\chapter[chap:description]{Hardware description}
- This chapter will provide an overview of the hardware description language
- that was created and the issues that have arisen in the process. It will
- focus on the issues of the language, not the implementation. The prototype
- implementation will be discussed in \in{chapter}[chap:prototype].
-
- \todo{Shortshort introduction to Cλash (Bit, Word, and, not, etc.)}
-
- When translating Haskell to hardware, we need to make choices about what kind
- of hardware to generate for which Haskell constructs. When faced with
- choices, we've tried to stick with the most obvious choice wherever
- possible. In a lot of cases, when you look at a hardware description it is
- comletely clear what hardware is described. We want our translator to
- generate exactly that hardware whenever possible, to minimize the amount of
- surprise for people working with it.
+ In this chapter an overview will be provided of the hardware
+ description language that was created and the issues that have arisen
+ in the process. The focus will be on the issues of the language, not
+ the implementation. The prototype implementation will be discussed in
+ \in{chapter}[chap:prototype].
+
+ To translate Haskell to hardware, every Haskell construct needs a
+ translation to \VHDL. There are often multiple valid translations
+ possible. When faced with choices, the most obvious choice has been
+ chosen wherever possible. In a lot of cases, when a programmer looks
+ at a functional hardware description it is completely clear what
+ hardware is described. We want our translator to generate exactly that
+ hardware whenever possible, to make working with Cλash as intuitive as
+ possible.
- In this chapter we try to describe how we interpret a Haskell program from a
+ \placeintermezzo{}{
+ \defref{reading examples}
+ \startframedtext[width=9cm,background=box,frame=no]
+ \startalignment[center]
+ {\tfa Reading the examples}
+ \stopalignment
+ \blank[medium]
+ In this thesis, a lot of functional code will be presented. Part of this
+ will be valid Cλash code, but others will just be small Haskell or Core
+ snippets to illustrate a concept.
+
+ In these examples, some functions and types will be used, without
+ properly defining every one of them. These functions (like \hs{and},
+ \hs{not}, \hs{add}, \hs{+}, etc.) and types (like \hs{Bit}, \hs{Word},
+ \hs{Bool}, etc.) are usually pretty self-explanatory.
+
+ The special type \hs{[t]} means \quote{list of \hs{t}'s}, where \hs{t}
+ can be any other type.
+
+ Of particular note is the use of the \hs{::} operator. It is used in
+ Haskell to explicitly declare the type of function or let binding. In
+ these examples and the text, we will occasionally use this operator to
+ show the type of arbitrary expressions, even where this would not
+ normally be valid. Just reading the \hs{::} operator as \quote{and also
+ note that \emph{this} expression has \emph{this} type} should work out.
+ \stopframedtext
+ }
+
+ In this chapter we describe how to interpret a Haskell program from a
hardware perspective. We provide a description of each Haskell language
element that needs translation, to provide a clear picture of what is
supported and how.
+
\section[sec:description:application]{Function application}
- \todo{Sidenote: Inputs vs arguments}
- The basic syntactic element of a functional program are functions and
- function application. These have a single obvious \small{VHDL} translation: Each
- function becomes a hardware component, where each argument is an input port
- and the result value is the (single) output port. This output port can have
- a complex type (such as a tuple), so having just a single output port does
- not pose a limitation.
+ The basic syntactic elements of a functional program are functions and
+ function application. These have a single obvious \small{VHDL}
+ translation: each top level function becomes a hardware component, where each
+ argument is an input port and the result value is the (single) output
+ port. This output port can have a complex type (such as a tuple), so
+ having just a single output port does not pose a limitation.
Each function application in turn becomes component instantiation. Here, the
result of each argument expression is assigned to a signal, which is mapped
to the corresponding input port. The output port of the function is also
mapped to a signal, which is used as the result of the application.
+ Since every top level function generates its own component, the
+ hierarchy of of function calls is reflected in the final \VHDL\ output
+ as well, creating a hierarchical \VHDL\ description of the hardware.
+ This separation in different components makes the resulting \VHDL\
+ output easier to read and debug.
+
\in{Example}[ex:And3] shows a simple program using only function
application and the corresponding architecture.
\startbuffer[And3]
--- | A simple function that returns
--- conjunction of three bits
+-- A simple function that returns
+-- conjunction of three bits
and3 :: Bit -> Bit -> Bit -> Bit
and3 a b c = and (and a b) c
\stopbuffer
-\todo{Mirror this image vertically}
\startuseMPgraphic{And3}
save a, b, c, anda, andb, out;
ncline(andb)(out);
\stopuseMPgraphic
- \placeexample[here][ex:And3]{Simple three input and gate.}
+ \startbuffer[And3VHDL]
+ entity and3Component_0 is
+ port (\azMyG2\ : in std_logic;
+ \bzMyI2\ : in std_logic;
+ \czMyK2\ : in std_logic;
+ \foozMySzMyS2\ : out std_logic;
+ clock : in std_logic;
+ resetn : in std_logic);
+ end entity and3Component_0;
+
+
+ architecture structural of and3Component_0 is
+ signal \argzMyMzMyM2\ : std_logic;
+ begin
+ \argzMyMzMyM2\ <= \azMyG2\ and \bzMyI2\;
+
+ \foozMySzMyS2\ <= \argzMyMzMyM2\ and \czMyK2\;
+ end architecture structural;
+ \stopbuffer
+
+ \placeexample[][ex:And3]{Simple three input and gate.}
\startcombination[2*1]
{\typebufferhs{And3}}{Haskell description using function applications.}
{\boxedgraphic{And3}}{The architecture described by the Haskell description.}
\stopcombination
- \note{This section should also mention hierarchy, top level functions and
- subcircuits}
+ \placeexample[][ex:And3VHDL]{\VHDL\ generated for \hs{and3} from \in{example}[ex:And3]}
+ {\typebuffervhdl{And3VHDL}}
+
+ \placeintermezzo{}{
+ \defref{top level binder}
+ \defref{top level function}
+ \startframedtext[width=8cm,background=box,frame=no]
+ \startalignment[center]
+ {\tfa Top level binders and functions}
+ \stopalignment
+ \blank[medium]
+ A top level binder is any binder (variable) that is declared in
+ the \quote{global} scope of a Haskell program (as opposed to a
+ binder that is bound inside a function.
+
+ In Haskell, there is no sharp distinction between a variable and a
+ function: a function is just a variable (binder) with a function
+ type. This means that a top level function is just any top level
+ binder with a function type.
+ As an example, consider the following Haskell snippet:
+
+ \starthaskell
+ foo :: Int -> Int
+ foo x = inc x
+ where
+ inc = \y -> y + 1
+ \stophaskell
+
+ Here, \hs{foo} is a top level binder, whereas \hs{inc} is a
+ function (since it is bound to a lambda extraction, indicated by
+ the backslash) but is not a top level binder or function. Since
+ the type of \hs{foo} is a function type, namely \hs{Int -> Int},
+ it is also a top level function.
+ \stopframedtext
+ }
\section{Choice}
Although describing components and connections allows us to describe a lot of
- hardware designs already, there is an obvious thing missing: Choice. We
+ hardware designs already, there is an obvious thing missing: choice. We
need some way to be able to choose between values based on another value.
- In Haskell, choice is achieved by \hs{case} expressions, \hs{if} expressions or
- pattern matching.
+ In Haskell, choice is achieved by \hs{case} expressions, \hs{if}
+ expressions, pattern matching and guards.
An obvious way to add choice to our language without having to recognize
any of Haskell's syntax, would be to add a primivite \quote{\hs{if}}
- function. This function would take three arguments: The condition, the
+ function. This function would take three arguments: the condition, the
value to return when the condition is true and the value to return when
the condition is false.
This \hs{if} function would then essentially describe a multiplexer and
- allows us to describe any architecture that uses multiplexers. \fxnote{Are
- there other mechanisms of choice in hardware?}
+ allows us to describe any architecture that uses multiplexers.
However, to be able to describe our hardware in a more convenient way, we
also want to translate Haskell's choice mechanisms. The easiest of these
simply be translated to a conditional assignment, where the conditions use
equality comparisons against the constructors in the \hs{case} expressions.
- \todo{Assignment vs multiplexers}
-
In \in{example}[ex:CaseInv] a simple \hs{case} expression is shown,
scrutinizing a boolean value. The corresponding architecture has a
comparator to determine which of the constructors is on the \hs{in}
for the output signals are just constants, but these could have been more
complex expressions (in which case also both of them would be working in
parallel, regardless of which output would be chosen eventually).
-
+
If we would translate a Boolean to a bit value, we could of course remove
- the comparator and directly feed 'in' into the multiplex (or even use an
+ the comparator and directly feed 'in' into the multiplexer (or even use an
inverter instead of a multiplexer). However, we will try to make a
general translation, which works for all possible \hs{case} expressions.
- Optimizations such as these are left for the \VHDL synthesizer, which
+ Optimizations such as these are left for the \VHDL\ synthesizer, which
handles them very well.
- \todo{Be more explicit about >2 alternatives}
+ \placeintermezzo{}{
+ \startframedtext[width=8cm,background=box,frame=no]
+ \startalignment[center]
+ {\tfa Arguments / results vs. inputs / outputs}
+ \stopalignment
+ \blank[medium]
+ Due to the translation chosen for function application, there is a
+ very strong relation between arguments, results, inputs and outputs.
+ For clarity, the former two will always refer to the arguments and
+ results in the functional description (either Haskell or Core). The
+ latter two will refer to input and output ports in the generated
+ \VHDL.
+
+ Even though these concepts seem to be nearly identical, when stateful
+ functions are introduces we will see arguments and results that will
+ not get translated into input and output ports, making this
+ distinction more important.
+ \stopframedtext
+ }
+
+ 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.
+
+ The architecture described by \in{example}[ex:PatternInv] is of course the
+ same one as the one in \in{example}[ex:CaseInv]. The general interpretation
+ of pattern matching is also similar to that of \hs{case} expressions: generate
+ hardware for each of the clauses (like each of the clauses of a \hs{case}
+ expression) and connect them to the function output through (a number of
+ nested) multiplexers. These multiplexers are driven by comparators and
+ other logic, that check each pattern in turn.
+
+ In these examples we have seen only binary case expressions and pattern
+ matches (\ie, with two alternatives). In practice, case expressions can
+ choose between more than two values, resulting in a number of nested
+ multiplexers.
\startbuffer[CaseInv]
inv :: Bool -> Bool
- inv True = False
- inv False = True
+ inv x = case x of
+ True -> False
+ False -> True
\stopbuffer
\startuseMPgraphic{CaseInv}
ncline(trueout)(mux) "posB(inpb)";
ncline(mux)(out) "posA(out)";
\stopuseMPgraphic
+
+ \startbuffer[CaseInvVHDL]
+ entity invComponent_0 is
+ port (\xzAMo2\ : in boolean;
+ \reszAMuzAMu2\ : out boolean;
+ clock : in std_logic;
+ resetn : in std_logic);
+ end entity invComponent_0;
+
+
+ architecture structural of invComponent_0 is
+ begin
+ \reszAMuzAMu2\ <= false when \xzAMo2\ = true else
+ true;
+ end architecture structural;
+ \stopbuffer
- \placeexample[here][ex:CaseInv]{Simple inverter.}
+ \placeexample[][ex:CaseInv]{Simple inverter.}
\startcombination[2*1]
{\typebufferhs{CaseInv}}{Haskell description using a Case expression.}
{\boxedgraphic{CaseInv}}{The architecture described by the Haskell description.}
\stopcombination
- 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.
+ \placeexample[][ex:CaseInvVHDL]{\VHDL\ generated for \hs{inv} from
+ \in{example}[ex:CaseInv] and \in{example}[ex:PatternInv]}
+ {\typebuffervhdl{CaseInvVHDL}}
\startbuffer[PatternInv]
inv :: Bool -> Bool
inv False = True
\stopbuffer
- \placeexample[here][ex:PatternInv]{Simple inverter using pattern matching.
+ \placeexample[][ex:PatternInv]{Simple inverter using pattern matching.
Describes the same architecture as \in{example}[ex:CaseInv].}
- {\typebufferhs{CaseInv}}
-
- The architecture described by \in{example}[ex:PatternInv] is of course the
- same one as the one in \in{example}[ex:CaseInv]. The general interpretation
- of pattern matching is also similar to that of \hs{case} expressions: Generate
- hardware for each of the clauses (like each of the clauses of a \hs{case}
- expression) and connect them to the function output through (a number of
- nested) multiplexers. These multiplexers are driven by comparators and
- other logic, that check each pattern in turn.
+ {\typebufferhs{PatternInv}}
\section{Types}
- We've seen the two most basic functional concepts translated to hardware:
- Function application and choice. Before we look further into less obvious
- concepts like higher order expressions and polymorphism, we will have a
- look at the types of the values we use in our descriptions.
-
- When working with values in our descriptions, we'll also need to provide
- some way to translate the values used to hardware equivalents. In
- particular, this means having to come up with a hardware equivalent for
- every \emph{type} used in our program.
-
- Since most functional languages have a lot of standard types that are
- hard to translate (integers without a fixed size, lists without a static
- length, etc.), we will start out by defining a number of \quote{builtin}
- types ourselves. These types are builtin in the sense that our compiler
- will have a fixed VHDL type for these. User defined types, on the other
- hand, will have their hardware type derived directly from their Haskell
- declaration automatically, according to the rules we sketch here.
+ 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 values used to its hardware
+ equivalents. In particular, this means a hardware equivalent for
+ every \emph{type} used in a hardware description is needed
+
+ Since most functional languages have a lot of standard types that
+ are hard to translate (integers without a fixed size, lists without
+ a static length, etc.), a number of \quote{built-in} types will be
+ defined first. These types are built-in in the sense that our
+ compiler will have a fixed VHDL type for these. User defined types,
+ on the other hand, will have their hardware type derived directly
+ from their Haskell declaration automatically, according to the rules
+ sketched here.
\todo{Introduce Haskell type syntax (type constructors, type application,
:: operator?)}
- \subsection{Builtin types}
- The language currently supports the following builtin types. Of these,
+ \subsection{Built-in types}
+ The language currently supports the following built-in types. Of these,
only the \hs{Bool} type is supported by Haskell out of the box (the
others are defined by the Cλash package, so they are user-defined types
from Haskell's point of view).
\todo{Sidenote bit vs stdlogic}
\stopdesc
\startdesc{\hs{Bool}}
- This is the only builtin Haskell type supported and is translated
+ This is the only built-in Haskell type supported and is translated
exactly like the 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
\stopdesc
\startdesc{\hs{Vector}}
This is a vector type, that can contain elements of any other type and
- has a fixed length. It has two type parameters: Its
+ has a fixed length. It has two type parameters: its
length and the type of the elements contained in it. By putting the
length parameter in the type, the length of a vector can be determined
at compile time, instead of only at runtime for conventional lists.
- The \hs{Vector} type constructor takes two type arguments: The length
+ The \hs{Vector} type constructor takes two type arguments: the length
of the vector and the type of the elements contained in it. The state
type of an 8 element register bank would then for example be:
A \hs{RangedWord} only has an upper bound, its lower bound is
implicitly zero. There is a lot of added implementation complexity
when adding a lower bound and having just an upper bound was enough
- for the primary purpose of this type: Typesafely indexing vectors.
+ for the primary purpose of this type: typesafely indexing vectors.
To define an index for the 8 element vector above, we would do:
\starthaskell
- type Register = RangedWord D7
+ type RegisterIndex = RangedWord D7
\stophaskell
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.
+ other words, this defines an unsigned word with values from
+ {\definedfont[Serif*normalnum]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 \type{unsigned} \small{VHDL} type.
\stopdesc
- \fxnote{There should be a reference to Christiaan's work here.}
+
+ The integer and vector built-in types are discussed in more detail
+ in \cite[baaij09].
\subsection{User-defined types}
- There are three ways to define new types in Haskell: Algebraic
+ There are three ways to define new types in Haskell: algebraic
datatypes with the \hs{data} keyword, type synonyms with the \hs{type}
- keyword and type renamings with the \hs{newtype} keyword. This
- explicitly excludes more advanced type creation from \GHC extensions
- such as type families, existential typing, \small{GADT}s, etc.
-
- The first of these actually introduces a new type, for which we provide
- the \VHDL translation below. The latter two only define new names for
+ 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 will be left outside the scope of this research.
+
+ 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 don't need any
- particular \VHDL translation, a synonym or renamed type will just use
- the same representation as the equivalent type.
+ 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:
A product type is an algebraic datatype with a single constructor with
two or more fields, denoted in practice like (a,b), (a,b,c), etc. This
is essentially a way to pack a few values together in a record-like
- structure. In fact, the builtin tuple types are just algebraic product
+ structure. In fact, the built-in tuple types are just algebraic product
types (and are thus supported in exactly the same way).
- The "product" in its name refers to the collection of values belonging
- to this type. The collection for a product type is the cartesian
+ The \quote{product} in its name refers to the collection of values belonging
+ to this type. The collection for a product type is the Cartesian
product of the collections for the types of its fields.
- These types are translated to \VHDL, record types, with one field for
+ These types are translated to \VHDL\ record types, with one field for
every field in the constructor. This translation applies to all single
- constructor algebraic datatypes, including those with no fields (unit
- types) and just one field (which are technically not a product).
+ constructor algebraic datatypes, including those with just one
+ field (which are technically not a product, but generate a VHDL
+ record for implementation simplicity).
\stopdesc
\startdesc{Enumerated types}
\defref{enumerated types}
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
+ 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.
\stopdesc
A sum type is an algebraic datatype with multiple constructors, where
the constructors have one or more fields. Technically, a type with
more than one field per constructor is a sum of products type, but
- for our purposes this distinction does not really make a difference,
- so we'll leave it out.
+ for our purposes this distinction does not really make a
+ difference, so this distinction is note made.
+
+ The \quote{sum} in its name refers again to the collection of values
+ belonging to this type. The collection for a sum type is the
+ union of the the collections for each of the constructors.
Sum types are currently not supported by the prototype, since there is
- no obvious \VHDL alternative. They can easily be emulated, however, as
+ no obvious \VHDL\ alternative. They can easily be emulated, however, as
we will see from an example:
\starthaskell
fields of the \hs{Sum} type are valid (the first two if \hs{A}, the
last one if \hs{B}), all the other ones have no useful value.
- An obvious problem with this naive approach is the space usage: The
- example above generates a fairly big \VHDL type. However, we can be
+ An obvious problem with this naive approach is the space usage: the
+ example above generates a fairly big \VHDL\ type. Since we can be
sure that the two \hs{Word}s in the \hs{Sum} type will never be valid
- at the same time, so this is a waste of space.
+ at the same time, this is a waste of space.
- Obviously, we could do some duplication detection here to reuse a
+ Obviously, duplication detection could be used to reuse a
particular field for another constructor, but this would only
- partially solve the problem. If I would, for example, have an array of
- 8 bits and a 8 bit unsiged word, these are different types and could
- not be shared. However, in the final hardware, both of these types
- would simply be 8 bit connections, so we have a 100\% size increase by
- not sharing these.
+ partially solve the problem. If two fields would be, for
+ example, an array of 8 bits and an 8 bit unsiged word, these are
+ different types and could not be shared. However, in the final
+ hardware, both of these types would simply be 8 bit connections,
+ so we have a 100\% size increase by not sharing these.
\stopdesc
Another interesting case is that of recursive types. In Haskell, an
- algebraic datatype can be recursive: Any of its field types can be (or
+ algebraic datatype can be recursive: any of its field types can be (or
contain) the type being defined. The most well-known recursive type is
probably the list type, which is defined is:
Note that \hs{Empty} is usually written as \hs{[]} and \hs{Cons} as
\hs{:}, but this would make the definition harder to read. This
- immediately shows the problem with recursive types: What hardware type
+ immediately shows the problem with recursive types: what hardware type
to allocate here?
- If we would use the naive approach for sum types we described above, we
- would create a record where the first field is an enumeration to
- distinguish \hs{Empty} from \hs{Cons}. Furthermore, we would add two
- more fields: One with the (\VHDL equivalent of) type \hs{t} (assuming we
- actually know what type this is at compile time, this should not be a
- problem) and a second one with type \hs{List t}. The latter one is of
- course a problem: This is exactly the type we were trying to translate
- in the first place.
+ If the naive approach for sum types described above would be used,
+ a record would be created where the first field is an enumeration
+ to distinguish \hs{Empty} from \hs{Cons}. Furthermore, two more
+ fields would be added: one with the (\VHDL\ equivalent of) type
+ \hs{t} (assuming this type is actually known at compile time, this
+ should not be a problem) and a second one with type \hs{List t}.
+ The latter one is of course a problem: this is exactly the type
+ that was to be translated in the first place.
- Our \VHDL type will thus become infinitely deep. In other words, there
- is no way to statically determine how long (deep) the list will be
- (it could even be infinite).
+ The resulting \VHDL\ type will thus become infinitely deep. In
+ other words, there is no way to statically determine how long
+ (deep) the list will be (it could even be infinite).
- In general, we can say we can never properly translate recursive types:
- All recursive types have a potentially infinite value (even though in
+ In general, recursive types can never be properly translated: all
+ recursive types have a potentially infinite value (even though in
practice they will have a bounded value, there is no way for the
- compiler to determine an upper bound on its size).
+ compiler to automatically determine an upper bound on its size).
\subsection{Partial application}
- Now we've seen how to to translate application, choice and types, we will
- get to a more complex concept: Partial applications. A \emph{partial
- application} is any application whose (return) type is (again) a function
- type.
+ Now the translation of application, choice and types has been
+ discussed, a more complex concept can be considered: partial
+ applications. A \emph{partial application} is any application whose
+ (return) type is (again) a function type.
- From this, we can see that the translation rules for full application do not
- apply to a partial application. \in{Example}[ex:Quadruple] shows an example
- use of partial application and the corresponding architecture.
+ From this, it should be clear that the translation rules for full
+ application does not apply to a partial application: there are not
+ enough values for all the input ports in the resulting \VHDL.
+ \in{Example}[ex:Quadruple] shows an example use of partial application
+ and the corresponding architecture.
\startbuffer[Quadruple]
--- | Multiply the input word by four.
+-- Multiply the input word by four.
quadruple :: Word -> Word
quadruple n = mul (mul n)
where
ncline(mulb)(out);
\stopuseMPgraphic
- \placeexample[here][ex:Quadruple]{Simple three port and.}
+ \placeexample[][ex:Quadruple]{Simple three port and.}
\startcombination[2*1]
{\typebufferhs{Quadruple}}{Haskell description using function applications.}
{\boxedgraphic{Quadruple}}{The architecture described by the Haskell description.}
\stopcombination
- Here, the definition of mul is a partial function application: It applies
+ Here, the definition of mul is a partial function application: it applies
the function \hs{(*) :: Word -> Word -> Word} to the value \hs{2 :: Word},
resulting in the expression \hs{(*) 2 :: Word -> Word}. Since this resulting
- expression is again a function, we can't generate hardware for it directly.
- This is because the hardware to generate for \hs{mul} depends completely on
- where and how it is used. In this example, it is even used twice.
-
- However, it is clear that the above hardware description actually describes
- valid hardware. In general, we can see that any partial applied function
- must eventually become completely applied, at which point we can generate
- hardware for it using the rules for function application given in
- \in{section}[sec:description:application]. It might mean that a partial
- application is passed around quite a bit (even beyond function boundaries),
- but eventually, the partial application will become completely applied.
- \todo{Provide a step-by-step example of how this works}
+ expression is again a function, hardware cannot be generated for it
+ directly. This is because the hardware to generate for \hs{mul}
+ depends completely on where and how it is used. In this example, it is
+ even used twice.
+
+ However, it is clear that the above hardware description actually
+ describes valid hardware. In general, any partial applied function
+ must eventually become completely applied, at which point hardware for
+ it can be generated using the rules for function application given in
+ \in{section}[sec:description:application]. It might mean that a
+ partial application is passed around quite a bit (even beyond function
+ boundaries), but eventually, the partial application will become
+ completely applied. An example of this principe is given in
+ \in{section}[sec:normalization:defunctionalization].
\section{Costless specialization}
Each (complete) function application in our description generates a
function is the same (of course, if a particular value, such as the result
of a function application, is used twice, it is not calculated twice).
- This is distinctly different from normal program compilation: Two separate
+ This is distinctly different from normal program compilation: two separate
calls to the same function share the same machine code. Having more
machine code has implications for speed (due to less efficient caching)
and memory usage. For normal compilation, it is therefore important to
- keep the amount of functions limited and maximize the code sharing.
+ keep the amount of functions limited and maximize the code sharing
+ (though there is a tradeoff between speed and memory usage here).
When generating hardware, this is hardly an issue. Having more \quote{code
sharing} does reduce the amount of \small{VHDL} output (Since different
component instantiations still share the same component), but after
- synthesis, the amount of hardware generated is not affected.
+ synthesis, the amount of hardware generated is not affected. This
+ means there is no tradeoff between speed and memory (or rather,
+ chip area) usage.
In particular, if we would duplicate all functions so that there is a
separate function for every application in the program (\eg, each function
\subsection{Specialization}
\defref{specialization}
- Given some function that has a \emph{domain} $D$ (\eg, the set of all
- possible arguments that could be applied), we create a specialized
- function with exactly the same behaviour, but with a domain $D' \subset
- D$. This subset can be chosen in all sorts of ways. Any subset is valid
- for the general definition of specialization, but in practice only some
- of them provide useful optimization opportunities.
+ Given some function that has a \emph{domain} $D$ (\eg, the set of
+ all possible arguments that the function could be applied to), we
+ create a specialized function with exactly the same behaviour, but
+ with a domain $D' \subset D$. This subset can be chosen in all
+ sorts of ways. Any subset is valid for the general definition of
+ specialization, but in practice only some of them provide useful
+ optimization opportunities.
Common subsets include limiting a polymorphic argument to a single type
(\ie, removing polymorphism) or limiting an argument to just a single
Specialization is useful for our hardware descriptions for functions
that contain arguments that cannot be translated to hardware directly
- (polymorphic or higher order arguments, for example). If we can create
+ (polymorphic or higher-order arguments, for example). If we can create
specialized functions that remove the argument, or make it translatable,
we can use specialization to make the original, untranslatable, function
obsolete.
\section{Higher order values}
What holds for partial application, can be easily generalized to any
- higher order expression. This includes partial applications, plain
+ higher-order expression. This includes partial applications, plain
variables (e.g., a binder referring to a top level function), lambda
expressions and more complex expressions with a function type (a \hs{case}
expression returning lambda's, for example).
become complete applications, applied lambda expressions disappear by
applying β-reduction, etc.
- So any higher order value will be \quote{pushed down} towards its
+ So any higher-order value will be \quote{pushed down} towards its
application just like partial applications. Whenever a function boundary
needs to be crossed, the called function can be specialized.
\fxnote{This section needs improvement and an example}
\section{Polymorphism}
- In Haskell, values can be \emph{polymorphic}: They can have multiple types. For
+ In Haskell, values can be \emph{polymorphic}: they can have multiple types. For
example, the function \hs{fst :: (a, b) -> a} is an example of a
- polymorphic function: It works for tuples with any two element types. Haskell
- typeclasses allow a function to work on a specific set of types, but the
+ polymorphic function: it works for tuples with any two element types. Haskell
+ type classes allow a function to work on a specific set of types, but the
general idea is the same. The opposite of this is a \emph{monomorphic}
value, which has a single, fixed, type.
% A type class is a collection of types for which some operations are
% defined. It is thus possible for a value to be polymorphic while having
-% any number of \emph{class constraints}: The value is not defined for
+% any number of \emph{class constraints}: the value is not defined for
% every type, but only for types in the type class. An example of this is
% the \hs{even :: (Integral a) => a -> Bool} function, which can map any
% value of a type that is member of the \hs{Integral} type class
- When generating hardware, polymorphism can't be easily translated. How
+ When generating hardware, polymorphism cannot be easily translated. How
many wires will you lay down for a value that could have any type? When
type classes are involved, what hardware components will you lay down for
a class method (whose behaviour depends on the type of its arguments)?
- Note that the language currently does not allow user-defined typeclasses,
- but does support partly some of the builtin typeclasses (like \hs{Num}).
+ Note that Cλash currently does not allow user-defined type classes,
+ but does partly support some of the built-in type classes (like \hs{Num}).
- Fortunately, we can again use the principle of specialization: Since every
+ Fortunately, we can again use the principle of specialization: since every
function application generates a separate piece of hardware, we can know
- the types of all arguments exactly. Provided that we don't use existential
- typing, all of the polymorphic types in a function must depend on the
- types of the arguments (In other words, the only way to introduce a type
- variable is in a lambda abstraction).
+ the types of all arguments exactly. Provided that existential typing
+ (which is a \GHC\ extension) is not used typing, all of the
+ polymorphic types in a function must depend on the types of the
+ arguments (In other words, the only way to introduce a type variable
+ is in a lambda abstraction).
If a function is monomorphic, all values inside it are monomorphic as
well, so any function that is applied within the function can only be
specialized to work just for these specific types, removing the
polymorphism from the applied functions as well.
- Our top level function must not have a polymorphic type (otherwise we
- wouldn't know the hardware interface to our top level function).
+ \defref{entry function}The entry function must not have a
+ polymorphic type (otherwise no hardware interface could be generated
+ for the entry function).
By induction, this means that all functions that are (indirectly) called
by our top level function (meaning all functions that are translated in
\section{State}
A very important concept in hardware designs is \emph{state}. In a
- stateless (or, \emph{combinatoric}) design, every output is directly and solely dependent on the
+ stateless (or, \emph{combinational}) design, every output is directly and solely dependent on the
inputs. In a stateful design, the outputs can depend on the history of
inputs, or the \emph{state}. State is usually stored in \emph{registers},
which retain their value during a clockcycle, and are typically updated at
\todo{Sidenote? Registers can contain any (complex) type}
- To make our hardware description language useful to describe more than
- simple combinatoric designs, we'll need to be able to describe state in
- some way.
+ To make Cλash useful to describe more than simple combinational
+ designs, it needs to be able to describe state in some way.
\subsection{Approaches to state}
In Haskell, functions are always pure (except when using unsafe
evaluate a given function with given inputs, it will always provide the
same output.
- \todo{Define pure}
-
- This is a perfect match for a combinatoric circuit, where the output
- also soley depends on the inputs. However, when state is involved, this
- no longer holds. Since we're in charge of our own language (or at least
- let's pretend we aren't using Haskell and we are), we could remove this
- purity constraint and allow a function to return different values
- depending on the cycle in which it is evaluated (or rather, the current
- state). However, this means that all kinds of interesting properties of
- our functional language get lost, and all kinds of transformations and
- optimizations might no longer be meaning preserving.
+ \placeintermezzo{}{
+ \defref{purity}
+ \startframedtext[width=8cm,background=box,frame=no]
+ \startalignment[center]
+ {\tfa Purity}
+ \stopalignment
+ \blank[medium]
+
+ A function is said to be pure if it satisfies two conditions:
+
+ \startitemize[KR]
+ \item When a pure function is called with the same arguments twice, it should
+ return the same value in both cases.
+ \item When a pure function is called, it should have not
+ observable side-effects.
+ \stopitemize
- Provided that we want to keep the function pure, the current state has
- to be present in the function's arguments in some way. There seem to be
- two obvious ways to do this: Adding the current state as an argument, or
- including the full history of each argument.
+ Purity is an important property in functional languages, since
+ it enables all kinds of mathematical reasoning and
+ optimizattions with pure functions, that are not guaranteed to
+ be correct for impure functions.
+
+ An example of a pure function is the square root function
+ \hs{sqrt}. An example of an impure function is the \hs{today}
+ function that returns the current date (which of course cannot
+ exist like this in Haskell).
+ \stopframedtext
+ }
+
+ This is a perfect match for a combinational circuit, where the output
+ also solely depends on the inputs. However, when state is involved, this
+ no longer holds. Of course this purity constraint cannot just be
+ removed from Haskell. But even when designing a completely new (hardware
+ description) language, it does not seem to be a good idea to
+ remove this purity. This would that all kinds of interesting properties of
+ the functional language get lost, and all kinds of transformations
+ and optimizations are no longer be meaning preserving.
+
+ So our functions must remain pure, meaning the current state has
+ to be present in the function's arguments in some way. There seem
+ to be two obvious ways to do this: adding the current state as an
+ argument, or including the full history of each argument.
\subsubsection{Stream arguments and results}
Including the entire history of each input (\eg, the value of that
An obvious downside of this solution is that on each cycle, all the
previous cycles must be resimulated to obtain the current state. To do
this, it might be needed to have a recursive helper function as well,
- wich might be hard to be properly analyzed by the compiler.
+ which might be hard to be properly analyzed by the compiler.
A slight variation on this approach is one taken by some of the other
functional \small{HDL}s in the field: \todo{References to Lava,
well (note that changing a single-element operation to a stream
operation can done with \hs{map}, \hs{zipwith}, etc.).
+ This also means that primitive operations from an existing
+ language such as Haskell cannot be used directly (including some
+ syntax constructs, like \hs{case} expressions and \hs{if}
+ expressions). This mkes this approach well suited for use in
+ \small{EDSL}s, since those already impose these same
+ limitations. \refdef{EDSL}
+
Note that the concept of \emph{state} is no more than having some way
to communicate a value from one cycle to the next. By introducing a
- \hs{delay} function, we can do exactly that: Delay (each value in) a
+ \hs{delay} function, we can do exactly that: delay (each value in) a
stream so that we can "look into" the past. This \hs{delay} function
simply outputs a stream where each value is the same as the input
value, but shifted one cycle. This causes a \quote{gap} at the
- beginning of the stream: What is the value of the delay output in the
- first cycle? For this, the \hs{delay} function has a second input
- (which is a value, not a stream!).
+ beginning of the stream: what is the value of the delay output in the
+ first cycle? For this, the \hs{delay} function has a second input, of
+ which only a single value is used.
\in{Example}[ex:DelayAcc] shows a simple accumulator expressed in this
style.
\stopuseMPgraphic
- \placeexample[here][ex:DelayAcc]{Simple accumulator architecture.}
+ \placeexample[][ex:DelayAcc]{Simple accumulator architecture.}
\startcombination[2*1]
{\typebufferhs{DelayAcc}}{Haskell description using streams.}
{\boxedgraphic{DelayAcc}}{The architecture described by the Haskell description.}
definition of out), but is essentially easy to interpret. There is a
single call to delay, resulting in a circuit with a single register,
whose input is connected to \hs{out} (which is the output of the
- adder), and it's output is the expression \hs{delay out 0} (which is
+ adder), and its output is the expression \hs{delay out 0} (which is
connected to one of the adder inputs).
- This notation has a number of downsides, amongst which are limited
- readability and ambiguity in the interpretation. \note{Reference
- Christiaan, who has done further investigation}
-
\subsubsection{Explicit state arguments and results}
A more explicit way to model state, is to simply add an extra argument
containing the current state value. This allows an output to depend on
the current state is now an argument.
In Haskell, this would look like
- \in{example}[ex:ExplicitAcc]\footnote[notfinalsyntax]{Note that this example is not in the final
- Cλash syntax}. \todo{Referencing notfinalsyntax from Introduction.tex doesn't
- work}
+ \in{example}[ex:ExplicitAcc]\footnote[notfinalsyntax]{This
+ example is not in the final Cλash syntax}. \todo{Referencing
+ notfinalsyntax from Introduction.tex doesn't work}
\startbuffer[ExplicitAcc]
-- input -> current state -> (new state, output)
s' = out
\stopbuffer
- \placeexample[here][ex:ExplicitAcc]{Simple accumulator architecture.}
+ \placeexample[][ex:ExplicitAcc]{Simple accumulator architecture.}
\startcombination[2*1]
{\typebufferhs{ExplicitAcc}}{Haskell description using explicit state arguments.}
% Picture is identical to the one we had just now.
variables are used by a function can be completely determined from its
type signature (as opposed to the stream approach, where a function
looks the same from the outside, regardless of what state variables it
- uses or whether it's stateful at all).
-
- This approach is the one chosen for Cλash and will be examined more
- closely below.
+ uses or whether it is stateful at all).
+
+ This approach to state has been one of the initial drives behind
+ this research. Unlike a stream based approach it is well suited
+ to completely use existing code and language features (like
+ \hs{if} and \hs{case} expressions) because it operates on normal
+ values. Because of these reasons, this is the approach chosen
+ for Cλash. It will be examined more closely below.
\subsection{Explicit state specification}
- We've seen the concept of explicit state in a simple example below, but
- what are the implications of this approach?
+ The concept of explicit state has been introduced with some
+ examples above, but what are the implications of this approach?
\subsubsection{Substates}
Since a function's state is reflected directly in its type signature,
has to somehow know the current state for these called functions. The
only way to do this, is to put these \emph{substates} inside the
caller's state. This means that a function's state is the sum of the
- states of all functions it calls, and its own state.
+ states of all functions it calls, and its own state. This sum
+ can be obtained using something simple like a tuple, or possibly
+ custom algebraic types for clarity.
This also means that the type of a function (at least the "state"
part) is dependent on its own implementation and of the functions it
calls.
- This is the major downside of this approach: The separation between
+ This is the major downside of this approach: the separation between
interface and implementation is limited. However, since Cλash is not
very suitable for separate compilation (see
\in{section}[sec:prototype:separate]) this is not a big problem in
deduce the statefulness of subfunctions by analyzing the flow of data
in the calling functions?
- To explore this matter, the following observeration is interesting: We
+ To explore this matter, the following observeration is interesting: we
get completely correct behaviour when we put all state registers in
the top level entity (or even outside of it). All of the state
arguments and results on subfunctions are treated as normal input and
\item accept that the generated hardware might not be exactly what we
intended, in some specific cases. In most cases, the hardware will be
what we intended.
- \item explicitely annotate state arguments and results in the input
+ \item explicitly annotate state arguments and results in the input
description.
\stopitemize
end up is easier to implement correctly with explicit annotations, so
for these reasons we will look at how this annotations could work.
- \todo{Sidenote: One or more state arguments?}
+ \todo{Sidenote: one or more state arguments?}
- \subsection{Explicit state annotation}
+ \subsection[sec:description:stateann]{Explicit state annotation}
To make our stateful descriptions unambigious and easier to translate,
we need some way for the developer to describe which arguments and
results are intended to become stateful.
result is stateful. This means that the annotation lives
\quote{outside} of the function, it is completely invisible when
looking at the function body.
- \item Use some kind of annotation on the type level, \ie give stateful
+ \item Use some kind of annotation on the type level, \ie\ give stateful
arguments and stateful (parts of) results a different type. This has the
potential to make this annotation visible inside the function as well,
such that when looking at a value inside the function body you can
- tell if it's stateful by looking at its type. This could possibly make
+ tell if it is stateful by looking at its type. This could possibly make
the translation process a lot easier, since less analysis of the
program flow might be required.
\stopitemize
\todo{Note about conditions on state variables and checking them}
\section[sec:recursion]{Recursion}
- An import concept in functional languages is recursion. In it's most basic
- form, recursion is a definition that is defined in terms of itself. A
+ An important concept in functional languages is recursion. In its most basic
+ form, recursion is a definition that is described in terms of itself. A
recursive function is thus a function that uses itself in its body. This
usually requires multiple evaluations of this function, with changing
arguments, until eventually an evaluation of the function no longer requires
itself.
- Recursion in a hardware description is a bit of a funny thing. Usually,
- recursion is associated with a lot of nondeterminism, stack overflows, but
- also flexibility and expressive power.
-
Given the notion that each function application will translate to a
component instantiation, we are presented with a problem. A recursive
function would translate to a component that contains itself. Or, more
problem for generating hardware.
This is expected for functions that describe infinite recursion. In that
- case, we can't generate hardware that shows correct behaviour in a single
+ case, we cannot generate hardware that shows correct behaviour in a single
cycle (at best, we could generate hardware that needs an infinite number of
cycles to complete).
- However, most recursive hardware descriptions will describe finite
+ \placeintermezzo{}{
+ \startframedtext[width=8cm,background=box,frame=no]
+ \startalignment[center]
+ {\tfa \hs{null}, \hs{head} and \hs{tail}}
+ \stopalignment
+ \blank[medium]
+ The functions \hs{null}, \hs{head} and \hs{tail} are common list
+ functions in Haskell. The \hs{null} function simply checks if a list is
+ empty. The \hs{head} function returns the first element of a list. The
+ \hs{tail} function returns containing everything \emph{except} the first
+ element of a list.
+
+ In Cλash, there are vector versions of these functions, which do exactly
+ the same.
+ \stopframedtext
+ }
+
+ However, most recursive definitions will describe finite
recursion. This is because the recursive call is done conditionally. There
is usually a \hs{case} expression where at least one alternative does not contain
the recursive call, which we call the "base case". If, for each call to the
False -> head xs + sum (tail xs)
\stophaskell
- However, the Haskell typechecker will now use the following reasoning (element
- type of the vector is left out). Below, we write conditions on type
- variables before the \hs{=>} operator. This is not completely valid Haskell
- syntax, but serves to illustrate the typechecker reasoning. Also note that a
- vector can never have a negative length, so \hs{Vector n} implicitly means
- \hs{(n >= 0) => Vector n}.
+ However, the Haskell typechecker will now use the following reasoning.
+ For simplicity, the element type of a vector is left out, all vectors
+ are assumed to have the same element type. Below, we write conditions
+ on type variables before the \hs{=>} operator. This is not completely
+ valid Haskell syntax, but serves to illustrate the typechecker
+ reasoning. Also note that a vector can never have a negative length,
+ so \hs{Vector n} implicitly means \hs{(n >= 0) => Vector n}.
\todo{This typechecker disregards the type signature}
\startitemize
\item tail has the type \hs{(n > 0) => Vector n -> Vector (n - 1)}
\item This means that xs must have the type \hs{(n > 0) => Vector n}
\item This means that sum must have the type \hs{(n > 0) => Vector n -> a}
+ (The type \hs{a} is can be anything at this stage, we will not try to finds
+ its actual type in this example).
\item sum is called with the result of tail as an argument, which has the
type \hs{Vector n} (since \hs{(n > 0) => Vector (n - 1)} is the same as \hs{(n >= 0)
=> Vector n}, which is the same as just \hs{Vector n}).
\stopitemize
As you can see, using a simple \hs{case} expression at value level causes
- the type checker to always typecheck both alternatives, which can't be done!
- This is a fundamental problem, that would seem perfectly suited for a type
- class. Considering that we need to switch between to implementations of the
- sum function, based on the type of the argument, this sounds like the
- perfect problem to solve with a type class. However, this approach has its
- own problems (not the least of them that you need to define a new typeclass
- for every recursive function you want to define).
-
- Another approach tried involved using GADTs to be able to do pattern
- matching on empty / non empty lists. While this worked partially, it also
- created problems with more complex expressions.
-
- \note{This should reference Christiaan}
-
- Evaluating all possible (and non-possible) ways to add recursion to our
- descriptions, it seems better to leave out list recursion alltogether. This
- allows us to focus on other interesting areas instead. By including
- (builtin) support for a number of higher order functions like map and fold,
- we can still express most of the things we would use list recursion for.
-
- \todo{Expand on this decision a bit}
-
+ the type checker to always typecheck both alternatives, which cannot be
+ done. The typechecker is unable to distinguish the two case
+ alternatives (this is partly possible using \small{GADT}s, but that
+ approach faced other problems \todo{ref christiaan?}).
+
+ This is a fundamental problem, that would seem perfectly suited for a
+ type class. Considering that we need to switch between to
+ implementations of the sum function, based on the type of the
+ argument, this sounds like the perfect problem to solve with a type
+ class. However, this approach has its own problems (not the least of
+ them that you need to define a new type class for every recursive
+ function you want to define).
+
+ \todo{This should reference Christiaan}
+
\subsection{General recursion}
Of course there are other forms of recursion, that do not depend on the
length (and thus type) of a list. For example, simple recursion using a
counter could be expressed, but only translated to hardware for a fixed
number of iterations. Also, this would require extensive support for compile
time simplification (constant propagation) and compile time evaluation
- (evaluation of constant comparisons), to ensure non-termination. Even then, it
- is hard to really guarantee termination, since the user (or GHC desugarer)
- might use some obscure notation that results in a corner case of the
- simplifier that is not caught and thus non-termination.
-
- Due to these complications and limited time available, we leave other forms
- of recursion as future work as well.
+ (evaluation of constant comparisons), to ensure non-termination.
+ Supporting general recursion will probably require strict conditions
+ on the input descriptions. Even then, it will be hard (if not
+ impossible) to really guarantee termination, since the user (or \GHC\
+ desugarer) might use some obscure notation that results in a corner
+ case of the simplifier that is not caught and thus non-termination.
+
+ Evaluating all possible (and non-possible) ways to add recursion to
+ our descriptions, it seems better to limit the scope of this research
+ to exclude recursion. This allows for focusing on other interesting
+ areas instead. By including (built-in) support for a number of
+ higher-order functions like \hs{map} and \hs{fold}, we can still
+ express most of the things we would use (list) recursion for.
+
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