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Induction and Recursion on Datatypes
, 1995
"... this paper we introduce a notion of induction over an arbitrary datatype and go on to show how the notion is used to establish unicity of a certain (broad) class of equations. Our overall goal is to develop a calculational theory of mathematical induction. That is we want to be able to calculate rel ..."
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Cited by 16 (7 self)
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this paper we introduce a notion of induction over an arbitrary datatype and go on to show how the notion is used to establish unicity of a certain (broad) class of equations. Our overall goal is to develop a calculational theory of mathematical induction. That is we want to be able to calculate relations on which inductive arguments may be based using laws that relate admitting induction to the mechanisms for constructing datatypes. We also want to incorporate such calculations into a methodology for calculating inductive hypotheses rather than leaving their creation to inspired guesswork. This is a bold aim, in view of the vast amount of knowledge and experience that already exists on proof by induction, but recent advances in the role played by Galois connections in the calculus of relations have led us to speculate that significant progress can be made in the short term. The theory developed in this paper is general and not specific to any particular datatype. We define a notion of F reductivity (so called in order to avoid confusion with existing notions of inductivity), where F stands for a "relator", and show that F reductive relations always exist, whatever the value of F . We also give laws for constructing reductive relations from existing reductive relations. We conclude the paper by introducing the dual notion of F inductivity and briefly contrast it with F reductivity. The organisation of this note is as follows. In section 2 we give a very brief introduction to the relational calculus. In section 3 the notion of reductivity is defined. This notion is a generalisation of wellfoundedness, or inductivity. Then in section 4 we define a class of equations and prove that an equation from that class has a unique solution if one of its components enjoys a red...
Knowledge Representation and Acquisition By Concept Lattices
, 1995
"... Concept lattices are practical representations of relations. Such a lattice is based on the category theoretical notion of a Galois connection. Due to their clear mathematical background, properties of concept lattices can be proved in a neat way, in fact, calculationally. We show how concept lattic ..."
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Cited by 5 (3 self)
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Concept lattices are practical representations of relations. Such a lattice is based on the category theoretical notion of a Galois connection. Due to their clear mathematical background, properties of concept lattices can be proved in a neat way, in fact, calculationally. We show how concept lattices can be used for representing knowledge, and how the recursive application of a Galois connection leads to a representation that may be useful for automatic acquisition of knowledge.
Demonic Operators and Monotype Factors
, 1993
"... This paper tackles the problem of constructing a compact, pointfree proof of the associativity of demonic composition of binary relations and its distributivity through demonic choice. In order to achieve this goal a definition of demonic composition is proposed in which angelic composition is r ..."
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Cited by 5 (1 self)
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This paper tackles the problem of constructing a compact, pointfree proof of the associativity of demonic composition of binary relations and its distributivity through demonic choice. In order to achieve this goal a definition of demonic composition is proposed in which angelic composition is restricted by means of a socalled "monotype factor". Monotype factors are characterised by a Galois connection similar to the Galois connection between composition and factorisation of binary relations. The identification of such a connection is argued to be highly conducive to the desired compactness of calculation. Nothing delights a mathematician more than to discover that two things, previously regarded as entirely distinct, are mathematicaly identical. W. W. Sawyer La math'ematique est l'art de donner le meme nom `a des choses diff'erentes. [J.] H. Poincar'e The term "Galois connexion" was coined by Oystein Ore [20] almost fifty years ago in order to describe a particularly si...
Domain Operators and Domain Kinds
, 1993
"... The notions of domain operator and domain kind are introduced. Several examples are presented. In particular, it is shown that the partial equivalence relations form a domain kind. The proof involves the construction of a Galois connection demonstrating that the partial equivalence relations form a ..."
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The notions of domain operator and domain kind are introduced. Several examples are presented. In particular, it is shown that the partial equivalence relations form a domain kind. The proof involves the construction of a Galois connection demonstrating that the partial equivalence relations form a complete lattice under the socalled domain ordering, thus providing another illustration of the importance of the early recognition of Galois connections. 1 Introduction This paper has been specially prepared for the workshop on Galois connections organised by the (Dutch) Mathematics of Programming group and held on 13th and 14th September, 1993. It is essentially a tutorial on how recognition of a Galois connection aids the construction of a complete lattice of partial equivalence relations. Additionally, the notions domain operator and domain kind are introduced. A central element of the calculus of datatypes [3, 2, 4, 1] developed by the authors and their students is the notion of a mon...