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Monadic Maps and Folds for Arbitrary Datatypes
 Memoranda Informatica, University of Twente
, 1994
"... Each datatype constructor comes equiped not only with a socalled map and fold (catamorphism), as is widely known, but, under some condition, also with a kind of map and fold that are related to an arbitrary given monad. This result follows from the preservation of initiality under lifting from the ..."
Abstract

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Each datatype constructor comes equiped not only with a socalled map and fold (catamorphism), as is widely known, but, under some condition, also with a kind of map and fold that are related to an arbitrary given monad. This result follows from the preservation of initiality under lifting from the category of algebras in a given category to a certain other category of algebras in the Kleisli category related to the monad.
Zstyle notation for Probabilities
, 2004
"... Abstract. A notation for probabilities is proposed that differs from the traditional, conventional notation by making explicit the domains and bound variables involved. The notation borrows from the Z notation, and lends itself well to calculational manipulations, with a smooth transition back and f ..."
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Abstract. A notation for probabilities is proposed that differs from the traditional, conventional notation by making explicit the domains and bound variables involved. The notation borrows from the Z notation, and lends itself well to calculational manipulations, with a smooth transition back and forth to set and predicate notation. 1 Introduction. The notation commonly used in applied probability theory suffers from two drawbacks: the domain of discourse is left implicit, and consequently in predicates the argument is left implicit. To say it in a crude way, the formulas have no meaning without a little verbal story along with them. As a consequence, it is hard to do machine assisted formal calculations (as striven for in, for example, transformational programming [1, 11, 2, 3,
Adjunctions
, 1993
"... ing from the particulars, the situation above is described as follows. ffl There are two categories A and B . [In the above example A = Set and B = Mon .] ffl There are two functors F : A ! B and G: B ! A . [Above Ff = Seq f and Gg = g for arrows f and g . For objects the functors act as follows: ..."
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ing from the particulars, the situation above is described as follows. ffl There are two categories A and B . [In the above example A = Set and B = Mon .] ffl There are two functors F : A ! B and G: B ! A . [Above Ff = Seq f and Gg = g for arrows f and g . For objects the functors act as follows: FA = join A , and G(\Phi) = the target set of \Phi .] ffl There are two transformations, ": FG ! I and j: I ! GF . [Above "\Omega =\Omega = and jA = tip A .] ffl There is the bijection bb cc A;B from arrow collection (FA ! B B) to (A ! A GB) (for arbitrary A in A and B in B ) defined by bb/cc = j ; G/ , with inverse dd ee defined by dd'ee = F' ; " . [Above bbgcc = `the restriction of g to the tip elements', and ddfee = `the extension of f to a homomorphism from join to\Omega '.] By definition, such data F; G; "; j; bb cc; and dd ee constitute an adjunction. It is rather easy to verify the given adjunction SeqAdj from the explicit definitions we have given for Mon; tip; join; Seq ; ...