Functorial Blog - Counterexamples of Type Classes
by Phil Freeman on 2015/12/06 In Haskell and PureScript, we are familiar with certain standard hierarchies of type classes. For example, here are some of the type classes provided by the PureScript core libraries: Semigroup, Monoid Semiring, Ring, CommutativeRing, EuclideanRing, Field Functor, Apply, Applicative, Bind, Monad Extend, Comonad Alt, Plus, Alternative Semigroupoid, Category Profunctor,
group-category
群から圏へ 集合 X から X への全単射を X 上の置換 (permutation) という． X 上の置換の集合 G でつぎをみたすものを，X 上の置換群という： 1. g, h∈G ⇒ gh∈G, 2. IdX∈G, 3. g∈G ⇒ g-1∈G. たとえば X 上の置換すべての集合は置換群である． G を X から自立させたものが群 (group) である． G の元が写像であったことをわすれる，といってもよい． すなわち集合 G にたいし， 1. 写像 m : G×G → G, (g, h)→gh が (gh)k=g(hk) をみたし， 2. 元 1∈G が 1g=g1=g をみたし， 3. 各 g∈G にたいし g-1∈G があって g-1g=gg-1=1 をみたすとき， (G, m, 1) を群という． 略してたんに群 G ともいう． m を群の演算，1 を単位元，g-1 を
apfelmus - Monoids and Finger Trees
In this article, we explain how monoids enable one and the same data structure, the finger tree, to implement virtually any other data structure; they arise by different choices of monoids. The technique is discussed using the examples of a list with random access and a priority queue. This post grew out of the big monoid discussion on the haskell-cafe mailing list. Introduction A very powerful ap
A Neighborhood of Infinity: Fast incremental regular expression matching with monoids
The Problem Consider this problem: Fix a regular expression R. Suppose you have a string of length N. There's not much you can do about it, you'll likely have to scan all N characters to test so see if the string matches R. But once you've performed the test, how fast can you test the string again if a small edit is made to it? It seems that in general you'd have to rescan the entire string, or at
A Neighborhood of Infinity: Haskell Monoids and their Uses
Haskell is a great language for constructing code modularly from small but orthogonal building blocks. One of these small blocks is the monoid. Although monoids come from mathematics (algebra in particular) they are found everywhere in computing. You probably use one or two monoids implicitly with every line of code you write, whatever the language, but you might not know it yet. By making them ex
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Beyond Mere Actors
