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\author{\IEEEauthorblockN{Christiaan P.R. Baaij, Matthijs Kooijman, Jan Kuper, Marco E.T. Gerards, Bert Molenkamp, Sabih H. Gerez}
\IEEEauthorblockA{University of Twente, Department of EEMCS\\
P.O. Box 217, 7500 AE, Enschede, The Netherlands\\
-c.p.r.baaij@@utwente.nl, matthijs@@stdin.nl}}
+c.p.r.baaij@@utwente.nl, matthijs@@stdin.nl, j.kuper@@utwente.nl}}
% \and
% \IEEEauthorblockN{Homer Simpson}
% \IEEEauthorblockA{Twentieth Century Fox\\
descriptions started in the early 1980s \cite{Cardelli1981, muFP,DAISY,FHDL},
a time which also saw the birth of the currently popular hardware description
languages such as \VHDL. The merit of using a functional language to describe
-hardware comes from the fact that basic combinatorial circuits are equivalent
-to mathematical functions and that functional languages are very good at
-describing and composing mathematical functions.
+hardware comes from the fact that combinatorial circuits can be directly
+modeled as mathematical functions and that functional languages are very good
+at describing and composing mathematical functions.
In an attempt to decrease the amount of work involved with creating all the
required tooling, such as parsers and type-checkers, many functional hardware
available in the functional hardware description languages that are embedded
in Haskell as a domain specific languages. As far as the authors know, such
extensive support for choice-elements is new in the domain of functional
-hardware description language. As the hardware descriptions are plain Haskell
-functions, these descriptions can be compiled for simulation using using the
-optimizing Haskell compiler \GHC.
+hardware description languages. As the hardware descriptions are plain Haskell
+functions, these descriptions can be compiled for simulation using an
+optimizing Haskell compiler such as the Glasgow Haskell Compiler (\GHC).
Where descriptions in a conventional hardware description language have an
explicit clock for the purpose state and synchronicity, the clock is implied
-in this research. The functions describe the behavior of the hardware between
+in this research. A developer describes the behavior of the hardware between
clock cycles, as such, only synchronous systems can be described. Many
-functional hardware description models signals as a stream of all values over
+functional hardware description model signals as a stream of all values over
time; state is then modeled as a delay on this stream of values. The approach
taken in this research is to make the current state of a circuit part of the
input of the function and the updated state part of the output.
netlist. This research also features a prototype translator called \CLaSH\
(pronounced: clash), which converts the Haskell code to equivalently behaving
synthesizable \VHDL\ code, ready to be converted to an actual netlist format
-by an optimizing \VHDL\ synthesis tools.
+by any (optimizing) \VHDL\ synthesis tool.
\section{Hardware description in Haskell}
\subsection{Function application}
The basic syntactic elements of a functional program are functions
and function application. These have a single obvious translation to a
- netlist: every function becomes a component, every function argument is an
- input port and the result value is of a function is an output port. This
- output port can have a complex type (such as a tuple), so having just a
- single output port does not create a limitation. Each function application
- in turn becomes a 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 itself.
+ netlist format:
+ \begin{inparaenum}
+ \item every function is translated to a component,
+ \item every function argument is translated to an input port,
+ \item the result value of a function is translated to an output port,
+ and
+ \item function applications are translated to component instantiations.
+ \end{inparaenum}
+ The output port can have a complex type (such as a tuple), so having just
+ a single output port does not pose any limitation. The arguments of a
+ function applications are assigned to a signal, which are then mapped to
+ the corresponding input ports of the component. The output port of the
+ function is also mapped to a signal, which is used as the result of the
+ application itself.
Since every top level function generates its own component, the
- hierarchy of function calls is reflected in the final netlist aswell,
+ hierarchy of function calls is reflected in the final netlist,% aswell,
creating a hierarchical description of the hardware. This separation in
different components makes the resulting \VHDL\ output easier to read and
debug.
As an example we can see the netlist of the |mac| function in
\Cref{img:mac-comb}; the |mac| function applies both the |mul| and |add|
function to calculate $a * b + c$:
+
\begin{code}
mac a b c = add (mul a b) c
\end{code}
+
\begin{figure}
\centerline{\includegraphics{mac}}
\caption{Combinatorial Multiply-Accumulate}
\label{img:mac-comb}
\end{figure}
+
The result of using a complex input type can be seen in
\cref{img:mac-comb-nocurry} where the |mac| function now uses a single
input tuple for the |a|, |b|, and |c| arguments:
+
\begin{code}
mac (a, b, c) = add (mul a b) c
\end{code}
+
\begin{figure}
\centerline{\includegraphics{mac-nocurry}}
\caption{Combinatorial Multiply-Accumulate (complex input)}
\label{img:mac-comb-nocurry}
\end{figure}
- \subsection{Choices}
- Although describing components and connections allows describing a
- lot of 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.
-
- The easiest of these are of course case expressions (and \hs{if}
- expressions, which can be very directly translated to \hs{case}
- expressions). A \hs{case} expression can in turn simply be
- translated to a conditional assignment in \VHDL, where the
- conditions use equality comparisons against the constructors in the
- \hs{case} expressions.
-
- A slightly more complex (but very powerful) form of choice is
- pattern matching. A function can be defined in multiple clauses,
- where each clause specifies a pattern. When the arguments match the
- pattern, the corresponding clause will be used.
-
- A pattern match (with optional guards) can also be implemented using
- conditional assignments in \VHDL, where the condition is the logical
- and of comparison results of each part of the pattern as well as the
- guard.
-
- Contrived example that sums two values when they are equal or
- non-equal (depending on the predicate given) and returns 0
- otherwise. This shows three implementations, one using and if
- expression, one using only case expressions and one using pattern
- matching and guards.
-
+ \subsection{Choice}
+ In Haskell, choice can be achieved by a large set of language constructs,
+ consisting of: \hs{case} constructs, \hs{if-then-else} constructs,
+ pattern matching, and guards. The easiest of these are the \hs{case}
+ constructs (\hs{if} expressions can be very directly translated to
+ \hs{case} expressions). A \hs{case} construct is translated to a
+ multiplexer, where the control value is linked to the selection port and
+ the output of each case is linked to the corresponding input port on the
+ multiplexer.
+ % A \hs{case} expression can in turn simply be translated to a conditional
+ % assignment in \VHDL, where the conditions use equality comparisons
+ % against the constructors in the \hs{case} expressions.
+ We can see two versions of a contrived example below, the first
+ using a \hs{case} construct and the other using a \hs{if-then-else}
+ constructs, in the code below. The example sums two values when they are
+ equal or non-equal (depending on the predicate given) and returns 0
+ otherwise. Both versions of the example roughly correspond to the same
+ netlist, which is depicted in \Cref{img:choice}.
+
\begin{code}
- sumif pred a b = if pred == Eq && a == b ||
- pred == Neq && a != b
- then a + b
- else 0
-
sumif pred a b = case pred of
Eq -> case a == b of
True -> a + b
Neq -> case a != b of
True -> a + b
False -> 0
-
- sumif Eq a b | a == b = a + b
- sumif Neq a b | a != b = a + b
- sumif _ _ _ = 0
\end{code}
- \begin{figure}
- \centerline{\includegraphics{choice-ifthenelse}}
- \caption{Choice - \emph{if-then-else}}
- \label{img:choice}
- \end{figure}
+ \begin{code}
+ sumif pred a b =
+ if pred == Eq then
+ if a == b then a + b else 0
+ else
+ if a != b then a + b else 0
+ \end{code}
\begin{figure}
\centerline{\includegraphics{choice-case}}
- \caption{Choice - \emph{case-statement / pattern matching}}
+ \caption{Choice - sumif}
\label{img:choice}
\end{figure}
+ A slightly more complex (but very powerful) form of choice is pattern
+ matching. A function can be defined in multiple clauses, where each clause
+ specifies a pattern. When the arguments match the pattern, the
+ corresponding clause will be used. Expressions can also contain guards,
+ where the expression is only executed if the guard evaluates to true. Like
+ \hs{if-then-else} constructs, pattern matching and guards have a
+ (straightforward) translation to \hs{case} constructs and can as such be
+ mapped to multiplexers. A third version of the earlier example, using both
+ pattern matching and guards, can be seen below. The version using pattern
+ matching and guards also has roughly the same netlist representation
+ (\Cref{img:choice}) as the earlier two versions of the example.
+
+ \begin{code}
+ sumif Eq a b | a == b = a + b
+ sumif Neq a b | a != b = a + b
+ sumif _ _ _ = 0
+ \end{code}
+
+ % \begin{figure}
+ % \centerline{\includegraphics{choice-ifthenelse}}
+ % \caption{Choice - \emph{if-then-else}}
+ % \label{img:choice}
+ % \end{figure}
+
\subsection{Types}
- Translation of two most basic functional concepts has been
- 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.
-
- Some way is needed to translate every value used to its hardware
- equivalents. In particular, this means a hardware equivalent for
- every \emph{type} used in a hardware description is needed.
-
- The following types are \emph{built-in}, meaning that their hardware
- translation is fixed into the \CLaSH\ compiler. A designer can also
- define his own types, which will be translated into hardware types
- using translation rules that are discussed later on.
-
- \subsection{Built-in types}
+ Haskell is a statically-typed language, meaning that the type of a
+ variable or function is determined at compile-time. Not all of Haskell's
+ typing constructs have a clear translation to hardware, as such this
+ section will only deal with the types that do have a clear correspondence
+ to hardware. The translatable types are divided into two categories:
+ \emph{built-in} types and \emph{user-defined} types. Built-in types are
+ those types for which a direct translation is defined within the \CLaSH\
+ compiler; the term user-defined types should not require any further
+ elaboration. The translatable types are also inferable by the compiler,
+ meaning that a developer does not have to annotate every function with a
+ type signature.
+
+ % Translation of two most basic functional concepts has been
+ % 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.
+ %
+ % Some way is needed to translate every value used to its hardware
+ % equivalents. In particular, this means a hardware equivalent for
+ % every \emph{type} used in a hardware description is needed.
+ %
+ % The following types are \emph{built-in}, meaning that their hardware
+ % translation is fixed into the \CLaSH\ compiler. A designer can also
+ % define his own types, which will be translated into hardware types
+ % using translation rules that are discussed later on.
+
+ \subsubsection{Built-in types}
+ The following types have direct translation defined within the \CLaSH\
+ compiler:
\begin{xlist}
- \item[\hs{Bit}]
+ \item[\bf{Bit}]
This is the most basic type available. It can have two values:
- \hs{Low} and \hs{High}. It is mapped directly onto the
- \texttt{std\_logic} \VHDL\ type.
- \item[\hs{Bool}]
+ \hs{Low} and \hs{High}.
+ % It is mapped directly onto the \texttt{std\_logic} \VHDL\ type.
+ \item[\bf{Bool}]
This is a basic logic type. It can have two values: \hs{True}
- and \hs{False}. It is translated to \texttt{std\_logic} exactly
- like the \hs{Bit} type (where a value of \hs{True} corresponds
- to a value of \hs{High}). Supporting the Bool type is
- particularly useful to support \hs{if ... then ... else ...}
- expressions, which always have a \hs{Bool} value for the
- condition.
- \item[\hs{SizedWord}, \hs{SizedInt}]
+ and \hs{False}.
+ % It is translated to \texttt{std\_logic} exactly like the \hs{Bit}
+ % type (where a value of \hs{True} corresponds to a value of
+ % \hs{High}).
+ Supporting the Bool type is required in order to support the
+ \hs{if-then-else} construct, which requires a \hs{Bool} value for
+ the condition.
+ \item[\bf{SizedWord}, \bf{SizedInt}]
These are types to represent integers. A \hs{SizedWord} is unsigned,
- while a \hs{SizedInt} is signed. These types are parametrized by a
- length type, so you can define an unsigned word of 32 bits wide as
- follows:
-
- \begin{code}
- type Word32 = SizedWord D32
- \end{code}
-
- Here, a type synonym \hs{Word32} is defined that is equal to the
- \hs{SizedWord} type constructor applied to the type \hs{D32}. \hs{D32}
- is the \emph{type level representation} of the decimal number 32,
- making the \hs{Word32} type a 32-bit unsigned word. These types are
- translated to the \VHDL\ \texttt{unsigned} and \texttt{signed}
- respectively.
- \item[\hs{Vector}]
- This is a vector type, that can contain elements of any other type and
+ while a \hs{SizedInt} is signed. Both are parametrizable in their
+ size.
+ % , so you can define an unsigned word of 32 bits wide as follows:
+
+ % \begin{code}
+ % type Word32 = SizedWord D32
+ % \end{code}
+
+ % Here, a type synonym \hs{Word32} is defined that is equal to the
+ % \hs{SizedWord} type constructor applied to the type \hs{D32}.
+ % \hs{D32} is the \emph{type level representation} of the decimal
+ % number 32, making the \hs{Word32} type a 32-bit unsigned word. These
+ % types are translated to the \VHDL\ \texttt{unsigned} and
+ % \texttt{signed} respectively.
+ \item[\bf{Vector}]
+ This is a vector type that can contain elements of any other type and
has a fixed 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:
-
- \begin{code}
- type RegisterState = Vector D8 Word32
- \end{code}
-
- Here, a type synonym \hs{RegisterState} is defined that is equal to
- the \hs{Vector} type constructor applied to the types \hs{D8} (The
- type level representation of the decimal number 8) and \hs{Word32}
- (The 32 bit word type as defined above). In other words, the
- \hs{RegisterState} type is a vector of 8 32-bit words. A fixed size
- vector is translated to a \VHDL\ array type.
- \item[\hs{RangedWord}]
+ contained in it. The short-hand notation used for the vector type in
+ the rest of paper is: \hs{[a|n]}. Where the \hs{a} is the element
+ type, and \hs{n} is the length of the vector.
+ % The state type of an 8 element register bank would then for example
+ % be:
+
+ % \begin{code}
+ % type RegisterState = Vector D8 Word32
+ % \end{code}
+
+ % Here, a type synonym \hs{RegisterState} is defined that is equal to
+ % the \hs{Vector} type constructor applied to the types \hs{D8} (The
+ % type level representation of the decimal number 8) and \hs{Word32}
+ % (The 32 bit word type as defined above). In other words, the
+ % \hs{RegisterState} type is a vector of 8 32-bit words. A fixed size
+ % vector is translated to a \VHDL\ array type.
+ \item[\bf{Index}]
This is another type to describe integers, but unlike the previous
two it has no specific bit-width, but an upper bound. This means that
its range is not limited to powers of two, but can be any number.
- A \hs{RangedWord} only has an upper bound, its lower bound is
- implicitly zero. The main purpose of the \hs{RangedWord} type is to be
+ An \hs{Index} only has an upper bound, its lower bound is
+ implicitly zero. The main purpose of the \hs{Index} type is to be
used as an index to a \hs{Vector}.
- \comment{TODO: Perhaps remove this example?} To define an index for
- the 8 element vector above, we would do:
+ % \comment{TODO: Perhaps remove this example?} To define an index for
+ % the 8 element vector above, we would do:
- \begin{code}
- type RegisterIndex = RangedWord D7
- \end{code}
+ % \begin{code}
+ % type RegisterIndex = RangedWord D7
+ % \end{code}
- Here, a type synonym \hs{RegisterIndex} is defined that is equal to
- the \hs{RangedWord} type constructor applied to the type \hs{D7}. In
- other words, this defines an unsigned word with values from
- 0 to 7 (inclusive). This word can be be used to index the
- 8 element vector \hs{RegisterState} above. This type is translated to
- the \texttt{unsigned} \VHDL type.
+ % Here, a type synonym \hs{RegisterIndex} is defined that is equal to
+ % the \hs{RangedWord} type constructor applied to the type \hs{D7}. In
+ % other words, this defines an unsigned word with values from
+ % 0 to 7 (inclusive). This word can be be used to index the
+ % 8 element vector \hs{RegisterState} above. This type is translated
+ % to the \texttt{unsigned} \VHDL type.
\end{xlist}
- \subsection{User-defined types}
+ \subsubsection{User-defined types}
There are three ways to define new types in Haskell: algebraic
data-types 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,
- existential typing, {\small{GADT}}s, etc.) which are not standard
- Haskell. These are not currently supported.
+ keyword and datatype renaming constructs with the \hs{newtype} keyword.
+ \GHC\ offers a few more advanced ways to introduce types (type families,
+ existential typing, {\small{GADT}}s, etc.) which are not standard Haskell.
+ As it is currently unclear how these advanced type constructs correspond
+ with hardware, they are for now unsupported by the \CLaSH\ compiler
Only an algebraic datatype declaration actually introduces a
- completely new type, for which we provide the \VHDL\ translation
- below. Type synonyms and renamings only define new names for
- existing types (where synonyms are completely interchangeable and
- renamings need explicit conversion). Therefore, these do not need
- any particular \VHDL\ translation, a synonym or renamed type will
- just use the same representation as the original type. The
- distinction between a renaming and a synonym does no longer matter
- in hardware and can be disregarded in the generated \VHDL.
-
- For algebraic types, we can make the following distinction:
+ completely new type. Type synonyms and renaming constructs only define new
+ names for existing types, where synonyms are completely interchangeable
+ and renaming constructs need explicit conversions. Therefore, these do not
+ need any particular translation, a synonym or renamed type will just use
+ the same representation as the original type. The distinction between a
+ renaming and a synonym does no longer matter in hardware and can be
+ disregarded in the translation process. For algebraic types, we can make
+ the following distinction:
\begin{xlist}
\item[\bf{Single constructor}]
Algebraic datatypes with a single constructor with one or more
fields, are essentially a way to pack a few values together in a
- record-like structure. An example of such a type is the following pair
- of integers:
-
-\begin{code}
-data IntPair = IntPair Int Int
-\end{code}
-
- Haskell's builtin tuple types are also defined as single
- constructor algebraic types and are translated according to this
- rule by the \CLaSH\ compiler. These types are translated to \VHDL\
- record types, with one field for every field in the constructor.
+ record-like structure. Haskell's built-in tuple types are also defined
+ as single constructor algebraic types An example of a single
+ constructor type is the following pair of integers:
+ \begin{code}
+ data IntPair = IntPair Int Int
+ \end{code}
+ % These types are translated to \VHDL\ record types, with one field
+ % for every field in the constructor.
\item[\bf{No fields}]
Algebraic datatypes with multiple constructors, but without any
fields are essentially a way to get an enumeration-like type
containing alternatives. Note that Haskell's \hs{Bool} type is also
defined as an enumeration type, but we have a fixed translation for
- that. These types are translated to \VHDL\ enumerations, with one
- value for each constructor. This allows references to these
- constructors to be translated to the corresponding enumeration value.
+ that. An example of such an enum type is the type that represents the
+ colors in a traffic light:
+ \begin{code}
+ data TrafficLight = Red | Orange | Green
+ \end{code}
+ % These types are translated to \VHDL\ enumerations, with one
+ % value for each constructor. This allows references to these
+ % constructors to be translated to the corresponding enumeration
+ % value.
\item[\bf{Multiple constructors with fields}]
Algebraic datatypes with multiple constructors, where at least
one of these constructors has one or more fields are not
currently supported.
\end{xlist}
+ \subsection{Polymorphic functions}
+ A powerful construct in most functional language is polymorphism.
+ This means the arguments of a function (and consequentially, values
+ within the function as well) do not need to have a fixed type.
+ Haskell supports \emph{parametric polymorphism}, meaning a
+ function's type can be parameterized with another type.
+
+ As an example of a polymorphic function, consider the following
+ \hs{append} function's type:
+
+ \comment{TODO: Use vectors instead of lists?}
+
+ \begin{code}
+ append :: [a|n] -> a -> [a|n + 1]
+ \end{code}
+
+ This type is parameterized by \hs{a}, which can contain any type at
+ all. This means that append can append an element to a list,
+ regardless of the type of the elements in the list (but the element
+ added must match the elements in the list, since there is only one
+ \hs{a}).
+
+ This kind of polymorphism is extremely useful in hardware designs to
+ make operations work on a vector without knowing exactly what elements
+ are inside, routing signals without knowing exactly what kinds of
+ signals these are, or working with a vector without knowing exactly
+ how long it is. Polymorphism also plays an important role in most
+ higher order functions, as we will see in the next section.
+
+ The previous example showed unconstrained polymorphism \comment{(TODO: How
+ is this really called?)}: \hs{a} can have \emph{any} type.
+ Furthermore,Haskell supports limiting the types of a type parameter to
+ specific class of types. An example of such a type class is the
+ \hs{Num} class, which contains all of Haskell's numerical types.
+
+ Now, take the addition operator, which has the following type:
+
+ \begin{code}
+ (+) :: Num a => a -> a -> a
+ \end{code}
+
+ This type is again parameterized by \hs{a}, but it can only contain
+ types that are \emph{instances} of the \emph{type class} \hs{Num}.
+ Our numerical built-in types are also instances of the \hs{Num}
+ class, so we can use the addition operator on \hs{SizedWords} as
+ well as on {SizedInts}.
+
+ In \CLaSH, unconstrained polymorphism is completely supported. Any
+ function defined can have any number of unconstrained type
+ parameters. The \CLaSH\ compiler will infer the type of every such
+ argument depending on how the function is applied. There is one
+ exception to this: The top level function that is translated, can
+ not have any polymorphic arguments (since it is never applied, so
+ there is no way to find out the actual types for the type
+ parameters).
+
+ \CLaSH\ does not support user-defined type classes, but does use some
+ of the builtin ones for its builtin functions (like \hs{Num} and
+ \hs{Eq}).
+
+ \subsection{Higher order}
+ Another powerful abstraction mechanism in functional languages, is
+ the concept of \emph{higher order functions}, or \emph{functions as
+ a first class value}. This allows a function to be treated as a
+ value and be passed around, even as the argument of another
+ function. Let's clarify that with an example:
+
+ \begin{code}
+ notList xs = map not xs
+ \end{code}
+
+ This defines a function \hs{notList}, with a single list of booleans
+ \hs{xs} as an argument, which simply negates all of the booleans in
+ the list. To do this, it uses the function \hs{map}, which takes
+ \emph{another function} as its first argument and applies that other
+ function to each element in the list, returning again a list of the
+ results.
+
+ As you can see, the \hs{map} function is a higher order function,
+ since it takes another function as an argument. Also note that
+ \hs{map} is again a polymorphic function: It does not pose any
+ constraints on the type of elements in the list passed, other than
+ that it must be the same as the type of the argument the passed
+ function accepts. The type of elements in the resulting list is of
+ course equal to the return type of the function passed (which need
+ not be the same as the type of elements in the input list). Both of
+ these can be readily seen from the type of \hs{map}:
+
+ \begin{code}
+ map :: (a -> b) -> [a] -> [b]
+ \end{code}
+
+ As an example from a common hardware design, let's look at the
+ equation of a FIR filter.
+
+ \begin{equation}
+ y_t = \sum\nolimits_{i = 0}^{n - 1} {x_{t - i} \cdot h_i }
+ \end{equation}
+
+ A FIR filter multiplies fixed constants ($h$) with the current and
+ a few previous input samples ($x$). Each of these multiplications
+ are summed, to produce the result at time $t$.
+
+ This is easily and directly implemented using higher order
+ functions. Consider that the vector \hs{hs} contains the FIR
+ coefficients and the vector \hs{xs} contains the current input sample
+ in front and older samples behind. How \hs{xs} gets its value will be
+ show in the next section about state.
+
+ \begin{code}
+ fir ... = foldl1 (+) (zipwith (*) xs hs)
+ \end{code}
+
+ Here, the \hs{zipwith} function is very similar to the \hs{map}
+ function: It takes a function two lists and then applies the
+ function to each of the elements of the two lists pairwise
+ (\emph{e.g.}, \hs{zipwith (+) [1, 2] [3, 4]} becomes
+ \hs{[1 + 3, 2 + 4]}.
+
+ The \hs{foldl1} function takes a function and a single list and applies the
+ function to the first two elements of the list. It then applies to
+ function to the result of the first application and the next element
+ from the list. This continues until the end of the list is reached.
+ The result of the \hs{foldl1} function is the result of the last
+ application.
+
+ As you can see, the \hs{zipwith (*)} function is just pairwise
+ multiplication and the \hs{foldl1 (+)} function is just summation.
+
+ To make the correspondence between the code and the equation even
+ more obvious, we turn the list of input samples in the equation
+ around. So, instead of having the the input sample received at time
+ $t$ in $x_t$, $x_0$ now always stores the current sample, and $x_i$
+ stores the $ith$ previous sample. This changes the equation to the
+ following (Note that this is completely equivalent to the original
+ equation, just with a different definition of $x$ that better suits
+ the \hs{x} from the code):
+
+ \begin{equation}
+ y_t = \sum\nolimits_{i = 0}^{n - 1} {x_i \cdot h_i }
+ \end{equation}
+
+ So far, only functions have been used as higher order values. In
+ Haskell, there are two more ways to obtain a function-typed value:
+ partial application and lambda abstraction. Partial application
+ means that a function that takes multiple arguments can be applied
+ to a single argument, and the result will again be a function (but
+ that takes one argument less). As an example, consider the following
+ expression, that adds one to every element of a vector:
+
+ \begin{code}
+ map ((+) 1) xs
+ \end{code}
+
+ Here, the expression \hs{(+) 1} is the partial application of the
+ plus operator to the value \hs{1}, which is again a function that
+ adds one to its argument.
+
+ A labmda expression allows one to introduce an anonymous function
+ in any expression. Consider the following expression, which again
+ adds one to every element of a list:
+
+ \begin{code}
+ map (\x -> x + 1) xs
+ \end{code}
+
+ Finally, higher order arguments are not limited to just builtin
+ functions, but any function defined in \CLaSH\ can have function
+ arguments. This allows the hardware designer to use a powerful
+ abstraction mechanism in his designs and have an optimal amount of
+ code reuse.
+
+ \comment{TODO: Describe ALU example (no code)}
+
\subsection{State}
A very important concept in hardware it the concept of state. In a
stateful design, the outputs depend on the history of the inputs, or the
valid option, as the language would then lose many of it mathematical
properties. In an effort to include the concept of state in pure
functions, the current value of the state is made an argument of the
- function; the updated state becomes part of the result.
+ function; the updated state becomes part of the result. A simple example
+ is adding an accumulator register to the earlier multiply-accumulate
+ circuit, of which the resulting netlist can be seen in
+ \Cref{img:mac-state}:
- A simple example is the description of an accumulator circuit:
\begin{code}
- acc :: Word -> State Word -> (State Word, Word)
- acc inp (State s) = (State s', outp)
+ macS a b (State c) = (State c', outp)
where
- outp = s + inp
- s' = outp
+ outp = mac a b c
+ c' = outp
\end{code}
- This approach makes the state of a function very explicit: which variables
+
+ \begin{figure}
+ \centerline{\includegraphics{mac-state}}
+ \caption{Stateful Multiply-Accumulate}
+ \label{img:mac-state}
+ \end{figure}
+
+ This approach makes the state of a circuit very explicit: which variables
are part of the state is completely determined by the type signature. This
approach to state is well suited to be used in combination with the
existing code and language features, such as all the choice constructs, as
projects / matthijs / master-project / dsd-paper.git / blobdiff