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explaining-theorems-for-free.lyx
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#LyX 2.3 created this file. For more info see http://www.lyx.org/
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\title[Theorems for free]{Explaining ``Theorems for Free'' and parametricity}
\subtitle{A tutorial, with code examples in Scala}
\author{Sergei Winitzki}
\date{2020-10-17}
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Beginning of slides.
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Parametricity: a theory about certain code refactorings
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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.
To formulate this fact rigorously, one writes an equation called a "naturality
law".
Programmers need to know that these laws exist and can be used for rigorous
reasoning about code.
I will show some examples of such reasoning.
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An inquisitive programmer would now ask how to write the naturality law
for any given function, and also how to prove that the law always holds
for purely functional code.
The main part of the talk will answer these questions.
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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
both contravariant in A.
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Examples of natural transformations in Scala:
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def headOption[A]: List[A] => Option[A] = ...
def reverse[A]: List[A] => List[A] = ...
def zipWithIndex[A]: List[A] => List[(A, Int)] = ...
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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).
I will introduce the notions of profunctors and dinatural transformations.
I will then show a recipe for writing naturality laws for all type signatures,
and give some examples.
\end_layout
\begin_layout Plain Layout
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.
\end_layout
\begin_layout Plain Layout
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:
\end_layout
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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:
\end_layout
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type T[A, B, C] = ...
// Define some type constructor, covariant with respect to 3 type parameters.
// Now define `fmap` methods with respect to each type parameter: def fmap_A[A,
B, C, Z](f: A => Z): T[A, B, C] => T[Z, B, C] = ??? def fmap_B[A, B, C,
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.
\end_layout
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[Difficulty level 2]
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(3) Any type constructor covariant with respect to a type parameter `A`
has a unique purely functional `fmap` method with respect to `A` satisfying
the functor laws.
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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 code that manipulates collections:
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First extract user information, then convert stream to list; or first convert
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First extract user information, then exclude invalid rows; or first exclude
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gives the same result as
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db.getRows.filter(getUserInfo andThen isValid).map(getUserInfo)
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These refactorings are guaranteed to be correct
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because
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...
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is also a natural transformation in disguise
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Natural transformations
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work the same way for all types
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...
and satisfy the
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naturality laws
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Refactoring rules written as equations
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\end_layout
\begin_deeper
\begin_layout Itemize
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
vspace{-0.1cm}
\end_layout
\end_inset
The refactoring involving
\begin_inset listings
inline true
status open
\begin_layout Plain Layout
toList
\end_layout
\end_inset
:
\end_layout
\begin_deeper
\begin_layout Standard
\begin_inset listings
inline true
status open
\begin_layout Plain Layout
def toList[A]: Stream[A] => List[A]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
For any function
\begin_inset listings
inline true
status open
\begin_layout Plain Layout
f: A => B
\end_layout
\end_inset
:
\end_layout
\begin_layout Standard
\begin_inset listings
inline true
status open
\begin_layout Plain Layout
_.toList.map(f) == _.map(f).toList
\end_layout
\end_inset
\end_layout
\end_deeper
\begin_layout Itemize
The refactoring involving
\begin_inset listings
inline true
status open
\begin_layout Plain Layout
filter
\end_layout
\end_inset
:
\end_layout
\begin_deeper
\begin_layout Standard
\begin_inset listings
inline true
status open
\begin_layout Plain Layout
def filter[A]: Stream[A] => (A => Boolean) => Stream[A]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
For any function
\begin_inset listings
inline true
status open
\begin_layout Plain Layout
f: A => B
\end_layout
\end_inset
and predicate
\begin_inset listings
inline true
status open
\begin_layout Plain Layout
p: B => Boolean
\end_layout
\end_inset
:
\end_layout
\begin_layout Standard
\begin_inset listings
inline true
status open
\begin_layout Plain Layout
_.filter(f andThen p).map(f) == _.map(f).filter(p)
\end_layout
\end_inset
\end_layout
\end_deeper
\begin_layout Itemize
Generally: applying
\begin_inset listings
inline true
status open
\begin_layout Plain Layout
t[A]: F[A] => G[A]
\end_layout
\end_inset
\emph on
before
\emph default
\begin_inset listings
inline true
status open
\begin_layout Plain Layout
_.map(f)
\end_layout
\end_inset
equals applying
\begin_inset listings
inline true
status open
\begin_layout Plain Layout
t[B]: F[B] => G[B]
\end_layout
\end_inset
\emph on
after
\emph default
\begin_inset listings
inline true
status open
\begin_layout Plain Layout
_.map(f)
\end_layout
\end_inset
, for any function
\begin_inset listings
inline true
status open
\begin_layout Plain Layout
f: A => B
\end_layout
\end_inset
\begin_inset Formula
\[
\xymatrix{\xyScaleY{2.8pc}\xyScaleX{5.5pc}\mathtt{F[A]}\ar[d]\sb(0.5){~\mathtt{\_.map(f)}\text{ for }\mathtt{F}}\ar[r]\sp(0.5){\mathtt{t[A]}} & \mathtt{G[A]}\ar[d]\sp(0.5){\mathtt{\_.map(f)}\text{ for }\mathtt{G}}\\
\mathtt{F[B]}\ar[r]\sp(0.5){\mathtt{t[B]}} & \mathtt{G[B]}
}
\]
\end_inset
\end_layout
\begin_deeper
\begin_layout Itemize
This is called a
\series bold
naturality law
\end_layout
\end_deeper
\end_deeper
\begin_layout Standard
\begin_inset Separator parbreak
\end_inset
\end_layout
\begin_layout Frame
\begin_inset Argument 4
status open
\begin_layout Plain Layout
Naturality laws: further examples
\end_layout
\end_inset
\begin_inset Separator latexpar
\end_inset
\end_layout
\begin_deeper
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
vspace{-0.1cm}
\end_layout
\end_inset
\series bold
Naturality law
\series default
for a function
\begin_inset listings
inline true
status open
\begin_layout Plain Layout
t[A]: F[A] => G[A]
\end_layout
\end_inset
is an equation that permutes the order of
\begin_inset listings
inline true
status open
\begin_layout Plain Layout
t
\end_layout
\end_inset
and of
\begin_inset listings
inline true
status open
\begin_layout Plain Layout
_.map(f)
\end_layout
\end_inset
, for an arbitrary function
\begin_inset listings
inline true
status open
\begin_layout Plain Layout
f: A => B
\end_layout
\end_inset
\end_layout
\begin_layout Itemize
We expect it to hold if the code works the same way for all types
\end_layout
\begin_layout Itemize
Intuition:
\family typewriter
t
\family default
rearranges data in a collection,
\begin_inset Quotes eld
\end_inset
not looking
\begin_inset Quotes erd
\end_inset
at values
\end_layout
\begin_layout Standard
Further examples:
\end_layout
\begin_layout Itemize
The headOption method:
\begin_inset listings
inline true
status open
\begin_layout Plain Layout
headOption: List[A] => Option[A]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\align center
\begin_inset listings
inline true
status open
\begin_layout Plain Layout
list.map(f).headOption == list.headOption.map(f)
\end_layout
\end_inset
\end_layout
\begin_layout Itemize
Reverse a list:
\begin_inset listings
inline true