Argumentation Theory for Mathematical Argument

To adequately model mathematical arguments the analyst must be able to represent the mathematical objects under discussion and the relationships between them, as well as inferences drawn about these objects and relationships as the discourse unfolds. We introduce a framework with these properties, which has been used to analyse mathematical dialogues and expository texts. The framework can recover salient elements of discourse at, and within, the sentence level, as well as the way mathematical content connects to form larger argumentative structures. We show how the framework might be used to support computational reasoning, and argue that it provides a more natural way to examine the process of proving theorems than do Lamport's structured proofs.Comment: 44 pages; to appear in Argumentatio

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

Tools for Tutoring Theoretical Computer Science Topics

This thesis introduces COMPLEXITY TUTOR, a tutoring system to assist in learning abstract proof-based topics, which has been specifically targeted towards the population of computer science students studying theoretical computer science. Existing literature has shown tremendous educational benefits produced by active learning techniques, student-centered pedagogy, gamification and intelligent tutoring systems. However, previously, there had been almost no research on adapting these ideas to the domain of theoretical computer science. As a population, computer science students receive immediate feedback from compilers and debuggers, but receive no similar level of guidance for theoretical coursework. One hypothesis of this thesis is that immediate feedback while working on theoretical problems would be particularly well-received by students, and this hypothesis has been supported by the feedback of students who used the system. This thesis makes several contributions to the field. It provides assistance for teaching proof construction in theoretical computer science. A second contribution is a framework that can be readily adapted to many other domains with abstract mathematical content. Exercises can be constructed in natural language and instructors with limited programming knowledge can quickly develop new subject material for COMPLEXITY TUTOR. A third contribution is a platform for writing algorithms in Python code that has been integrated into this framework, for constructive proofs in computer science. A fourth contribution is development of an interactive environment that uses a novel graphical puzzle-like platform and gamification ideas to teach proof concepts. The learning curve for students is reduced, in comparison to other systems that use a formal language or complex interface. A multi-semester evaluation of 101 computer science students using COMPLEXITY TUTOR was conducted. An additional 98 students participated in the study as part of control groups. COMPLEXITY TUTOR was used to help students learn the topics of NP-completeness in algorithms classes and prepositional logic proofs in discrete math classes. Since this is the first significant study of using a computerized tutoring system in theoretical computer science, results from the study not only provide evidence to support the suitability of using tutoring systems in theoretical computer science, but also provide insights for future research directions

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

Foundations for a self-reflective, context-aware semantic representation of mathematical specifications

Das Projekt "a modeling system for mathematics" (MoSMath), das zur Zeit and der Universität Wien durchgeführt wird, hat die Erstellung eines Systems zur Spezifikation von numerischen Modellen zum Ziel, in einer Form wie sie für Mathematiker natürlich ist. Das spezifizierte Modell soll innerhalb des Systems repräsentiert und bearbeitet werden, und dann zu numerischen Solvern, die nicht Teil des System sind, übermittelt werden können. Als ein erster Schritt zu einer universal einsetzbaren Software für die Repräsentation und bearbeitung von Mathematik auf dem Computer (das FMathL Projekt) entwickeln wir eine Repräsentation von Mathematik in einem Semantischen Netz (das "semantic memory"), zusammen mit einem Typsystem das die Gültigkeit der Repräsentation prüft, und einer virtuellen Maschine, die Algorithmen ausführen kann. Der Benutzer profitiert von so einem System auf mehrfache Weise: Der offensichtlichste Vorteil ist dass der Benutzer nicht gezwungen ist eine Modellierungssprache zu erlernen und kann stattdessen die natürliche Sprache der Mathematik verwenden, welche von jedem Mathematiker, Informatiker, Physker, etc. erlernt und praktiziert wird. Zusätzlich ist diese Art der Spezifizierung eines Modells am wenigsten Fehleranfällig, und die natürlichste Art ein Modell zu kommunizieren. Einmal in dem System repräsentiert, können ohne zusätzlichen Aufwand Ausgaben des Modells in verschiedenen Modellierungssprachen, und verschiednenen natürlichen Sprachen erzeugt werden, vorausgesetzt dass passende Transformationsroutinen verfügbar sind.The project "a modeling system for mathematics" (MoSMath), currently carried out at the University of Vienna, aims to create a modeling system for the specification of models for the numerical work in optimization in a form that is natural for the working mathematician. The specified model is represented and processed inside a framework and can then be communicated to numerical solvers or other systems. As a first step towards a general purpose tool for representing and interfacing general mathematics on the computer (the FMathL project), we developed a representation of mathematics in a semantic network called the semantic memory, together with a type system that checks validity of the representation and a virtual machine that can execute algorithms. The user benefits from this input format in multiple ways: The most obvious advantage is that a user is not forced to learn an algebraic modeling language and can use the usual natural mathematical language, which is learned and practiced by every mathematician, computer scientist, physicist, and engineer. In addition, this kind of specification of a model is the least error prone, and the most natural way to communicate a model. Once represented in the framework, multiple outputs in different modeling languages (or even descriptions in different natural languages) would not mean extra work for the user if appropriate transformation modules are available

Aspects of the theory of containers within automated theorem proving

This thesis explores applications of the theory of containers within automated theorem proving. Container theory provides a foundational analysis of data types as containers, specified by a type $S$ of shapes and a function P assigning to each shape its set of positions for data.More importantly, a representation theorem guarantees that polymorphic functions between container data types are given by container morphisms, which are characterised by mappings between shapes and positions. Container theory is interesting, in this context, for the following reasons. A mechanism for representing and reasoning with ellipsis (the dots in x_1, x_2, ... , x_n) in lists, existing in the literature, has proved to be very useful for formalisations involving abstractions. Success with this mechanism came by means of a meta-level representation through which many functions that normally require recursive definitions can be given explicit ones. As a result, not only can induction and generalisation be eliminated from proofs but, by means of an associated portrayal system, the resulting proofs are also intuitive and much closer to informal mathematical proofs. This ellipsis mechanism, however, is not based on any formal theory, making it rather exiguous in comparison with rival techniques. There also remains questions about its scope and applications. Our aim is to improve this ellipsis mechanism. In this connection, we hypothesize that the theory of containers provides a formal underpinning for such representations. In order to test our hypothesis, we identify limitations of the ellipsis mechanism and show how they can be addressed within the theory of containers. We subsequently develop a new reasoning system based on containers, which does not suffer from these limitations. This judicious container-based system endorses representations of polymorphic rewrite rules using arithmetic, which naturally lends itself to applications of arithmetic decision procedures. We exploit this facet to develop a new technique for deciding properties of lists. Our technique is developed within a quasi-container setting: shape maps are given as piecewise-linear functions, while a new representation is derived for re-indexing functions that obviates the need for dependent types, which are fundamental in a judicious container approach. We show that this new setting enables us to represent and reason about a large class of properties

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

Aspects of the theory of containers within automated theorem proving

This thesis explores applications of the theory of containers within automated theorem proving. Container theory provides a foundational analysis of data types as containers, specified by a type $S$ of shapes and a function P assigning to each shape its set of positions for data.More importantly, a representation theorem guarantees that polymorphic functions between container data types are given by container morphisms, which are characterised by mappings between shapes and positions. Container theory is interesting, in this context, for the following reasons. A mechanism for representing and reasoning with ellipsis (the dots in x_1, x_2, ... , x_n) in lists, existing in the literature, has proved to be very useful for formalisations involving abstractions. Success with this mechanism came by means of a meta-level representation through which many functions that normally require recursive definitions can be given explicit ones. As a result, not only can induction and generalisation be eliminated from proofs but, by means of an associated portrayal system, the resulting proofs are also intuitive and much closer to informal mathematical proofs. This ellipsis mechanism, however, is not based on any formal theory, making it rather exiguous in comparison with rival techniques. There also remains questions about its scope and applications. Our aim is to improve this ellipsis mechanism. In this connection, we hypothesize that the theory of containers provides a formal underpinning for such representations. In order to test our hypothesis, we identify limitations of the ellipsis mechanism and show how they can be addressed within the theory of containers. We subsequently develop a new reasoning system based on containers, which does not suffer from these limitations. This judicious container-based system endorses representations of polymorphic rewrite rules using arithmetic, which naturally lends itself to applications of arithmetic decision procedures. We exploit this facet to develop a new technique for deciding properties of lists. Our technique is developed within a quasi-container setting: shape maps are given as piecewise-linear functions, while a new representation is derived for re-indexing functions that obviates the need for dependent types, which are fundamental in a judicious container approach. We show that this new setting enables us to represent and reason about a large class of properties

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