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denotational-design.lhs
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denotational-design.lhs
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%% -*- latex -*-
%% %let atwork = True
% Presentation
% \documentclass{beamer}
\documentclass[handout]{beamer}
%% % Printed, 2-up
%% \documentclass[serif,handout]{beamer}
%% \usepackage{pgfpages}
%% \pgfpagesuselayout{2 on 1}[border shrink=1mm]
%% % Printed, 4-up
%% \documentclass[serif,handout,landscape]{beamer}
%% \usepackage{pgfpages}
%% \pgfpagesuselayout{4 on 1}[border shrink=1mm]
\usefonttheme{serif}
\usepackage{hyperref}
\usepackage{color}
\definecolor{linkColor}{rgb}{0.62,0,0}
\hypersetup{colorlinks=true,urlcolor=linkColor}
%% \usepackage{beamerthemesplit}
%% % http://www.latex-community.org/forum/viewtopic.php?f=44&t=16603
%% \makeatletter
%% \def\verbatim{\small\@verbatim \frenchspacing\@vobeyspaces \@xverbatim}
%% \makeatother
\usepackage{graphicx}
\usepackage{color}
\DeclareGraphicsExtensions{.pdf,.png,.jpg}
%% \usepackage{wasysym}
\usepackage{mathabx}
\usepackage{setspace}
\usepackage{enumerate}
\useinnertheme[shadow]{rounded}
% \useoutertheme{default}
\useoutertheme{shadow}
\useoutertheme{infolines}
% Suppress navigation arrows
\setbeamertemplate{navigation symbols}{}
\input{macros}
%include polycode.fmt
%include forall.fmt
%include greek.fmt
%include mine.fmt
\title{Denotational Design}
\subtitle{from meanings to programs}
\author{\href{http://conal.net}{Conal Elliott}}
% \institute{\href{http://tabula.com/}{Tabula}}
% Abbreviate date/venue to fit in infolines space
\date{LambdaJam 2015}
% \date{\emph{Draft of \today}}
\setlength{\itemsep}{2ex}
\setlength{\parskip}{1ex}
% \setlength{\blanklineskip}{1.5ex}
\nc\pitem{\pause \item}
%%%%
% \setbeameroption{show notes} % un-comment to see the notes
\setstretch{1.2}
\begin{document}
\rnc\quote[2]{
\begin{center}\begin{minipage}[t]{0.7\textwidth}\begin{center}
\emph{#1}
\end{center}
\begin{flushright}
\vspace{-1.2ex}
- #2\hspace{2ex}~
\end{flushright}
\end{minipage}\end{center}
}
\nc\pquote{\pause\quote}
\frame{\titlepage}
\framet{Abstraction}{
\pause
\large \setstretch{1.5}
\quote{The purpose of abstraction is not to be vague,\\
but to create a new semantic level\\
in which one can be absolutely precise.}{Edsger Dijkstra}
}
\framet{Goals}{
\begin{itemize}\parskip 4ex
\item \emph{Abstractions}: precise, elegant, reusable.
\item \emph{Implementations}: correct, efficient, maintainable.
\item \emph{Documentation}: clear, simple, accurate.
\end{itemize}
}
\framet{Not even wrong}{\parskip 2ex
Conventional programming is precise only about how, not what.
\pquote{It is not only not right, it is not even wrong.}{Wolfgang Pauli}
\pquote{Everything is vague to a degree you do not realize till you have tried to make it precise.}{Bertrand Russell}
\pquote{What we wish, that we readily believe.}{Demosthenes}
}
\framet{Denotative programming}{\parskip 4ex
Peter Landin recommended ``denotative'' to replace ill-defined ``functional'' and ``declarative''.
Properties:\vspace{-4ex}
\begin{itemize}
\item Nested expression structure.
\item Each expression \emph{denotes} something,
\item depending only on denotations of subexpressions.
\end{itemize}
``\ldots gives us a test for whether the notation is genuinely functional or merely masquerading.''
(\href{http://www.scribd.com/doc/12878059/The-Next-700-Programming-Languages}{\emph{The
Next 700 Programming Languages}}, 1966)
}
\framet{Denotational design}{ % \parskip 2ex
Design methodology for ``genuinely functional'' programming:
\begin{itemize}\itemsep 1.5ex
\item Precise, simple, and compelling specification.
\item Informs \emph{use} and \emph{implementation} without entangling them.
\item Standard algebraic abstractions.
\item Free of abstraction leaks.
\item Laws for free.
\item Principled construction of correct implementation.
\end{itemize}
}
\framet{Overview}{
\begin{itemize}\parskip 2ex
\item Broad outline:
\begin{itemize}\parskip 2ex
\item Example, informally
\item \emph{Pretty pictures}
\item Principles
\item More examples
\item Reflection
\end{itemize}
\pitem Discussion throughout
\pitem Try it on.
\end{itemize}
}
\framet{Example: image synthesis/manipulation}{
\begin{itemize}\parskip4ex
\item How to start?
\item What is success?
\end{itemize}
}
\framet{Functionality}{
\pause
\begin{itemize}\parskip 1.25ex
\item Import \& export
\item Spatial transformation:
\begin{itemize}\parskip 1.2ex
\item Affine: translate, scale, rotate
\item Non-affine: swirls, lenses, inversions, \ldots{}
\end{itemize}
\item Cropping
\item Monochrome
\item Overlay
\item Blend
\item Blur \& sharpen
\item Geometry, gradients, \ldots.
\end{itemize}
}
\framet{API first pass}{
\pause
> type Image
> over :: Image -> Image -> Image
> transform :: Transform -> Image -> Image
> crop :: Region -> Image -> Image
> monochrome :: Color -> Image
> -- shapes, gradients, etc.
{}
> fromBitmap :: Bitmap -> Image
> toBitmap :: Image -> Bitmap
}
\framet{How to implement?}{
\pause
\begin{center}
\emph{wrong first question}
\end{center}
}
\framet{\emph{What} to implement?}{
\begin{itemize}\parskip4ex
\pitem What do these operations mean?
\pitem More centrally: What do the \emph{types} mean?
\end{itemize}
}
\framet{What is an image?}{\parskip3ex
\pause
Specification goals:
\pause
\begin{itemize}\parskip2ex
\item Adequate
\item Simple
\item Precise
\end{itemize}
\pause
\vspace{4ex}
Why these properties?
}
\framet{What is an image?}{
\pause
My answer:
assignment of colors to 2D locations.
\pause How to make precise?
> type Image
\pause
Model:
> meaning :: Image -> (Loc -> Color)
\pause
What about regions?
\pause
> meaning :: Region -> (Loc -> Bool)
}
\framet{Specifying |Image| operations}{
> meaning (over top bot) == ...
>
> meaning (crop reg im) == ...
>
> meaning (monochrome c) == ...
>
> meaning (transform tr im) == ...
\vspace{13ex}
}
\framet{Specifying |Image| operations}{
> meaning (over top bot) == \ p -> overC (meaning top p) (meaning bot p)
>
> meaning (crop reg im) == \ p -> if meaning reg p then meaning im p else clear
>
> meaning (monochrome c) == \ p -> c
>
> meaning (transform tr im) == -- coming up
> overC :: Color -> Color -> Color
\pause
Note compositionality of |meaning|.
}
\framet{Compositional semantics}{
Make more explicit:
> meaning (over top bot) == overS (meaning top) (meaning bot)
> meaning (crop reg im) == cropS (meaning reg) (meaning im)
> overS :: (Loc -> Color) -> (Loc -> Color) -> (Loc -> Color)
> overS f g = \ p -> overC (f p) (g p)
> SPACE
> cropS :: (Loc -> Bool) -> (Loc -> Color) -> (Loc -> Color)
> cropS f g = \ p -> if f p then g p else clear
}
\framet{Generalize and simplify}{\parskip3ex
\begin{itemize}\itemsep2ex
\item What about transforming \emph{regions}?
\item Other pointwise combinations (lerp, threshold)?
\end{itemize}
\pause
Generalize:
> type Image a
> type ImageC = Image Color
> type Region = Image Bool
Now some operations become more general.
}
\framet{Generalize and simplify}{
> transform :: Transform -> Image a -> Image a
> cond :: Image Bool -> Image a -> Image a -> Image a
\pause\vspace{-4ex}
> lift0 :: a -> Image a
> lift1 :: (a -> b) -> (Image a -> Image b)
> lift2 :: (a -> b -> c) -> (Image a -> Image b -> Image c)
> ...
\pause
Specializing,
> monochrome = lift0
> over = lift2 overC
> crop r im = cond r im emptyIm
> cond = lift3 ifThenElse
}
\framet{Spatial transformation}{
> meaning :: Transform -> ??
> meaning (transform tr im) == ??
\vspace{28.55ex}
}
\framet{Spatial transformation}{
> meaning :: Transform -> ??
> meaning (transform tr im) == transformS (meaning tr) (meaning im)
where
> transformS :: ?? -> (Loc -> Color) -> (Loc -> Color)
\vspace{17ex}
}
\framet{Spatial transformation}{
> meaning :: Transform -> (Loc -> Loc)
> meaning (transform tr im) == transformS (meaning tr) (meaning im)
where
> transformS :: (Loc -> Loc) -> (Loc -> Color) -> (Loc -> Color)
\pause\vspace{-8ex}
> transformS h f = \ p -> f (h p)
Subtle implications.
\pause
What is |Loc|?
\pause
My answer: continuous, infinite 2D space.
> type Loc = R2
}
\framet{Why continuous \& infinite (vs discrete/finite) space?}{
\pause
Same benefits as for time (FRP):
%\\ \hspace{6ex} \ldots and for pure, non-strict functional programming.
\pause
\begin{itemize}\itemsep0.3ex
\item Transformation flexibility with simple \& precise semantics.
\item Modularity/reusability/composability:
\begin{itemize}
\item Fewer assumptions, more uses (resolution-independence).
\item More info available for extraction.
\end{itemize}
\item Integration and differentiation: natural, accurate, efficient.
% \item Simplicity: eliminate non-essential details.
\pause
\item Quality/accuracy.
\item Efficiency (adapative).
\item Reconcile differing input sampling rates.
\end{itemize}
\pause
% \fbox{\emph{Principle:} Approximations/prunings compose badly, so postpone.}
\vspace{1ex}
{\color{blue}
\fbox{\normalcolor\emph{Principle:} Approximations/prunings compose badly, so postpone.}
}
See \href{http://www.cse.chalmers.se/~rjmh/Papers/whyfp.html}{\emph{Why~Functional~Programming~Matters}}.
}
\framet{Examples}{
\begin{center}
\href{http://conal.net/Pan/Gallery/}{Pan gallery}
\end{center}
}
\framet{Using standard vocabulary}{
\pause
\begin{itemize}\parskip1.3ex
\item We've created a domain-specific vocabulary.
\item Can we reuse standard vocabularies instead?
\item Why would we want to?
\pause
\begin{itemize}\parskip1.5ex
\item User knowledge.
\item Ecosystem support (multiplicative power).
\item Laws as sanity check.
\item Tao check.
\item Specification and laws for free, as we'll see.
\end{itemize}
\pitem In Haskell, standard type classes.
\end{itemize}
}
\framet{Monoid}{
Interface:
> class Monoid m where
> mempty :: m -- ``mempty''
> (<>) :: m -> m -> m -- ``mappend''
\pause
Laws:
> a <> mempty == a
> mempty <> b == b
> a <> (b <> c) == (a <> b) <> c
\pause Why do laws |matter|?
\pause Compositional (modular) reasoning.
\pause What monoids have we seen today?
}
\framet{Image monoid}{
\pause
> instance Monoid ImageC where
> mempty = lift0 clear
> (<>) = over
\pause
Is there a more general form on |Image a|?
\pause
> instance Monoid a => Monoid (Image a) where
> mempty = lift0 mempty
> (<>) = lift2 (<>)
\vspace{-1ex}
\pause
Do these instances satisfy the |Monoid| laws?
}
\framet{|Functor|}{
> class Functor f where
> fmap :: (a -> b) -> (f a -> f b)
{}
\pause
For images?
\vspace{2ex}
\pause
> instance Functor Image where
> fmap = lift1
{}
\pause Laws?
}
\framet{|Applicative|}{
> class Functor f => Applicative f where
> pure :: a -> f a
> (<*>) :: f (a -> b) -> f a -> f b
\pause
For images?
\pause
> instance Applicative Image where
> pure = lift0
> (<*>) = lift2 ($)
From |Applicative|, where |(<$>) = fmap|:
> liftA2 f p q = f <$> p <*> q
> liftA3 f p q r = f <$> p <*> q <*> r
> -- etc
\pause Laws?
}
\framet{Instance semantics}{
\pause
|Monoid|:\vspace{-2ex}
> mu mempty == \ p -> mempty
> mu (top <> bot) == \ p -> mu top p <> mu bot p
\pause
|Functor|:\vspace{-2ex}
> mu (fmap f im) == \ p -> f (mu im p)
> == f . mu im
\pause
|Applicative|:\vspace{-2ex}
> mu (pure a) == \ p -> a
> mu (imf <*> imx) == \ p -> (mu imf p) (mu imx p)
}
\framet{|Monad| and |Comonad|}{
> class Monad f where
> return :: a -> f a
> join :: f (f a) -> f a
> class Functor f => Comonad f where
> coreturn :: f a -> a
> cojoin :: f a -> f (f a)
|Comonad| gives us neighborhood operations.
}
\framet{Monoid specification, revisited}{
Image monoid specification:
\vspace{-1.5ex}
> mu mempty == \ p -> mempty
> mu (top <> bot) == \ p -> mu top p <> mu bot p
\pause
Instance for the semantic model:
\vspace{-1.5ex}
> instance Monoid m => Monoid (z -> m) where
> mempty = \ z -> mempty
> f <> g = \ z -> f z <> g z
\pause
Refactoring,
\vspace{-1.5ex}
> mu mempty == mempty
> mu (top <> bot) == mu top <> mu bot
\pause
So |mu| \emph{distributes} over monoid operations\pause, i.e., a monoid homomorphism.
}
\framet{Functor specification, revisited}{
Functor specification:
\vspace{-1.5ex}
> mu (fmap f im) == f . mu im
\pause
Instance for the semantic model:
\vspace{-1.5ex}
> instance Functor ((->) u) where
> fmap f h = f . h
Refactoring,
\vspace{-1.5ex}
> mu (fmap f im) == fmap f (mu im)
So |mu| is a \emph{functor} homomorphism.
}
\framet{Applicative specification, revisited}{
Applicative specification:
\vspace{-1.5ex}
> mu (pure a) == \ p -> a
> mu (imf <*> imx) == \ p -> (mu imf p) (mu imx p)
\pause
Instance for the semantic model:
\vspace{-1.5ex}
> instance Applicative ((->) u) where
> pure a = \ u -> a
> fs <*> xs = \ u -> (fs u) (xs u)
Refactoring,
\vspace{-1.5ex}
> mu (pure a) == pure a
> mu (imf <*> imx) == mu imf <*> mu imx
So |mu| is an \emph{applicative} homomorphism.
}
\framet{Specifications for free}{\parskip3ex
Semantic type class morphism (TCM) principle:
\begin{quotation}
\emph{The instance's meaning follows the meaning's instance.}
\end{quotation}
That is, the type acts like its meaning.
Every TCM failure is an abstraction leak.
Strong design principle.
Class laws necessarily hold, as we'll see.
}
\setlength{\fboxsep}{-1.7ex}
\framet{Laws for free}{
%% Semantic homomorphisms guarantee class laws. For `Monoid`,
\begin{center}
\fbox{\begin{minipage}[c]{0.4\textwidth}
> meaning mempty == mempty
> meaning (a <> b) == meaning a <> meaning b
\end{minipage}}
\begin{minipage}[c]{0.07\textwidth}\begin{center}$\Rightarrow$\end{center}\end{minipage}
\fbox{\begin{minipage}[c]{0.45\textwidth}
> a <> mempty == a
> mempty <> b == b
> a <> (b <> c) == (a <> b) <> c
\end{minipage}}
\end{center}
\vspace{-1ex}
where equality is \emph{semantic}.
\pause
Proofs:
\begin{center}
\fbox{\begin{minipage}[c]{0.3\textwidth}
> meaning (a <> mempty)
> == meaning a <> meaning mempty
> == meaning a <> mempty
> == meaning a
\end{minipage}}
\fbox{\begin{minipage}[c]{0.3\textwidth}
> meaning (mempty <> b)
> == meaning mempty <> meaning b
> == mempty <> meaning b
> == meaning b
\end{minipage}}
\fbox{\begin{minipage}[c]{0.39\textwidth}
> meaning (a <> (b <> c))
> == meaning a <> (meaning b <> meaning c)
> == (meaning a <> meaning b) <> meaning c
> == meaning ((a <> b) <> c)
\end{minipage}}
\end{center}
Works for other classes as well.
}
%if True
\framet{Example: functional reactive programming}{
See previous talks:
\begin{itemize}\itemsep2ex
\item \href{https://github.com/conal/talk-2015-essence-and-origins-of-frp/}{\emph{The essence and origins of FRP}}
\item \href{https://github.com/conal/talk-2015-more-elegant-frp}{\emph{A more elegant specification for FRP}}
\end{itemize}
}
%else
\framet{Example: functional reactive programming}{
\pause
Two essential properties:
\begin{itemize}
\item \emph{Continuous} time!
(Natural \& composable.)
\item Denotational design.
(Elegant \& rigorous.)
\end{itemize}
{\parskip 3ex
\pause
Deterministic, continuous ``concurrency''.
More aptly, \emph{``Denotative continuous-time programming''} (DCTP).
Warning: many modern ``FRP'' systems have neither property.
}
}
\framet{Denotational design}{
Central type:
> type Behavior a
Model:
> meaning :: Behavior a -> (T -> a)
\pause
Suggests API and semantics (via morphisms).
What standard algebraic abstractions does the model inhabit?
\pause
|Monoid|, |Functor|, |Applicative|, |Monad|, |Comonad|.
}
\framet{Functor}{
> instance Functor ((->) t) where
> fmap f h = f . h
Morphism:
> meaning (fmap f b)
> == fmap f (meaning b)
> SPACE
> == f . meaning b
}
\framet{Applicative}{
> instance Applicative ((->) t) where
> pure a = \ t -> a
> g <*> h = \ t -> (g t) (h t)
Morphisms:
\begin{center}
\fbox{\begin{minipage}[c]{0.48\textwidth}
> meaning (pure a)
> == pure a
> SPACE
> == \ t -> a
\end{minipage}}
\hspace{0.02\textwidth}
\fbox{\begin{minipage}[c]{0.48\textwidth}
> meaning (fs <*> xs)
> == meaning fs <*> meaning xs
> SPACE
> == \ t -> (meaning' fs t) (meaning' xs t)
\end{minipage}}
\end{center}
Corresponds exactly to the original FRP denotation.
}
\framet{Monad}{
> instance Monad ((->) t) where
> join ff = \ t -> ff t t
Morphism:
\begin{center}
\fbox{\begin{minipage}[c]{0.48\textwidth}
> meaning (join bb)
> == join (fmap meaning (meaning bb))
> SPACE
> == join (meaning . meaning bb)
> == \ t -> (meaning . meaning bb) t t
> == \ t -> meaning (meaning bb t) t
\end{minipage}}
\end{center}
}
\framet{Comonad}{
> class Comonad w where
> coreturn :: w a -> a
> cojoin :: w a -> w (w a)
Functions:
> instance Monoid t => Comonad ((->) t) where
> coreturn :: (t -> a) -> a
> coreturn f = f mempty
> cojoin f = \ t t' -> f (t <> t')
Suggest a relative time model.
}
\framet{Why continuous \& infinite (vs discrete/finite) time?}{
\pause
\begin{itemize}\parskip0.3ex
\item Transformation flexibility with simple \& precise semantics
\item Efficiency (adapative)
\item Quality/accuracy
\item Modularity/composability:
\begin{itemize}
\item Fewer assumptions, more uses (resolution-independence).
\item More info available for extraction.
\item Same benefits as pure, non-strict functional programming.\\
See \href{http://www.cse.chalmers.se/~rjmh/Papers/whyfp.html}{\emph{Why Functional Programming Matters}}.
\end{itemize}
\pitem Integration and differentiation: natural, accurate, efficient.
\pitem Reconcile differing input sampling rates.
\end{itemize}
\pause
% Approximations/prunings compose badly, so postpone.
{\color{blue}
\fbox{\normalcolor\emph{Principle:} Approximations/prunings compose badly, so postpone.}
}
%% \item Strengthen induction hypothesis
}
%endif
\framet{Example: uniform pairs}{
Type:
> data Pair a = a :# a
~
API: |Monoid|, |Functor|, |Applicative|, |Monad|, |Foldable|, |Traversable|.
\out{
\begin{itemize}
\item How to implement the methods? \pause (Answer: ``Correctly''.)
\item More fundamentally, what should the methods mean?
\end{itemize}
}
\pause\vspace{6ex}
Specification follows from simple \& precise denotation.
% What is it?
}
\framet{Uniform pairs --- denotation}{
\pause
|Pair| is an \emph{indexable} container.
What's the index type?
\pause
> type P a = Bool -> a
> meaning :: Pair a -> P a
\pause\vspace{-8ex}
> meaning (u :# v) False = u
> meaning (u :# v) True = v
\out{
Equivalently,
> meaning (u :# v) = \ b -> if b then v else u
}
API specification? \pause Homomorphisms, as usual!
}
\framet{Uniform pairs --- monoid}{
Monoid homomorphism:
> meaning mempty == mempty
> meaning (u <> v) == meaning u <> meaning v
\pause
In this case,
> instance Monoid m => Monoid (z -> m) where
> mempty = \ z -> mempty
> f <> g = \ z -> f z <> g z
\pause
so
> meaning mempty == \ z -> mempty
> meaning (u <> v) == \ z -> meaning u z <> meaning v z
Implementation: solve for |mempty| and |(<>)| on the left.
\pause
Hint: find |meaningInv|.
}
\framet{Uniform pairs --- other classes}{
Exercise: apply the same principle for
\begin{itemize}
\item |Functor|
\item |Applicative|
\item |Monad|
\item |Foldable|
\item |Traversable|
\end{itemize}
}
\framet{Example: streams}{
> data Stream a = Cons a (Stream a)
API: same classes as with |Pair|.
Denotation?
\pause
Hint: |Stream| is also an indexable type.
\pause
> data S a = Nat -> a
>
> data Nat = Zero | Succ Nat
Interpret |Stream| as |S|: