language of a mathematical text

Proceedings of the Conference on Natural Language Processing 2010

This book contains state-of-the-art contributions to the 10th conference on Natural Language Processing, KONVENS 2010 (Konferenz zur Verarbeitung natürlicher Sprache), with a focus on semantic processing. The KONVENS in general aims at offering a broad perspective on current research and developments within the interdisciplinary field of natural language processing. The central theme draws specific attention towards addressing linguistic aspects ofmeaning, covering deep as well as shallow approaches to semantic processing. The contributions address both knowledgebased and data-driven methods for modelling and acquiring semantic information, and discuss the role of semantic information in applications of language technology. The articles demonstrate the importance of semantic processing, and present novel and creative approaches to natural language processing in general. Some contributions put their focus on developing and improving NLP systems for tasks like Named Entity Recognition or Word Sense Disambiguation, or focus on semantic knowledge acquisition and exploitation with respect to collaboratively built ressources, or harvesting semantic information in virtual games. Others are set within the context of real-world applications, such as Authoring Aids, Text Summarisation and Information Retrieval. The collection highlights the importance of semantic processing for different areas and applications in Natural Language Processing, and provides the reader with an overview of current research in this field

Students´ language in computer-assisted tutoring of mathematical proofs

Truth and proof are central to mathematics. Proving (or disproving) seemingly simple statements often turns out to be one of the hardest mathematical tasks. Yet, doing proofs is rarely taught in the classroom. Studies on cognitive difficulties in learning to do proofs have shown that pupils and students not only often do not understand or cannot apply basic formal reasoning techniques and do not know how to use formal mathematical language, but, at a far more fundamental level, they also do not understand what it means to prove a statement or even do not see the purpose of proof at all. Since insight into the importance of proof and doing proofs as such cannot be learnt other than by practice, learning support through individualised tutoring is in demand. This volume presents a part of an interdisciplinary project, set at the intersection of pedagogical science, artificial intelligence, and (computational) linguistics, which investigated issues involved in provisioning computer-based tutoring of mathematical proofs through dialogue in natural language. The ultimate goal in this context, addressing the above-mentioned need for learning support, is to build intelligent automated tutoring systems for mathematical proofs. The research presented here has been focused on the language that students use while interacting with such a system: its linguistic propeties and computational modelling. Contribution is made at three levels: first, an analysis of language phenomena found in students´ input to a (simulated) proof tutoring system is conducted and the variety of students´ verbalisations is quantitatively assessed, second, a general computational processing strategy for informal mathematical language and methods of modelling prominent language phenomena are proposed, and third, the prospects for natural language as an input modality for proof tutoring systems is evaluated based on collected corpora

A change-oriented architecture for mathematical authoring assistance

The computer-assisted authoring of mathematical documents using a scientific text-editor requires new mathematical knowledge management and transformation techniques to organize the overall workflow of anassistance system like the ΩMEGAsystem.The challenge is that, throughout the system, various kinds of given and derived knowledge units occur in different formats and with different dependencies. If changes occur in these pieces of knowledge, they need to be effectively propagated. We present a Change-Oriented Architecture for mathematical authoring assistance. Thereby, documents are used as interfaces and the components of the architecture interact by actively changing the interface documents and by reacting on changes. In order to optimize this style of interaction, we present two essential methods in this thesis. First, we develop an efficient method for the computation of weighted semantic changes between two versions of a document. Second, we present an invertible grammar formalism for the automated bidirectional transformation between interface documents. The presented architecture provides an adequate basis for the computer-assisted authoring of mathematical documents with semantic annotations and a controlled mathematical language

Q(sqrt(-3))-Integral Points on a Mordell Curve

We use an extension of quadratic Chabauty to number fields,recently developed by the author with Balakrishnan, Besser and M ̈uller,combined with a sieving technique, to determine the integral points overQ(√−3) on the Mordell curve y2 = x3 − 4

Parsing and Disambiguation of Symbolic Mathematics in the Naproche System

Abstract. The Naproche system is a system for linguistically analysing and proof-checking mathematical texts written in a controlled natural language. The aim is to have an input language that is as close as possible to the language that mathematicians actually use when writing textbooks or papers. Mathematical texts consist of a combination of natural language and symbolic mathematics, with symbolic mathematics obeying its own syntactic rules. We discuss the difficulties that a program for parsing and disambiguating symbolic mathematics must face and present how these difficulties have been tackled in the Naproche system. One of these difficulties is the fact that information provided in the preceding context – including information provided in natural language – can influence the way a symbolic expression has to be disambiguated