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parametricity-meetup-talk.lyx
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\title[Parametricity properties]{Parametricity properties of purely functional code}
\subtitle{``Theorems for free'' demystified. A tutorial, with code examples in Scala}
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\date{2020-03-24}
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Refactoring code by permuting the order of operations
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To introduce the topic, I begin with practical use examples.
It is useful for a programmer to know that purely functional code can be
refactored in ways that are guaranteed to be correct.
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An inquisitive programmer would now ask how to write the naturality law
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[Difficulty level 1]
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The name "naturality" comes from "natural transformations" of category theory.
However, a software engineer needs only a simplified definition of natural
transformations in the context of functional programming.
We just need to learn to recognize a natural transformation by its type
signature: a function between two functors sharing a type parameter.
After that, the corresponding naturality law is straightforward to write.
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The type signature of a natural transformation is of this form:
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def nt[A]: F[A] => G[A] = ??? // Both F and G must be covariant in A, or
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Examples of natural transformations in Scala:
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def headOption[A]: List[A] => Option[A] = ...
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However, not all widely used functions have the type signature of a natural
transformation; for example, `find`, `filter`, or `foldLeft` do not.
A more general approach is necessary.
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[Difficulty level 2]
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One completely general approach is based on the Reynolds-Wadler parametricity
theorem (known from Wadler's "theorems for free" paper).
The Reynolds-Wadler theorem is usually formulated using specially defined
type and value "relations", which are hard to use and to understand for
programmers who always work with functions and never with arbitrary relations.
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I will explain a different approach that derives parametricity properties
directly, without using relations.
In that approach, Wadler's "free theorems" are just naturality laws satisfied
by any purely functional code with a given type signature.
For this to work with arbitrary type signatures, we need to extend naturality
laws from natural transformations to more general "dinatural" transformations
(functions between two "profunctors" sharing a type parameter).
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I will then show a recipe for writing naturality laws for all type signatures,
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The recipe depends on being able to derive the code of `map` or `contramap`
for any given covariant or contravariant type constructor.
I will explain an algorithm for deriving that code mechanically.
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In most cases, it is enough for a programmer to know how to write the naturality
law.
Proofs are more complicated.
Without going through the full details, I will outline the proofs of three
parametricity properties that apply to purely functional code:
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[Difficulty level 3]
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(1) The `fmap` methods defined with respect to any two different type parameters
will always commute.
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Example in Scala:
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type T[A, B, C] = ...
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Z](f: B => Z): T[A, B, C] => T[A, Z, C] = ??? def fmap_C[A, B, C, Z](f:
C => Z): T[A, B, C] => T[A, B, Z] = ??? Then fmap_A, fmap_B, and fmap_C
pairwise commute, for example:
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fmap_A(f) andThen fmap_B(g) == fmap_B(g) andThen fmap_A(f)
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Proof is by induction on the type structure of T.
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(2) Any purely functional expression with type parameters will satisfy one
naturality law per type parameter.
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This applies not only to functions, but generally to expressions of any
type.
The naturality law is written via the "recipe" derived earlier in the talk.
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Proof is by induction on the code structure of the given expression.
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[Difficulty level 2]
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(3) Any type constructor covariant with respect to a type parameter `A`
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Proof is a short symbolic calculation that uses statement (2) and the functor
laws.
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Expected properties of refactored code:
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These refactorings are guaranteed to be correct
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Summary of the type notation
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The short type notation helps in symbolic reasoning about types
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inline true
status open
\begin_layout Plain Layout
x: Int
\end_layout
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\size footnotesize
\begin_inset Formula $x^{:\text{Int}}$
\end_inset
or
\begin_inset Formula $x:\text{Int}$
\end_inset
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\size footnotesize
Unit type
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\size footnotesize
\begin_inset listings
inline true
status open
\begin_layout Plain Layout
Unit
\end_layout
\end_inset
,
\begin_inset listings
inline true
status open
\begin_layout Plain Layout
Nil
\end_layout
\end_inset
,
\begin_inset listings
inline true
status open
\begin_layout Plain Layout
None
\end_layout
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\size footnotesize
\begin_inset Formula $\bbnum 1$
\end_inset
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\size footnotesize
Type parameter
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\size footnotesize
\begin_inset listings
inline true
status open
\begin_layout Plain Layout
A
\end_layout
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\size footnotesize
\begin_inset Formula $A$
\end_inset
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\size footnotesize
Product type
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\size footnotesize
\begin_inset listings
inline true
status open
\begin_layout Plain Layout
(A, B)
\end_layout
\end_inset
or
\begin_inset listings
inline true
status open
\begin_layout Plain Layout
case class P(x: A, y: B)
\end_layout
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\size footnotesize
\begin_inset Formula $A\times B$
\end_inset
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\size footnotesize
Co-product type
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\size footnotesize
\begin_inset listings
inline true
status open
\begin_layout Plain Layout
Either[A, B]
\end_layout
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\size footnotesize
\begin_inset Formula $A+B$
\end_inset
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\size footnotesize
Function type
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\size footnotesize
\begin_inset listings
inline true
status open
\begin_layout Plain Layout
A => B
\end_layout
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\size footnotesize
\begin_inset Formula $A\rightarrow B$
\end_inset
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\size footnotesize
Type constructor
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\size footnotesize
\begin_inset listings
inline true
status open
\begin_layout Plain Layout
List[A]
\end_layout
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\size footnotesize
\begin_inset Formula $\text{List}^{A}$
\end_inset
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\size footnotesize
Universal quantifier
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\size footnotesize
\begin_inset listings
inline true
status open
\begin_layout Plain Layout
trait P
\backslash
{ def f[A]: Q[A]
\backslash
}
\end_layout
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\size footnotesize
\begin_inset Formula $P\triangleq\forall A.\,Q^{A}$
\end_inset
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\size footnotesize
Existential quantifier
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\size footnotesize
\begin_inset Box Frameless
position "t"
hor_pos "c"
has_inner_box 1
inner_pos "t"