\chapter{Hardware description}
This chapter will provide an overview of the hardware description language
that was created and the issues that have arisen in the process. It will
focus on the issues of the language, not the implementation.
When translating Haskell to hardware, we need to make choices in what kind
of hardware to generate for what Haskell constructs. When faced with
choices, we've tried to stick with the most obvious choice wherever
possible. In a lot of cases, when you look at a hardware description it is
comletely clear what hardware is described. We want our translator to
generate exactly that hardware whenever possible, to minimize the amount of
surprise for people working with it.
In this chapter we try to describe how we interpret a Haskell program from a
hardware perspective. We provide a description of each Haskell language
element that needs translation, to provide a clear picture of what is
supported and how.
\section{Function application}
The basic syntactic element of a functional program are functions and
function application. These have a single obvious VHDL translation: Each
function becomes a hardware component, where each argument is an input port
and the result value is the output port.
Each function application in turn becomes component instantiation. Here, the
result of each argument expression is assigned to a signal, which is mapped
to the corresponding input port. The output port of the function is also
mapped to a signal, which is used as the result of the application.
An example of a simple program using only function application would be:
\starthaskell
-- | A simple function that returns the and of three bits
and3 :: Bit -> Bit -> Bit -> Bit
and3 a b c = and (and a b) c
\stophaskell
This results in the following hardware:
TODO: Pretty picture
\subsection{Partial application}
It should be obvious that we cannot generate hardware signals for all
expressions we can express in Haskell. The most obvious criterium for this
is the type of an expression. We will see more of this below, but for now it
should be obvious that any expression of a function type cannot be
represented as a signal or i/o port to a component.
From this, we can see that the above translation rules do not apply to a
partial application. Let's look at an example:
\starthaskell
-- | Multiply the input word by four.
quadruple :: Word -> Word
quadruple n = mul (mul n)
where
mul = (*) 2
\stophaskell
It should be clear that the above code describes the following hardware:
TODO: Pretty picture
Here, the definition of mul is a partial function application: It applies
\hs{2 :: Word} to the function \hs{(*) :: Word -> Word -> Word} resulting in
the expression \hs{(*) 2 :: Word -> Word}. Since this resulting expression
is again a function, we can't generate hardware for it directly. This is
because the hardware to generate for \hs{mul} depends completely on where
and how it is used. In this example, it is even used twice!
However, it is clear that the above hardware description actually describes
valid hardware. In general, we can see that any partial applied function
must eventually become completely applied, at which point we can generate
hardware for it using the rules for function application above. It might
mean that a partial application is passed around quite a bit (even beyond
function boundaries), but eventually, the partial application will become
completely applied.
\section{Recursion}
An import concept in functional languages is recursion. In it's most basic
form, recursion is a function that is defined in terms of itself. This
usually requires multiple evaluations of this function, with changing
arguments, until eventually an evaluation of the function no longer requires
itself.
Recursion in a hardware description is a bit of a funny thing. Usually,
recursion is associated with a lot of nondeterminism, stack overflows, but
also flexibility and expressive power.
Given the notion that each function application will translate to a
component instantiation, we are presented with a problem. A recursive
function would translate to a component that contains itself. Or, more
precisely, that contains an instance of itself. This instance would again
contain an instance of itself, and again, into infinity. This is obviously a
problem for generating hardware.
This is expected for functions that describe infinite recursion. In that
case, we can't generate hardware that shows correct behaviour in a single
cycle (at best, we could generate hardware that needs an infinite number of
cycles to complete).
However, most recursive hardware descriptions will describe finite
recursion. This is because the recursive call is done conditionally. There
is usually a case statement where at least one alternative does not contain
the recursive call, which we call the "base case". If, for each call to the
recursive function, we would be able to detect which alternative applies,
we would be able to remove the case expression and leave only the base case
when it applies. This will ensure that expanding the recursive functions
will terminate after a bounded number of expansions.
This does imply the extra requirement that the base case is detectable at
compile time. In particular, this means that the decision between the base
case and the recursive case must not depend on runtime data.
\subsection{List recursion}
The most common deciding factor in recursion is the length of a list that is
passed in as an argument. Since we represent lists as vectors that encode
the length in the vector type, it seems easy to determine the base case. We
can simply look at the argument type for this. However, it turns out that
this is rather non-trivial to write down in Haskell in the first place. As
an example, we would like to write down something like this:
\starthaskell
sum :: Vector n Word -> Word
sum xs = case null xs of
True -> 0
False -> head xs + sum (tail xs)
\stophaskell
However, the typechecker will now use the following reasoning (element type
of the vector is left out):
\startitemize
\item tail has the type \hs{(n > 0) => Vector n -> Vector (n - 1)}
\item This means that xs must have the type \hs{(n > 0) => Vector n}
\item This means that sum must have the type \hs{(n > 0) => Vector n -> a}
\item sum is called with the result of tail as an argument, which has the
type \hs{Vector n} (since \hs{(n > 0) => n - 1 == m}).
\item This means that sum must have the type \hs{Vector n -> a}
\item This is a contradiction between the type deduced from the body of sum
(the input vector must be non-empty) and the use of sum (the input vector
could have any length).
\stopitemize
As you can see, using a simple case at value level causes the type checker
to always typecheck both alternatives, which can't be done! This is a
fundamental problem, that would seem perfectly suited for a type class.
Considering that we need to switch between to implementations of the sum
function, based on the type of the argument, this sounds like the perfect
problem to solve with a type class. However, this approach has its own
problems (not the least of them that you need to define a new typeclass for
every recursive function you want to define).
Another approach tried involved using GADTs to be able to do pattern
matching on empty / non empty lists. While this worked partially, it also
created problems with more complex expressions.
TODO: How much detail should there be here? I can probably refer to
Christiaan instead.
Evaluating all possible (and non-possible) ways to add recursion to our
descriptions, it seems better to leave out list recursion alltogether. This
allows us to focus on other interesting areas instead. By including
(builtin) support for a number of higher order functions like map and fold,
we can still express most of the things we would use list recursion for.
\subsection{General recursion}
Of course there are other forms of recursion, that do not depend on the
length (and thus type) of a list. For example, simple recursion using a
counter could be expressed, but only translated to hardware for a fixed
number of iterations. Also, this would require extensive support for compile
time simplification (constant propagation) and compile time evaluation
(evaluation constant comparisons), to ensure non-termination. Even then, it
is hard to really guarantee termination, since the user (or GHC desugarer)
might use some obscure notation that results in a corner case of the
simplifier that is not caught and thus non-termination.
Due to these complications, we leave other forms of recursion as
future work as well.