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Rough Concept Analysis – theory development in the Mizar system
 Proc. of MKM 2004, Lecture Notes in Computer Science 3119
, 2004
"... Abstract. 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 ..."
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Abstract. 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. In this paper we investigate the facilities of the Mizar system concerning extending and combing theories based on structure and attribute definitions. As an example we consider the formation of rough concept analysis out of formal concept analysis and rough sets. 1
Mizar Attributes: A Technique to Encode Mathematical Knowledge into Type Systems
"... Abstract. At first glance Mizar attributes look like unary predicates over mathematical objects enabling a more natural writing and reading. Attributes in Mizar, however, serve additional, more important purposes concerning typing of mathematical objects: Using attributes not only new (sub)types can ..."
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Abstract. At first glance Mizar attributes look like unary predicates over mathematical objects enabling a more natural writing and reading. Attributes in Mizar, however, serve additional, more important purposes concerning typing of mathematical objects: Using attributes not only new (sub)types can be introduced, but also the user can characterize further relations between types and in this way make available existing notations for new objects. Thereby it should be stressed that these type relations can stand for elaborated mathematical theorems. This paper describes the properties and benefits of Mizar attributes from a user’s perspective. We comprehend the development of Mizar attributes, and give examples highlighting their use — essentially in the area of algebra. Concluding we discuss their impact on building mathematical repositories. 1
Knowledge Archives in Theorema: A LogicInternal Approach
"... Abstract. Archives are implemented as an extension of Theorema for representing mathematical repositories in a natural way. An archive can be conceived as one large formula in a language consisting of higherorder predicate logic together with a few constructs for structuring knowledge: attaching la ..."
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Abstract. Archives are implemented as an extension of Theorema for representing mathematical repositories in a natural way. An archive can be conceived as one large formula in a language consisting of higherorder predicate logic together with a few constructs for structuring knowledge: attaching labels to subhierarchies, disambiguating symbols by the use of namespaces, importing symbols from other namespaces and specifying the domains of categories and functors as namespaces with variable operations. All these constructs are logicinternal in the sense that they have a natural translation to higherorder logic so that certain aspects of Mathematical Knowledge Management can be realized in the object logic itself. There are a variety of operations on archives, though in this paper we can only sketch a few of them: knowledge retrieval and theory exploration, merging and splitting, insertion and translation to predicate logic.
Efficient Rough Set Theory Merging
"... Abstract. Theory exploration is a term describing the development of a formal (i.e. with the help of an automated proofassistant) approach to selected topic, usually within mathematics or computer science. This activityhoweverusuallydoesn’t reflecttheviewofscience consideredasa whole, notas separat ..."
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Abstract. Theory exploration is a term describing the development of a formal (i.e. with the help of an automated proofassistant) approach to selected topic, usually within mathematics or computer science. This activityhoweverusuallydoesn’t reflecttheviewofscience consideredasa whole, notas separated islands ofknowledge. Merging theoriesessentially has its primary aim of bridging these gaps between specific disciplines. As we provided formal apparatus for basic notions within rough set theory (as e.g. approximation operators and membership functions), we try to reuse the knowledge which is already contained in available repositories of computerchecked mathematical knowledge, or which can be obtained in a relatively easy way. We can point out at least three topics here: topological aspects of rough sets – as approximation operators have properties of the topological interior and closure; latticetheoretic approach giving the algebraic viewpoint (e.g. Stone algebras); possible connections with formal concept analysis. In such a way we can give the formal characterization of rough sets in terms of topologies or orders. Although fully formal, still the approach can be revised to keep the uniformity all the time.
Reading an Algebra Textbook
"... Abstract. We report on a formalisation experiment where excerpts from an algebra textbook are compared to their translation into formal texts of the Isabelle/Isar prover, and where an attempt is made in the formal text to stick as closely as possible with the structure of the informal counterpart. T ..."
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Abstract. We report on a formalisation experiment where excerpts from an algebra textbook are compared to their translation into formal texts of the Isabelle/Isar prover, and where an attempt is made in the formal text to stick as closely as possible with the structure of the informal counterpart. The purpose of the exercise is to gain understanding on how adequately a modern algebra text can be represented using the module facilities of Isabelle. Our initial results are promising. 1