\documentclass{article}
\usepackage{haskell}

\hscommand{\vect}[1]{(#1)_v}
\hscommand{\lift}[2]{(#1)^{\uparrow#2}}
\hscommand{\liftn}[1]{(#1)^{\uparrow{n}}}

\hscommand{\patype}[1]{\mathsf{patype}\langle#1\rangle}
\hscommand{\pa}[1]{\mathsf{pa}\langle#1\rangle}

\hscommand{\capp}{\mathbin{{\$}{:}}}
\hscommand{\cappP}{\mathbin{{\$}{:}^\uparrow}}

\hscommand{\parr}[1]{[{:}{:}#1{:}{:}]}
\hscommand{\pparr}[1]{[{:}#1{:}]}

\hscommand{\Data}{\hskwd{data}}
\hscommand{\DataF}{\hskwd{data~family}}
\hscommand{\DataI}{\hskwd{data~instance}}
\hscommand{\NewtypeI}{\hskwd{newtype~instance}}

\setlength{\parindent}{0cm}

\begin{document}

\section*{Library}

\subsection*{Parallel arrays}

We distinguish between user-visible, parametric arrays (\<\pparr{\cdot}\>) and 
flattened parallel arrays (\<\parr{\cdot}\>) which are internal to our
implementation. Operations on the former have purely sequential semantics.
During vectorisation, they are replaced by parallel arrays and operations.

\begin{haskell}
\Data \pparr{\alpha} = Array Int \alpha  \hscom{or similar} \\
\DataF \parr{\alpha}
\end{haskell}

\subsection*{\<PA\> dictionaries}

To avoid problems with using typeclasses in Core, we use explicit records for
representing dictionaries of type-dependent operations on parallel arrays:

\begin{haskell}
\Data PA \alpha = & PA \{
  \hsbody{lengthP & :: \parr{\alpha}\to{Int} \\
          replicateP & :: Int\to\alpha\to\parr{\alpha} \\
          \ldots \\ \}}
\end{haskell}

In vectorised code, polymorphic functions must be supplied with a \<PA\>
dictionary for each type variable. For instance, \<\Lambda\alpha.e\> turns
into \<\Lambda\alpha.\lambda{dPA_\alpha}::PA \alpha.e'\>.

For higher-kinded type variables, we expect a function of appropriate type
which computes the dictionary for a saturated application of the type
variable from the dictionaries of the arguments. For instance,
\<\Lambda{m}::{*}\to{*}.e\> turns into
\<\Lambda{m}::{*}\to{*}.\lambda{dPA_m}::(\forall\alpha::{*}.PA \alpha\to{PA}
(m a)).e'\>.
In general, the type of the \<dPA\> argument for a type \<\sigma::\kappa\> is
given by

\begin{haskell}
\patype{\sigma:{*}} & = PA \sigma \\
\patype{\sigma:\kappa_1\to\kappa_2} & =
\forall\alpha:\kappa_1.\patype{\alpha:\kappa_1}\to\patype{\sigma \alpha:\kappa_2}
\end{haskell}

For each user-defined or built-in type constructor \<T\> we
automatically define its dictionary \<dPA_T::\patype{T}\>. Moreover, for every
in-scope type variable \<\alpha\> we have its dictionary
\<dPA_\alpha::\patype{\alpha}\>. This allows us to compute the dictionary for
an arbitrary type as follows:

\begin{haskell}
\pa{T}           & = dPA_T \\
\pa{\alpha}      & = dPA_{\alpha} \\
\pa{\sigma \phi} & = \pa{\sigma}[\phi] \pa{\phi} \\
\pa{\forall\alpha::\kappa.\sigma} & =
\Lambda\alpha::\kappa.\lambda{dPA_{\alpha}}::\patype{\alpha::\kappa}.\pa{\sigma}
\end{haskell}

\subsection*{Closures}

Closures are represented as follows:

\begin{haskell}
\Data & Clo \alpha \beta & = \exists\gamma. Clo & (PA \gamma) \gamma
 & (\gamma\to\alpha\to\beta) (\parr{\gamma}\to\parr{\alpha}\to\parr{\beta}) \\
\DataI & \parr{Clo \alpha \beta} & = \exists\gamma. AClo & (PA \gamma)
  \parr{\gamma}
  & (\gamma\to\alpha\to\beta) (\parr{\gamma}\to\parr{\alpha}\to\parr{\beta}) 
\end{haskell}

Closure application is implemented by the following two operators:

\begin{haskell}
({\capp}) & :: Clo \alpha \beta \to \alpha \to \beta \\
({\cappP}) & :: \parr{Clo \alpha \beta} \to \parr{\alpha} \to \parr{\beta}
\end{haskell}

Note that \<({\cappP})\> is effectively the lifted version of \<({\capp})\>.

\subsection*{Flat array representation}

Some important instances of the \<\parr{\cdot}\> family:

\begin{haskell}
\hskwd{data} & \hskwd{instance} \parr{(\alpha_1,\ldots,\alpha_n)}
  & = ATup_n !Int \parr{\alpha_1} \ldots \parr{\alpha_n} \\
\hskwd{newtype}~ & \hskwd{instance} \parr{\alpha\to\beta}
  & = AFn (\parr{\alpha}\to\parr{\beta}) \\
\hskwd{newtype} & \hskwd{instance} \parr{PA \alpha}
  & = APA (PA \alpha)
\end{haskell}

The last two definitions are discussed later.

\section*{Vectorisation}

\subsection*{Types}

We assume that for each type constructor \<T\>, there exists a vectorised
version \<T_v\> (which can be the same as \<T\>). In particular, we have
\begin{haskell}
({\to}_v)       & = Clo \\
\pparr{\cdot}_v & = \parr{\cdot}
\end{haskell}

Vectorisation of types is defined as follows:

\begin{haskell}
\vect{T}      & = T_v \\
\vect{\alpha} & = \alpha \\
\vect{\sigma \phi} & = \vect{\sigma} \vect{\phi} \\
\vect{\forall\alpha:\kappa.\sigma} & = \forall\alpha:\kappa.\patype{\alpha:\kappa}\to\vect{\sigma}
\end{haskell}

\subsection*{Data type declarations}

\begin{haskell}
\vect{\hskwd{data} T = \overline{C t_1 \ldots t_n}} = \hskwd{data} T_v =
\overline{C_v \vect{t_1} \ldots \vect{t_n}}
\end{haskell}

\subsection*{Terms}

We distinguish between local variables (\<x\>) and global variables and
literals \<c\>.  We assume that we have the following typing judgement:

\begin{haskell}
\Delta,\Gamma\vdash{e}:\sigma
\end{haskell}

Here, \<\Delta\> assigns types to globals and \<\Gamma\> to locals. Moreover,
we assume that for each global variable \<c\>, there exists a
vectorised version \<c_v\>, i.e.,

\begin{haskell}
c:\sigma\in\Delta \Longrightarrow c_v:\vect{\sigma}\in\Delta
\end{haskell}

\subsubsection*{Vectorisation}

\begin{haskell}
\vect{c} & = c_v & c is global \\
\vect{x} & = x   & x is local \\
\vect{\Lambda\alpha:\kappa.e} & = 
\Lambda\alpha:\kappa.\lambda{dPA_{\alpha}}:\patype{\alpha:\kappa}.\vect{e} \\
\vect{e[\sigma]} & = \vect{e}[\vect{\sigma}] \pa{\vect{\sigma}} \\
\vect{e_1 e_2} & = \vect{e_1}\capp\vect{e_2} \\
\vect{\lambda{x}:\sigma.e} & = Clo \vect{\sigma} \vect{\phi} \tau \pa{\tau}
                                   (y_1,\dots,y_n) \\
  &
\quad\quad(\lambda{ys}:\tau.
           \lambda{x}:\vect{\sigma}.
           \hskwd{case} ys \hskwd{of} (y_1,\dots,y_n) \to \vect{e}) \\
  &
\quad\quad(\lambda{ys}:\parr{\tau}.
           \lambda{x}:\parr{\vect{\sigma}}.
           \hskwd{case} ys \hskwd{of} ATup_n l y_1 \dots y_n \to \lift{e}{l})
\\
 \hswhere{e has type \phi \\
          \{y_1:\tau_1,\dots,y_n:\tau_n\} & = FVS(e)\setminus{x} \\
          \tau & = (\vect{\tau_1},\dots,\vect{\tau_n})}
% \\
%          e has type \phi}
\end{haskell}

Vectorisation maintains the following invariant:

\begin{haskell}
\Delta,\Gamma\vdash{e}:\sigma \Longrightarrow
      \Delta,\Gamma_v\vdash\vect{e}:\vect{\sigma}
\end{haskell}

where \<\Gamma_v\> is defined by

\begin{haskell}
x:\sigma\in\Gamma \Longleftrightarrow x:\vect{\sigma}\in\Gamma_v
\end{haskell}

\subsubsection*{Lifting}
\begin{haskell}
\liftn{c:\sigma} & = replicateP \pa{\vect{\sigma}} n c_v \\
\liftn{x}        & = x \\
\liftn{\Lambda\alpha:\kappa.e} & =
\Lambda\alpha:\kappa.\lambda{dPA_{\alpha}}:\patype{\alpha:\kappa}.\liftn{e} \\
\liftn{e[\sigma]} &  = \liftn{e}[\vect{\sigma}] \pa{\vect{\sigma}} \\
\liftn{e_1 e_2} & = \liftn{e_1} \cappP \liftn{e_2} \\
\liftn{\lambda{x}:\sigma.e} & = AClo \vect{\sigma} \vect{\phi} \vect{\tau}
                                   \pa{\vect{\tau}}
                                   (ATup_n y_1 \dots y_n) \\
  &
\quad\quad(\lambda{ys}:\vect{\tau}.
           \lambda{x}:\vect{\sigma}.
           \hskwd{case} ys \hskwd{of} (y_1,\dots,y_n) \to \vect{e}) \\
  &
\quad\quad(\lambda{ys}:\parr{\vect{\tau}}.
           \lambda{x}:\parr{\vect{\sigma}}.
           \hskwd{case} ys \hskwd{of} ATup_n l y_1 \dots y_n \to \lift{e}{l})
 \hswhere{e has type \phi \\
          \{y_1:\tau_1,\dots,y_n:\tau_n\} & = FVS(e)\setminus{x} \\
          \tau & = (\tau_1,\dots,\tau_n)}
\end{haskell}

Lifting maintains the following invariant:

\begin{haskell}
\Delta,\Gamma\vdash{e}:\sigma \Longrightarrow
  \Delta,\Gamma^\uparrow\vdash\liftn{e} : \parr{\sigma_v}
\end{haskell}

where

\begin{haskell}
x:\sigma\in\Gamma \Longleftrightarrow x:\parr{\vect{\sigma}}\in\Gamma^\uparrow
\end{haskell}

Note that this is precisely the reason for the \<\parr{\cdot}\> instances for
\<\alpha\to\beta\> and \<PA \alpha\>. A term of type \<\forall\alpha.\sigma\>
will be lifted to a term of type
\<\parr{\forall\alpha.PA \alpha\to\vect{\sigma}}\> which requires the
instances. Apart from closures, these are the only occurrences of \<({\to})\> in
the transformed program, however.


\section*{What to vectorise?}

An expression is vectorisable if it only mentions globals and type constructors
which have a vectorised form. When vectorising, we look for maximal
vectorisable subexpressions and transform those. For instance, assuming that
\<print\> hasn't been vectorised, in

\begin{haskell}
main = \hsdo{
         print (sumP \pparr{\ldots}) \\
         print (mapP \ldots \pparr{\ldots})}
\end{haskell}

we would vectorise the arguments to \<print\>.  Note that this implies that we
never call non-vectorised code from vectorised code (is that a problem?).

Whenever we come out of a vectorised ``bubble'', we have to convert between
vectorised and unvectorised data types. The examples above would roughly
translate to

\begin{haskell}
main = \hsdo{
  print (unvect (sumP_v \parr{\ldots})) \\
  print (unvect (mapP_v \ldots \parr{\ldots}))}
\end{haskell}

For this, we have to have the following functions:

\begin{haskell}
vect_\sigma & :: \sigma\to\vect{\sigma} \\
unvect_\sigma & :: \vect{\sigma}\to\sigma
\end{haskell}

It is sufficient to have these functions only for a restricted set of types as
we can always vectorise less if the conversions becomes too complex.

Sometimes, it doesn't make sense to vectorise things. For instance, in

\begin{haskell}
foo f xs = print (f xs)
\end{haskell}

we wouldn't vectorise \<f xs\>. Unfortunately, this means that
\<foo (mapP f) xs\> will be purely sequential.

For each expression, the vectoriser gives one of the following answers.

\begin{tabular}{lp{10cm}}
\textbf{Yes} &
the expression can be (and has been) vectorised \\
\textbf{Maybe} &
the expression can be vectorised but it doesn't make sense to do so
unless it is used in a vectrorisable context (e.g., for \<f xs\> in \<foo\>)
\\
\textbf{No} &
the expression can't be vectorised (although parts of it can, so we
still get back a transformed expression)
\end{tabular}

\subsection*{Top-level definitions}

For a top-level definition of the form

\begin{haskell}
f :: \sigma = e
\end{haskell}

vectorisation proceeds as follows.

\begin{itemize}
\item If \<e\> can be fully vectorised, we generate
\begin{haskell}
f_v :: \vect{\sigma} = \vect{e}
\end{haskell}

\item If it doesn't always make sense to vectorise \<e\>, i.e., the vectoriser
returned \textbf{Maybe}, we leave the definition of \<f\> unchanged. Thus, we
would not change
\begin{haskell}
({\$}) = \lambda{f}.\lambda{x}.f x
\end{haskell}
but would additionally generate
\begin{haskell}
({\$}_v) = Clo \ldots
\end{haskell}

\item Otherwise (if the vectoriser said \textbf{Yes}) and we have
\<unconv_\sigma\>, we change the definition of \<f\> to
\begin{haskell}
f :: \sigma = unconv_\sigma f_v
\end{haskell}

\item Otherwise (the vectoriser said \textbf{Yes} but we do not have
\<unconv_\sigma\> or if \<e\> couldn't be fully vectorised), we change the
definition of \<f\> to
\begin{haskell}
f :: \sigma = e'
\end{haskell}
where \<e'\> is obtaining by vectorising \<e\> as much as possible without
changing its type. For instance, for
\begin{haskell}
f = \lambda{g}.\lambda{x}.mapP (\ldots) (g x)
\end{haskell}
we would generate
\begin{haskell}
f & = \lambda{g}.\lambda{x}.unvect (mapP_v (\ldots) (vect (g x))) \\
f_v & = Clo \ldots
\end{haskell}
assuming we have the necessary conversions but cannot convert functions (i.e.,
\<g\>).
\end{itemize}

\end{document}

