Catálogo de publicaciones - libros

The relationship between theorems and lemmas in mathematical reasoning is often vague. No system exists that formalizes the structure of theorems in a mathematical library. Nevertheless, the decisions we make in creating lemmas provide an inherent hierarchical structure to the statements we prove. In this paper, we develop a formal system that organizes theorems based on . Lemmas are simply theorems with a local scope. We develop a representation of proofs that captures scope and present a set of proof rules to create and reorganize the scopes of theorems and lemmas. The representation and rules allow systems for formalized mathematics to more accurately reflect the natural structure of mathematical knowledge.

Although knowledge is a central topic for MKM there is little explicit discussion on what ‘knowledge’ might actually be. There are specific intuitions about form and content of knowledge, about its structure, and epistemological nature that shape the MKM systems, but a conceptual model is missing.

In this paper we try to rationalize this discussion to give MKM a firmer footing, to start a discussion among MKM researchers and help relate the MKM intuitions and discourses to other communities.

Starting from the observation that many concrete realizations of mathematical knowledge objects are considered equivalent, we propose a conceptual model of the space of (mathematical) knowledge objects graded by levels of abstraction and presentational explicitness and draw conclusions for MKM markup formats.

Some mathematical objects can have more than one notation. When a system compiles mathematical material from multiple sources, a management effort to maintain uniform and appropriate notations becomes necessary. Additionally, the need arises to facilitate the notations editing of the mathematical objects with authoring tools. In this paper, we present our work towards those needs. We have designed a framework that defines an authoring cycle supported by series of tools, which eases the creation of notations for the symbols in the process of publishing mathematics for the web.

Defining functions is a major topic when building mathematical repositories. Though relatively easy in mathematical vernacular, function definitions rise a number of questions and problems in fully formal languages (see [4]). This becomes even more important for repositories in which properties of the defined functions are not only stated, but also proved correct. In this paper we investigate function definitions in the Mizar system. Though most of them are straightforward and follow the intuition, we also found a number of examples differing from mathematical vernacular or where different solutions seem equally reasonable. Sometimes there even do not seem to exist solutions not somehow “ignoring mathematical vernacular”. So the question is: Should we seek for some kind of standard, that is a “formal mathematical vernacular”, or should we accept that different authors prefer different styles?

Recently, significant advances have been made in formalised mathematical texts for large, demanding proofs. But although such large developments are possible, they still take an inordinate amount of effort and time, and there is a significant gap between the resulting formalised machine-checkable proof scripts and the corresponding human-readable mathematical texts. We present an authoring system for formal proof which addresses these concerns. It is based on a format which, in the tradition of literate programming, allows one to extract either a formal proof script or a human-readable document; the two may have differing structure and detail levels, but are developed together in a synchronised way. Additionally, we introduce ways to assist production of the central document, by allowing tools to contribute to update and extend it. Our authoring system builds on the new PG Kit architecture for Proof General, bringing the extra advantage that it works in a uniform interface, generically across various interactive theorem provers.

In this contribution we address two related questions. Firstly, we want to shed light on the question how to use a representation formalism to represent a given problem. Secondly, we want to find out how different formalizations are related and in particular how it is possible to check that one formalization entails another. The latter question is a tough nut for mathematical knowledge management systems, since it amounts to the question, how a system can recognize that a solution to a problem is already available, although possibly in disguise. As our starting point we take McCarthy’s 1964 mutilated checkerboard challenge problem for proof procedures and compare some of its different formalizations.

Unlike computer algebra systems, automated theorem provers have not yet achieved considerable recognition and relevance in mathematical practice. A significant shortcoming of mathematical proof assistance systems is that they require the fully formal representation of mathematical content, whereas in mathematical practice an informal, natural-language-like representation where obvious parts are omitted is common. We aim to support mathematical paper writing by integrating a scientific text editor and mathematical assistance systems such that mathematical derivations authored by human beings in a mathematical document can be automatically checked. To this end, we first define a calculus-independent representation language for formal mathematics that allows for underspecified parts. Then we provide two systems of rules that check if a proof is correct and at an acceptable level of granularity. These checks are done by decomposing the proof into basic steps that are then passed on to proof assistance systems for formal verification. We illustrate our approach using an example textbook proof.

In mathematical textbooks matrices are often represented as objects of indefinite size containing abbreviations. To make the knowledge implicitly given in these representations available in electronic form they have to be interpreted correctly. We present an algorithm that provides the interface between the textbook style representation of matrix expressions and their concrete interpretation as formal mathematical objects. Given an underspecified matrix containing ellipses and fill symbols, our algorithm extracts the semantic information contained. Matrices are interpreted as a collection of regions that can be interpolated with a particular term structure. The effectiveness of our procedure is demonstrated with an implementation in the computer algebra system Maple.

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.

In a problem solving process, a step may not result in the expected progress or may not be applicable as expected. Hence, knowledge how to overcome and react to impasses and other failures is an important ingredient of successful mathematical problem solving. To employ such knowledge in a proving system requires a variety of behaviors and a flexible control. Multi-strategy proof planning is a knowledge-based theorem proving approach that provides a variety of strategies and knowledge-based guidance for search at different levels. This paper introduces reasoning about impasses as a natural ingredient of meta-reasoning at a strategic level and illustrates the use of knowledge about failure handling in the proof planner .