Keith Pinson
I am a software developer who is getting into speaking.
What is an Algebraic Data Type?
A construct for enforcing the logic of a system at compile-time
Something you should care about
Sorry math nerds. Hand-waving to follow.
What good are mathy types?
Statement from Domain as a Type
Translate this into a data type:
"An entree has a sauce"
public class Entree
{
public string Sauce { get; set; }
}Statement with "and" as a type
Translate this into a data type:
"An entree consists of pasta and sauce"
public class Entree
{
public string Pasta { get; set; }
public string Sauce { get; set; }
}We can put logic in a type
Not Boolean
Propositions are logical
Created a type to express a proposition
Type level AND...and OR?
Saying "and" for types is indispensable
What about "or"? Would that be useful?
OR could be useful to...
Make "no value" explicit instead of...
null
throw
bool TryGetValue(A, out B)
Clearly represent known finite possibilities
So logical types are useful. But are they algebraic?
We all agree this is math
2 \times 3 = 6
2x = 6
xy = z
Remember multiplication tables?
| Multiply | 4 | 5 | 6 |
|---|---|---|---|
| 2 | 8 | 10 | 12 |
| 3 | 12 | 15 | 18 |
Or we could do addition
| Add | 4 | 5 | 6 |
|---|---|---|---|
| 2 | 6 | 7 | 8 |
| 3 | 7 | 8 | 9 |
Now moving into algebra land...
| Add | x | y | z |
|---|---|---|---|
| a | a + x | a + y | a + z |
| b | b + x | b + y | b + z |
| Multiply | x | y | z |
|---|---|---|---|
| a |
a * x |
a * y |
a * z |
| b | b * x |
b * y |
b * z |
Abstract away the operation...
| Operation | x | y | z |
|---|---|---|---|
| a | a op x | a op y | a op z |
| b | b op x | b op y | b op z |
Pop Quiz
An Italian restaurant offers an entree of pasta and sauce. Pasta options are spaghetti, tortellini and gnocchi. Sauce options are marinara and alfredo. What are the entree options?
This is a logic problem
Pasta = Spaghetti or Tortellini or Gnocchi
Sauce = Marinara or Alfredo
Entree = Pasta and Sauce
This is an algebra problem
| Spaghetti | Tortellini | Gnocchi | |
|---|---|---|---|
| Marinara | Spaghetti with Marinara | Tortellini with Marinara | Gnocchi with Marinara |
| Alfredo | Spaghetti with Alfredo | Tortellini with Alfredo | Gnocchi with Alfredo |
This is a problem for ADTs
type Pasta = Spaghetti | Tortellini | Gnocchi
type Sauce = Marinara | Alfredo
type Entree = (Pasta, Sauce)Sum Types
type Pasta = Spaghetti | Tortellini | GnocchiSauce has 1 + 1 = 2 possible values
"Or" types behave like addition!
type Sauce = Marinara | AlfredoPasta has 1 + 1 + 1 = 3 possible values
Product Types
type Entree = (Pasta, Sauce)Entree has 2 x 3 = 6 possible values
"And" types behave like multiplication!
Algebraic types compose!
type ParseResult =
| Success of Token * string
| Failure of ErrorConsider: one-line ADT in F#...
type Maybe<'a> = Nothing | Just of 'a
Equivalent* Makeshift
ADT in C#
public abstract class Maybe<a>
{
public static class Tags
{
public const int Nothing = 0;
public const int Just = 1;
}
internal class _Nothing : Du.Maybe<a>
{
}
public class Just : Du.Maybe<a>
{
internal readonly a item;
public a Item
{
get { return this.item; }
}
internal Just(a item)
{
this.item = item;
}
}
internal static readonly Du.Maybe<a> _unique_Nothing = new Du.Maybe<a>._Nothing();
public int Tag
{
get { return (!(this is Du.Maybe<a>.Just)) ? 0 : 1; }
}
public bool IsJust
{
get { return this is Du.Maybe<a>.Just; }
}
public static Du.Maybe<a> Nothing
{
get { return Du.Maybe<a>._unique_Nothing; }
}
public bool IsNothing
{
get { return this is Du.Maybe<a>._Nothing; }
}
internal Maybe()
{
}
public static Du.Maybe<a> NewJust(a item)
{
return new Du.Maybe<a>.Just(item);
}
}*Lacks equality methods
*Lacks string conversion methods
Algebra is inherent in the domains we model
Make friends with the compiler
Tell computers and humans the same story
Algebraic types are computable types
Who let algebra get funky with my data types?
By Keith Pinson
Who let algebra get funky with my data types?
A hand-waving introduction to how data types can be mathematical. This is the version I will present at LambdaConf 2016.
