Interested in formalizing the generation of fast running code for linear algebra applications, the authors show how an index-free, calculational approach to matrix algebra can be developed by regarding matrices as morphisms of a category with biproducts. This shifts the traditional view of matrices as indexed structures to a type-level perspective analogous to that of the pointfree algebra of programming. The derivation of fusion, cancellation and abide laws from the biproduct equations makes it easy to calculate algorithms implementing matrix multiplication, the central operation of matrix algebra, ranging from its divide-and-conquer version to its vectorization implementation.From errant attempts to learn how particular products and coproducts emerge from biproducts, not only blocked matrix algebra is rediscovered but also a way of extending other operations (e.g. Gaussian elimination) blockwise, in a calculational style, is found.The prospect of building biproduct-based type checkers for computer algebra systems such as MATLAB™ is also considered.Keywords: Linear algebra, categories of matrices, algebra of programming

A two-level data transformation consists of a type-level transformation of a data format coupled with value-level transformations of data instances corresponding to that format. Examples of two-level data transformations include XML schema evolution coupled with document migration, and data mappings used for interoperability and persistence. We provide a formal treatment of two-level data transformations that is typesafe in the sense that the well-formedness of the value-level transformations with respect to the type-level transformation is guarded by a strong type system. We rely on various techniques for generic functional programming to implement the formalization in Haskell. The formalization addresses various two-level transformation scenarios, covering fully automated as well as user-driven transformations, and allowing transformations that are information-preserving or not. In each case, two-level transformations are disciplined by one-step transformation rules and type-level transformations induce value-level transformations. We demonstrate an example hierarchicalrelational mapping and subsequent migration of relational data induced by hierarchical format evolution.

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The pointfree transform offers to the predicate calculus what the Laplace transform offers to the differential/integral calculus: the possibility of changing the underlying mathematical space so as to enable agile algebraic calculation. This paper addresses the foundations of the transform and its application to a calculational approach to extended static checking (ESC) in the context of abstract modeling. In particular, a calculus is given whose rules help in breaking the complexity of the proof obligations involved in static checking arguments. The close connection between such calculus and that of weakest preconditions makes it possible to use the latter in ESC proof obligation discharge, where pointfree notation is again used, this time to calculate with invariant properties to be maintained. A connection with the "everything is a relation" lemma of Alloy is established, showing how close to each other the pointfree and Alloy notations are. The main advantage of this connection is that of complementing pen-and-paper pointfree calculations with model checking support wherever validating sizable abstract models.

A new enantioselective Heck-Matsuda desymmetrization reaction was accomplished by using 3-cyclopentenol to produce chiral five-membered 4-aryl cyclopentenol scaffolds in good yields and high ee's, together with some 3-aryl-cyclopentanones as minor products. Mechanistically, the hydroxyl group of 3-cyclopentenol acts as a directing group and is responsible for the cis- arrangement in the formation of the 4-aryl-cyclopentenols.

Abstract. This paper addresses the foundations of data-model transformation. A catalog of data mappings is presented which includes abstraction and representation relations and associated constraints. These are justified in an algebraic style via the pointfree-transform, a technique whereby predicates are lifted to binary relation terms (of the algebra of programming) in a two-level style encompassing both data and operations. This approach to data calculation, which also includes transformation of recursive data models into "flat" database schemes, is offered as alternative to standard database design from abstract models. The calculus is also used to establish a link between the proposed transformational style and bidirectional lenses developed in the context of the classical view-update problem.

Abstract.The standard operation refinement ordering is a kind of "meet of opposites": non-determinism reduction suggests "smaller" behaviour while increase of definition suggests "larger" behaviour. Groves' factorization of this ordering into two simpler relations, one per refinement concern, makes it more mathematically tractable but is far from fully exploited in the literature. We present a pointfree theory for this factorization which is more agile and calculational than the standard set-theoretic approach. In particular, we show that factorization leads to a simple proof of structural refinement for arbitrary parametric types and exploit factor instantiation across different subclasses of (relational) operation. The prospect of generalizing the factorization to coalgebraic refinement is discussed.

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