% *** PDF, URL AND HYPERLINK PACKAGES ***
%
-%\usepackage{url}
+\usepackage{url}
% url.sty was written by Donald Arseneau. It provides better support for
% handling and breaking URLs. url.sty is already installed on most LaTeX
% systems. The latest version can be obtained at:
% Macro for certain acronyms in small caps. Doesn't work with the
% default font, though (it contains no smallcaps it seems).
\def\acro#1{{\small{#1}}}
+\def\acrop#1{\acro{#1}s}
\def\acrotiny#1{{\scriptsize{#1}}}
\def\VHDL{\acro{VHDL}}
\def\GHC{\acro{GHC}}
%include polycode.fmt
%include clash.fmt
+\newcounter{Codecount}
+\setcounter{Codecount}{0}
+
+\newenvironment{example}
+ {
+ \refstepcounter{equation}
+ }
+ {
+ \begin{flushright}
+ (\arabic{equation})
+ \end{flushright}
+ }
+
\begin{document}
%
% paper title
\begin{abstract}
%\boldmath
\CLaSH\ is a functional hardware description language that borrows both its
-syntax and semantics from the functional programming language Haskell. Due to
-the abstraction and generality offered by polymorphism and higher-order
-functions, a circuit designer can describe circuits in a more natural way than
-he could in the traditional hardware description languages.
+syntax and semantics from the functional programming language Haskell.
+Polymorphism and higher-order functions provide a level of abstraction and
+generality that allow a circuit designer to describe circuits in a more
+natural way than possible in a traditional hardware description language.
Circuit descriptions can be translated to synthesizable VHDL using the
prototype \CLaSH\ compiler. As the circuit descriptions, simulation code, and
-test input are also valid Haskell, complete simulations can be compiled to an
-executable binary by a Haskell compiler allowing high-speed simulation and
-analysis.
-
-Stateful descriptions are supported by explicitly making the current state an
-argument of the function, and the updated state part of the result. In this
-sense, the descriptions made in \CLaSH\ are the combinational parts of a mealy
-machine.
+test input are also valid Haskell, complete simulations can be done by a
+Haskell compiler allowing high-speed simulation and analysis.
+
+% \CLaSH\ supports stateful descriptions by explicitly making the current
+% state an argument of the function, and the updated state part of the result.
+% This makes \CLaSH\ descriptions in essence the combinational parts of a
+% mealy machine.
\end{abstract}
% IEEEtran.cls defaults to using nonbold math in the Abstract.
% This preserves the distinction between vectors and scalars. However,
\IEEEpeerreviewmaketitle
\section{Introduction}
-Hardware description languages have allowed the productivity of hardware
-engineers to keep pace with the development of chip technology. Standard
-Hardware description languages, like \VHDL~\cite{VHDL2008} and
-Verilog~\cite{Verilog}, allowed an engineer to describe circuits using a
-programming language. These standard languages are very good at describing
-detailed hardware properties such as timing behavior, but are generally
-cumbersome in expressing higher-level abstractions. In an attempt to raise the
-abstraction level of the descriptions, a great number of approaches based on
-functional languages has been proposed \cite{T-Ruby,Hydra,HML2,Hawk1,Lava,
-ForSyDe1,Wired,reFLect}. The idea of using functional languages for hardware
-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. Functional languages are especially suited to
-describe hardware because combinational circuits can be directly modeled
-as mathematical functions. Furthermore, functional languages are very good at
-describing and composing mathematical functions.
-
-In an attempt to decrease the amount of work involved in creating all the
-required tooling, such as parsers and type-checkers, many functional
-hardware description languages \cite{Hydra,Hawk1,Lava,ForSyDe1,Wired}
-are embedded as a domain specific language inside the functional
-language Haskell \cite{Haskell}. This means that a developer is given a
-library of Haskell functions and types that together form the language
-primitives of the domain specific language. The primitive functions used
-to describe a circuit do not actually process any signals, but instead
-compose a large domain-specific datatype (which is usually hidden from
-the designer). This datatype is then further processed by an embedded
-circuit compiler. This circuit compiler actually runs in the same
-environment as the description; as a result compile-time and run-time
-become hard to define, as the embedded circuit compiler is usually
-compiled by the same Haskell compiler as the circuit description itself.
-
-The approach taken in this research is not to make another domain specific
-language embedded in Haskell, but to use (a subset of) the Haskell language
-\emph{itself} for the purpose of describing hardware. By taking this approach,
-we can capture certain language constructs, such as Haskell's choice elements
-(if-expressions, case-expressions, pattern matching, etc.), which are not
-available in the functional hardware description languages that are embedded
-in Haskell as a domain specific language. As far as the authors know, such
-extensive support for choice-elements is new in the domain of functional
-hardware description languages. As the hardware descriptions are plain Haskell
-functions, these descriptions can be compiled to an executable binary
-for simulation using an optimizing Haskell compiler such as the Glasgow
-Haskell Compiler (\GHC)~\cite{ghc}.
-
-Where descriptions in a conventional hardware description language have an
-explicit clock for the purposes state and synchronicity, in the research
-presented in this paper the clock is implied. A developer describes the
-behavior of the hardware between clock cycles. Many 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. The current abstraction of
-state and time limits the descriptions to synchronous hardware, there however
-is room within the language to eventually add a different abstraction
-mechanism that will allow for the modeling of asynchronous systems.
-
-Like the standard hardware description languages, descriptions made in a
-functional hardware description language must eventually be converted into a
-netlist. This research also features a prototype translator, which has the
-same name as the language: \CLaSH\footnote{\CLaSHtiny: \acrotiny{CAES}
-Language for Synchronous Hardware} (pronounced: clash). This compiler converts
-the Haskell code to equivalently behaving synthesizable \VHDL\ code, ready to
-be converted to an actual netlist format by an (optimizing) \VHDL\ synthesis
-tool.
-
-Besides trivial circuits such as variants of both the \acro{FIR} filter and
-the simple \acro{CPU} shown in \Cref{sec:usecases}, the \CLaSH\ compiler has
-also been shown to work for non-trivial descriptions. \CLaSH\ has been able to
-successfully translate the functional description of a streaming reduction
-circuit~\cite{reductioncircuit} for floating point numbers.
+Hardware description languages (\acrop{HDL}) have not allowed the productivity
+of hardware engineers to keep pace with the development of chip technology.
+While traditional \acrop{HDL}, like \VHDL~\cite{VHDL2008} and
+Verilog~\cite{Verilog}, are very good at describing detailed hardware
+properties such as timing behavior, they are generally cumbersome in
+expressing the higher-level abstractions needed for today's large and complex
+circuit designs. In an attempt to raise the abstraction level of the
+descriptions, a great number of approaches based on functional languages has
+been proposed \cite{Cardelli1981,muFP,DAISY,T-Ruby,HML2,Hydra,Hawk1,Lava,
+Wired,ForSyDe1,reFLect}. The idea of using functional languages for hardware
+descriptions started in the early 1980s \cite{Cardelli1981,muFP,DAISY}, a
+time which also saw the birth of the currently popular \acrop{HDL}, such as
+\VHDL. Functional languages are especially well suited to describe hardware
+because combinational circuits can be directly modeled as mathematical
+functions and functional languages are very good at describing and composing
+these functions.
+
+In an attempt to ease the prototyping process of the language, such as
+creating all the required tooling like parsers and type-checkers, many
+functional \acrop{HDL} \cite{Hydra,Hawk1,Lava,Wired} are embedded as a domain
+specific language (\acro{DSL}) within the functional language Haskell
+\cite{Haskell}. This means that a developer is given a library of Haskell
+functions and types that together form the language primitives of the
+\acro{DSL}. The primitive functions used to describe a circuit do not actually
+process any signals, they instead compose a large domain-specific graph
+(which is usually hidden from the designer). This graph is then further
+processed by an embedded circuit compiler which can perform e.g. simulation or
+synthesis. As Haskell's choice elements (\hs{case}-expressions,
+pattern-matching, etc.) are evaluated at the time the domain-specific graph is
+being build, they are no longer visible to the embedded compiler that
+processes the datatype. Consequently, it is impossible to capture Haskell's
+choice elements within a circuit description when taking the embedded language
+approach. This does not mean that circuits specified in an embedded language
+can not contain choice, just that choice elements only exists as functions,
+e.g. a multiplexer function, and not as syntactic elements of the language
+itself.
+
+The approach taken in this research is to use (a subset of) the Haskell
+language \emph{itself} for the purpose of describing hardware. By taking this
+approach, this research \emph{can} capture certain language constructs, like
+all of Haskell's choice elements, within circuit descriptions. The more
+advanced features of Haskell, such as polymorphic typing and higher-order
+functions, are also supported.
+
+% supporting polymorphism, higher-order functions and such an extensive array
+% of choice-elements, combined with a very concise way of specifying circuits
+% is new in the domain of (functional) \acrop{HDL}.
+% As the hardware descriptions are plain Haskell
+% functions, these descriptions can be compiled to an executable binary
+% for simulation using an optimizing Haskell compiler such as the Glasgow
+% Haskell Compiler (\GHC)~\cite{ghc}.
+
+Where descriptions in a conventional \acro{HDL} have an explicit clock for the
+purposes state and synchronicity, the clock is implicit for the descriptions
+and research presented in this paper. A circuit designer describes the
+behavior of the hardware between clock cycles. Many functional \acrop{HDL}
+model signals as a stream of all values over time; state is then modeled as a
+delay on this stream of values. Descriptions presented in this research make
+the current state an additional input and the updated state a part of their
+output. This abstraction of state and time limits the descriptions to
+synchronous hardware, there is however room within the language to eventually
+add a different abstraction mechanism that will allow for the modeling of
+asynchronous systems.
+
+Likewise as with the traditional \acrop{HDL}, descriptions made in a functional \acro{HDL} must eventually be converted into a netlist. This research also features a prototype compiler, which has the same name as the language: \CLaSH\footnote{\CLaSHtiny: \acrotiny{CAES} Language for Synchronous Hardware, where \acrotiny{CAES} is the acronyom of our chair.} (pronounced: clash). This compiler converts the Haskell code to equivalently behaving synthesizable \VHDL\ code, ready to be converted to an actual netlist format by an (optimizing) \VHDL\ synthesis tool.
+
+To the best knowledge of the authors, \CLaSH\ is the only (functional)
+\acro{HDL} that allows circuit specification to be written in a very concise
+way and at the same time support such advanced features as polymorphic typing,
+user-defined higher-order functions and pattern matching.
\section{Hardware description in Haskell}
+This section describes the basic language elements of \CLaSH\ and the support
+of these elements within the \CLaSH\ compiler. In various subsections, the
+relation between the language elements and their eventual netlist
+representation is also highlighted.
\subsection{Function application}
- Two basic syntactic elements of a functional program are functions
- and function application. These have a single obvious translation to a
- netlist format:
+ Two basic elements of a functional program are functions and function
+ application. These have a single obvious translation to a netlist format:
\begin{inparaenum}
\item every function is translated to a component,
\item every function argument is translated to an input port,
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, creating a hierarchical description of the hardware.
+ final netlist. %, creating a hierarchical description of the hardware.
% The separation in different components makes it easier for a developer
% to understand and possibly hand-optimize the resulting \VHDL\ output of
% the \CLaSH\ compiler.
- The short example demonstrated below gives an indication of the level of
- conciseness that can be achieved with functional hardware description
- languages when compared with the more traditional hardware description
- languages. The example is a combinational multiply-accumulate circuit that
- works for \emph{any} word length (this type of polymorphism will be
- further elaborated in \Cref{sec:polymorhpism}). The corresponding netlist
- is depicted in \Cref{img:mac-comb}.
+ The short example below (\ref{code:mac}) gives a demonstration of
+ the conciseness that can be achieved with \CLaSH\ when compared with
+ other (more traditional) \acrop{HDL}. The example is a combinational
+ multiply-accumulate circuit that works for \emph{any} word length (this
+ type of polymorphism will be further elaborated in
+ \Cref{sec:polymorhpism}). The corresponding netlist is depicted in
+ \Cref{img:mac-comb}.
+ \hspace{-1.7em}
+ \begin{minipage}{0.93\linewidth}
\begin{code}
mac a b c = add (mul a b) c
\end{code}
+ \end{minipage}
+ \begin{minipage}{0.07\linewidth}
+ \begin{example}
+ \label{code:mac}
+ \end{example}
+ \end{minipage}
\begin{figure}
\centerline{\includegraphics{mac.svg}}
- \caption{Combinatorial Multiply-Accumulate}
+ \caption{Combinational Multiply-Accumulate}
\label{img:mac-comb}
+ \vspace{-1.5em}
\end{figure}
- The use of a composite result value is demonstrated in the next example,
- where the multiply-accumulate circuit not only returns the accumulation
- result, but also the intermediate multiplication result. Its corresponding
- netlist can be see in \Cref{img:mac-comb-composite}.
+ The use of a composite result value is demonstrated in the next example
+ (\ref{code:mac-composite}), where the multiply-accumulate circuit not only
+ returns the accumulation result, but also the intermediate multiplication
+ result (see \Cref{img:mac-comb-composite}, where the double arrow suggests
+ the composite output).
+ \hspace{-1.7em}
+ \begin{minipage}{0.93\linewidth}
\begin{code}
mac a b c = (z, add z c)
where
z = mul a b
\end{code}
+ \end{minipage}
+ \begin{minipage}{0.07\linewidth}
+ \begin{example}
+ \label{code:mac-composite}
+ \end{example}
+ \end{minipage}
+ \vspace{-1.5em}
\begin{figure}
+ \vspace{1em}
\centerline{\includegraphics{mac-nocurry.svg}}
- \caption{Combinatorial Multiply-Accumulate (composite output)}
+ \caption{Combinational Multiply-Accumulate (composite output)}
\label{img:mac-comb-composite}
+ \vspace{-1.5em}
\end{figure}
\subsection{Choice}
In Haskell, choice can be achieved by a large set of syntactic elements,
consisting of: \hs{case} expressions, \hs{if-then-else} expressions,
pattern matching, and guards. The most general of these are the \hs{case}
- expressions (\hs{if} expressions can be very directly translated to
- \hs{case} expressions). A \hs{case} expression is translated to a
- multiplexer, where the control value is fed into a number of
- comparators and their output is used to compose the selection port
- of the multiplexer. The result of each alternative is linked to the
- corresponding input port on the multiplexer.
+ expressions (\hs{if} expressions can be directly translated to
+ \hs{case} expressions). When transforming a \CLaSH\ description to a
+ netlist, a \hs{case} expression is translated to a multiplexer. The
+ control value of the \hs{case} expression is fed into a number of
+ comparators, and their combined output forms the selection port of the
+ multiplexer. The result of each alternative in the \hs{case} expression is
+ linked to the corresponding input port of 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} expression and the other using an \hs{if-then-else}
- expression. Both examples sums two values when they are
- equal or non-equal (depending on the given predicate, the \hs{pred}
- variable) and returns 0 otherwise. The \hs{pred} variable has the
- following, user-defined, enumeration datatype:
+
+ % Two versions of a contrived example are displayed below, the first
+ % (\ref{lst:code3}) using a \hs{case} expression and the second
+ % (\ref{lst:code4}) using an \hs{if-then-else} expression. Both examples
+ % sum two values when they are equal or non-equal (depending on the given
+ % predicate, the \hs{pred} variable) and return 0 otherwise.
+
+ A code example (\ref{code:counter1}) that uses a \hs{case} expression and
+ \hs{if-then-else} expressions is shown below. The function counts up or
+ down depending on the \hs{direction} variable, and has a \hs{bound}
+ variable that determines both the upper bound and wrap-around point of the
+ counter. The \hs{direction} variable is of the following, user-defined,
+ enumeration datatype:
\begin{code}
- data Pred = Equal | NotEqual
+ data Direction = Up | Down
\end{code}
- The naive netlist corresponding to both versions of the example is
- depicted in \Cref{img:choice}. Note that the \hs{pred} variable is only
- compared to the \hs{Equal} value, as an inequality immediately implies
- that the \hs{pred} variable has a \hs{NotEqual} value.
+ The naive netlist corresponding to this example is depicted in
+ \Cref{img:counter}. Note that the \hs{direction} variable is only
+ compared to \hs{Up}, as an inequality immediately implies that
+ \hs{direction} is \hs{Down} (as derived by the compiler).
+ \hspace{-1.7em}
+ \begin{minipage}{0.93\linewidth}
\begin{code}
- sumif pred a b = case pred of
- Equal -> case a == b of
- True -> a + b
- False -> 0
- NotEqual -> case a != b of
- True -> a + b
- False -> 0
+ counter bound direction x = case direction of
+ Up -> if x < bound then
+ x + 1 else
+ 0
+ Down -> if x > 0 then
+ x - 1 else
+ bound
\end{code}
+ \end{minipage}
+ \begin{minipage}{0.07\linewidth}
+ \begin{example}
+ \label{code:counter1}
+ \end{example}
+ \end{minipage}
+
+ % \hspace{-1.7em}
+ % \begin{minipage}{0.93\linewidth}
+ % \begin{code}
+ % sumif pred a b =
+ % if pred == Equal then
+ % if a == b then a + b else 0
+ % else
+ % if a != b then a + b else 0
+ % \end{code}
+ % \end{minipage}
+ % \begin{minipage}{0.07\linewidth}
+ % \begin{example}
+ % \label{lst:code4}
+ % \end{example}
+ % \end{minipage}
- \begin{code}
- sumif pred a b =
- if pred == Equal then
- if a == b then a + b else 0
- else
- if a != b then a + b else 0
- \end{code}
+ % \begin{figure}
+ % \vspace{1em}
+ % \centerline{\includegraphics{choice-case.svg}}
+ % \caption{Choice - sumif}
+ % \label{img:choice}
+ % \vspace{-1.5em}
+ % \end{figure}
\begin{figure}
- \centerline{\includegraphics{choice-case.svg}}
- \caption{Choice - sumif}
- \label{img:choice}
+ \centerline{\includegraphics{counter.svg}}
+ \caption{Counter netlist}
+ \label{img:counter}
+ \vspace{-2em}
\end{figure}
A user-friendly and also very powerful form of choice that is not found in
clause if the guard evaluates to false. Like \hs{if-then-else}
expressions, pattern matching and guards have a (straightforward)
translation to \hs{case} expressions 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 guard is the expression that
- follows the vertical bar (\hs{|}) and precedes the assignment operator
- (\hs{=}). The \hs{otherwise} guards always evaluate to \hs{true}.
+ multiplexers. A second version (\ref{code:counter2}) of the earlier
+ example, now using both pattern matching and guards, can be seen below.
+ The guard is the expression that follows the vertical bar (\hs{|}) and
+ precedes the assignment operator (\hs{=}). The \hs{otherwise} guards
+ always evaluate to \hs{true}.
- The version using pattern matching and guards corresponds to the same
- naive netlist representation (\Cref{img:choice}) as the earlier two
- versions of the example.
+ The second version corresponds to the same naive netlist representation
+ (\Cref{img:counter}) as the earlier example.
+ \hspace{-1.7em}
+ \begin{minipage}{0.93\linewidth}
\begin{code}
- sumif Equal a b | a == b = a + b
- | otherwise = 0
- sumif NotEqual a b | a != b = a + b
- | otherwise = 0
+ counter bound Up x | x < bound = x + 1
+ | otherwise = 0
+
+ counter bound Down x | x > 0 = x - 1
+ | otherwise = bound
\end{code}
+ \end{minipage}
+ \begin{minipage}{0.07\linewidth}
+ \begin{example}
+ \label{code:counter2}
+ \end{example}
+ \end{minipage}
% \begin{figure}
% \centerline{\includegraphics{choice-ifthenelse}}
\subsection{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, this section will
- therefore 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 fixed translation is defined within the \CLaSH\
- compiler. The \CLaSH\ compiler has generic translation rules to
- translated the user-defined types described below.
-
- The \CLaSH\ compiler is able to infer unspecified types,
+ variable or function is determined at compile-time. Not all of
+ Haskell's typing constructs have a clear translation to hardware, this
+ section therefor only deals 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 fixed translation is defined within the
+ \CLaSH\ compiler. The \CLaSH\ compiler has generic translation rules to
+ translate the user-defined types, which are described later on.
+
+ The \CLaSH\ compiler is able to infer unspecified (polymorphic) types,
meaning that a developer does not have to annotate every function with a
- type signature (even if it is good practice to do so).
+ type signature. Given that the top-level entity of a circuit design is
+ annotated with specific types, the \CLaSH\ compiler can specialize
+ polymorphic functions to functions with specific types.
% Translation of two most basic functional concepts has been
% discussed: function application and choice. Before looking further
% 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} expression, 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. Both are parametrizable in their
- size.
+ \hs{if-then-else} expression.
+ \item[\bf{Signed}, \bf{Unsigned}]
+ these are types to represent integers, and both are parametrizable in
+ their size. The overflow behavior of the numeric operators defined for
+ these types is \emph{wrap-around}.
% , so you can define an unsigned word of 32 bits wide as follows:
% \begin{code}
% 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 short-hand notation used for the vector type in
- the rest of paper is: \hs{[a|n]}, here \hs{a} is the element
- type, and \hs{n} is the length of the vector. Note that this is
- a notation used in this paper only, vectors are slightly more
- verbose in real \CLaSH\ descriptions.
+ this type can contain elements of any type and has a static length.
+ The \hs{Vector} type constructor takes two arguments: the length of
+ the vector and the type of the elements contained in it. The
+ short-hand notation used for the vector type in the rest of paper is:
+ \hs{[a|n]}, where \hs{a} is the element type, and \hs{n} is the length
+ of the vector.
+ % Note that this is a notation used in this paper only, vectors are
+ % slightly more verbose in real \CLaSH\ descriptions.
% The state type of an 8 element register bank would then for example
% be:
% \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.
- 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}.
+ the main purpose of the \hs{Index} type is to be used as an index into
+ a \hs{Vector}, and has an integer range from zero to a specified upper
+ bound.
+ % This means that its range is not limited to powers of two, but
+ % can be any number.
+ If a value of this type exceeds either bounds, an error will be thrown
+ during simulation.
% \comment{TODO: Perhaps remove this example?} To define an index for
% the 8 element vector above, we would do:
\end{xlist}
\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 datatype renaming constructs with the \hs{newtype} keyword.
- \GHC\ offers a few more advanced ways to introduce types (type families,
- existential typing, {\acro{GADT}}s, etc.) which are not standard Haskell.
- As it is currently unclear how these advanced type constructs correspond
- to hardware, they are for now unsupported by the \CLaSH\ compiler.
-
- Only an algebraic datatype declaration actually introduces a
- completely new type. Type synonyms and type renaming only define new
- names for existing types, where synonyms are completely interchangeable
- and type renaming requires 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.
+ % 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 datatype renaming constructs with the \hs{newtype} keyword.
+ % \GHC\ offers a few more advanced ways to introduce types (type families,
+ % existential typing, {\acro{GADT}}s, etc.) which are not standard
+ % Haskell. As it is currently unclear how these advanced type constructs
+ % correspond to hardware, they are for now unsupported by the \CLaSH\
+ % compiler.
+ A designer may define a completely new type by an algebraic datatype
+ declaration using the \hs{data} keyword. Type synonyms can be introduced
+ using the \hs{type} keyword.
+ % Only an algebraic datatype declaration actually introduces a
+ % completely new type. Type synonyms and type renaming only define new
+ % names for existing types, where synonyms are completely interchangeable
+ % and a type renaming requires an explicit conversion.
+ Type synonyms do not need any particular translation, as a synonym will
+ use the same representation as the original type.
- For algebraic types, we can make the following distinctions:
+ Algebraic datatypes can be categorized as follows:
\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. Haskell's built-in tuple types are also defined
- as single constructor algebraic types (but with a bit of
- syntactic sugar). An example of a single constructor type is the
- following pair of integers:
+ datatypes with a single constructor with one or more fields allow
+ values to be packed together in a record-like structure. Haskell's
+ built-in tuple types are also defined as single constructor algebraic
+ types (using a bit of syntactic sugar). An example of a single
+ constructor type with multiple fields 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 that there is a fixed translation
- for that type within the \CLaSH\ compiler. An example of such an
- enumeration type is the type that represents the colors in a traffic
- light:
+ \item[\bf{Multiple constructors, No fields}]
+ datatypes with multiple constructors, but without any
+ fields are essentially enumeration types.
+ % Note that Haskell's \hs{Bool} type is also defined as an enumeration
+ % type, but that there is a fixed translation for that type within the
+ % \CLaSH\ compiler.
+ An example of an enumeration type definition is:
\begin{code}
data TrafficLight = Red | Orange | Green
\end{code}
% constructors to be translated to the corresponding enumeration
% value.
\item[\bf{Multiple constructors with fields}]
- Algebraic datatypes with multiple constructors, where at least
+ datatypes with multiple constructors, where at least
one of these constructors has one or more fields are currently not
- supported.
+ supported. Additional research is required to optimize the overlap of
+ fields belonging to the different constructors.
\end{xlist}
\subsection{Polymorphism}\label{sec:polymorhpism}
- A powerful feature of most (functional) programming languages is
- polymorphism, it allows a function to handle values of different data
- types in a uniform way. Haskell supports \emph{parametric
- polymorphism}~\cite{polymorphism}, meaning functions can be written
- without mention of any specific type and can be used transparently with
- any number of new types.
+ A powerful feature of some programming languages is polymorphism, it
+ allows a function to handle values of different data types in a uniform
+ way. Haskell supports \emph{parametric polymorphism}, meaning that
+ functions can be written without mentioning specific types, and they can
+ be used for arbitrary types.
As an example of a parametric polymorphic function, consider the type of
- the following \hs{append} function, which appends an element to a
- vector:\footnote{The \hs{::} operator is used to annotate a function
+ the following \hs{first} function, which returns the first element of a
+ tuple:\footnote{The \hs{::} operator is used to annotate a function
with its type.}
\begin{code}
- append :: [a|n] -> a -> [a|n + 1]
+ first :: (a,b) -> a
\end{code}
- This type is parameterized by \hs{a}, which can contain any type at
- all. This means that \hs{append} can append an element to a vector,
- regardless of the type of the elements in the list (as long as the type of
- the value to be added is of the same type as the values in the vector).
- 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.
-
- Another type of polymorphism is \emph{ad-hoc
- polymorphism}~\cite{polymorphism}, which refers to polymorphic
- functions which can be applied to arguments of different types, but which
- behave differently depending on the type of the argument to which they are
- applied. In Haskell, ad-hoc polymorphism is achieved through the use of
- type classes, where a class definition provides the general interface of a
- function, and class instances define the functionality for the specific
- types. An example of such a type class is the \hs{Num} class, which
- contains all of Haskell's numerical operations. A designer can make use
- of this ad-hoc polymorphism by adding a constraint to a parametrically
- polymorphic type variable. Such a constraint indicates that the type
- variable can only be instantiated to a type whose members supports the
- overloaded functions associated with the type class.
+ This type is parameterized in \hs{a} and \hs{b}, which can both
+ represent any type at all, as long as that type is supported by the
+ \CLaSH\ compiler. This means that \hs{first} works for any tuple,
+ regardless of what elements it contains. This kind of polymorphism is
+ extremely useful in hardware designs, for example when routing signals
+ without knowing their exact type, or specifying vector operations that
+ work on vectors of any length and element type. Polymorphism also plays an
+ important role in most higher order functions, as will be shown in the
+ next section.
+
+ % Another type of polymorphism is \emph{ad-hoc
+ % polymorphism}~\cite{polymorphism}, which refers to polymorphic
+ % functions which can be applied to arguments of different types, but
+ % which behave differently depending on the type of the argument to which
+ % they are applied. In Haskell, ad-hoc polymorphism is achieved through
+ % the use of \emph{type classes}, where a class definition provides the
+ % general interface of a function, and class \emph{instances} define the
+ % functionality for the specific types. An example of such a type class is
+ % the \hs{Num} class, which contains all of Haskell's numerical
+ % operations. A designer can make use of this ad-hoc polymorphism by
+ % adding a \emph{constraint} to a parametrically polymorphic type
+ % variable. Such a constraint indicates that the type variable can only be
+ % instantiated to a type whose members supports the overloaded functions
+ % associated with the type class.
+
+ Another type of polymorphism is \emph{ad-hoc polymorphism}, which refers
+ to function that can be applied to arguments of a limited set to types.
+ Furthermore, how such functions work may depend on the type of their
+ arguments. For example, addition only works for numeric types, and it
+ works differently for e.g. integers and complex numbers.
+
+ In Haskell, ad-hoc polymorphism is achieved through the use of \emph{type
+ classes}, where a class definition provides the general interface of a
+ function, and class \emph{instances} define the functionality for the
+ specific types. For example, all numeric operators are gathered in the
+ \hs{Num} class, so every type that wants to use those operators must be
+ made an instance of \hs{Num}.
+
+ By prefixing a type signature with class constraints, the constrained type
+ parameters are forced to belong to that type class. For example, the
+ arguments of the \hs{add} function must belong to the \hs{Num} type class
+ because the \hs{add} function adds them with the (+) operator:
- As an example we will take a look at type signature of the function
- \hs{sum}, which sums the values in a vector:
\begin{code}
- sum :: Num a => [a|n] -> a
+ add :: Num a => a -> a -> a
+ add a b = a + b
\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}, so that
- we know that the addition (+) operator is defined for that type.
- \CLaSH's built-in numerical types are also instances of the \hs{Num}
- class, so we can use the addition operator (and thus the \hs{sum}
- function) with \hs{SizedWords} as well as with \hs{SizedInts}.
-
- In \CLaSH, parametric 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 however one constraint: the top level
- function that is being translated can not have any polymorphic arguments.
- The arguments can not be polymorphic as the function is never applied and
- consequently there is no way to determine the actual types for the type
- parameters.
-
- \CLaSH\ does \emph{currently} not support\emph{ user-defined} type
- classes, but does use some of the standard Haskell type classes for its
- built-in function, such as: \hs{Num} for numerical operations, \hs{Eq} for
- the equality operators, and \hs{Ord} for the comparison/order operators.
+
+ % An example of a type signature that includes such a constraint if the
+ % signature of the \hs{sum} function, which sums the values in a vector:
+ % \begin{code}
+ % sum :: Num a => [a|n] -> 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}, so
+ % that the compiler knows that the addition (+) operator is defined for
+ % that type.
+
+ % A place where class constraints also play a role is in the size and
+ % range parameters of the \hs{Vector} and numeric types. The reason being
+ % that these parameters have to be limited to types that can represent
+ % \emph{natural} numbers. The complete type of for example the \hs{Vector}
+ % type is:
+ % \begin{code}
+ % Natural n => Vector n a
+ % \end{code}
+
+ % \CLaSH's built-in numerical types are also instances of the \hs{Num}
+ % class.
+ % so we can use the addition operator (and thus the \hs{sum}
+ % function) with \hs{Signed} as well as with \hs{Unsigned}.
+
+ \CLaSH\ supports both parametric polymorphism and ad-hoc polymorphism. A
+ circuit designer can specify his own type classes and corresponding
+ instances. The \CLaSH\ compiler will infer the type of every polymorphic
+ argument depending on how the function is applied. There is however one
+ constraint: the top level function that is being translated can not have
+ polymorphic arguments. The arguments of the top-level can not be
+ polymorphic as there is no way to infer the \emph{specific} types of the
+ arguments.
+
+ With regard to the built-in types, it should be noted that members of
+ some of the standard Haskell type classes are supported as built-in
+ functions. These include: the numerial operators of \hs{Num}, the equality
+ operators of \hs{Eq}, and the comparison (order) operators of \hs{Ord}.
\subsection{Higher-order functions \& values}
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. The following example should clarify this concept:
+ the concept of \emph{functions as a first class value} and
+ \emph{higher-order functions}. These concepts allows a function to be
+ treated as a value and be passed around, even as the argument of another
+ function. The following example clarifies this concept:
-%format not = "\mathit{not}"
+ \hspace{-1.7em}
+ \begin{minipage}{0.93\linewidth}
+ %format not = "\mathit{not}"
\begin{code}
- negateVector xs = map not xs
+ negate{-"\!\!\!"-}Vector xs = map not xs
\end{code}
-
- The code above defines the \hs{negateVector} function, which takes a
- vector of booleans, \hs{xs}, and returns a vector where all the values are
- negated. It achieves this by calling the \hs{map} function, and passing it
- \emph{another function}, boolean negation, and the vector of booleans,
- \hs{xs}. The \hs{map} function applies the negation function to all the
- elements in the vector.
+ \end{minipage}
+ \begin{minipage}{0.07\linewidth}
+ \begin{example}
+ \label{code:negatevector}
+ \end{example}
+ \end{minipage}
+
+ The code above defines the \hs{negate{-"\!\!\!"-}Vector} function, which
+ takes a vector of booleans, \hs{xs}, and returns a vector where all the
+ values are negated. It achieves this by calling the \hs{map} function, and
+ passing it another \emph{function}, boolean negation, and the vector of
+ booleans, \hs{xs}. The \hs{map} function applies the negation function to
+ all the elements in the vector.
The \hs{map} function is called a higher-order function, since it takes
another function as an argument. Also note that \hs{map} is again a
type as the first argument of the function passed to \hs{map}. The element
type of the resulting vector is equal to the return type of the function
passed, which need not necessarily be the same as the element type of the
- input vector. All of these characteristics can readily be inferred from
- the type signature belonging to \hs{map}:
+ input vector. All of these characteristics can be inferred from the type
+ signature belonging to \hs{map}:
\begin{code}
map :: (a -> b) -> [a|n] -> [b|n]
\end{code}
- 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:
+ 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 takes one argument
+ less. As an example, consider the following expression, that adds one to
+ every element of a vector:
+ \hspace{-1.7em}
+ \begin{minipage}{0.93\linewidth}
\begin{code}
map (add 1) xs
\end{code}
+ \end{minipage}
+ \begin{minipage}{0.07\linewidth}
+ \begin{example}
+ \label{code:partialapplication}
+ \end{example}
+ \end{minipage}
Here, the expression \hs{(add 1)} is the partial application of the
addition function to the value \hs{1}, which is again a function that
- adds one to its (next) argument. A lambda expression allows one to
- introduce an anonymous function in any expression. Consider the following
- expression, which again adds one to every element of a vector:
+ adds 1 to its (next) argument.
+
+ A lambda expression allows a designer to introduce an anonymous function
+ in any expression. Consider the following expression, which again adds 1
+ to every element of a vector:
+ \hspace{-1.7em}
+ \begin{minipage}{0.93\linewidth}
\begin{code}
map (\x -> x + 1) xs
\end{code}
-
- Finally, not only built-in functions can have higher order
- arguments, 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. The only exception is again the top-level function: if a
- function-typed argument is not applied with an actual function, no
- hardware can be generated.
-
- % \comment{TODO: Describe ALU example (no code)}
+ \end{minipage}
+ \begin{minipage}{0.07\linewidth}
+ \begin{example}
+ \label{code:lambdaexpression}
+ \end{example}
+ \end{minipage}
+
+ Finally, not only built-in functions can have higher order arguments (such
+ as the \hs{map} function), but any function defined in \CLaSH\ may have
+ functions as arguments. This allows the circuit designer to apply a
+ large amount of code reuse. The only exception is again the top-level
+ function: if a function-typed argument is not instantiated with an actual
+ function, no hardware can be generated.
+
+ An example of a common circuit where higher-order functions and partial
+ application lead to a very concise and natural description is a crossbar.
+ The code (\ref{code:crossbar}) for this example can be seen below:
+
+ \hspace{-1.7em}
+ \begin{minipage}{0.93\linewidth}
+ \begin{code}
+ crossbar inputs selects = map (mux inputs) selects
+ where
+ mux inp x = (inp ! x)
+ \end{code}
+ \end{minipage}
+ \begin{minipage}{0.07\linewidth}
+ \begin{example}
+ \label{code:crossbar}
+ \end{example}
+ \end{minipage}
+
+ The the \hs{crossbar} function selects those values from \hs{inputs} that
+ are indicated by the indexes in the vector \hs{selects}. The crossbar is
+ polymorphic in the width of the input (defined by the length of
+ \hs{inputs}), the width of the output (defined by the length of
+ \hs{selects}), and the signal type (defined by the element type of
+ \hs{inputs}). The type-checker can also automatically infer that
+ \hs{selects} is a vector of \hs{Index} values due to the use of the vector
+ indexing operator (\hs{!}).
\subsection{State}
- A very important concept in hardware is the concept of state. In a
- stateful design, the outputs depend on the history of the inputs, or the
- state. State is usually stored in registers, which retain their value
- during a clock cycle. As we want to describe more than simple
- combinational designs, \CLaSH\ needs an abstraction mechanism for state.
+ In a stateful design, the outputs depend on the history of the inputs, or
+ the state. State is usually stored in registers, which retain their value
+ during a clock cycle. As \CLaSH\ has to be able to describe more than
+ plain combinational designs, there is a need for an abstraction mechanism
+ for state.
- An important property in Haskell, and in most other functional languages,
+ An important property in Haskell, and in many other functional languages,
is \emph{purity}. A function is said to be \emph{pure} if it satisfies two
conditions:
\begin{inparaenum}
% correct for impure functions.
Pure functions are as such a perfect match for combinational circuits,
where the output solely depends on the inputs. When a circuit has state
- however, it can no longer be simply described by a pure function.
+ however, it can no longer be described by a pure function.
% Simply removing the purity property is not a valid option, as the
% language would then lose many of it mathematical properties.
- In \CLaSH\ we deal with the concept of state in pure functions by making
- current value of the state an additional argument of the function and the
- updated state part of result. In this sense the descriptions made in
- \CLaSH\ are the combinational parts of a mealy machine.
+ \CLaSH\ deals with the concept of state by making the current state an
+ additional argument of the function, and the updated state part of the
+ result. In this sense the descriptions made in \CLaSH\ are the
+ combinational parts of a mealy machine.
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}:
+ \hspace{-1.7em}
+ \begin{minipage}{0.93\linewidth}
\begin{code}
macS (State c) a b = (State c', c')
where
c' = mac a b c
\end{code}
+ \end{minipage}
+ \begin{minipage}{0.07\linewidth}
+ \begin{example}
+ \label{code:macstate}
+ \end{example}
+ \end{minipage}
- \begin{figure}
- \centerline{\includegraphics{mac-state.svg}}
- \caption{Stateful Multiply-Accumulate}
- \label{img:mac-state}
- \end{figure}
-
- Note that the \hs{macS} function returns bot the new state and the value
- of the output port. The \hs{State} keyword indicates which arguments are
+ Note that the \hs{macS} function returns both the new state and the value
+ of the output port. The \hs{State} wrapper indicates which arguments are
part of the current state, and what part of the output is part of the
updated state. This aspect will also be reflected in the type signature of
the function. Abstracting the state of a circuit in this way makes it 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 elements, as state values are just normal values. We can simulate
- stateful descriptions using the recursive \hs{run} function:
+ choice elements, as state values are just normal values from Haskell's
+ point of view. Stateful descriptions are simulated using the recursive
+ \hs{run} function:
+ \hspace{-1.7em}
+ \begin{minipage}{0.93\linewidth}
\begin{code}
run f s (i : inps) = o : (run f s' inps)
where
(s', o) = f s i
\end{code}
+ \end{minipage}
+ \begin{minipage}{0.07\linewidth}
+ \begin{example}
+ \label{code:run}
+ \end{example}
+ \end{minipage}
The \hs{(:)} operator is the list concatenation operator, where the
left-hand side is the head of a list and the right-hand side is the
and the updated state \hs{s'}. The next iteration of the \hs{run} function
is then called with the updated state, \hs{s'}, and the rest of the
inputs, \hs{inps}. For the time being, and in the context of this paper,
- It is assumed that there is one input per clock cycle.
- Also note how the order of the input, output, and state in the \hs{run}
- function corresponds with the order of the input, output and state of the
- \hs{macS} function described earlier.
+ it is assumed that there is one input per clock cycle. Note that the order
+ of \hs{s',o,s,i} in the where clause of the \hs{run} functions corresponds
+ with the order of the input, output and state of the \hs{macS} function
+ described earlier. Thus, in Haskell the expression \hs{run macS 0 inputs}
+ simulates \hs{macS} on \hs{inputs} starting with the value \hs{0}
+
+ \begin{figure}
+ \centerline{\includegraphics{mac-state.svg}}
+ \caption{Stateful Multiply-Accumulate}
+ \label{img:mac-state}
+ \vspace{-1.5em}
+ \end{figure}
- As the \hs{run} function, the hardware description, and the test
- inputs are also valid Haskell, the complete simulation can be compiled to
- an executable binary by an optimizing Haskell compiler, or executed in an
- Haskell interpreter. Both simulation paths are much faster than first
+ The complete simulation can be compiled to an executable binary by a
+ Haskell compiler, or executed in an Haskell interpreter. Both
+ simulation paths require less effort from a circuit designer than first
translating the description to \VHDL\ and then running a \VHDL\
- simulation.
+ simulation; it is also very likely that both simulation paths are much
+ faster.
\section{The \CLaSH\ compiler}
An important aspect in this research is the creation of the prototype
compiler, which allows us to translate descriptions made in the \CLaSH\
-language as described in the previous section to synthesizable \VHDL, allowing
-a designer to actually run a \CLaSH\ design on an \acro{FPGA}.
+language as described in the previous section to synthesizable \VHDL.
+% , allowing a designer to actually run a \CLaSH\ design on an \acro{FPGA}.
-The Glasgow Haskell Compiler (\GHC) is an open-source Haskell compiler that
-also provides a high level API to most of its internals. The availability of
-this high-level API obviated the need to design many of the tedious parts of
-the prototype compiler, such as the parser, semantic checker, and especially
-the type-checker. These parts together form the front-end of the prototype compiler pipeline, as seen in \Cref{img:compilerpipeline}.
+The Glasgow Haskell Compiler (\GHC)~\cite{ghc} is an open-source Haskell
+compiler that also provides a high level \acro{API} to most of its internals.
+The availability of this high-level \acro{API} obviated the need to design
+many of the tedious parts of the prototype compiler, such as the parser,
+semantics checker, and especially the type-checker. These parts together form
+the front-end of the prototype compiler pipeline, as seen in
+\Cref{img:compilerpipeline}.
\begin{figure}
+\vspace{1em}
\centerline{\includegraphics{compilerpipeline.svg}}
\caption{\CLaSHtiny\ compiler pipeline}
\label{img:compilerpipeline}
+\vspace{-1.5em}
\end{figure}
-The output of the \GHC\ front-end consists of the translation of the original Haskell description in \emph{Core}~\cite{Sulzmann2007}, which is a smaller, typed, functional language. This \emph{Core} language is relatively easy to process compared to the larger Haskell language. A description in \emph{Core} can still contain elements which have no direct translation to hardware, such as polymorphic types and function-valued arguments. Such a description needs to be transformed to a \emph{normal form}, which only contains elements that have a direct translation. The second stage of the compiler, the \emph{normalization} phase, exhaustively applies a set of \emph{meaning-preserving} transformations on the \emph{Core} description until this description is in a \emph{normal form}. This set of transformations includes transformations typically found in reduction systems and lambda calculus~\cite{lambdacalculus}, such as $\beta$-reduction and $\eta$-expansion. It also includes self-defined transformations that are responsible for the reduction of higher-order functions to `regular' first-order functions.
+The output of the \GHC\ front-end consists of the translation of the original
+Haskell description to \emph{Core}~\cite{Sulzmann2007}, which is a small
+typed functional language. This \emph{Core} language is relatively easy to
+process compared to the larger Haskell language. A description in \emph{Core}
+can still contain elements which have no direct translation to hardware, such
+as polymorphic types and function-valued arguments. Such a description needs
+to be transformed to a \emph{normal form}, which only contains elements that
+have a direct translation. The second stage of the compiler, the
+\emph{normalization} phase, exhaustively applies a set of
+\emph{meaning-preserving} transformations on the \emph{Core} description until
+this description is in a \emph{normal form}. This set of transformations
+includes transformations typically found in reduction systems and lambda
+calculus~\cite{lambdacalculus}, such as $\beta$-reduction and
+$\eta$-expansion. It also includes self-defined transformations that are
+responsible for the reduction of higher-order functions to `regular'
+first-order functions, and specializing polymorphic types to concrete types.
The final step in the compiler pipeline is the translation to a \VHDL\
-\emph{netlist}, which is a straightforward process due to resemblance of a
-normalized description and a set of concurrent signal assignments. We call the
-end-product of the \CLaSH\ compiler a \VHDL\ \emph{netlist} as the resulting
-\VHDL\ resembles an actual netlist description and not idiomatic \VHDL.
+\emph{netlist}, which is a straightforward process due to the resemblance of a
+normalized description and a set of concurrent signal assignments. The
+end-product of the \CLaSH\ compiler is called a \VHDL\ \emph{netlist} as the
+result resembles an actual netlist description, and the fact that it is \VHDL\
+is only an implementation detail; e.g., the output could have been Verilog.
\section{Use cases}
\label{sec:usecases}
\subsection{FIR Filter}
-As an example of a common hardware design where the use of higher-order
-functions leads to a very natural description is a \acro{FIR} filter, which is
-basically the dot-product of two vectors:
+As an example of a common hardware design where the relation between
+functional languages and mathematical functions, combined with the use of
+higher-order functions leads to a very natural description is a \acro{FIR}
+filter:
\begin{equation}
y_t = \sum\nolimits_{i = 0}^{n - 1} {x_{t - i} \cdot h_i }
A \acro{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 $t$. The equation of a \acro{FIR}
-filter is indeed equivalent to the equation of the dot-product, which is
-shown below:
+filter is equivalent to the equation of the dot-product of two vectors, which
+is shown below:
\begin{equation}
\mathbf{a}\bullet\mathbf{b} = \sum\nolimits_{i = 0}^{n - 1} {a_i \cdot b_i }
\end{equation}
-We can easily and directly implement the equation for the dot-product
-using higher-order functions:
+The equation for the dot-product is easily and directly implemented using
+higher-order functions:
+\hspace{-1.7em}
+\begin{minipage}{0.93\linewidth}
\begin{code}
-as *+* bs = foldl1 (+) (zipWith (*) as bs)
+as *+* bs = fold (+) (zipWith (*) as bs)
\end{code}
+\end{minipage}
+\begin{minipage}{0.07\linewidth}
+ \begin{example}
+ \label{code:dotproduct}
+ \end{example}
+\end{minipage}
The \hs{zipWith} function is very similar to the \hs{map} function seen
earlier: It takes a function, two vectors, and then applies the function to
each of the elements in the two vectors pairwise (\emph{e.g.}, \hs{zipWith (*)
[1, 2] [3, 4]} becomes \hs{[1 * 3, 2 * 4]}).
-The \hs{foldl1} function takes a binary function, a single vector, and applies
+The \hs{fold} function takes a binary function, a single vector, and applies
the function to the first two elements of the vector. It then applies the
function to the result of the first application and the next element in the
vector. This continues until the end of the vector is reached. The result of
-the \hs{foldl1} function is the result of the last application. It is obvious
+the \hs{fold} function is the result of the last application. It is obvious
that the \hs{zipWith (*)} function is pairwise multiplication and that the
-\hs{foldl1 (+)} function is summation.
+\hs{fold (+)} function is summation.
% Returning to the actual \acro{FIR} filter, we will slightly change the
% equation describing it, so as to make the translation to code more obvious and
% concise. What we do is change the definition of the vector of input samples
% \begin{equation}
% y_t = \sum\nolimits_{i = 0}^{n - 1} {x_i \cdot h_i }
% \end{equation}
-The complete definition of the \acro{FIR} filter in code then becomes:
+The complete definition of the \acro{FIR} filter in \CLaSH\ is:
+\hspace{-1.7em}
+\begin{minipage}{0.93\linewidth}
\begin{code}
fir (State (xs,hs)) x =
- (State (x >> xs,hs), (x +> xs) *+* hs)
+ (State (shiftInto x xs,hs), (x +> xs) *+* hs)
\end{code}
-
-Where the vector \hs{xs} contains the previous input samples, the vector \hs{hs} contains the \acro{FIR} coefficients, and \hs{x} is the current input sample. The concatenate operator (\hs{+>}) creates a new vector by placing the current sample (\hs{x}) in front of the previous samples vector (\hs{xs}). The code for the shift (\hs{>>}) operator, that adds the new input sample (\hs{x}) to the list of previous input samples (\hs{xs}) and removes the oldest sample, is shown below:
-
+\end{minipage}
+\begin{minipage}{0.07\linewidth}
+ \begin{example}
+ \label{code:fir}
+ \end{example}
+\end{minipage}
+
+where the vector \hs{xs} contains the previous input samples, the vector
+\hs{hs} contains the \acro{FIR} coefficients, and \hs{x} is the current input
+sample. The concatenate operator (\hs{+>}) creates a new vector by placing the
+current sample (\hs{x}) in front of the previous samples vector (\hs{xs}). The
+code for the \hs{shiftInto} function, that adds the new input sample (\hs{x})
+to the list of previous input samples (\hs{xs}) and removes the oldest sample,
+is shown below:
+
+\hspace{-1.7em}
+\begin{minipage}{0.93\linewidth}
\begin{code}
-x >> xs = x +> init xs
+shiftInto x xs = x +> init xs
\end{code}
-
-Where the \hs{init} function returns all but the last element of a vector.
+\end{minipage}
+\begin{minipage}{0.07\linewidth}
+ \begin{example}
+ \label{code:shiftinto}
+ \end{example}
+\end{minipage}
+
+where the \hs{init} function returns all but the last element of a vector.
The resulting netlist of a 4-taps \acro{FIR} filter, created by specializing
the vectors of the \acro{FIR} code to a length of 4, is depicted in
\Cref{img:4tapfir}.
\centerline{\includegraphics{4tapfir.svg}}
\caption{4-taps \acrotiny{FIR} Filter}
\label{img:4tapfir}
+\vspace{-1.5em}
\end{figure}
\subsection{Higher-order CPU}
-The following simple CPU is an example of user-defined higher order
-functions and pattern matching. The CPU consists of four function units,
-of which three have a fixed function and one can perform some less
-common operations.
-
-The CPU contains a number of data sources, represented by the horizontal
-lines in figure TODO:REF. These data sources offer the previous outputs
-of each function units, along with the single data input the cpu has and
-two fixed intialization values.
-
-Each of the function units has both its operands connected to all data
-sources, and can be programmed to select any data source for either
-operand. In addition, the leftmost function unit has an additional
-opcode input to select the operation it performs. Its output is also the
-output of the entire cpu.
-
-Looking at the code, the function unit is the most simple. It arranges
-the operand selection for the function unit. Note that it does not
-define the actual operation that takes place inside the function unit,
-but simply accepts the (higher order) argument \hs{op} which is a function
-of two arguments that defines the operation.
+%format fun x = "\textit{fu}_" x
+In this section discusses a somewhat more serious example in which
+user-defined higher-order function, partial application, lambda expressions,
+and pattern matching are exploited. The example concerns a \acro{CPU} which
+consists of four function unites \hs{fun 0,{-"\ldots"-},fun 3} (see
+\Cref{img:highordcpu}) that each perform some binary operation.
+
+\begin{figure}
+\centerline{\includegraphics{highordcpu.svg}}
+\caption{CPU with higher-order Function Units}
+\label{img:highordcpu}
+\vspace{-1.5em}
+\end{figure}
+
+Every function unit has seven data inputs (of type \hs{Word}), and two address
+inputs (of type \hs{Index 6}) which indicate which data inputs have to be
+chosen as arguments for the the binary operation that the unit performs. These
+data inputs consists of one external input \hs{x}, two fixed initialization
+values (0 and 1), and the previous outputs of the four function units. The
+output of the \acro{CPU} as a whole is the previous output of \hs{fun 3}.
+
+The function units \hs{fun 1, fun 2, fun 3} can perform a fixed binary
+operation, whereas \hs{fun 0} has an additional input for an opcode to choose
+a binary operation out of a few possibilities.
+
+Each function unit outputs its result into a register, i.e., the state of the
+\acro{CPU}. This can can e.g. be defined as follows:
+
+\begin{code}
+type CpuState = State [Word | 4]
+\end{code}
+
+Every function unit can now be defined by the following higher-order function
+\hs{fu}, which takes three arguments: the operation \hs{op} that the function
+unit performs, the seven \hs{inputs}, and the pair \hs{(a1,a2)} of two
+addresses:
+\hspace{-1.7em}
+\begin{minipage}{0.93\linewidth}
\begin{code}
-fu op inputs (addr1, addr2) = regIn
+fu op inputs (a1, a2) = regIn
where
- in1 = inputs!addr1
- in2 = inputs!addr2
- regIn = op in1 in2
+ arg1 = inputs!a1
+ arg2 = inputs!a2
+ regIn = op arg1 arg2
\end{code}
+\end{minipage}
+\begin{minipage}{0.07\linewidth}
+ \begin{example}
+ \label{code:functionunit}
+ \end{example}
+\end{minipage}
-The multiop function defines the operation that takes place in the
-leftmost function unit. It is essentially a simple three operation alu
-that makes good use of pattern matching and guards in its description.
-The \hs{shift} function used here shifts its first operand by the number
-of bits indicated in the second operand, the \hs{xor} function produces
-the bitwise xor of its operands.
+Using partial application we now define:
+\hspace{-1.7em}
+\begin{minipage}{0.93\linewidth}
+\begin{code}
+fun 1 = fu add
+fun 2 = fu sub
+fun 3 = fu mul
+\end{code}
+\end{minipage}
+\begin{minipage}{0.07\linewidth}
+ \begin{example}
+ \label{code:functionunits1to3}
+ \end{example}
+\end{minipage}
+
+In order to define \hs{fun 0} we first define the type \hs{Opcode} for the
+opcode and the function \hs{multiop} that chooses a specific operation given
+the opcode. We assume that the functions \hs{shifts} (which shifts its first
+operand by the number of bits indicate in the second operand), \hs{xor} (for
+the bitwise \hs{xor}), and (==) (for equality) already exits.
+
+\hspace{-1.7em}
+\begin{minipage}{0.93\linewidth}
\begin{code}
data Opcode = Shift | Xor | Equal
-multiop :: Opcode -> Word -> Word -> Word
-multiop opc a b = case opc of
- Shift -> shift a b
- Xor -> xor a b
- Equal | a == b -> 1
- | otherwise -> 0
+multiop Shift = shift
+multiop Xor = xor
+multiop Equal = \a b -> if a == b then 1 else 0
\end{code}
+\end{minipage}
+\begin{minipage}{0.07\linewidth}
+ \begin{example}
+ \label{code:multiop}
+ \end{example}
+\end{minipage}
+
+Note that the result of \hs{multiop} is a binary function; this is supported
+by \CLaSH. We can now define \hs{fun 0} as a function which takes an opcode as
+additional argument:
+
+\hspace{-1.7em}
+\begin{minipage}{0.93\linewidth}
+\begin{code}
+fun 0 c = fu (multiop c)
+\end{code}
+\end{minipage}
+\begin{minipage}{0.07\linewidth}
+ \begin{example}
+ \label{code:functionunit0}
+ \end{example}
+\end{minipage}
-The cpu function ties everything together. It applies the \hs{fu}
-function four times, to create a different function unit each time. The
-first application is interesting, because it does not just pass a
-function to \hs{fu}, but a partial application of \hs{multiop}. This
-shows how the first funcition unit effectively gets an extra input,
-compared to the others.
-
-The vector \hs{inputs} is the set of data sources, which is passed to
-each function unit for operand selection. The cpu also receives a vector
-of address pairs, which are used by each function unit to select their
-operand. The application of the function units to the \hs{inputs} and
-\hs{addrs} arguments seems quite repetive and could be rewritten to use
-a combination of the \hs{map} and \hs{zipwith} functions instead.
-However, the prototype does not currently support working with lists of
-functions, so the more explicit version of the code is given instead).
+Now we come to the definition \hs{cpu} of the full \acro{CPU}. Its type is:
\begin{code}
-type CpuState = State [Word | 4]
+cpu :: CpuState
+ -> (Word, Opcode, [(Index 6, Index 6) | 4])
+ -> (CpuState, Word)
+\end{code}
+
+Note that this type fits the requirements of the function \hs{run}. The
+definition of the \hs{cpu} now is:
-cpu :: CpuState -> Word -> [(Index 6, Index 6) | 4]
- -> Opcode -> (CpuState, Word)
-cpu (State s) input addrs opc = (State s', out)
+\hspace{-1.7em}
+\begin{minipage}{0.93\linewidth}
+\begin{code}
+cpu (State s) (x,opc,addrs) = (State s', out)
where
- s' = [ fu (multiop opc) inputs (addrs!0)
- , fu add inputs (addrs!1)
- , fu sub inputs (addrs!2)
- , fu mul inputs (addrs!3)
- ]
- inputs = 0 +> (1 +> (input +> s))
- out = head s'
+ inputs = x +> (0 +> (1 +> s))
+ s' = [{-"\;"-}fun 0 opc inputs (addrs!0)
+ ,{-"\;"-}fun 1 inputs (addrs!1)
+ ,{-"\;"-}fun 2 inputs (addrs!2)
+ ,{-"\;"-}fun 3 inputs (addrs!3)
+ ]
+ out = last s
\end{code}
-
-Of course, this is still a simple example, but it could form the basis
-of an actual design, in which the same techniques can be reused.
+\end{minipage}
+\begin{minipage}{0.07\linewidth}
+ \begin{example}
+ \label{code:cpu}
+ \end{example}
+\end{minipage}
+
+While this is still a simple (and maybe not very useful) design, it
+illustrates some possibilities that \CLaSH\ offers and suggests how to write
+actual designs.
+
+% Each of the function units has both its operands connected to all data
+% sources, and can be programmed to select any data source for either
+% operand. In addition, the leftmost function unit has an additional
+% opcode input to select the operation it performs. The previous output of the
+% rightmost function unit is the output of the entire \acro{CPU}.
+%
+% The code of the function unit (\ref{code:functionunit}), which arranges the
+% operand selection for the function unit, is shown below. Note that the actual
+% operation that takes place inside the function unit is supplied as the
+% (higher-order) argument \hs{op}, which is a function that takes two arguments.
+%
+%
+%
+% The \hs{multiop} function (\ref{code:multiop}) defines the operation that takes place in the leftmost function unit. It is essentially a simple three operation \acro{ALU} that makes good use of pattern matching and guards in its description. The \hs{shift} function used here shifts its first operand by the number of bits indicated in the second operand, the \hs{xor} function produces
+% the bitwise xor of its operands.
+%
+%
+% The \acro{CPU} function (\ref{code:cpu}) ties everything together. It applies
+% the function unit (\hs{fu}) to several operations, to create a different
+% function unit each time. The first application is interesting, as it does not
+% just pass a function to \hs{fu}, but a partial application of \hs{multiop}.
+% This demonstrates how one function unit can effectively get extra inputs
+% compared to the others.
+%
+% The vector \hs{inputs} is the set of data sources, which is passed to
+% each function unit as a set of possible operants. The \acro{CPU} also receives
+% a vector of address pairs, which are used by each function unit to select
+% their operand.
+% The application of the function units to the \hs{inputs} and
+% \hs{addrs} arguments seems quite repetitive and could be rewritten to use
+% a combination of the \hs{map} and \hs{zipwith} functions instead.
+% However, the prototype compiler does not currently support working with
+% lists of functions, so a more explicit version of the code is given instead.
+
+% While this is still a simple example, it could form the basis of an actual
+% design, in which the same techniques can be reused.
\section{Related work}
This section describes the features of existing (functional) hardware
description languages and highlights the advantages that this research has
over existing work.
-Many functional hardware description languages have been developed over the
-years. Early work includes such languages as $\mu$\acro{FP}~\cite{muFP}, an
-extension of Backus' \acro{FP} language to synchronous streams, designed
-particularly for describing and reasoning about regular circuits. The
-Ruby~\cite{Ruby} language uses relations, instead of functions, to describe
-circuits, and has a particular focus on layout.
+% Many functional hardware description languages have been developed over the
+% years. Early work includes such languages as $\mu$\acro{FP}~\cite{muFP}, an
+% extension of Backus' \acro{FP} language to synchronous streams, designed
+% particularly for describing and reasoning about regular circuits. The
+% Ruby~\cite{Ruby} language uses relations, instead of functions, to describe
+% circuits, and has a particular focus on layout.
\acro{HML}~\cite{HML2} is a hardware modeling language based on the strict
functional language \acro{ML}, and has support for polymorphic types and
-higher-order functions. Published work suggests that there is no direct
-simulation support for \acro{HML}, but that a description in \acro{HML} has to
-be translated to \VHDL\ and that the translated description can then be
-simulated in a \VHDL\ simulator. Also not all of the mentioned language
-features of \acro{HML} could be translated to hardware. The \CLaSH\ compiler
-on the other hand can correctly translate all of the language constructs
-mentioned in this paper to a netlist format.
-
-Like the work presented in this paper, many functional hardware description languages have some sort of foundation in the functional programming language Haskell. Hawk~\cite{Hawk1} uses Haskell to describe system-level executable
-specifications used to model the behavior of superscalar microprocessors. Hawk
-specifications can be simulated; to the best knowledge of the authors there is however no support for automated circuit synthesis.
+higher-order functions. There is no direct simulation support for \acro{HML},
+so a description in \acro{HML} has to be translated to \VHDL\ and the
+translated description can then be simulated in a \VHDL\ simulator. Certain
+aspects of HML, such as higher-order functions are however not supported by
+the \VHDL\ translator~\cite{HML3}. The \CLaSH\ compiler on the other hand can
+correctly translate all of its language constructs.
+
+Like the research presented in this paper, many functional hardware
+description languages have some sort of foundation in the functional
+programming language Haskell. Hawk~\cite{Hawk1} is a hardware modeling
+language embedded in Haskell and has sequential environments that make it
+easier to specify stateful computation (by using the \acro{ST} monad). Hawk
+specifications can be simulated; to the best knowledge of the authors there is
+however no support for automated circuit synthesis.
The ForSyDe~\cite{ForSyDe2} system uses Haskell to specify abstract system
-models, which can (manually) be transformed into an implementation model using
-semantic preserving transformations. A designer can model systems using
-heterogeneous models of computation, which include continuous time,
-synchronous and untimed models of computation. Using so-called domain
-interfaces a designer can simulate electronic systems which have both analog
-as digital parts. ForSyDe has several backends including simulation and
-automated synthesis, though automated synthesis is restricted to the
-synchronous model of computation within ForSyDe. Unlike \CLaSH\ there is no
-support for the automated synthesis of descriptions that contain polymorphism
-or higher-order functions.
-
-Lava~\cite{Lava} is a hardware description language that focuses on the
-structural representation of hardware. Besides support for simulation and
-circuit synthesis, Lava descriptions can be interfaced with formal method
-tools for formal verification. Lava descriptions are actually circuit
-generators when viewed from a synthesis viewpoint, in that the language
-elements of Haskell, such as choice, can be used to guide the circuit
-generation. If a developer wants to insert a choice element inside an actual
-circuit he will have to explicitly instantiate a multiplexer-like component.
-In this respect \CLaSH\ differs from Lava, in that all the choice elements,
-such as case-statements and pattern matching, are synthesized to choice
-elements in the eventual circuit. As such, richer control structures can both
-be specified and synthesized in \CLaSH\ compared to any of the embedded
-languages such as Hawk, ForSyDe and Lava.
-
-The merits of polymorphic typing, combined with higher-order functions, are
-now also recognized in the `main-stream' hardware description languages,
-exemplified by the new \VHDL-2008 standard~\cite{VHDL2008}. \VHDL-2008 support
-for generics has been extended to types and subprograms, allowing a developer
-to describe components with polymorphic ports and function-valued arguments.
-Note that the types and subprograms still require an explicit generic map,
-whereas types can be automatically inferred, and function-values can be
-automatically propagated by the \CLaSH\ compiler. There are also no (generally
-available) \VHDL\ synthesis tools that currently support the \VHDL-2008
-standard, and thus the synthesis of polymorphic types and function-valued
-arguments.
+models. A designer can model systems using heterogeneous models of
+computation, which include continuous time, synchronous and untimed models of
+computation. Using so-called domain interfaces a designer can simulate
+electronic systems which have both analog and digital parts. ForSyDe has
+several backends including simulation and automated synthesis, though
+automated synthesis is restricted to the synchronous model of computation.
+Though ForSyDe offers higher-order functions and polymorphism, ForSyDe's
+choice elements are limited to \hs{if} and \hs{case} expressions. ForSyDe's
+explicit conversions, where functions have to be wrapped in processes and
+processes have to be wrapped in systems, combined with the explicit
+instantiations of components, also makes ForSyDe far more verbose than \CLaSH.
+
+Lava~\cite{Lava,kansaslava} is a hardware description language embedded in
+Haskell which focuses on the structural representation of hardware. Like
+\CLaSH, Lava has support for polymorphic types and higher-order functions.
+Besides support for simulation and circuit synthesis, Lava descriptions can be
+interfaced with formal method tools for formal verification. As discussed in
+the introduction, taking the embedded language approach does not allow for
+Haskell's choice elements to be captured within the circuit descriptions. In
+this respect \CLaSH\ differs from Lava, in that all of Haskell's choice
+elements, such as \hs{case}-expressions and pattern matching, are synthesized
+to choice elements in the eventual circuit. Consequently, descriptions
+containing rich control structures can be specified in a more user-friendly
+way in \CLaSH\ than possible within Lava, and hence are less error-prone.
+
+Bluespec~\cite{Bluespec} is a high-level synthesis language that features
+guarded atomic transactions and allows for the automated derivation of control
+structures based on these atomic transactions. Bluespec, like \CLaSH, supports
+polymorphic typing and function-valued arguments. Bluespec's syntax and
+language features \emph{had} their basis in Haskell. However, in order to
+appeal to the users of the traditional \acrop{HDL}, Bluespec has adapted
+imperative features and a syntax that resembles Verilog. As a result, Bluespec
+is (unnecessarily) verbose when compared to \CLaSH.
+
+The merits of polymorphic typing and function-valued arguments are now also
+recognized in the traditional \acrop{HDL}, exemplified by the new \VHDL-2008
+standard~\cite{VHDL2008}. \VHDL-2008 support for generics has been extended to
+types and subprograms, allowing a designer to describe components with
+polymorphic ports and function-valued arguments. Note that the types and
+subprograms still require an explicit generic map, while the \CLaSH\ compiler
+automatically infers types, and automatically propagates function-valued
+arguments. There are also no (generally available) \VHDL\ synthesis tools that
+currently support the \VHDL-2008 standard.
% Wired~\cite{Wired},, T-Ruby~\cite{T-Ruby}, Hydra~\cite{Hydra}.
%
\section{Conclusion}
This research demonstrates once more that functional languages are well suited
for hardware descriptions: function applications provide an elegant notation
-for component instantiation. Where this research goes beyond the existing
-(functional) hardware descriptions languages is the inclusion of various
-choice elements, such as patter matching, that are well suited to describe the
-conditional assignments in control-oriented hardware. Besides being able to
-translate these basic constructs to synthesizable \VHDL, the prototype
-compiler can also correctly translate descriptions that contain both
-polymorphic types and function-valued arguments.
-
-Where recent functional hardware description languages have mostly opted to
-embed themselves in an existing functional language, this research features a
-`true' compiler. As a result there is a clear distinction between compile-time
-and run-time, which allows a myriad of choice constructs to be part of the
-actual circuit description; a feature the embedded hardware description
-languages do not offer.
+for component instantiation. While circuit descriptions made in \CLaSH\ are
+very concise when compared to other (traditional) \acrop{HDL}, their intended
+functionality remains clear. \CLaSH\ goes beyond the existing (functional)
+hardware descriptions languages by including advanced choice elements, such as
+pattern matching and guards, which are well suited to describe the conditional
+assignments in control-oriented circuits. Besides being able to translate
+these basic constructs to synthesizable \VHDL, the prototype compiler can also
+correctly translate descriptions that contain both polymorphic types and
+user-defined higher-order functions.
+
+% Where recent functional hardware description languages have mostly opted to
+% embed themselves in an existing functional language, this research features
+% a `true' compiler. As a result there is a clear distinction between
+% compile-time and run-time, which allows a myriad of choice constructs to be
+% part of the actual circuit description; a feature the embedded hardware
+% description languages do not offer.
+
+Besides simple circuits such as variants of both the \acro{FIR} filter and
+the higher-order \acro{CPU} shown in \Cref{sec:usecases}, the \CLaSH\ compiler
+has also been able to translate non-trivial functional descriptions such as a
+streaming reduction circuit~\cite{reductioncircuit} for floating point
+numbers.
\section{Future Work}
-The choice of describing state explicitly as extra arguments and results can
-be seen as a mixed blessing. Even though the description that use state are
-usually very clear, one finds that dealing with unpacking, passing, receiving
-and repacking can become tedious and even error-prone, especially in the case
-of sub-states. Removing this boilerplate, or finding a more suitable
-abstraction mechanism would make \CLaSH\ easier to use.
-
-The transformations in normalization phase of the prototype compiler were
+The choice of describing state explicitly as and extra argument and result can
+be seen as a mixed blessing. Even though descriptions that use state are
+usually very clear, distributing and collecting substate can become tedious
+and even error-prone. Automating the required distribution and collection, or
+finding a more suitable abstraction mechanism for state would make \CLaSH\
+easier to use. Currently, one of the examined approaches to suppress state in
+the specification is by using Haskell's arrow-abstraction.
+
+The transformations in the normalization phase of the prototype compiler are
developed in an ad-hoc manner, which makes the existence of many desirable
properties unclear. Such properties include whether the complete set of
-transformations will always lead to a normal form or if the normalization
-process always terminates. Though various use cases suggests that these
-properties usually hold, they have not been formally proven. A systematic
-approach to defining the set of transformations allows one to proof that the
-earlier mentioned properties do indeed exist.
+transformations will always lead to a normal form or whether the normalization
+process always terminates. Though extensive use of the compiler suggests that
+these properties usually hold, they have not been formally proven. A
+systematic approach to defining the set of transformations allows one to proof
+that the earlier mentioned properties do indeed hold.
% conference papers do not normally have an appendix
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