Dominik Dietrich

A generic modular data structure for proof attempts alternating on ideas and granularity

A practically useful mathematical assistant system requires the sophisticated combination of interaction and automation. Central in such a system is the proof data structure, which has to maintain the current proof state and which has to allow the flexible interplay of various components including the human user. We describe a parameterized proof data structure for the management of proofs, which includes our experience with the development of two proof assistants. It supports and bridges the gap between abstract level proof explanation and low-level proof verification. The proof data structure enables, in particular, the flexible handling of lemmas, the maintenance of different proof alternatives, and the representation of different granularities of proof attempts.

A tactic language for declarative proofs

Influenced by the success of the Mizar system many declarative proof languages have been developed in the theorem prover community, as declarative proofs are more readable, easier to modify and to maintain than their procedural counterparts. However, despite their advantages, many users still prefer the procedural style of proof, because procedural proofs are faster to write. In this paper we show how to define a declarative tactic language on top of a declarative proof language. The language comes along with a rich facility to declaratively specify conditions on proof states in the form of sequent patterns, as well as ellipses (dot notation) to provide a limited form of iteration. As declarative tactics are specified using the declarative proof language, they offer the same advantages as declarative proof languages. At the same time, they also produce declarative justifications in the form of a declarative proof script and can thus be seen as an attempt to reduce the gap between procedural and declarative proofs.

Synthesizing proof planning methods and Ω-ants agents from mathematical knowledge

In this paper we investigate how to extract proof procedural information contained in declarative representations of mathematical knowledge, such as axioms, definitions, lemmas and theorems (collectively called assertions) and how to effectively include it into automated proof search techniques. In the context of the proof planner Multi and the agent-based reasoning system Ω-Ants, we present techniques to automatically synthesize proof planning methods and Ω-Ants-agents from assertions such that they can be actively used by these systems. This in turn enables a user to effectively use these systems without having to know the peculiarities of coding methods and agents.

Authoring Verified Documents by Interactive Proof Construction and Verification in Text-Editors

Aiming at a document-centric approach to formalizing and verifying mathematics and software we integrated the proof assistance system i¾? mega with the standard scientific text-editor MACS . The author writes her mathematical document entirely inside the text-editor in a controlled language with formulas in style. The notation specified in such a document is used for both parsing and rendering formulas in the document. To make this approach effectively usable as a real-time application we present an efficient hybrid parsing technique that is able to deal with the scalability problem resulting from modifying or extending notation dynamically. Furthermore, we present incremental methods to quickly verify constructed or modified proof steps by i¾? mega . If the system detects incomplete or underspecified proof steps, it tries to automatically repair them. For collaborative authoring we propose to manage partially or fully verified documents together with its justifications and notational information centrally in a mathematics repository using an extension of OMDoc .

Proof Step Analysis for Proof Tutoring -- A Learning Approach to Granularity

We present a proof step diagnosis module based on the mathematical assistant system Ωmega. The task of this module is to evaluate proof steps as typically uttered by students in tutoring sessions on mathematical proofs. In particular, we categorise the step size of proof steps performed by the student, in order to recognise if they are appropriate with respect to the student model. We propose an approach which builds on reconstructions of the proof in question via automated proof search using a cognitively motivated proof calculus. Our approach employs learning techniques and incorporates a student model, and our diagnosis module can be adjusted to different domains and users. We present a first evaluation based on empirical data.

ΩMEGA: Resource-Adaptive Processes in an Automated Reasoning System

The ΩMEGA project and its predecessor, the MKRP-system, grew out of the principal dissatisfaction with the methodology and lack of success of the search-based “logic engines” of the 1960s and 1970s.

Organization, Transformation, and Propagation of Mathematical Knowledge in Ωmega

Abstract.Mathematical assistance systems and proof assistance systems in general have traditionally been developed as large, monolithic systems which are often hard to maintain and extend. In this article we propose a component network architecture as a means to design and implement such systems. Under this view a mathematical assistance system is an integrated knowledge-based system composed as a network of individual, specialized components. These components manipulate and mutually exchange different kinds of mathematical knowledge encoded within different document formats. Consequently, several units of mathematical knowledge coexist throughout the system within these components and this knowledge changes non-monotonically over time. Our approach has resulted in a lean and maintainable system code and makes the system open for extensions. Moreover, it naturally decomposes the global and complex reasoning and truth maintenance task into local reasoning and truth maintenance tasks inside the system components. The interplay between neighboring components in the network is thereby realized by nonmonotonic updates over agreed interface representations encoding different kinds of mathematical knowledge.

Deep Inference for Automated Proof Tutoring

i¾?MEGA [7], a mathematical assistant environment comprising an interactive proof assistant, a proof planner, a structured knowledge base, a graphical user interface, access to external reasoners, etc., is being developed since the early 90s at Saarland University. Similar to HOL4, Isabelle/HOL, Coq, or Mizar, the overall goal of the project is to develop a system platform for formal methods (not only) in mathematics and computer science. In i¾?MEGA, user and system interact in order to produce verifiable and trusted proofs. By continously improving (not only) automation and interaction support in the system we want to ease the usually very tedious formalization and proving task for the user.

Workflows for the management of change in science, technologies, engineering and mathematics

Mathematical knowledge is a central component in science, engineering, and technology (documentation). Most of it is represented informally, and - in contrast to published research mathematics - subject to continual change. Unfortunately, machine support for change management has either been very coarse grained and thus barely useful, or restricted to formal languages, where automation is possible. In this paper, we report on an effort to extend change management to collections of semi-formal documents which flexibly intermix mathematical formulas and natural language and to integrate it into a semantic publishing system for mathematical knowledge. We validate the long-standing assumption that the semantic annotations in these flexiformal documents that drive the machine-supported interaction with documents can support semantic impact analyses at the same time. But in contrast to the fully formal setting, where adaptations of impacted documents can be automated to some degree, the flexiformal setting requires much more user interaction and thus a much tighter integration into document management workflows.

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.