\section{Function application}
The basic syntactic element of a functional program are functions and
- function application. These have a single obvious VHDL translation: Each
+ 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.
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.
- An example of a simple program using only function application would be:
+ \in{Example}[ex:And3] shows a simple program using only function
+ application and the corresponding architecture.
- \starthaskell
- -- | A simple function that returns the and of three bits
- and3 :: Bit -> Bit -> Bit -> Bit
- and3 a b c = and (and a b) c
- \stophaskell
+\startbuffer[And3]
+-- | A simple function that returns
+-- the and of three bits
+and3 :: Bit -> Bit -> Bit -> Bit
+and3 a b c = and (and a b) c
+\stopbuffer
- This results in the following hardware:
-
- TODO: Pretty picture
+ \startuseMPgraphic{And3}
+ save a, b, c, anda, andb, out;
+
+ % I/O ports
+ newCircle.a(btex $a$ etex) "framed(false)";
+ newCircle.b(btex $b$ etex) "framed(false)";
+ newCircle.c(btex $c$ etex) "framed(false)";
+ newCircle.out(btex $out$ etex) "framed(false)";
+
+ % Components
+ newCircle.anda(btex $and$ etex);
+ newCircle.andb(btex $and$ etex);
+
+ a.c = origin;
+ b.c = a.c + (0cm, 1cm);
+ c.c = b.c + (0cm, 1cm);
+ anda.c = midpoint(a.c, b.c) + (2cm, 0cm);
+ andb.c = midpoint(b.c, c.c) + (4cm, 0cm);
+
+ out.c = andb.c + (2cm, 0cm);
+
+ % Draw objects and lines
+ drawObj(a, b, c, anda, andb, out);
+
+ ncarc(a)(anda) "arcangle(-10)";
+ ncarc(b)(anda);
+ ncarc(anda)(andb);
+ ncarc(c)(andb);
+ ncline(andb)(out);
+ \stopuseMPgraphic
+
+ \placeexample[here][ex:And3]{Simple three port and.}
+ \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.
\subsection{Partial application}
It should be obvious that we cannot generate hardware signals for all
represented as a signal or i/o port to a component.
From this, we can see that the above translation rules do not apply to a
- partial application. Let's look at an example:
+ partial application. \in{Example}[ex:Quadruple] shows an example use of
+ partial application and the corresponding architecture.
- \starthaskell
- -- | Multiply the input word by four.
- quadruple :: Word -> Word
- quadruple n = mul (mul n)
- where
- mul = (*) 2
- \stophaskell
+\startbuffer[Quadruple]
+-- | Multiply the input word by four.
+quadruple :: Word -> Word
+quadruple n = mul (mul n)
+ where
+ mul = (*) 2
+\stopbuffer
+
+ \startuseMPgraphic{Quadruple}
+ save in, two, mula, mulb, out;
+
+ % I/O ports
+ newCircle.in(btex $n$ etex) "framed(false)";
+ newCircle.two(btex $2$ etex) "framed(false)";
+ newCircle.out(btex $out$ etex) "framed(false)";
+
+ % Components
+ newCircle.mula(btex $\times$ etex);
+ newCircle.mulb(btex $\times$ etex);
- It should be clear that the above code describes the following hardware:
+ two.c = origin;
+ in.c = two.c + (0cm, 1cm);
+ mula.c = in.c + (2cm, 0cm);
+ mulb.c = mula.c + (2cm, 0cm);
+ out.c = mulb.c + (2cm, 0cm);
- TODO: Pretty picture
+ % Draw objects and lines
+ drawObj(in, two, mula, mulb, out);
+
+ nccurve(two)(mula) "angleA(0)", "angleB(45)";
+ nccurve(two)(mulb) "angleA(0)", "angleB(45)";
+ ncline(in)(mula);
+ ncline(mula)(mulb);
+ ncline(mulb)(out);
+ \stopuseMPgraphic
+
+ \placeexample[here][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
\hs{2 :: Word} to the function \hs{(*) :: Word -> Word -> Word} resulting in
function boundaries), but eventually, the partial application will become
completely applied.
- \section{Recursion}
+ \section{Costless specialization}
+ Each (complete) function application in our description generates a
+ component instantiation, or a specific piece of hardware in the final
+ 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.
+
+ 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.
+
+ 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.
+
+ 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.
+
+ TODO: Perhaps these next two 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
+ 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).
+
+ 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
+ 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
+ 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
+ compared to the non-specialized version).
+
+ 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
+ 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
+ variables (e.g., a binder referring to a top level function), lambda
+ expressions and more complex expressions with a function type (a case
+ expression returning lambda's, for example).
+
+ Each of these values cannot be directly represented in hardware (just like
+ partial applications). Also, to make them representable, they need to be
+ applied: function variables and partial applications will then eventually
+ become complete applications, applied lambda expressions disappear by
+ applying β-reduction, etc.
+
+ 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
+
+ \section{Polymorphism}
+ In Haskell, values can be 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
+ typeclasses allow a function to work on a specific set of types, but the
+ general idea is the same.
+
+% 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
+% 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
+ many wire 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)?
+
+ Fortunately, we can again use the principle of specialization: Since every
+ function application generates separate pieces 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).
+
+ 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
+ applied to monomorphic values. The applied functions can then be
+ specialized to work just for these specific types, removing the
+ polymorphism from the applied functions as well.
+
+ 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
+ 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}.
+
+ To make our hardware description language useful to describe more that
+ 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.
+
+ 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.
+
+ 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.
+
+ \subsubsection{Stream arguments and results}
+ Including the entire history of each input (\eg, the value of that
+ input for each previous clockcycle) is an obvious way to make outputs
+ depend on all previous input. This is easily done by making every
+ input a list instead of a single value, containing all previous values
+ as well as the current value.
+
+ 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.
+
+ 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
+ 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.
+
+ Since our inputs and outputs are streams, all other (intermediate)
+ values must be streams. All of our primitive operators (\eg, addition,
+ substraction, bitwise operations, etc.) must operate on streams as
+ well (note that changing a single-element operation to a stream
+ operation can done with \hs{map}, \hs{zipwith}, etc.).
+
+ 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
+ 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!).
+
+ \in{Example}[ex:DelayAcc] shows a simple accumulator expressed in this
+ style.
+
+\startbuffer[DelayAcc]
+acc :: Stream Word -> Stream Word
+acc in = out
+ where
+ out = (delay out 0) + in
+\stopbuffer
+
+\startuseMPgraphic{DelayAcc}
+ save in, out, add, reg;
+
+ % I/O ports
+ newCircle.in(btex $in$ etex) "framed(false)";
+ newCircle.out(btex $out$ etex) "framed(false)";
+
+ % Components
+ newReg.reg("") "dx(4mm)", "dy(6mm)", "reflect(true)";
+ newCircle.add(btex + etex);
+
+ in.c = origin;
+ add.c = in.c + (2cm, 0cm);
+ out.c = add.c + (2cm, 0cm);
+ reg.c = add.c + (0cm, 2cm);
+
+ % Draw objects and lines
+ drawObj(in, out, add, reg);
+
+ nccurve(add)(reg) "angleA(0)", "angleB(180)", "posB(d)";
+ nccurve(reg)(add) "angleA(180)", "angleB(-45)", "posA(out)";
+ ncline(in)(add);
+ ncline(add)(out);
+\stopuseMPgraphic
+
+
+ \placeexample[here][ex:DelayAcc]{Simple accumulator architecture.}
+ \startcombination[2*1]
+ {\typebufferhs{DelayAcc}}{Haskell description using streams.}
+ {\boxedgraphic{DelayAcc}}{The architecture described by the Haskell description.}
+ \stopcombination
+
+
+ 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).
+
+ This notation has a number of downsides, amongst which are limited
+ readability and ambiguity in the interpretation. TODO: Reference
+ Christiaan.
+
+ \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
+ both the inputs as well as the current state while keeping the
+ 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].
+
+\startbuffer[ExplicitAcc]
+-- input -> current state -> (new state, output)
+acc :: Word -> Word -> (Word, Word)
+acc in (State s) = (State s', out)
+ where
+ out = s + in
+ s' = out
+\stopbuffer
+
+ \placeexample[here][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.
+ {\boxedgraphic{DelayAcc}}{The architecture described by the Haskell description.}
+ \stopcombination
+
+ This approach makes a function's state very explicit, which state
+ 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).
+
+ This approach is the one chosen for Cλash and 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?
+
+ \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
+ 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.
+ 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
+ of each function, we can still provide explicit type signatures
+ while keeping the state specification for a function near its
+ definition only.
+
+ \subsubsection{...}
+ 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
+ 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
+ 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?
+
+ 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.
+
+ 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
+ subcircuit, we remove the corresponding input and output port and put
+ the register inside the subcircuit instead. This is slightly less
+ trivial when looking at the Haskell code instead of the resulting
+ circuit, but the idea is still the same.
+
+ 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
+ in the caller's state and pass the current value of that state
+ variable to that same function, the register might get pushed down too
+ far. It is impossible to distinguish this case from similar code where
+ the called function is in fact stateful. From this we can conclude
+ that we have to either:
+
+ \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
+ what we intended.
+ \item explicitely annotate state arguments and results in the input
+ description.
+ \stopitemize
+
+ The first option causes (non-obvious) exceptions in the language
+ intepretation. Also, automatically determining where registers should
+ 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.
+
+ \subsection{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.
+
+ Roughly, we have two ways to achieve this:
+ \startitemize[KR]
+ \item Use some kind of annotation method or syntactic construction in
+ the language to indicate exactly which argument and (part of the)
+ 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
+ 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
+ the translation process a lot easier, since less analysis of the
+ program flow might be required.
+ \stopitemize
+
+ From these approaches, the type level \quote{annotations} have been
+ 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
+
+ \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
usually requires multiple evaluations of this function, with changing
Due to these complications, we leave other forms of recursion as
future work as well.
+
+ \section{Supported types}
+ TODO
projects / matthijs / master-project / report.git / blobdiff