Free Algebras

Author: kryptt

algebraic-data-types/src/main/tut/index.html

@@ -99,7 +99,7 @@
 
 # Generalized Algebraic Data Type
 
-As long as you trust inheritance
+As long as you trust inheritance and dynamic dispatch you would probably also trust:
 
 <small>
 ```tut
@@ -116,11 +116,13 @@
 ```
 </small>
 
+<small>Paul Chiusano (scalaz-stream) has documented his attempts [here](https://github.com/scalaz/scalaz-stream/blob/topic/0.8-redesign/implementation-notes.markdown#gadts)
+</small>
 ---
 
 # Sum Types and Product Types
 
-Structurally:
+Speaking structurally:
 
 ```tut
 type Sum[A]      = Lam[A] // Lift ∨ Tup ∨ Lambda ∨ App ∨ Fix
@@ -179,14 +181,42 @@
 
 ---
 
+# Free Algebras
+
+The group \\((Z,+)\\) of integers is free; we can take \\(S = {1}\\).
+
+A monoid is free:
+
+```tut:invisible
+trait Equal[F] { def equal(a: F, b: F): Boolean }
+```
+
+```tut:silent
+trait Monoid[F] {
+  def zero: F
+  def append(a: F, b: F): F
+  trait MonoidLaw {
+    def associative(f1: F, f2: F, f3: F)(implicit F: Equal[F]) =
+      F.equal(append(f1, append(f2, f3)), append(append(f1, f2), f3))
+    def leftIdentity(a: F)(implicit F: Equal[F]) = F.equal(a, append(zero, a))
+    def rightIdentity(a: F)(implicit F: Equal[F]) = F.equal(a, append(a, zero))
+  }
+}
+```
+
+`List[G]` is the canonical implementation of the Monoid Algebra.
+
+`List[Set[G]]` is the canonical implementation of Classical Linear Logic (CLL).
+
+---
+
 # References
 
-* [Algebraic Data Types](http://tpolecat.github.io/presentations/algebraic_types.html)
-by Rob Norris (@tpolecat)
-* [Shapeless 2.1.0 Feature Overview](https://github.com/milessabin/shapeless/wiki/Feature-overview:-shapeless-2.1.0)
-* [I am not a monad, I am a free algebra](http://biosimilarity.blogspot.com/2011/08/i-am-not-monad-i-am-free-algebra-pt-1.html)
-* [Sweeping crap APIs under the rug](http://tpolecat.github.io/presentations/lambdaconf14.html)
-by Rob Norris (@tpolecat)
+* [Algebraic Data Types](http://tpolecat.github.io/presentations/algebraic_types.html) by Rob Norris (@tpolecat)
+* [Encoding of GADTs in Scala](https://github.com/scalaz/scalaz-stream/blob/topic/0.8-redesign/implementation-notes.markdown#gadts) by Paul Chiusano (@pchiusano)
+* [Shapeless 2.1.0 Feature Overview](https://github.com/milessabin/shapeless/wiki/Feature-overview:-shapeless-2.1.0) by Miles Sabin (@milessabin)
+* [I am not a monad, I am a free algebra](http://biosimilarity.blogspot.com/2011/08/i-am-not-monad-i-am-free-algebra-pt-1.html) by Greg Meredith
+* [Sweeping crap APIs under the rug](http://tpolecat.github.io/presentations/lambdaconf14.html) by Rob Norris (@tpolecat)
 
     </textarea>
     <script src="http://gnab.github.io/remark/downloads/remark-latest.min.js" type="text/javascript"></script>