Towards an Intelligent Tutor for Mathematical Proofs

Computer-supported learning is an increasingly important form of study since it allows for independent learning and individualized instruction. In this paper, we discuss a novel approach to developing an intelligent tutoring system for teaching textbook-style mathematical proofs. We characterize the particularities of the domain and discuss common ITS design models. Our approach is motivated by phenomena found in a corpus of tutorial dialogs that were collected in a Wizard-of-Oz experiment. We show how an intelligent tutor for textbook-style mathematical proofs can be built on top of an adapted assertion-level proof assistant by reusing representations and proof search strategies originally developed for automated and interactive theorem proving. The resulting prototype was successfully evaluated on a corpus of tutorial dialogs and yields good results.Comment: In Proceedings THedu'11, arXiv:1202.453

Assertion level proof planning with compiled strategies

This book presents new techniques that allow the automatic verification and generation of abstract human-style proofs. The core of this approach builds an efficient calculus that works directly by applying definitions, theorems, and axioms, which reduces the size of the underlying proof object by a factor of ten. The calculus is extended by the deep inference paradigm which allows the application of inference rules at arbitrary depth inside logical expressions and provides new proofs that are exponentially shorter and not available in the sequent calculus without cut. In addition, a strategy language for abstract underspecified declarative proof patterns is developed. Together, the complementary methods provide a framework to automate declarative proofs. The benefits of the techniques are illustrated by practical applications.Die vorliegende Arbeit beschäftigt sich damit, das Formalisieren von Beweisen zu vereinfachen, indem Methoden entwickelt werden, um informale Beweise formal zu verifizieren und erzeugen zu können. Dazu wird ein abstrakter Kalkül entwickelt, der direkt auf der Faktenebene arbeitet, welche von Menschen geführten Beweisen relativ nahe kommt. Anhand einer Fallstudie wird gezeigt, dass die abstrakte Beweisführung auf der Fakteneben vorteilhaft für automatische Suchverfahren ist. Zusätzlich wird eine Strategiesprache entwickelt, die es erlaubt, unterspezifizierte Beweismuster innerhalb des Beweisdokumentes zu spezifizieren und Beweisskizzen automatisch zu verfeinern. Fallstudien zeigen, dass komplexe Beweismuster kompakt in der entwickelten Strategiesprache spezifiziert werden können. Zusammen bilden die einander ergänzenden Methoden den Rahmen zur Automatisierung von deklarativen Beweisen auf der Faktenebene, die bisher überwiegend manuell entwickelt werden mussten

Assertion level proof planning with compiled strategies

This book presents new techniques that allow the automatic verification and generation of abstract human-style proofs. The core of this approach builds an efficient calculus that works directly by applying definitions, theorems, and axioms, which reduces the size of the underlying proof object by a factor of ten. The calculus is extended by the deep inference paradigm which allows the application of inference rules at arbitrary depth inside logical expressions and provides new proofs that are exponentially shorter and not available in the sequent calculus without cut. In addition, a strategy language for abstract underspecified declarative proof patterns is developed. Together, the complementary methods provide a framework to automate declarative proofs. The benefits of the techniques are illustrated by practical applications.Die vorliegende Arbeit beschäftigt sich damit, das Formalisieren von Beweisen zu vereinfachen, indem Methoden entwickelt werden, um informale Beweise formal zu verifizieren und erzeugen zu können. Dazu wird ein abstrakter Kalkül entwickelt, der direkt auf der Faktenebene arbeitet, welche von Menschen geführten Beweisen relativ nahe kommt. Anhand einer Fallstudie wird gezeigt, dass die abstrakte Beweisführung auf der Fakteneben vorteilhaft für automatische Suchverfahren ist. Zusätzlich wird eine Strategiesprache entwickelt, die es erlaubt, unterspezifizierte Beweismuster innerhalb des Beweisdokumentes zu spezifizieren und Beweisskizzen automatisch zu verfeinern. Fallstudien zeigen, dass komplexe Beweismuster kompakt in der entwickelten Strategiesprache spezifiziert werden können. Zusammen bilden die einander ergänzenden Methoden den Rahmen zur Automatisierung von deklarativen Beweisen auf der Faktenebene, die bisher überwiegend manuell entwickelt werden mussten

Defining interfaces between hardware and software: Quality and performance

One of the most important interfaces in a computer system is the interface between hardware and software. This interface is the contract between the hardware designer and the programmer that defines the functional behaviour of the hardware. This thesis examines two critical aspects of defining the hardware-software interface: quality and performance. The first aspect is creating a high quality specification of the interface as conventionally defined in an instruction set architecture. The majority of this thesis is concerned with creating a specification that covers the full scope of the interface; that is applicable to all current implementations of the architecture; and that can be trusted to accurately describe the behaviour of implementations of the architecture. We describe the development of a formal specification of the two major types of Arm processors: A-class (for mobile devices such as phones and tablets) and M-class (for micro-controllers). These specifications are unparalleled in their scope, applicability and trustworthiness. This thesis identifies and illustrates what we consider the key ingredient in achieving this goal: creating a specification that is used by many different user groups. Supporting many different groups leads to improved quality as each group finds different problems in the specification; and, by providing value to each different group, it helps justify the considerable effort required to create a high quality specification of a major processor architecture. The work described in this thesis led to a step change in Arm's ability to use formal verification techniques to detect errors in their processors; enabled extensive testing of the specification against Arm's official architecture conformance suite; improved the quality of Arm's architecture conformance suite based on measuring the architectural coverage of the tests; supported earlier, faster development of architecture extensions by enabling animation of changes as they are being made; and enabled early detection of problems created from architecture extensions by performing formal validation of the specification against semi-structured natural language specifications. As far as we are aware, no other mainstream processor architecture has this capability. The formal specifications are included in Arm's publicly released architecture reference manuals and the A-class specification is also released in machine-readable form. The second aspect is creating a high performance interface by defining the hardware-software interface of a software-defined radio subsystem using a programming language. That is, an interface that allows software to exploit the potential performance of the underlying hardware. While the hardware-software interface is normally defined in terms of machine code, peripheral control registers and memory maps, we define it using a programming language instead. This higher level interface provides the opportunity for compilers to hide some of the low-level differences between different systems from the programmer: a potentially very efficient way of providing a stable, portable interface without having to add hardware to provide portability between different hardware platforms. We describe the design and implementation of a set of extensions to the C programming language to support programming high performance, energy efficient, software defined radio systems. The language extensions enable the programmer to exploit the pipeline parallelism typically present in digital signal processing applications and to make efficient use of the asymmetric multiprocessor systems designed to support such applications. The extensions consist primarily of annotations that can be checked for consistency and that support annotation inference in order to reduce the number of annotations required. Reducing the number of annotations does not just save programmer effort, it also improves portability by reducing the number of annotations that need to be changed when porting an application from one platform to another. This work formed part of a project that developed a high-performance, energy-efficient, software defined radio capable of implementing the physical layers of the 4G cellphone standard (LTE), 802.11a WiFi and Digital Video Broadcast (DVB) with a power and silicon area budget that was competitive with a conventional custom ASIC solution. The Arm architecture is the largest computer architecture by volume in the world. It behooves us to ensure that the interface it describes is appropriately defined

On the Formal Flexibility of Syntactic Categories

This dissertation explores the formal flexibility of syntactic categories. The main proposal is that Universal Grammar (UG) only provides templatic guidance for syntactic category formation and organization but leaves many other issues open, including issues internal to a single category and issues at the intercategorial, system level: these points that UG "does not care about" turn out to enrich the categorial ontology of human language in important ways. The dissertation consists of seven chapters. After a general introduction in Chapter 1, I lay out some foundational issues regarding features and categories in Chapter 2 and delineate a featural metalanguage comprising four components: specification, valuation, typing, and granularity. Based on that I put forward a templatic definition for syntactic categories, which unifies the combinatorial and taxonomic perspectives under the notion mergeme. Then, a detailed overview of the "categorial universe" I work with is presented, which shows that the syntactic category system (SCS) is an intricate web structured by five layers of abstraction divided into three broad levels of concern: the individual level (layers 1–2), the global level (layers 3–4), and the supraglobal level (layer 5). In the subsequent chapters I explore the template-flexibility pairs at each abstraction layer, with Chapters 3–4 focusing on the first layer, Chapter 5 on the second layer, and Chapter 6 on the third and fourth layers; the fifth layer is not in the scope of this dissertation. Chapter 3 examines a special type of category defined by an underspecified mergeme, the defective category, which behaves like a "chameleon" in that it gets assimilated into whatever nondefective category it merges with. This characteristic makes it potentially useful in analyzing certain adjunction structures, and I explore this potential by two case studies, one focusing on modifier-head compounds and the other on sentence-final particles. Chapter 4 examines another special type of category defined by the absence of a mergeme, the Root category. Deductive reasoning leads me to propose a generalized root syntax, according to which roots are not confined to lexical categorial environments but may legally merge with and hence "support" any non-Root category. I demonstrate the empirical consequences of this theory by a comprehensive study of the half-lexical–half-functional vocabulary items in Chinese. Chapter 5 ascends to the second abstraction layer and raises the question of whether the categorial sequences (or projection hierarchies) in human language are necessarily totally ordered, as certain analytical devices (e.g., "flavored" categories) can only be theoretically maintained if we also allow categorial sequences to be partially ordered. After a diachronic study of the flavored verbalizer $v_{BE}$ (stative) in Chinese resultative compounds, I conclude that while "flavoring" is indeed a possible type of flexibility in the SCS, it is the deviation rather than the norm due to non-UG or "third" factors and hence should be cautiously used in syntactic analyses. Chapter 6 ascends even higher on the ladder of abstraction and examines the global interconnection in the SCS ontology with the aid of mathematical Category theory. I formalize the functional parallelism across major parts of speech and the inheritance-based relations across granularity levels as Category-theoretic structures, which reveal further and more abstract templates and flexibility types in the SCS. A crucial mathematical concept in the formalization is epi-Adjunction. Finally, in Chapter 7 I summarize the main results of this dissertation and briefly discuss some potential directions of future research.My PhD is funded by Cambridge Trust and China Scholarship Council. I have also received travel grants and financial aids from Gonville and Caius College and the Faculty of Modern and Medieval Languages

Barry Smith an sich

Festschrift in Honor of Barry Smith on the occasion of his 65th Birthday. Published as issue 4:4 of the journal Cosmos + Taxis: Studies in Emergent Order and Organization. Includes contributions by Wolfgang Grassl, Nicola Guarino, John T. Kearns, Rudolf Lüthe, Luc Schneider, Peter Simons, Wojciech Żełaniec, and Jan Woleński

Requirements specification using concrete scenarios

The precision of formal specifications allows us to prove program correctness. Even if formal methods are not used throughout the software project, formalisation improves our understanding of the problem. Formal specifications are amenable to automated analysis and consistency checking. However using them is challenging. Customers do not understand formal notations. Specifiers have difficulty tackling large problems. Once systems are built, formal specifications quickly become outdated during software maintenance. A method of developing formal specifications using concrete scenarios is proposed to tackle the disadvantages just mentioned. A concrete scenario describes system behaviour with successive steps. The pre- and post-states of scenario steps are expressed with actual data rather than variables. Concrete scenarios are expressed in a natural language or formal notation. They increase customer involvement in the creation of formal specifications. Scenarios may be ranked by priorities allowing specifiers to focus on a small part of the system. Formal specifications are constructed incrementally. New requirements are also captured in concrete scenarios which guide the modification of formal specifications. On one hand, concrete scenarios assist the creation and maintenance of formal specifications. On the other hand, they facilitate program correctness proofs without using conventional formal specifications. This is achieved by adding implementation details to customer scenarios. The resulting developer scenarios, encapsulating decisions of data structures and algorithms, are generalised to operation schemas. With the implementation details, the schemas written in formal notations are programs rather than specifications.EThOS - Electronic Theses Online ServiceGBUnited Kingdo

Deductive Verification of Safety-Critical Java Programs

This work investigates the application of deductive verification techniques to safety critical Java programs, in particular RTSJ programs. A focus is put on the formalization of the RTSJ memory model in dynamic logic, the utilization of a region-based memory model for ensuring non-interference and a design-by-contract based approach for the formal specification and verification of worst case memory consumption