\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.
+ focus on the issues of the language, not the implementation. The prototype
+ implementation will be discussed in \in{chapter}[chap:prototype].
- When translating Haskell to hardware, we need to make choices in what kind
- of hardware to generate for what Haskell constructs. When faced with
+ \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
element that needs translation, to provide a clear picture of what is
supported and how.
- \section{Function application}
+ \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 output 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
\startbuffer[And3]
-- | A simple function that returns
--- the and of three bits
+-- 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 port and.}
+ \placeexample[here][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
- TODO: Define top level function and subfunctions/circuits.
+ \note{This section should also mention hierarchy, top level functions and
+ subcircuits}
\section{Choice}
- Since describing components and connections allows us to describe a lot of
+ Although describing components and connections allows us to describe a lot of
hardware designs already, there is an obvious thing missing: Choice. We
need some way to be able to choose between values based on another value.
- In Haskell, choice is achieved by case expressions, if expressions or
- pattern matching (all of which can be reduced to case expressions).
+ In Haskell, choice is achieved by \hs{case} expressions, \hs{if} expressions or
+ pattern matching.
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}}
the condition is false.
This \hs{if} function would then essentially describe a multiplexer and
- allows us to describe any architecture that uses multiplexers. (TODO: Are
- there other mechanisms of choice in hardware?)
+ allows us to describe any architecture that uses multiplexers. \fxnote{Are
+ there other mechanisms of choice in hardware?}
However, to be able to describe our hardware in a more convenient way, we
- also need to translate Haskell's choice mechanisms. The easiest of these
- are of course case expressions (and if expressions, which can be very
- directly translated to case expressions). A case expression can in turn
+ also want to translate Haskell's choice mechanisms. The easiest of these
+ are of course case expressions (and \hs{if} expressions, which can be very
+ directly translated to \hs{case} expressions). A \hs{case} expression can in turn
simply be translated to a conditional assignment, where the conditions use
- equality comparisons against the constructors in the case expressions.
+ equality comparisons against the constructors in the \hs{case} expressions.
+
+ \todo{Assignment vs multiplexers}
- In \in{example}[ex:CaseInv] a simple case statement is shown,
- scrutinizing a boolean value. 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 inverter instead of a multiplexer).
- However, we will try to make a general translation, which works for all
- possible case statements. Optimizations such as these are left for the
- \VHDL synthesizer, which handles them very well.
+ 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}
+ input. There is a multiplexer to select the output signal. The two options
+ 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
+ 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
+ handles them very well.
+
+ \todo{Be more explicit about >2 alternatives}
\startbuffer[CaseInv]
inv :: Bool -> Bool
- inv in = case in of
- True -> False
- False -> True
+ inv True = False
+ inv False = True
\stopbuffer
\startuseMPgraphic{CaseInv}
{\boxedgraphic{CaseInv}}{The architecture described by the Haskell description.}
\stopcombination
- Slightly more complex (but very powerful) forms of choice are pattern
- matches. A function can be defined in multiple clauses, where each clause
+ 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.
inv False = True
\stopbuffer
- \placeexample[here][ex:PatternInv]{Simple inverter using pattern matching.}
+ \placeexample[here][ex:PatternInv]{Simple inverter using pattern matching.
+ Describes the same architecture as \in{example}[ex:CaseInv].}
{\typebufferhs{CaseInv}}
- The architecture described in \in{example}[ex:PatternInv] is of course the
- same one as the one in \in{example}[ex:CaseInv]. The general interpration
- of pattern matching is also similar to that of case expressions: Generate
- hardware for each of the clause (like each of the clauses of a case
+ 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.
\section{Types}
- Apart from giving structure to our hardware, we'll also need to provide
+ 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.
hand, will have their hardware type derived directly from their Haskell
declaration automatically, according to the rules we sketch here.
+ \todo{Introduce Haskell type syntax (type constructors, type application,
+ :: operator?)}
+
\subsection{Builtin types}
- \startdesc{Bit}
+ The language currently supports the following builtin 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).
+
+ \startdesc{\hs{Bit}}
This is the most basic type available. It is mapped directly onto
the \type{std_logic} \small{VHDL} type. Mapping this to the
\type{bit} type might make more sense (since the Haskell version
only has two values), but using \type{std_logic} is more standard
(and allowed for some experimentation with don't care values)
+
+ \todo{Sidenote bit vs stdlogic}
\stopdesc
- \startdesc{Bool}
- This is a builtin Haskell type, but is translated exactly like the
- Bit type. Supporting the Bool type is particularly useful to support
- \hs{if ... then ... else ...} expressions, which always have a
- \hs{Bool} value for the condition.
+ \startdesc{\hs{Bool}}
+ This is the only builtin 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
+ always have a \hs{Bool} value for the condition.
A \hs{Bool} is translated to a \type{std_logic}, just like \hs{Bit}.
\stopdesc
- \startdesc{SizedWord, SizedInt}
- These are types to represent integers. \hs{SizedWord} is unsigned,
- while \hs{SizedInt} is signed. These types are parameterized by a
+ \startdesc{\hs{SizedWord}, \hs{SizedInt}}
+ These are types to represent integers. A \hs{SizedWord} is unsigned,
+ while a \hs{SizedInt} is signed. These types are parameterized by a
length type, so you can define an unsigned word of 32 bits wide as
follows:
type Word32 = SizedWord D32
\stophaskell
- Here, \hs{D32} is the \emph{type level representation} of the number
- 32. TODO: Introduce dependent types somewhere.
+ Here, a type synonym \hs{Word32} is defined that is equal to the
+ \hs{SizedWord} type constructor applied to the type \hs{D32}. \hs{D32}
+ is the \emph{type level representation} of the decimal number 32,
+ making the \hs{Word32} type a 32-bit unsigned word.
These types are translated to the \small{VHDL} \type{unsigned} and
\type{signed} respectively.
+ \todo{Sidenote on dependent typing?}
\stopdesc
- \startdesc{RangedWord}
+ \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
+ 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
+ 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:
+
+ \starthaskell
+ type RegisterState = Vector D8 Word32
+ \stophaskell
+
+ Here, a type synonym \hs{RegisterState} is defined that is equal to
+ the \hs{Vector} type constructor applied to the types \hs{D8} (The type
+ level representation of the decimal number 8) and \hs{Word32} (The 32
+ bit word type as defined above). In other words, the
+ \hs{RegisterState} type is a vector of 8 32-bit words.
+
+ A fixed size vector is translated to a \small{VHDL} array type.
+ \stopdesc
+ \startdesc{\hs{RangedWord}}
This is another type to describe integers, but unlike the previous
- two it has no specific width, but an upper bound. This means that
+ two it has no specific bitwidth, but an upper bound. This means that
its range is not limited to powers of two, but can be any number.
A \hs{RangedWord} only has an upper bound, its lower bound is
implicitly zero. 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.
- This type is translated to the \type{unsigned} \small{VHDL} type.
- \stopdesc
- \startdesc{TFVec}
- This is a vector type, with a fixed length. It has two type
- parameters: Its length and the type of the elements contained in it.
+ To define an index for the 8 element vector above, we would do:
- A fixed size vector is translated to a \small{VHDL} array type.
-
- The \hs{TF} in its name is a reference to the implementation used in
- the prototype, which uses \emph{type families}.
+ \starthaskell
+ type Register = 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.
+
+ This type is translated to the \type{unsigned} \small{VHDL} type.
\stopdesc
+ \fxnote{There should be a reference to Christiaan's work here.}
- TODO: Reference Christiaan / describe dependent typing
\subsection{User-defined types}
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
- expclitely excludes more advanced type creation from \GHC extensions
+ 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
\startdesc{Product types}
A product type is an algebraic datatype with a single constructor with
- two or more fields. This is essentially a way to pack a few values
- together in a record-like structure. In fact, the builtin tuple types
- are just product types (and are thus supported in exactly the same
- way).
+ 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
+ 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 cartesic
+ 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).
for our purposes this distinction does not really make a difference,
so we'll leave it out.
- Sum types are currently not supported by the prototype, since its
- translation is not so obvious as before. The can easily be emulated,
- as we will see from an example:
+ Sum types are currently not supported by the prototype, since there is
+ no obvious \VHDL alternative. They can easily be emulated, however, as
+ we will see from an example:
\starthaskell
data Sum = A Bit Word | B Word
type Sum = (SumC, Bit, Word, Word)
\stophaskell
- Here, the \hs{SumC} type effectively signals which of the fields of
- the \hs{Sum} type are valid, all the other ones have no useful value.
+ Here, the \hs{SumC} type effectively signals which of the latter three
+ 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
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!
+ not sharing these.
\stopdesc
Another interesting case is that of recursive types. In Haskell, an
probably the list type, which is defined is:
\starthaskell
- data List a = Empty | Cons a (List a)
+ data List t = Empty | Cons t (List t)
\stophaskell
- Where \hs{Empty} is usually written as \hs{[]} and \hs{Cons} as \hs{:}.
- This immediately shows the problem with recursive types: What hardware
- type to allocate here? There is no way to statically determine how long
- the list will be
+ 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
+ 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.
- In general, we can say we can never properly translated recursive types:
+ 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).
+
+ In general, we can say we can never properly translate recursive types:
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 determine an upper bound on its size).
\subsection{Partial application}
- It should be obvious that we cannot generate hardware signals for all
- expressions we can express in Haskell. The most obvious criterium for this
- is the type of an expression. We will see more of this below, but for now it
- should be obvious that any expression of a function type cannot be
- represented as a signal or i/o port to a component.
+ 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.
- From this, we can see that the above translation rules do not apply to a
- partial application. \in{Example}[ex:Quadruple] shows an example use of
- partial application and the corresponding architecture.
+ 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.
\startbuffer[Quadruple]
-- | Multiply the input word by four.
\stopcombination
Here, the definition of mul is a partial function application: It applies
- \hs{2 :: Word} to the function \hs{(*) :: Word -> Word -> Word} resulting in
- the expression \hs{(*) 2 :: Word -> Word}. Since this resulting expression
- is again a function, we can't generate hardware for it directly. This is
- because the hardware to generate for \hs{mul} depends completely on where
- and how it is used. In this example, it is even used twice!
+ 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 above. It might
- mean that a partial application is passed around quite a bit (even beyond
- function boundaries), but eventually, the partial application will become
- completely applied.
+ 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}
\section{Costless specialization}
Each (complete) function application in our description generates a
design. It is interesting to note that each application of a function
generates a \emph{separate} piece of hardware. In the final design, none
of the hardware is shared between applications, even when the applied
- function is the same.
+ 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
calls to the same function share the same machine code. Having more
synthesis, the amount of hardware generated is not affected.
In particular, if we would duplicate all functions so that there is a
- duplicate for every application in the program (\eg, each function is then
- only applied exactly once), there would be no increase in hardware size
- whatsoever.
+ separate function for every application in the program (\eg, each function
+ is then only applied exactly once), there would be no increase in hardware
+ size whatsoever.
+
+ Because of this, a common optimization technique called
+ \emph{specialization} can be applied to hardware generation without any
+ performance or area cost (unlike for software).
- TODO: Perhaps these next two sections are a bit too
- implementation-oriented?
+ \fxnote{Perhaps these next three sections are a bit too
+ implementation-oriented?}
\subsection{Specialization}
- Because of this, a common optimization technique called
- \emph{specialization} is as good as free for hardware generation. Given
- some function that has a \emph{domain} of $D$ (\eg, the set of all
+ \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 an domain of $D'
- \subset D$. This subset can be derived in all sort of ways, but commonly
- this is done by limiting a polymorphic argument to a single type (\eg,
- removing polymorphism) or by limiting an argument to just a single value
- (\eg, cross-function constant propagation, effectively removing the
- argument).
+ 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
+ value (\ie, cross-function constant propagation, effectively removing
+ the argument).
Since we limit the argument domain of the specialized function, its
definition can often be optimized further (since now more types or even
- values of arguments are already know). By replacing any application of
+ values of arguments are already known). By replacing any application of
the function that falls within the reduced domain by an application of
the specialized version, the code gets faster (but the code also gets
- bigger, since we now have two versions instead of one!). If we apply
+ bigger, since we now have two versions instead of one). If we apply
this technique often enough, we can often replace all applications of a
function by specialized versions, allowing the original function to be
removed (in some cases, this can even give a net reduction of the code
What holds for partial application, can be easily generalized to any
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 case
+ expressions and more complex expressions with a function type (a \hs{case}
expression returning lambda's, for example).
Each of these values cannot be directly represented in hardware (just like
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.
-
- TODO: This is section should be improved
+
+ \fxnote{This section needs improvement and an example}
\section{Polymorphism}
- In Haskell, values can be 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 element types. Haskell
+ 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
- general idea is the same.
+ 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
% value of a type that is member of the \hs{Integral} type class
When generating hardware, polymorphism can't be easily translated. How
- many wire will you lay down for a value that could have any type? When
+ 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}).
Fortunately, we can again use the principle of specialization: Since every
- function application generates separate pieces of hardware, we can know
+ 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). Our top level function must not have
- a polymorphic type (otherwise we wouldn't know the hardware interface to
- our top level function).
+ 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).
+
By induction, this means that all functions that are (indirectly) called
by our top level function (meaning all functions that are translated in
the final hardware) become monomorphic.
\section{State}
A very important concept in hardware designs is \emph{state}. In a
- stateless (or, \emph{combinatoric}) design, every output is a directly and solely dependent on the
+ stateless (or, \emph{combinatoric}) 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
the start of every clockcycle. Since the updating of the state is tightly
coupled (synchronized) to the clock signal, these state updates are often
- called \emph{synchronous}.
+ called \emph{synchronous} behaviour.
+
+ \todo{Sidenote? Registers can contain any (complex) type}
- To make our hardware description language useful to describe more that
+ 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.
\subsection{Approaches to state}
In Haskell, functions are always pure (except when using unsafe
- functions like \hs{unsafePerformIO}, which should be prevented whenever
- possible). This means that the output of a function solely depends on
- its inputs. If you evaluate a given function with given inputs, it will
- always provide the same output.
+ functions with the \hs{IO} monad, which is not supported by Cλash). This
+ means that the output of a function solely depends on its inputs. If you
+ evaluate a given function with given inputs, it will always provide the
+ same output.
- TODO: Define pure
+ \todo{Define pure}
This is a perfect match for a combinatoric circuit, where the output
- also soley depend on the inputs. However, when state is involved, this
- no longer holds. Since we're in charge of our own language, 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.
+ 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.
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
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 properly analyze by the compiler.
+ wich 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,
- ForSyDe, ...): Make functions operate on complete streams. This means
+ functional \small{HDL}s in the field: \todo{References to Lava,
+ ForSyDe, ...} Make functions operate on complete streams. This means
that a function is no longer called on every cycle, but just once. It
takes stream as inputs instead of values, where each stream contains
all the values for every clockcycle since system start. This is easily
modeled using an (infinite) list, with one element for each clock
- cycle. Since the funciton is only evaluated once, its output is also a
- stream. Note that, since we are working with infinite lists and still
- want to be able to simulate the system cycle-by-cycle, this relies
- heavily on the lazy semantics of Haskell.
+ cycle. Since the function is only evaluated once, its output must also
+ be a stream. Note that, since we are working with infinite lists and
+ still want to be able to simulate the system cycle-by-cycle, this
+ relies heavily on the lazy semantics of Haskell.
Since our inputs and outputs are streams, all other (intermediate)
values must be streams. All of our primitive operators (\eg, addition,
This notation can be confusing (especially due to the loop in the
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{outl (which is the output of the
- adder)}, and it's output is the \hs{delay out 0} (which is connected
- to one of the adder inputs).
+ 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
+ connected to one of the adder inputs).
This notation has a number of downsides, amongst which are limited
- readability and ambiguity in the interpretation. TODO: Reference
- Christiaan.
+ 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
function pure (letting the result depend only on the arguments), since
the current state is now an argument.
- In Haskell, this would look like \in{example}[ex:ExplicitAcc].
+ 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}
\startbuffer[ExplicitAcc]
-- input -> current state -> (new state, output)
acc :: Word -> Word -> (Word, Word)
-acc in (State s) = (State s', out)
+acc in s = (s', out)
where
out = s + in
s' = out
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 wether it's stateful at all).
+ uses (or whether it's stateful at all).
This approach is the one chosen for Cλash and will be examined more
closely below.
\subsubsection{Substates}
Since a function's state is reflected directly in its type signature,
- if a function calls other stateful functions (\eg, has subcircuits) it
+ if a function calls other stateful functions (\eg, has subcircuits), it
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.
This also means that the type of a function (at least the "state"
- part) is dependent on its implementation and the functions it calls.
+ part) is dependent on its own implementation and of the functions it
+ calls.
+
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
- practice. Additionally, when using a type synonym for the state type
+ practice.
+
+ Additionally, when using a type synonym for the state type
of each function, we can still provide explicit type signatures
while keeping the state specification for a function near its
definition only.
+ \todo{Example}
- \subsubsection{...}
+ \subsubsection{Which arguments and results are stateful?}
+ \fxnote{This section should get some examples}
We need some way to know which arguments should become input ports and
which argument(s?) should become the current state (\eg, be bound to
- the register outputs). This does not hold holds not just for the top
- level function, but also for any subfunctions. Or could we perhaps
+ the register outputs). This does not hold just for the top
+ level function, but also for any subfunction. Or could we perhaps
deduce the statefulness of subfunctions by analyzing the flow of data
in the calling functions?
- To explore this matter, we make an interesting observation: 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 output
- ports. Effectively, a stateful function results in a stateless
+ 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
+ output ports. Effectively, a stateful function results in a stateless
hardware component that has one of its input ports connected to the
output of a register and one of its output ports connected to the
input of the same register.
- TODO: Example?
+ \todo{Example}
Of course, even though the hardware described like this has the
correct behaviour, unless the layout tool does smart optimizations,
there will be a lot of extra wire in the design (since registers will
not be close to the component that uses them). Also, when working with
the generated \small{VHDL} code, there will be a lot of extra ports
- just to pass one state values, which can get quite confusing.
+ just to pass on state values, which can get quite confusing.
To fix this, we can simply \quote{push} the registers down into the
subcircuits. When we see a register that is connected directly to a
trivial when looking at the Haskell code instead of the resulting
circuit, but the idea is still the same.
- TODO: Example?
+ \todo{Example}
However, when applying this technique, we might push registers down
too far. When you intend to store a result of a stateless subfunction
the called function is in fact stateful. From this we can conclude
that we have to either:
+ \todo{Example of wrong downpushing}
+
\startitemize
\item accept that the generated hardware might not be exactly what we
intended, in some specific cases. In most cases, the hardware will be
end up is easier to implement correctly with explicit annotations, so
for these reasons we will look at how this annotations could work.
-
- TODO: Note about conditions on state variables and checking them.
+ \todo{Sidenote: One or more state arguments?}
\subsection{Explicit state annotation}
To make our stateful descriptions unambigious and easier to translate,
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, \eg give stateful
- arguments and (part of) results a different type. This has the
+ \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
implemented in Cλash. \in{Section}[sec:prototype:statetype] expands on
the possible ways this could have been implemented.
- TODO: Say something about dependent types and fixed size vectors
+ \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 function that is defined in terms of itself. This
+ form, recursion is a definition that is defined 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.
However, most recursive hardware descriptions will describe finite
recursion. This is because the recursive call is done conditionally. There
- is usually a case statement where at least one alternative does not contain
+ 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
- recursive function, we would be able to detect which alternative applies,
- we would be able to remove the case expression and leave only the base case
- when it applies. This will ensure that expanding the recursive functions
- will terminate after a bounded number of expansions.
+ recursive function, we would be able to detect at compile time which
+ alternative applies, we would be able to remove the \hs{case} expression and
+ leave only the base case when it applies. This will ensure that expanding
+ the recursive functions will terminate after a bounded number of expansions.
This does imply the extra requirement that the base case is detectable at
compile time. In particular, this means that the decision between the base
passed in as an argument. Since we represent lists as vectors that encode
the length in the vector type, it seems easy to determine the base case. We
can simply look at the argument type for this. However, it turns out that
- this is rather non-trivial to write down in Haskell in the first place. As
- an example, we would like to write down something like this:
+ this is rather non-trivial to write down in Haskell already, not even
+ looking at translation. As an example, we would like to write down something
+ like this:
\starthaskell
sum :: Vector n Word -> Word
False -> head xs + sum (tail xs)
\stophaskell
- However, the typechecker will now use the following reasoning (element type
- of the vector is left out):
+ 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}.
+ \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}
\item sum is called with the result of tail as an argument, which has the
- type \hs{Vector n} (since \hs{(n > 0) => n - 1 == m}).
+ 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}).
\item This means that sum must have the type \hs{Vector n -> a}
\item This is a contradiction between the type deduced from the body of sum
(the input vector must be non-empty) and the use of sum (the input vector
could have any length).
\stopitemize
- As you can see, using a simple case at value level causes the type checker
- to always typecheck both alternatives, which can't be done! This is a
- fundamental problem, that would seem perfectly suited for a type class.
- Considering that we need to switch between to implementations of the sum
- function, based on the type of the argument, this sounds like the perfect
- problem to solve with a type class. However, this approach has its own
- problems (not the least of them that you need to define a new typeclass for
- every recursive function you want to define).
+ 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.
- TODO: How much detail should there be here? I can probably refer to
- Christiaan instead.
+ \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}
\subsection{General recursion}
Of course there are other forms of recursion, that do not depend on the
counter could be expressed, but only translated to hardware for a fixed
number of iterations. Also, this would require extensive support for compile
time simplification (constant propagation) and compile time evaluation
- (evaluation constant comparisons), to ensure non-termination. Even then, it
+ (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, we leave other forms of recursion as
- future work as well.
+ Due to these complications and limited time available, we leave other forms
+ of recursion as future work as well.
this does not concern the \emph{implementation language} of the compiler,
just the language \emph{translated by} the compiler).
- On the highest level, we have two choices:
+ Initially, we have two choices:
\startitemize
\item Create a new functional language from scratch. This has the
advantage of having a language that contains exactly those elements that
are convenient for describing hardware and can contain special
- constructs that might.
+ constructs that allows our hardware descriptions to be more powerful or
+ concise.
\item Use an existing language and create a new backend for it. This has
the advantage that existing tools can be reused, which will speed up
development.
\stopitemize
+ \todo{Sidenote: No EDSL}
+
Considering that we required a prototype which should be working quickly,
and that implementing parsers, semantic checkers and especially
- typcheckers isn't exactly the core of this research (but it is lots and
+ typcheckers is not exactly the core of this research (but it is lots and
lots of work!), using an existing language is the obvious choice. This
also has the advantage that a large set of language features is available
to experiment with and it is easy to find which features apply well and
just the useful features (and possibly extra features that are specific to
the domain of hardware description as well).
- The second choice is to pick one of the many existing languages. As
- mentioned before, this language is Haskell. This choice has not been the
+ The second choice is which of the many existing languages to use. As
+ mentioned before, the chosen language is Haskell. This choice has not been the
result of a thorough comparison of languages, for the simple reason that
the requirements on the language were completely unclear at the start of
- this language. The fact that Haskell is a language with a broad spectrum
+ this research. The fact that Haskell is a language with a broad spectrum
of features, that it is commonly used in research projects and that the
- primary compiler, GHC, provides a high level API to its internals, made
+ primary compiler, \GHC, provides a high level API to its internals, made
Haskell an obvious choice.
- TODO: Was Haskell really a good choice? Perhaps say this somewhere else?
+ \note{There should be evaluation of the choice of Haskell and \VHDL}
\section[sec:prototype:output]{Output format}
The second important question is: What will be our output format? Since
However, \small{EDIF} is not completely tool-independent. It specifies a
meta-format, but the hardware components that can be used vary between
various tool and hardware vendors, as well as the interpretation of the
- \small{EDIF} standard (TODO Is this still true? Reference:
- http://delivery.acm.org/10.1145/80000/74534/p803-li.pdf?key1=74534\&key2=8370537521\&coll=GUIDE\&dl=GUIDE\&CFID=61207158\&CFTOKEN=61908473).
+ \small{EDIF} standard. \todo{Is this still true? Reference:
+ http://delivery.acm.org/10.1145/80000/74534/p803-li.pdf?key1=74534\&key2=8370537521\&coll=GUIDE\&dl=GUIDE\&CFID=61207158\&CFTOKEN=61908473}
- This means that when working with EDIF, our prototype would become
+ This means that when working with \small{EDIF}, our prototype would become
technology dependent (\eg only work with \small{FPGA}s of a specific
vendor, or even only with specific chips). This limits the applicability
of our prototype. Also, the tools we'd like to use for verifying,
simulating and draw pretty pictures of our output (like Precision, or
- QuestaSim) work on \small{VHDL} or Verilog input (TODO: Is this really
- true?).
+ QuestaSim) are designed for \small{VHDL} or Verilog input.
- For these reasons, we will use \small{VHDL} as our output language.
- Verilog is not used simply because we are familiar with \small{VHDL}
- already. The differences between \small{VHDL} and Verilog are on the
- higher level, while we will be using \small{VHDL} mainly to write low
- level, netlist-like descriptions anyway.
+ For these reasons, we will not use \small{EDIF}, but \small{VHDL} as our
+ output language. We choose \VHDL over Verilog simply because we are
+ familiar with \small{VHDL} already. The differences between \small{VHDL}
+ and Verilog are on the higher level, while we will be using \small{VHDL}
+ mainly to write low level, netlist-like descriptions anyway.
An added advantage of using VHDL is that we can profit from existing
optimizations in VHDL synthesizers. A lot of optimizations are done on the
Note that we will be using \small{VHDL} as our output language, but will
not use its full expressive power. Our output will be limited to using
simple, structural descriptions, without any behavioural descriptions
- (which might not be supported by all tools).
+ (which might not be supported by all tools). This ensures that any tool
+ that works with \VHDL will understand our output (most tools don't support
+ synthesis of more complex \VHDL). This also leaves open the option to
+ switch to \small{EDIF} in the future, with minimal changes to the
+ prototype.
\section{Prototype design}
- As stated above, we will use the Glasgow Haskell Compiler (\small{GHC}) to
+ As suggested above, we will use the Glasgow Haskell Compiler (\small{GHC}) to
implement our prototype compiler. To understand the design of the
compiler, we will first dive into the \small{GHC} compiler a bit. It's
compilation consists of the following steps (slightly simplified):
% Create objects
save inp, front, desugar, simpl, back, out;
newEmptyBox.inp(0,0);
- newBox.front(btex Parser etex);
+ newBox.front(btex Fronted etex);
newBox.desugar(btex Desugarer etex);
newBox.simpl(btex Simplifier etex);
newBox.back(btex Backend etex);
\emph{Core} language. Core is a very small functional language with lazy
semantics, that can still express everything Haskell can express. Its
simpleness makes Core very suitable for further simplification and
- translation. Core is the language we will be working on as well.
+ translation. Core is the language we will be working with as well.
\stopdesc
\startdesc{Simplification}
Through a number of simplification steps (such as inlining, common
representation as well.
However, we will use the normal core representation, not the simplified
- core. Reasons for this are detailed below.
+ core. Reasons for this are detailed below. \todo{Ref}
The final prototype roughly consists of three steps:
- \startuseMPgraphic{ghc-pipeline}
+ \startuseMPgraphic{clash-pipeline}
% Create objects
save inp, front, norm, vhdl, out;
newEmptyBox.inp(0,0);
% Draw the objects (and deferred labels)
drawObj (inp, front, norm, vhdl, out);
\stopuseMPgraphic
- \placefigure[right]{GHC compiler pipeline}{\useMPgraphic{ghc-pipeline}}
+ \placefigure[right]{Cλash compiler pipeline}{\useMPgraphic{clash-pipeline}}
\startdesc{Frontend}
This is exactly the frontend and desugarer from the \small{GHC}
\in{chapter}[chap:normalization].
\section{The Core language}
+ \defreftxt{core}{the Core language}
Most of the prototype deals with handling the program in the Core
language. In this section we will show what this language looks like and
how it works.
Each Core expression consists of one of these possible expressions.
\startdesc{Variable reference}
+ \defref{variable reference}
\startlambda
- a
+ x
\stoplambda
- This is a simple reference to a binder. It's written down as the
+ This is a reference to a binder. It's written down as the
name of the binder that is being referred to, which should of course be
bound in a containing scope (including top level scope, so a reference
to a top level function is also a variable reference). Additionally,
\stopdesc
\startdesc{Literal}
+ \defref{literal}
\startlambda
10
\stoplambda
- This is a simple literal. Only primitive types are supported, like
+ This is a literal. Only primitive types are supported, like
chars, strings, ints and doubles. The types of these literals are the
\quote{primitive} versions, like \lam{Char\#} and \lam{Word\#}, not the
normal Haskell versions (but there are builtin conversion functions).
\stopdesc
\startdesc{Application}
+ \defref{application}
\startlambda
func arg
\stoplambda
- This is simple function application. Each application consists of two
+ This is function application. Each application consists of two
parts: The function part and the argument part. Applications are used
for normal function \quote{calls}, but also for applying type
abstractions and data constructors.
\stopdesc
\startdesc{Lambda abstraction}
+ \defref{lambda abstraction}
\startlambda
λbndr.body
\stoplambda
\stopdesc
\startdesc{Non-recursive let expression}
+ \defref{let expression}
\startlambda
let bndr = value in body
\stoplambda
The main difference with the normal let expression is that each of the
binders is in scope in each of the values, in addition to the body. This
- allows for self-recursive definitions or mutually recursive definitions.
+ allows for self-recursive or mutually recursive definitions.
It should also be possible to express a recursive let using normal
lambda calculus, if we use the \emph{least fixed-point operator},
- \lam{Y}.
+ \lam{Y}. This falls beyond the scope of this report, since it is not
+ needed for this research.
\stopdesc
\startdesc{Case expression}
+ \defref{case expression}
\startlambda
- case scrut of bndr
+ case scrutinee of bndr
DEFAULT -> defaultbody
C0 bndr0,0 ... bndr0,m -> body0
\vdots
Cn bndrn,0 ... bndrn,m -> bodyn
\stoplambda
- TODO: Define WHNF
+ \todo{Define WHNF}
- A case expression is the only way in Core to choose between values. A case
- expression evaluates its scrutinee, which should have an algebraic
- datatype, into weak head normal form (\small{WHNF}) and (optionally) binds
- it to \lam{bndr}. It then chooses a body depending on the constructor of
- its scrutinee. If none of the constructors match, the \lam{DEFAULT}
- alternative is chosen.
+ A case expression is the only way in Core to choose between values. All
+ \hs{if} expressions and pattern matchings from the original Haskell
+ PRogram have been translated to case expressions by the desugarer.
+
+ A case expression evaluates its scrutinee, which should have an
+ algebraic datatype, into weak head normal form (\small{WHNF}) and
+ (optionally) binds it to \lam{bndr}. It then chooses a body depending on
+ the constructor of its scrutinee. If none of the constructors match, the
+ \lam{DEFAULT} alternative is chosen. A case expression must always be
+ exhaustive, \ie it must cover all possible constructors that the
+ scrutinee can have (if all of them are covered explicitly, the
+ \lam{DEFAULT} alternative can be left out).
Since we can only match the top level constructor, there can be no overlap
in the alternatives and thus order of alternatives is not relevant (though
binders after the constructor and are in scope only in the corresponding
body.
- To support strictness, the scrutinee is always evaluated into WHNF, even
- when there is only a \lam{DEFAULT} alternative. This allows a strict
- function argument to be written like:
+ To support strictness, the scrutinee is always evaluated into
+ \small{WHNF}, even when there is only a \lam{DEFAULT} alternative. This
+ allows aplication of the strict function \lam{f} to the argument \lam{a}
+ to be written like:
\startlambda
- function (case argument of arg
- DEFAULT -> arg)
+ f (case a of arg DEFAULT -> arg)
\stoplambda
- This seems to be the only use for the extra binder to which the scrutinee
- is bound. When not using strictness annotations (which is rather pointless
- in hardware descriptions), \small{GHC} seems to never generate any code
- making use of this binder. The current prototype does not handle it
- either, which probably means that code using it would break.
+ According to the \GHC documentation, this is the only use for the extra
+ binder to which the scrutinee is bound. When not using strictness
+ annotations (which is rather pointless in hardware descriptions),
+ \small{GHC} seems to never generate any code making use of this binder.
+ In fact, \GHC has never been observed to generate code using this
+ binder, even when strictness was involved. Nonetheless, the prototype
+ handles this binder as expected.
Note that these case statements are less powerful than the full Haskell
case statements. In particular, they do not support complex patterns like
- in Haskell. Only the constructor of an expression can be matched, complex
- patterns are implemented using multiple nested case expressions.
+ in Haskell. Only the constructor of an expression can be matched,
+ complex patterns are implemented using multiple nested case expressions.
Case statements are also used for unpacking of algebraic datatypes, even
when there is only a single constructor. For examples, to add the elements
\stopdesc
\startdesc{Cast expression}
+ \defref{cast expression}
\startlambda
body :: targettype
\stoplambda
In our hardware descriptions, we typically see casts to change between a
Haskell newtype and its contained type, since those are effectively
- different representations of the same type.
+ different types (so a cast is needed) with the same representation (but
+ no work is done by the cast).
More complex are types that are proven to be equal by the typechecker,
but look different at first glance. To ensure that, once the typechecker
has proven equality, this information sticks around, explicit casts are
added. In our notation we only write the target type, but in reality a
cast expressions carries around a \emph{coercion}, which can be seen as a
- proof of equality. TODO: Example
+ proof of equality. \todo{Example}
The value of a cast is the value of its body, unchanged. The type of this
value is equal to the target type, not the type of its body.
+ \todo{Move this paragraph}
Note that this syntax is also used sometimes to indicate that a particular
expression has a particular type, even when no cast expression is
involved. This is then purely informational, since the only elements that
\stopdesc
\startdesc{Note}
-
The Core language in \small{GHC} allows adding \emph{notes}, which serve
as hints to the inliner or add custom (string) annotations to a core
expression. These shouldn't be generated normally, so these are not
\stopdesc
\startdesc{Type}
+ \defref{type expression}
\startlambda
@type
\stoplambda
- It is possibly to use a Core type as a Core expression. This is done to
- allow for type abstractions and applications to be handled as normal
- lambda abstractions and applications above. This means that a type
- expression in Core can only ever occur in the argument position of an
- application, and only if the type of the function that is applied to
- expects a type as the first argument. This happens for all polymorphic
- functions, for example, the \lam{fst} function:
+ It is possibly to use a Core type as a Core expression. For the actual
+ types supported by Core, see \in{section}[sec:prototype:coretypes]. This
+ \quote{lifting} of a type into the value domain is done to allow for
+ type abstractions and applications to be handled as normal lambda
+ abstractions and applications above. This means that a type expression
+ in Core can only ever occur in the argument position of an application,
+ and only if the type of the function that is applied to expects a type
+ as the first argument. This happens for all polymorphic functions, for
+ example, the \lam{fst} function:
\startlambda
fst :: \forall a. \forall b. (a, b) -> a
\lam{fst @Int @Int :: (Int, Int) -> Int}).
\stopdesc
- \subsection{Core type system}
+ \subsection[sec:prototype:coretypes]{Core type system}
Whereas the expression syntax of Core is very simple, its type system is
a bit more complicated. It turns out it is harder to \quote{desugar}
Haskell's complex type system into something more simple. Most of the
We will slightly limit our view on Core's type system, since the more
complicated parts of it are only meant to support Haskell's (or rather,
- \GHC's) type extensions, such as existential types, type families and
- other non-standard Haskell stuff which we don't (plan to) support.
+ \GHC's) type extensions, such as existential types, \small{GADT}s, type
+ families and other non-standard Haskell stuff which we don't (plan to)
+ support.
In Core, every expression is typed. The translation to Core happens
after the typechecker, so types in Core are always correct as well
\startdesc{A type variable}
\startlambda
- a
+ t
\stoplambda
This is a reference to a type defined elsewhere. This can either be a
- polymorphic type (like \hs{a} in \hs{id :: a -> a}), or a type
- constructor (like \hs{Bool} in \hs{null :: [a] -> Bool}).
+ polymorphic type (like the latter two \lam{t}'s in \lam{id :: \forall t.
+ t -> t}), or a type constructor (like \lam{Bool} in \lam{not :: Bool ->
+ Bool}). Like in Haskell, polymorphic type variables always
+ start with a lowercase letter, while type constructors always start
+ with an uppercase letter.
+
+ \todo{How to define (new) type constructors?}
A special case of a type constructor is the \emph{function type
- constructor}, \hs{->}. This is a type constructor taking two arguments
+ constructor}, \lam{->}. This is a type constructor taking two arguments
(using application below). The function type constructor is commonly
- written inline, so we write \hs{a -> b} when we really mean \hs{-> a
- b}, the function type constructor applied to a and b.
+ written inline, so we write \lam{a -> b} when we really mean \lam{-> a
+ b}, the function type constructor applied to \lam{a} and \lam{b}.
Polymorphic type variables can only be defined by a lambda
abstraction, see the forall type below.
This applies a some type to another type. This is particularly used to
apply type variables (type constructors) to their arguments.
- As mentioned above, some type applications have special notation. In
- particular, these are applications of the \emph{function type
- constructor} and \emph{tuple type constructors}:
- \starthaskell
+ As mentioned above, applications of some type constructors have
+ special notation. In particular, these are applications of the
+ \emph{function type constructor} and \emph{tuple type constructors}:
+ \startlambda
foo :: a -> b
foo' :: -> a b
bar :: (a, b, c)
bar' :: (,,) a b c
- \stophaskell
+ \stoplambda
\stopdesc
\startdesc{The forall type}
\startlambda
- id :: \forall a . a -> a
+ id :: \forall a. a -> a
\stoplambda
The forall type introduces polymorphism. It is the only way to
introduce new type variables, which are completely unconstrained (Any
id = λa.λx.x
\stoplambda
- here, type type of the binder \lam{x} is \lam{a}, referring to the
+ Here, the type of the binder \lam{x} is \lam{a}, referring to the
binder in the topmost lambda.
When using a value with a forall type, the actual type
used must be applied first. For example haskell expression \hs{id
- True} translates to the following Core:
+ True} (the function \hs{id} appleid to the dataconstructor \hs{True})
+ translates to the following Core:
\startlambda
id @Bool True
\stoplambda
Here, id is first applied to the type to work with. Note that the type
- then changes from \lam{id :: \forall a . a -> a} to \lam{id @Bool ::
+ then changes from \lam{id :: \forall a. a -> a} to \lam{id @Bool ::
Bool -> Bool}. Note that the type variable \lam{a} has been
substituted with the actual type.
+
+ In Haskell, forall types are usually not explicitly specified (The use
+ of a lowercase type variable implicitly introduces a forall type for
+ that variable). In fact, in standard Haskell there is no way to
+ explicitly specify forall types. Through a language extension, the
+ \hs{forall} keyword is available, but still optional for normal forall
+ types (it is needed for \emph{existentially quantified types}, which
+ Cλash does not support).
\stopdesc
\startdesc{Predicate type}
\startlambda
- show :: \forall a. Show s => s -> String
+ show :: \forall a. Show s ⇒ s → String
\stoplambda
- TODO: Introduce type classes?
+ \todo{Sidenote: type classes?}
A predicate type introduces a constraint on a type variable introduced
by a forall type (or type lambda). In the example above, the type
variable \lam{a} can only contain types that are an \emph{instance} of
- the \emph{type class} \lam{Show}.
+ the \emph{type class} \lam{Show}. \refdef{type class}
There are other sorts of predicate types, used for the type families
extension, which we will not discuss here.
instance declaration. This dictionary, as well as the binder
introduced by a lambda that introduces a dictionary, have the
predicate type as their type. These binders are usually named starting
- with a \lam{\$}. Usually the name of the type concerned is not
+ with a \lam{$}. Usually the name of the type concerned is not
reflected in the name of the dictionary, but the name of the type
class is. The Haskell expression \hs{show True} thus becomes:
\stopdesc
Using this set of types, all types in basic Haskell can be represented.
+
+ \todo{Overview of polymorphism with more examples (or move examples
+ here)}.
\section[sec:prototype:statetype]{State annotations in Haskell}
- TODO: This section should be reviewed and expanded.
+ \fxnote{This entire section on state annotations should be reviewed}
Ideal: Type synonyms, since there is no additional code overhead for
packing and unpacking. Downside: there is no explicit conversion in Core
Implementation issues: state splitting, linking input to output state,
checking usage constraints on state variables.
- TODO: Implementation issues
- TODO: \subsection[sec:prototype:separate]{Separate compilation}
- TODO: Simplified core?
+ \todo{Implementation issues: Separate compilation, simplified core.}
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