[PDF] Fun with semirings: a functional pearl on the abuse of linear algebra | Semantic Scholar

A derivation of a purely functional version of Kleene’s closure algorithm for Kleene algebras (with tests) that contain a subset where the closure is already known that is applicable to the Kleene algebra of square matrices over a given Kleenegebra.

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Pilatus is a polymorphic iterative linear algebra language applicable to various types of data analytics workloads, and allows us to use multi-stage programming and rewrite-based optimisation to recover the performance of specialised code.

An axiom system for path problems on directed graphs which are solvable by a method similar to Gaussian elimination is given, and a decomposition method which solves a path problem by breaking it into subproblems, solving each subproblem by elimination, and combining the solutions is presented.

The procedure was originally programmed in FORTRAN for the Control Data 160 desk-size computer and was limited to te t ra t ion because subroutine recursiveness in CONTROL Data 160 FORTRan has been held down to four levels in the interests of economy.