Adam Grabowski

Mizar in a Nutshell

This paper is intended to be a practical reference manual for basic Mizar terminology which may be helpful to get started using the system. The paper describes most important aspects of the Mizar language as well as some features of the verification software.

Mizar: State-of-the-art and Beyond

Mizari¾?is one of the pioneering systems for mathematics formalization, which still has ani¾?active user community. The project has been in constant development since 1973, when Andrzej Trybulec designed the fundamentals of ai¾?language capable of rigorously encoding mathematical knowledge in ai¾?computerized environment which guarantees its full logical correctness. Since then, the system with its feature-rich language devised to approximate mathematics writing has influenced other formalization projects and has given rise to ai¾?number of Mizari¾?modes implemented on top of other systems. However, the information about the system as ai¾?whole is not readily available to developers of other systems. Various papers describing Mizari¾?features have been rather incremental and focused only on particular newly implemented Mizari¾?aspects. The objective of the current paper is to give ai¾?survey of the most important Mizari¾?features that distinguish it from other popular proof checkers. We also go ai¾?step further and describe most important current trends and lines of development that go beyond the state-of-the-art system.

Mechanizing Complemented Lattices Within Mizar Type System

Recently some longstanding open lattice theory problems were solved with the help of automated theorem provers. The question which may be posed is how to cope with such results to improve their presentation for human without loss of machine-readability, not only at the proof level, which should be rather straightforward, but also at the stage of rebuilding appropriate data structure. We describe the framework extending already existed in the Mizar library for Boolean algebras to cover more general cases of lattice with complements. The efficiency of this approach was tested e.g. on short axiom systems for Boolean algebras based on negation and disjunction. We also proved Nachbin theorem for spectra of distributive lattices.

Automated Discovery of Properties of Rough Sets

The computer certification of rough sets the translation in a way understandable by machines seems to be far beyond the test phase. To assure the feasibility of the approach, we try to encode selected problems within rough set theory and as the testbed of already developed foundations --and in the same time as a payoff of the established framework --we shed some new light on the well-known question of generalization of rough sets and the axiomatization of approximation operators in terms of various types of binary relations. We show how much the human work can be enhanced with the use of automatic tools, without loosing too much time for the translation. Although the syntax is understandable by the computer, it offers relative flexibility and expressive power of the formal language.

Translating mathematical vernacular into knowledge repositories

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?

On the Computer-Assisted Reasoning about Rough Sets

The paper presents some of the issues concerning a formal description of rough sets. We require the indiscernibility relation to be a tolerance of the carrier, not an equivalence relation, as in the Pawlak’s classical approach. As a tool for formalization we use the Mizar system, which is equipped with the largest formalized library of mathematical facts. This uniform and computer-checked for correctness framework seems to present a satisfactory level of generality and may be used by other systems as well as it is easily readable for humans.

Rough Set Theory from a Math-Assistant Perspective

In the paper, we draw a perspective of the computer-assisted theory exploration within rough set theory. We examine two well-known approaches to the topic, drawing some paradigms for a machine math-assistant to be feasible tool any researcher can use to verify his own results. Some features of a Mizar language chosen for the verification task are also presented.

Rough Concept Analysis – Theory Development in the Mizar System

Theories play an important role in building mathematical knowledge repositories. Organizing knowledge in theories is an obvious approach to cope with the growing number of definitions, theorems, and proofs. However, they are also a matter of subject on their own: developing a new piece of mathematics often relies on extending or combining already developed theories in this way reusing definitions as well as theorems. We believe that this aspect of theory development is crucial for mathematical knowledge management.

On duplication in mathematical repositories

Building a repository of proof-checked mathematical knowledge is without any doubt a lot of work, and besides the actual formalization process there is also the task of maintaining the repository. Thus it seems obvious to keep a repository as small as possible, in particular each piece of mathematical knowledge should be formalized only once. In this paper, however, we claim that it might be reasonable or even necessary to duplicate knowledge in a mathematical repository. We analyze different situations and reasons for doing so, provide a number of examples supporting our thesis and discuss some implications for building mathematical repositories.

Equality in computer proof-assistants

Equality is fundamental notion of logic and mathematics as a whole. If computer-supported formalization of knowledge is taken into account, sooner or later one should precisely declare the intended meaning/interpretation of the primitive predicate symbol of equality. In the paper we draw some issues how computerized proof-assistants can deal with this notion, and at the same time, we propose solutions, which are not contradictory with mathematical tradition and readability of source code. Our discussion is illustrated with examples taken from the implementation of the MIZAR system.