1 \chapter[chap:description]{Hardware description}
2 This chapter will provide an overview of the hardware description language
3 that was created and the issues that have arisen in the process. It will
4 focus on the issues of the language, not the implementation.
6 When translating Haskell to hardware, we need to make choices in what kind
7 of hardware to generate for what Haskell constructs. When faced with
8 choices, we've tried to stick with the most obvious choice wherever
9 possible. In a lot of cases, when you look at a hardware description it is
10 comletely clear what hardware is described. We want our translator to
11 generate exactly that hardware whenever possible, to minimize the amount of
12 surprise for people working with it.
14 In this chapter we try to describe how we interpret a Haskell program from a
15 hardware perspective. We provide a description of each Haskell language
16 element that needs translation, to provide a clear picture of what is
19 \section{Function application}
20 The basic syntactic element of a functional program are functions and
21 function application. These have a single obvious \small{VHDL} translation: Each
22 function becomes a hardware component, where each argument is an input port
23 and the result value is the output port.
25 Each function application in turn becomes component instantiation. Here, the
26 result of each argument expression is assigned to a signal, which is mapped
27 to the corresponding input port. The output port of the function is also
28 mapped to a signal, which is used as the result of the application.
30 \in{Example}[ex:And3] shows a simple program using only function
31 application and the corresponding architecture.
34 -- | A simple function that returns
35 -- the and of three bits
36 and3 :: Bit -> Bit -> Bit -> Bit
37 and3 a b c = and (and a b) c
40 \startuseMPgraphic{And3}
41 save a, b, c, anda, andb, out;
44 newCircle.a(btex $a$ etex) "framed(false)";
45 newCircle.b(btex $b$ etex) "framed(false)";
46 newCircle.c(btex $c$ etex) "framed(false)";
47 newCircle.out(btex $out$ etex) "framed(false)";
50 newCircle.anda(btex $and$ etex);
51 newCircle.andb(btex $and$ etex);
54 b.c = a.c + (0cm, 1cm);
55 c.c = b.c + (0cm, 1cm);
56 anda.c = midpoint(a.c, b.c) + (2cm, 0cm);
57 andb.c = midpoint(b.c, c.c) + (4cm, 0cm);
59 out.c = andb.c + (2cm, 0cm);
61 % Draw objects and lines
62 drawObj(a, b, c, anda, andb, out);
64 ncarc(a)(anda) "arcangle(-10)";
71 \placeexample[here][ex:And3]{Simple three port and.}
72 \startcombination[2*1]
73 {\typebufferhs{And3}}{Haskell description using function applications.}
74 {\boxedgraphic{And3}}{The architecture described by the Haskell description.}
77 \subsection{Partial application}
78 It should be obvious that we cannot generate hardware signals for all
79 expressions we can express in Haskell. The most obvious criterium for this
80 is the type of an expression. We will see more of this below, but for now it
81 should be obvious that any expression of a function type cannot be
82 represented as a signal or i/o port to a component.
84 From this, we can see that the above translation rules do not apply to a
85 partial application. \in{Example}[ex:Quadruple] shows an example use of
86 partial application and the corresponding architecture.
88 \startbuffer[Quadruple]
89 -- | Multiply the input word by four.
90 quadruple :: Word -> Word
91 quadruple n = mul (mul n)
96 \startuseMPgraphic{Quadruple}
97 save in, two, mula, mulb, out;
100 newCircle.in(btex $n$ etex) "framed(false)";
101 newCircle.two(btex $2$ etex) "framed(false)";
102 newCircle.out(btex $out$ etex) "framed(false)";
105 newCircle.mula(btex $\times$ etex);
106 newCircle.mulb(btex $\times$ etex);
109 in.c = two.c + (0cm, 1cm);
110 mula.c = in.c + (2cm, 0cm);
111 mulb.c = mula.c + (2cm, 0cm);
112 out.c = mulb.c + (2cm, 0cm);
114 % Draw objects and lines
115 drawObj(in, two, mula, mulb, out);
117 nccurve(two)(mula) "angleA(0)", "angleB(45)";
118 nccurve(two)(mulb) "angleA(0)", "angleB(45)";
124 \placeexample[here][ex:Quadruple]{Simple three port and.}
125 \startcombination[2*1]
126 {\typebufferhs{Quadruple}}{Haskell description using function applications.}
127 {\boxedgraphic{Quadruple}}{The architecture described by the Haskell description.}
130 Here, the definition of mul is a partial function application: It applies
131 \hs{2 :: Word} to the function \hs{(*) :: Word -> Word -> Word} resulting in
132 the expression \hs{(*) 2 :: Word -> Word}. Since this resulting expression
133 is again a function, we can't generate hardware for it directly. This is
134 because the hardware to generate for \hs{mul} depends completely on where
135 and how it is used. In this example, it is even used twice!
137 However, it is clear that the above hardware description actually describes
138 valid hardware. In general, we can see that any partial applied function
139 must eventually become completely applied, at which point we can generate
140 hardware for it using the rules for function application above. It might
141 mean that a partial application is passed around quite a bit (even beyond
142 function boundaries), but eventually, the partial application will become
146 A very important concept in hardware designs is \emph{state}. In a
147 stateless (or, \emph{combinatoric}) design, every output is a directly and solely dependent on the
148 inputs. In a stateful design, the outputs can depend on the history of
149 inputs, or the \emph{state}. State is usually stored in \emph{registers},
150 which retain their value during a clockcycle, and are typically updated at
151 the start of every clockcycle. Since the updating of the state is tightly
152 coupled (synchronized) to the clock signal, these state updates are often
153 called \emph{synchronous}.
155 To make our hardware description language useful to describe more that
156 simple combinatoric designs, we'll need to be able to describe state in
159 \subsection{Approaches to state}
160 In Haskell, functions are always pure (except when using unsafe
161 functions like \hs{unsafePerformIO}, which should be prevented whenever
162 possible). This means that the output of a function solely depends on
163 its inputs. If you evaluate a given function with given inputs, it will
164 always provide the same output.
168 This is a perfect match for a combinatoric circuit, where the output
169 also soley depend on the inputs. However, when state is involved, this
170 no longer holds. Since we're in charge of our own language, we could
171 remove this purity constraint and allow a function to return different
172 values depending on the cycle in which it is evaluated (or rather, the
173 current state). However, this means that all kinds of interesting
174 properties of our functional language get lost, and all kinds of
175 transformations and optimizations might no longer be meaning preserving.
177 Provided that we want to keep the function pure, the current state has
178 to be present in the function's arguments in some way. There seem to be
179 two obvious ways to do this: Adding the current state as an argument, or
180 including the full history of each argument.
182 \subsubsection{Stream arguments and results}
183 Including the entire history of each input (\eg, the value of that
184 input for each previous clockcycle) is an obvious way to make outputs
185 depend on all previous input. This is easily done by making every
186 input a list instead of a single value, containing all previous values
187 as well as the current value.
189 An obvious downside of this solution is that on each cycle, all the
190 previous cycles must be resimulated to obtain the current state. To do
191 this, it might be needed to have a recursive helper function as well,
192 wich might be hard to properly analyze by the compiler.
194 A slight variation on this approach is one taken by some of the other
195 functional \small{HDL}s in the field (TODO: References to Lava,
196 ForSyDe, ...): Make functions operate on complete streams. This means
197 that a function is no longer called on every cycle, but just once. It
198 takes stream as inputs instead of values, where each stream contains
199 all the values for every clockcycle since system start. This is easily
200 modeled using an (infinite) list, with one element for each clock
201 cycle. Since the funciton is only evaluated once, its output is also a
202 stream. Note that, since we are working with infinite lists and still
203 want to be able to simulate the system cycle-by-cycle, this relies
204 heavily on the lazy semantics of Haskell.
206 Since our inputs and outputs are streams, all other (intermediate)
207 values must be streams. All of our primitive operators (\eg, addition,
208 substraction, bitwise operations, etc.) must operate on streams as
209 well (note that changing a single-element operation to a stream
210 operation can done with \hs{map}, \hs{zipwith}, etc.).
212 Note that the concept of \emph{state} is no more than having some way
213 to communicate a value from one cycle to the next. By introducing a
214 \hs{delay} function, we can do exactly that: Delay (each value in) a
215 stream so that we can "look into" the past. This \hs{delay} function
216 simply outputs a stream where each value is the same as the input
217 value, but shifted one cycle. This causes a \quote{gap} at the
218 beginning of the stream: What is the value of the delay output in the
219 first cycle? For this, the \hs{delay} function has a second input
220 (which is a value, not a stream!).
222 \in{Example}[ex:DelayAcc] shows a simple accumulator expressed in this
225 \startbuffer[DelayAcc]
226 acc :: Stream Word -> Stream Word
229 out = (delay out 0) + in
232 \startuseMPgraphic{DelayAcc}
233 save in, out, add, reg;
236 newCircle.in(btex $in$ etex) "framed(false)";
237 newCircle.out(btex $out$ etex) "framed(false)";
240 newReg.reg("") "dx(4mm)", "dy(6mm)", "reflect(true)";
241 newCircle.add(btex + etex);
244 add.c = in.c + (2cm, 0cm);
245 out.c = add.c + (2cm, 0cm);
246 reg.c = add.c + (0cm, 2cm);
248 % Draw objects and lines
249 drawObj(in, out, add, reg);
251 nccurve(add)(reg) "angleA(0)", "angleB(180)", "posB(d)";
252 nccurve(reg)(add) "angleA(180)", "angleB(-45)", "posA(out)";
258 \placeexample[here][ex:DelayAcc]{Simple accumulator architecture.}
259 \startcombination[2*1]
260 {\typebufferhs{DelayAcc}}{Haskell description using streams.}
261 {\boxedgraphic{DelayAcc}}{The architecture described by the Haskell description.}
265 This notation can be confusing (especially due to the loop in the
266 definition of out), but is essentially easy to interpret. There is a
267 single call to delay, resulting in a circuit with a single register,
268 whose input is connected to \hs{outl (which is the output of the
269 adder)}, and it's output is the \hs{delay out 0} (which is connected
270 to one of the adder inputs).
272 This notation has a number of downsides, amongst which are limited
273 readability and ambiguity in the interpretation. TODO: Reference
276 \subsubsection{Explicit state arguments and results}
277 A more explicit way to model state, is to simply add an extra argument
278 containing the current state value. This allows an output to depend on
279 both the inputs as well as the current state while keeping the
280 function pure (letting the result depend only on the arguments), since
281 the current state is now an argument.
283 In Haskell, this would look like \in{example}[ex:ExplicitAcc].
285 \startbuffer[ExplicitAcc]
286 acc :: Word -> (State Word) -> (State Word, Word)
287 acc in (State s) = (State s', out)
293 \placeexample[here][ex:ExplicitAcc]{Simple accumulator architecture.}
294 \startcombination[2*1]
295 {\typebufferhs{ExplicitAcc}}{Haskell description using explicit state arguments.}
296 % Picture is identical to the one we had just now.
297 {\boxedgraphic{DelayAcc}}{The architecture described by the Haskell description.}
300 This approach makes a function's state very explicit, which state
301 variables are used by a function can be completely determined from its
302 type signature (as opposed to the stream approach, where a function
303 looks the same from the outside, regardless of what state variables it
304 uses (or wether it's stateful at all).
306 A direct consequence of this, is that if a function calls other
307 stateful functions (\eg, has subcircuits), it has to somehow know the
308 current state for these called functions. The only way to do this, is
309 to put these \emph{substates} inside the caller's state. This means
310 that a function's state is the sum of the states of all functions it
311 calls, and its own state.
313 This approach is the one chosen for Cλash and will be examined more
316 \subsection{Explicit state specification}
317 Note about semantic correctness of top level state.
319 Note about automatic ``down-pushing'' of state.
321 Note about explicit state specification as the best solution.
325 Note about conditions on state variables and checking them.
327 \subsection{Explicit state implementation}
328 Recording state variables at the type level.
330 Ideal: Type synonyms, since there is no additional code overhead for
331 packing and unpacking. Downside: there is no explicit conversion in Core
332 either, so type synonyms tend to get lost in expressions (they can be
333 preserved in binders, but this makes implementation harder, since that
334 statefulness of a value must be manually tracked).
336 Less ideal: Newtype. Requires explicit packing and unpacking of function
337 arguments. If you don't unpack substates, there is no overhead for
338 (un)packing substates. This will result in many nested State constructors
339 in a nested state type. \eg:
342 State (State Bit, State (State Word, Bit), Word)
345 Alternative: Provide different newtypes for input and output state. This
346 makes the code even more explicit, and typechecking can find even more
347 errors. However, this requires defining two type synomyms for each
348 stateful function instead of just one. \eg:
350 type AccumStateIn = StateIn Bit
351 type AccumStateOut = StateOut Bit
353 This also increases the possibility of having different input and output
354 states. Checking for identical input and output state types is also
355 harder, since each element in the state must be unpacked and compared
358 Alternative: Provide a type for the entire result type of a stateful
359 function, not just the state part. \eg:
362 newtype Result state result = Result (state, result)
365 This makes it easy to say "Any stateful function must return a
366 \type{Result} type, without having to sort out result from state. However,
367 this either requires a second type for input state (similar to
368 \type{StateIn} / \type{StateOut} above), or requires the compiler to
369 select the right argument for input state by looking at types (which works
370 for complex states, but when that state has the same type as an argument,
371 things get ambiguous) or by selecting a fixed (\eg, the last) argument,
372 which might be limiting.
374 \subsubsection{Example}
375 As an example of the used approach, a simple averaging circuit, that lets
376 the accumulation of the inputs be done by a subcomponent.
379 newtype State s = State s
381 type AccumState = State Bit
382 accum :: Word -> AccumState -> (AccumState, Word)
383 accum i (State s) = (State (s + i), s + i)
385 type AvgState = (AccumState, Word)
386 avg :: Word -> AvgState -> (AvgState, Word)
387 avg i (State s) = (State s', o)
390 -- Pass our input through the accumulator, which outputs a sum
391 (accums', sum) = accum i accums
392 -- Increment the count (which will be our new state)
394 -- Compute the average
396 s' = (accums', count')
399 And the normalized, core-like versions:
402 accum i spacked = res
404 s = case spacked of (State s) -> s
412 s = case spacked of (State s) -> s
413 accums = case s of (accums, \_) -> accums
414 count = case s of (\_, count) -> count
415 accumres = accum i accums
416 accums' = case accumres of (accums', \_) -> accums'
417 sum = case accumres of (\_, sum) -> sum
420 s' = (accums', count')
427 As noted above, any component of a function's state that is a substate,
428 \eg passed on as the state of another function, should have no influence
429 on the hardware generated for the calling function. Any state-specific
430 \small{VHDL} for this component can be generated entirely within the called
431 function. So,we can completely leave out substates from any function.
433 From this observation, we might think to remove the substates from a
434 function's states alltogether, and leave only the state components which
435 are actual states of the current function. While doing this would not
436 remove any information needed to generate \small{VHDL} from the function, it would
437 cause the function definition to become invalid (since we won't have any
438 substate to pass to the functions anymore). We could solve the syntactic
439 problems by passing \type{undefined} for state variables, but that would
440 still break the code on the semantic level (\ie, the function would no
441 longer be semantically equivalent to the original input).
443 To keep the function definition correct until the very end of the process,
444 we will not deal with (sub)states until we get to the \small{VHDL} generation.
445 Here, we are translating from Core to \small{VHDL}, and we can simply not generate
446 \small{VHDL} for substates, effectively removing the substate components
449 There are a few important points when ignore substates.
451 First, we have to have some definition of "substate". Since any state
452 argument or return value that represents state must be of the \type{State}
453 type, we can simply look at its type. However, we must be careful to
454 ignore only {\em substates}, and not a function's own state.
456 In the example above, this means we should remove \type{accums'} from
457 \type{s'}, but not throw away \type{s'} entirely. We should, however,
458 remove \type{s'} from the output port of the function, since the state
459 will be handled by a \small{VHDL} procedure within the function.
461 When looking at substates, these can appear in two places: As part of an
462 argument and as part of a return value. As noted above, these substates
463 can only be used in very specific ways.
465 \desc{State variables can appear as an argument.} When generating \small{VHDL}, we
466 completely ignore the argument and generate no input port for it.
468 \desc{State variables can be extracted from other state variables.} When
469 extracting a state variable from another state variable, this always means
470 we're extracting a substate, which we can ignore. So, we simply generate no
471 \small{VHDL} for any extraction operation that has a state variable as a result.
473 \desc{State variables can be passed to functions.} When passing a
474 state variable to a function, this always means we're passing a substate
475 to a subcomponent. The entire argument can simply be ingored in the
478 \desc{State variables can be returned from functions.} When returning a
479 state variable from a function (probably as a part of an algebraic
480 datatype), this always mean we're returning a substate from a
481 subcomponent. The entire state variable should be ignored in the resulting
482 port map. The type binder of the binder that the function call is bound
483 to should not include the state type either.
485 \startdesc{State variables can be inserted into other variables.} When inserting
486 a state variable into another variable (usually by constructing that new
487 variable using its constructor), we can identify two cases:
490 \item The state is inserted into another state variable. In this case,
491 the inserted state is a substate, and can be safely left out of the
492 constructed variable.
493 \item The state is inserted into a non-state variable. This happens when
494 building up the return value of a function, where you put state and
495 retsult variables together in an algebraic type (usually a tuple). In
496 this case, we should leave the state variable out as well, since we
497 don't want it to be included as an output port.
500 So, in both cases, we can simply leave out the state variable from the
501 resulting value. In the latter case, however, we should generate a state
502 proc instead, which assigns the state variable to the input state variable
506 \desc{State variables can appear as (part of) a function result.} When
507 generating \small{VHDL}, we can completely ignore any part of a function result
508 that has a state type. If the entire result is a state type, this will
509 mean the entity will not have an output port. Otherwise, the state
510 elements will be removed from the type of the output port.
513 Now, we know how to handle each use of a state variable separately. If we
514 look at the whole, we can conclude the following:
517 \item A state unpack operation should not generate any \small{VHDL}. The binder
518 to which the unpacked state is bound should still be declared, this signal
519 will become the register and will hold the current state.
520 \item A state pack operation should not generate any \small{VHDL}. The binder th
521 which the packed state is bound should not be declared. The binder that is
522 packed is the signal that will hold the new state.
523 \item Any values of a State type should not be translated to \small{VHDL}. In
524 particular, State elements should be removed from tuples (and other
525 datatypes) and arguments with a state type should not generate ports.
526 \item To make the state actually work, a simple \small{VHDL} proc should be
527 generated. This proc updates the state at every clockcycle, by assigning
528 the new state to the current state. This will be recognized by synthesis
529 tools as a register specification.
533 When applying these rules to the example program (in normal form), we will
534 get the following result. All the parts that don't generate any value are
535 crossed out, leaving some very boring assignments here and there.
539 avg i --spacked-- = res
541 s = --case spacked of (State s) -> s--
542 --accums = case s of (accums, \_) -> accums--
543 count = case s of (--\_,-- count) -> count
544 accumres = accum i --accums--
545 --accums' = case accumres of (accums', \_) -> accums'--
546 sum = case accumres of (--\_,-- sum) -> sum
549 s' = (--accums',-- count')
550 --spacked' = State s'--
551 res = (--spacked',-- o)
554 When we would really leave out the crossed out parts, we get a slightly
555 weird program: There is a variable \type{s} which has no value, and there
556 is a variable \type{s'} that is never used. Together, these two will form
557 the state proc of the function. \type{s} contains the "current" state,
558 \type{s'} is assigned the "next" state. So, at the end of each clock
559 cycle, \type{s'} should be assigned to \type{s}.
561 Note that the definition of \type{s'} is not removed, even though one
562 might think it as having a state type. Since the state type has a single
563 argument constructor \type{State}, some type that should be the resulting
564 state should always be explicitly packed with the State constructor,
565 allowing us to remove the packed version, but still generate \small{VHDL} for the
566 unpacked version (of course with any substates removed).
568 As you can see, the definition of \type{s'} is still present, since it
569 does not have a state type (The State constructor. The \type{accums'} substate has been removed,
570 leaving us just with the state of \type{avg} itself.
571 \subsection{Initial state}
572 How to specify the initial state? Cannot be done inside a hardware
573 function, since the initial state is its own state argument for the first
574 call (unless you add an explicit, synchronous reset port).
576 External init state is natural for simulation.
578 External init state works for hardware generation as well.
580 Implementation issues: state splitting, linking input to output state,
581 checking usage constraints on state variables.
583 \section[sec:recursion]{Recursion}
584 An import concept in functional languages is recursion. In it's most basic
585 form, recursion is a function that is defined in terms of itself. This
586 usually requires multiple evaluations of this function, with changing
587 arguments, until eventually an evaluation of the function no longer requires
590 Recursion in a hardware description is a bit of a funny thing. Usually,
591 recursion is associated with a lot of nondeterminism, stack overflows, but
592 also flexibility and expressive power.
594 Given the notion that each function application will translate to a
595 component instantiation, we are presented with a problem. A recursive
596 function would translate to a component that contains itself. Or, more
597 precisely, that contains an instance of itself. This instance would again
598 contain an instance of itself, and again, into infinity. This is obviously a
599 problem for generating hardware.
601 This is expected for functions that describe infinite recursion. In that
602 case, we can't generate hardware that shows correct behaviour in a single
603 cycle (at best, we could generate hardware that needs an infinite number of
606 However, most recursive hardware descriptions will describe finite
607 recursion. This is because the recursive call is done conditionally. There
608 is usually a case statement where at least one alternative does not contain
609 the recursive call, which we call the "base case". If, for each call to the
610 recursive function, we would be able to detect which alternative applies,
611 we would be able to remove the case expression and leave only the base case
612 when it applies. This will ensure that expanding the recursive functions
613 will terminate after a bounded number of expansions.
615 This does imply the extra requirement that the base case is detectable at
616 compile time. In particular, this means that the decision between the base
617 case and the recursive case must not depend on runtime data.
619 \subsection{List recursion}
620 The most common deciding factor in recursion is the length of a list that is
621 passed in as an argument. Since we represent lists as vectors that encode
622 the length in the vector type, it seems easy to determine the base case. We
623 can simply look at the argument type for this. However, it turns out that
624 this is rather non-trivial to write down in Haskell in the first place. As
625 an example, we would like to write down something like this:
628 sum :: Vector n Word -> Word
629 sum xs = case null xs of
631 False -> head xs + sum (tail xs)
634 However, the typechecker will now use the following reasoning (element type
635 of the vector is left out):
638 \item tail has the type \hs{(n > 0) => Vector n -> Vector (n - 1)}
639 \item This means that xs must have the type \hs{(n > 0) => Vector n}
640 \item This means that sum must have the type \hs{(n > 0) => Vector n -> a}
641 \item sum is called with the result of tail as an argument, which has the
642 type \hs{Vector n} (since \hs{(n > 0) => n - 1 == m}).
643 \item This means that sum must have the type \hs{Vector n -> a}
644 \item This is a contradiction between the type deduced from the body of sum
645 (the input vector must be non-empty) and the use of sum (the input vector
646 could have any length).
649 As you can see, using a simple case at value level causes the type checker
650 to always typecheck both alternatives, which can't be done! This is a
651 fundamental problem, that would seem perfectly suited for a type class.
652 Considering that we need to switch between to implementations of the sum
653 function, based on the type of the argument, this sounds like the perfect
654 problem to solve with a type class. However, this approach has its own
655 problems (not the least of them that you need to define a new typeclass for
656 every recursive function you want to define).
658 Another approach tried involved using GADTs to be able to do pattern
659 matching on empty / non empty lists. While this worked partially, it also
660 created problems with more complex expressions.
662 TODO: How much detail should there be here? I can probably refer to
665 Evaluating all possible (and non-possible) ways to add recursion to our
666 descriptions, it seems better to leave out list recursion alltogether. This
667 allows us to focus on other interesting areas instead. By including
668 (builtin) support for a number of higher order functions like map and fold,
669 we can still express most of the things we would use list recursion for.
671 \subsection{General recursion}
672 Of course there are other forms of recursion, that do not depend on the
673 length (and thus type) of a list. For example, simple recursion using a
674 counter could be expressed, but only translated to hardware for a fixed
675 number of iterations. Also, this would require extensive support for compile
676 time simplification (constant propagation) and compile time evaluation
677 (evaluation constant comparisons), to ensure non-termination. Even then, it
678 is hard to really guarantee termination, since the user (or GHC desugarer)
679 might use some obscure notation that results in a corner case of the
680 simplifier that is not caught and thus non-termination.
682 Due to these complications, we leave other forms of recursion as