Typing linear algebra: A biproduct-oriented approach

  title={Typing linear algebra: A biproduct-oriented approach},
  author={Hugo Daniel Macedo and Jos{\'e} Nuno Oliveira},
Typed Linear Algebra for Weigthed (Probabilistic) Automata
This paper shows typed LA at work in describing weighted (probabilistic) automata and some attention is paid to the interface between the index-free language of matrix combinators and the corresponding index-wise notation, so as to blend with traditional set theoretic notation.
Weighted Automata as Coalgebras in Categories of Matrices
  • J. Oliveira
  • Computer Science
    Int. J. Found. Comput. Sci.
  • 2013
The evolution from non-deterministic to weighted automata represents a shift from qualitative to quantitative methods in computer science. The trend calls for a language able to reconcile
Relational style laws and constructs of linear algebra
Type your matrices for great good: a Haskell library of typed matrices and applications (functional pearl)
A simple inductive data type for representing correct-by-construction matrices that can be used to implement matrix-manipulation algorithms efficiently and safely, performing in some cases faster than existing alternatives even though the algorithms are written in a direct and purely functional style.
A linear algebra approach to OLAP
This paper investigates, in particular, how aggregation operations such as cross tabulations and data cubes essential to quantitative analysis of data can be expressed solely in terms of matrix multiplication, transposition and the Khatri–Rao variant of the Kronecker product.
The data cube as a typed linear algebra operator
This paper shows that a typed notation for linear algebra exists and can be useful in formalizing and reasoning about data aggregation operations, and one such operation - the construction of a data cube - is shown to be easily expressible as a linear algebra operator.
Gaussian elimination is not optimal, revisited
  • H. D. Macedo
  • Computer Science
    J. Log. Algebraic Methods Program.
  • 2016
Compiling quantamorphisms for the IBM Q Experience
This paper aims at building correct-by-construction quantum circuits to be deployed on quantum devices such as those available at the IBM Q Experience, to contribute to extending the laws of classical program algebra to quantum programming.
Relations in linear algebra
Calculational Proofs in Relational Graphical Linear Algebra
This work proposes a minimal framework of six axioms that highlight the dualities and symmetries of linear algebra, and uses the resulting diagrammatic calculus as a convenient tool to prove a number of diverse theorems.


Matrices as Arrows!
It is shown how an index-free, calculational approach to matrix algebra can be developed by regarding matrices as morphisms of a category with biproducts, which shifts the traditional view ofMatrices as indexed structures to a type-level perspective analogous to that of the pointfree algebra of programming.
Linear Algebra
The project before us is to introduce specialized vector-matrix notation and to extend the methods used to solve linear algebraic equations to include a full study of rank, nullity and basis from the vector-Matrix viewpoint.
Matrices, machines and behaviors
A bicategory whose objects are natural numbers, in which an arrow M: n→p is a finite state automaton with n input states, p output states, and some additional internal states is described, which gives a compositional semantics to a primitive notion of concurrent processes.
Quantitative Kleene coalgebras
Do the middle letters of \OLAP" stand for Linear Algebra (\LA")?
This paper investigates, in particular, how the generation of aggregation operations such as cross tabulations and data cubes in OLAP can be expressed in terms of matrix multiplication, transposition and the Khatri-Rao variant of the Kronecker product.
Lectures on Constructive Functional Programming
This work hopes to show that a functional approach to the problem of systematically calculating programs from their specifications can take its place alongside other methodologies.
It is well-known that, given a Dedekind category R the category of (typed) matrices with coefficients from R is a Dedekind category with arbitrary relational sums. In this paper we show that under
Fork Algebras in Algebra, Logic and Computer Science
This paper study fork algebras from the points of view of their algebraic and logical properties and applications, which will prove to be essential for the definition of a wide-spectrum calculus for program construction.
Encyclopedia of Parallel Computing
The highly-structured essays in this work comprise synonyms, a definition and discussion of the topic, bibliographies, and links to related literature support efficient, user-friendly searchers for immediate access to useful information.
Transforming Data by Calculation
A catalog of data mappings is presented which includes abstraction and representation relations and associated constraints which are justified in an algebraic style via the pointfree-transform, a technique whereby predicates are lifted to binary relation terms in a two-level style encompassing both data and operations.