binder that is bound inside a function.
In Haskell, there is no sharp distinction between a variable and a
- function: A function is just a variable (binder) with a function
+ function: a function is just a variable (binder) with a function
type. This means that a top level function is just any top level
binder with a function type.
\section[sec:description:application]{Function application}
The basic syntactic elements of a functional program are functions and
function application. These have a single obvious \small{VHDL}
- translation: Each top level function becomes a hardware component, where each
+ translation: each top level function becomes a hardware component, where each
argument is an input port and the result value is the (single) output
port. This output port can have a complex type (such as a tuple), so
having just a single output port does not pose a limitation.
\section{Choice}
Although describing components and connections allows us to describe a lot of
- hardware designs already, there is an obvious thing missing: Choice. We
+ hardware designs already, there is an obvious thing missing: choice. We
need some way to be able to choose between values based on another value.
In Haskell, choice is achieved by \hs{case} expressions, \hs{if}
expressions, pattern matching and guards.
An obvious way to add choice to our language without having to recognize
any of Haskell's syntax, would be to add a primivite \quote{\hs{if}}
- function. This function would take three arguments: The condition, the
+ function. This function would take three arguments: the condition, the
value to return when the condition is true and the value to return when
the condition is false.
The architecture described by \in{example}[ex:PatternInv] is of course the
same one as the one in \in{example}[ex:CaseInv]. The general interpretation
- of pattern matching is also similar to that of \hs{case} expressions: Generate
+ of pattern matching is also similar to that of \hs{case} expressions: generate
hardware for each of the clauses (like each of the clauses of a \hs{case}
expression) and connect them to the function output through (a number of
nested) multiplexers. These multiplexers are driven by comparators and
\section{Types}
Translation of two most basic functional concepts has been
- discussed: Function application and choice. Before looking further
+ discussed: function application and choice. Before looking further
into less obvious concepts like higher-order expressions and
polymorphism, the possible types that can be used in hardware
descriptions will be discussed.
\stopdesc
\startdesc{\hs{Vector}}
This is a vector type, that can contain elements of any other type and
- has a fixed length. It has two type parameters: Its
+ has a fixed length. It has two type parameters: its
length and the type of the elements contained in it. By putting the
length parameter in the type, the length of a vector can be determined
at compile time, instead of only at runtime for conventional lists.
- The \hs{Vector} type constructor takes two type arguments: The length
+ The \hs{Vector} type constructor takes two type arguments: the length
of the vector and the type of the elements contained in it. The state
type of an 8 element register bank would then for example be:
A \hs{RangedWord} only has an upper bound, its lower bound is
implicitly zero. There is a lot of added implementation complexity
when adding a lower bound and having just an upper bound was enough
- for the primary purpose of this type: Typesafely indexing vectors.
+ for the primary purpose of this type: typesafely indexing vectors.
To define an index for the 8 element vector above, we would do:
in \cite[baaij09].
\subsection{User-defined types}
- There are three ways to define new types in Haskell: Algebraic
+ There are three ways to define new types in Haskell: algebraic
datatypes with the \hs{data} keyword, type synonyms with the \hs{type}
keyword and type renamings with the \hs{newtype} keyword. \GHC\
offers a few more advanced ways to introduce types (type families,
fields of the \hs{Sum} type are valid (the first two if \hs{A}, the
last one if \hs{B}), all the other ones have no useful value.
- An obvious problem with this naive approach is the space usage: The
+ An obvious problem with this naive approach is the space usage: the
example above generates a fairly big \VHDL\ type. Since we can be
sure that the two \hs{Word}s in the \hs{Sum} type will never be valid
at the same time, this is a waste of space.
\stopdesc
Another interesting case is that of recursive types. In Haskell, an
- algebraic datatype can be recursive: Any of its field types can be (or
+ algebraic datatype can be recursive: any of its field types can be (or
contain) the type being defined. The most well-known recursive type is
probably the list type, which is defined is:
Note that \hs{Empty} is usually written as \hs{[]} and \hs{Cons} as
\hs{:}, but this would make the definition harder to read. This
- immediately shows the problem with recursive types: What hardware type
+ immediately shows the problem with recursive types: what hardware type
to allocate here?
If the naive approach for sum types described above would be used,
a record would be created where the first field is an enumeration
to distinguish \hs{Empty} from \hs{Cons}. Furthermore, two more
- fields would be added: One with the (\VHDL\ equivalent of) type
+ fields would be added: one with the (\VHDL\ equivalent of) type
\hs{t} (assuming this type is actually known at compile time, this
should not be a problem) and a second one with type \hs{List t}.
- The latter one is of course a problem: This is exactly the type
+ The latter one is of course a problem: this is exactly the type
that was to be translated in the first place.
The resulting \VHDL\ type will thus become infinitely deep. In
other words, there is no way to statically determine how long
(deep) the list will be (it could even be infinite).
- In general, recursive types can never be properly translated: All
+ In general, recursive types can never be properly translated: all
recursive types have a potentially infinite value (even though in
practice they will have a bounded value, there is no way for the
compiler to automatically determine an upper bound on its size).
\subsection{Partial application}
Now the translation of application, choice and types has been
- discussed, a more complex concept can be considered: Partial
+ discussed, a more complex concept can be considered: partial
applications. A \emph{partial application} is any application whose
(return) type is (again) a function type.
From this, it should be clear that the translation rules for full
- application does not apply to a partial application: There are not
+ application does not apply to a partial application: there are not
enough values for all the input ports in the resulting \VHDL.
\in{Example}[ex:Quadruple] shows an example use of partial application
and the corresponding architecture.
{\boxedgraphic{Quadruple}}{The architecture described by the Haskell description.}
\stopcombination
- Here, the definition of mul is a partial function application: It applies
+ Here, the definition of mul is a partial function application: it applies
the function \hs{(*) :: Word -> Word -> Word} to the value \hs{2 :: Word},
resulting in the expression \hs{(*) 2 :: Word -> Word}. Since this resulting
expression is again a function, hardware cannot be generated for it
function is the same (of course, if a particular value, such as the result
of a function application, is used twice, it is not calculated twice).
- This is distinctly different from normal program compilation: Two separate
+ This is distinctly different from normal program compilation: two separate
calls to the same function share the same machine code. Having more
machine code has implications for speed (due to less efficient caching)
and memory usage. For normal compilation, it is therefore important to
\fxnote{This section needs improvement and an example}
\section{Polymorphism}
- In Haskell, values can be \emph{polymorphic}: They can have multiple types. For
+ In Haskell, values can be \emph{polymorphic}: they can have multiple types. For
example, the function \hs{fst :: (a, b) -> a} is an example of a
- polymorphic function: It works for tuples with any two element types. Haskell
+ polymorphic function: it works for tuples with any two element types. Haskell
type classes allow a function to work on a specific set of types, but the
general idea is the same. The opposite of this is a \emph{monomorphic}
value, which has a single, fixed, type.
% A type class is a collection of types for which some operations are
% defined. It is thus possible for a value to be polymorphic while having
-% any number of \emph{class constraints}: The value is not defined for
+% any number of \emph{class constraints}: the value is not defined for
% every type, but only for types in the type class. An example of this is
% the \hs{even :: (Integral a) => a -> Bool} function, which can map any
% value of a type that is member of the \hs{Integral} type class
Note that Cλash currently does not allow user-defined type classes,
but does partly support some of the built-in type classes (like \hs{Num}).
- Fortunately, we can again use the principle of specialization: Since every
+ Fortunately, we can again use the principle of specialization: since every
function application generates a separate piece of hardware, we can know
the types of all arguments exactly. Provided that existential typing
(which is a \GHC\ extension) is not used typing, all of the
So our functions must remain pure, meaning the current state has
to be present in the function's arguments in some way. There seem
- to be two obvious ways to do this: Adding the current state as an
+ to be two obvious ways to do this: adding the current state as an
argument, or including the full history of each argument.
\subsubsection{Stream arguments and results}
stream so that we can "look into" the past. This \hs{delay} function
simply outputs a stream where each value is the same as the input
value, but shifted one cycle. This causes a \quote{gap} at the
- beginning of the stream: What is the value of the delay output in the
+ beginning of the stream: what is the value of the delay output in the
first cycle? For this, the \hs{delay} function has a second input, of
which only a single value is used.
part) is dependent on its own implementation and of the functions it
calls.
- This is the major downside of this approach: The separation between
+ This is the major downside of this approach: the separation between
interface and implementation is limited. However, since Cλash is not
very suitable for separate compilation (see
\in{section}[sec:prototype:separate]) this is not a big problem in
deduce the statefulness of subfunctions by analyzing the flow of data
in the calling functions?
- To explore this matter, the following observeration is interesting: We
+ To explore this matter, the following observeration is interesting: we
get completely correct behaviour when we put all state registers in
the top level entity (or even outside of it). All of the state
arguments and results on subfunctions are treated as normal input and
end up is easier to implement correctly with explicit annotations, so
for these reasons we will look at how this annotations could work.
- \todo{Sidenote: One or more state arguments?}
+ \todo{Sidenote: one or more state arguments?}
\subsection[sec:description:stateann]{Explicit state annotation}
To make our stateful descriptions unambigious and easier to translate,
have a direct hardware interpretation.
\section{Normal form}
- The transformations described here have a well-defined goal: To bring the
+ The transformations described here have a well-defined goal: to bring the
program in a well-defined form that is directly translatable to
\VHDL, while fully preserving the semantics of the program. We refer
to this form as the \emph{normal form} of the program. The formal
EBNF-like description captures most of the intended structure (and
generates a subset of \GHC's core format).
- There are two things missing: Cast expressions are sometimes
+ There are two things missing: cast expressions are sometimes
allowed by the prototype, but not specified here and the below
definition allows uses of state that cannot be translated to \VHDL\
properly. These two problems are discussed in
The type of this expression is \lam{Word}, so it does not match \lam{a -> b}
and the transformation does not apply. Next, we have two options for the
- next expression to look at: The function position and argument position of
+ next expression to look at: the function position and argument position of
the application. The expression in the argument position is \lam{b}, which
has type \lam{Word}, so the transformation does not apply. The expression in
the function position is:
function position (which makes the second condition false). In the same
way the transformation does not apply to both components of this
expression (\lam{case opcode of Low -> (+); High -> (-)} and \lam{a}), so
- we will skip to the components of the case expression: The scrutinee and
+ we will skip to the components of the case expression: the scrutinee and
both alternatives. Since the opcode is not a function, it does not apply
here.
A \emph{hardware representable} (or just \emph{representable}) type or value
is (a value of) a type that we can generate a signal for in hardware. For
example, a bit, a vector of bits, a 32 bit unsigned word, etc. Values that are
- not runtime representable notably include (but are not limited to): Types,
+ not runtime representable notably include (but are not limited to): types,
dictionaries, functions.
\defref{representable}
\stoplambda
This is obviously not what was supposed to happen! The root of this problem is
- the reuse of binders: Identical binders can be bound in different,
+ the reuse of binders: identical binders can be bound in different,
but overlapping scopes. Any variable reference in those
overlapping scopes then refers to the variable bound in the inner
(smallest) scope. There is not way to refer to the variable in the
outer scope. This effect is usually referred to as
- \emph{shadowing}: When a binder is bound in a scope where the
+ \emph{shadowing}: when a binder is bound in a scope where the
binder already had a value, the inner binding is said to
\emph{shadow} the outer binding. In the example above, the \lam{c}
binder was bound outside of the expression and in the inner lambda
\transexample{unusedlet}{Unused let binding removal}{from}{to}
\subsubsection{Empty let removal}
- This transformation is simple: It removes recursive lets that have no bindings
+ This transformation is simple: it removes recursive lets that have no bindings
(which usually occurs when unused let binding removal removes the last
binding from it).
This transformation makes all non-recursive lets recursive. In the
end, we want a single recursive let in our normalized program, so all
non-recursive lets can be converted. This also makes other
- transformations simpler: They only need to be specified for recursive
+ transformations simpler: they only need to be specified for recursive
let expressions (and simply will not apply to non-recursive let
expressions until this transformation has been applied).
values used in our expression representable. There are two main
transformations that are applied to \emph{all} unrepresentable let
bindings and function arguments. These are meant to address three
- different kinds of unrepresentable values: Polymorphic values,
+ different kinds of unrepresentable values: polymorphic values,
higher-order values and literals. The transformation are described
- generically: They apply to all non-representable values. However,
+ generically: they apply to all non-representable values. However,
non-representable values that do not fall into one of these three
categories will be moved around by these transformations but are
unlikely to completely disappear. They usually mean the program was not
take care of exactly this.
There is one case where polymorphism cannot be completely
- removed: Built-in functions are still allowed to be polymorphic
+ removed: built-in functions are still allowed to be polymorphic
(Since we have no function body that we could properly
specialize). However, the code that generates \VHDL\ for built-in
functions knows how to handle this, so this is not a problem.
\hs{Bool} Haskell type, which is just an enumerated type.
There is, however, a second type of literal that does not have a
- representable type: Integer literals. Cλash supports using integer
+ representable type: integer literals. Cλash supports using integer
literals for all three integer types supported (\hs{SizedWord},
\hs{SizedInt} and \hs{RangedWord}). This is implemented using
Haskell's \hs{Num} type class, which offers a \hs{fromInteger} method
in y + z
\stoplambda
- Looking at this, we could imagine an alternative approach: Create a
+ Looking at this, we could imagine an alternative approach: create a
transformation that removes let bindings that bind identical values.
In the above expression, the \lam{y} and \lam{z} variables could be
merged together, resulting in the more efficient expression:
expanding some expression.
\item[q:soundness] Is our system \emph{sound}? Since our transformations
continuously modify the expression, there is an obvious risk that the final
- normal form will not be equivalent to the original program: Its meaning could
+ normal form will not be equivalent to the original program: its meaning could
have changed.
\item[q:completeness] Is our system \emph{complete}? Since we have a complex
system of transformations, there is an obvious risk that some expressions will
not end up in our intended normal form, because we forgot some transformation.
- In other words: Does our transformation system result in our intended normal
+ In other words: does our transformation system result in our intended normal
form for all possible inputs?
\item[q:determinism] Is our system \emph{deterministic}? Since we have defined
no particular order in which the transformation should be applied, there is an
\emph{different} normal forms. They might still both be intended normal forms
(if our system is \emph{complete}) and describe correct hardware (if our
system is \emph{sound}), so this property is less important than the previous
- three: The translator would still function properly without it.
+ three: the translator would still function properly without it.
\stopitemize
Unfortunately, the final transformation system has only been
Since each of the transformations can be applied to any
subexpression as well, there is a constraint on our meaning
- definition: The meaning of an expression should depend only on the
+ definition: the meaning of an expression should depend only on the
meaning of subexpressions, not on the expressions themselves. For
example, the meaning of the application in \lam{f (let x = 4 in
x)} should be the same as the meaning of the application in \lam{f
Fortunately, we can also prove the complement (which is
equivalent, since $A \subseteq B \Leftrightarrow \overline{B}
- \subseteq \overline{A}$): Show that the set of nodes not in
+ \subseteq \overline{A}$): show that the set of nodes not in
intended normal form is a subset of the set of nodes not in normal
form. In other words, show that for every expression that is not
in intended normal form, that there is at least one transformation
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