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 An example of a simple program using only function application would be:
33 -- | A simple function that returns the and of three bits
34 and3 :: Bit -> Bit -> Bit -> Bit
35 and3 a b c = and (and a b) c
38 This results in the following hardware:
42 \subsection{Partial application}
43 It should be obvious that we cannot generate hardware signals for all
44 expressions we can express in Haskell. The most obvious criterium for this
45 is the type of an expression. We will see more of this below, but for now it
46 should be obvious that any expression of a function type cannot be
47 represented as a signal or i/o port to a component.
49 From this, we can see that the above translation rules do not apply to a
50 partial application. Let's look at an example:
53 -- | Multiply the input word by four.
54 quadruple :: Word -> Word
55 quadruple n = mul (mul n)
60 It should be clear that the above code describes the following hardware:
64 Here, the definition of mul is a partial function application: It applies
65 \hs{2 :: Word} to the function \hs{(*) :: Word -> Word -> Word} resulting in
66 the expression \hs{(*) 2 :: Word -> Word}. Since this resulting expression
67 is again a function, we can't generate hardware for it directly. This is
68 because the hardware to generate for \hs{mul} depends completely on where
69 and how it is used. In this example, it is even used twice!
71 However, it is clear that the above hardware description actually describes
72 valid hardware. In general, we can see that any partial applied function
73 must eventually become completely applied, at which point we can generate
74 hardware for it using the rules for function application above. It might
75 mean that a partial application is passed around quite a bit (even beyond
76 function boundaries), but eventually, the partial application will become
80 \subsection{Introduction}
84 \subsection{Approaches to state}
85 Explain impact of state (or rather, temporal behaviour) on function signature.
86 \subsubsection{Stream arguments and results}
87 \subsubsection{Explicit state arguments and results}
88 Nested state for called functions.
90 \subsection{Explicit state specification}
91 Note about semantic correctness of top level state.
93 Note about automatic ``down-pushing'' of state.
95 Note about explicit state specification as the best solution.
99 Note about conditions on state variables and checking them.
101 \subsection{Explicit state implementation}
102 Recording state variables at the type level.
104 Ideal: Type synonyms, since there is no additional code overhead for
105 packing and unpacking. Downside: there is no explicit conversion in Core
106 either, so type synonyms tend to get lost in expressions (they can be
107 preserved in binders, but this makes implementation harder, since that
108 statefulness of a value must be manually tracked).
110 Less ideal: Newtype. Requires explicit packing and unpacking of function
111 arguments. If you don't unpack substates, there is no overhead for
112 (un)packing substates. This will result in many nested State constructors
113 in a nested state type. \eg:
116 State (State Bit, State (State Word, Bit), Word)
119 Alternative: Provide different newtypes for input and output state. This
120 makes the code even more explicit, and typechecking can find even more
121 errors. However, this requires defining two type synomyms for each
122 stateful function instead of just one. \eg:
124 type AccumStateIn = StateIn Bit
125 type AccumStateOut = StateOut Bit
127 This also increases the possibility of having different input and output
128 states. Checking for identical input and output state types is also
129 harder, since each element in the state must be unpacked and compared
132 Alternative: Provide a type for the entire result type of a stateful
133 function, not just the state part. \eg:
136 newtype Result state result = Result (state, result)
139 This makes it easy to say "Any stateful function must return a
140 \type{Result} type, without having to sort out result from state. However,
141 this either requires a second type for input state (similar to
142 \type{StateIn} / \type{StateOut} above), or requires the compiler to
143 select the right argument for input state by looking at types (which works
144 for complex states, but when that state has the same type as an argument,
145 things get ambiguous) or by selecting a fixed (\eg, the last) argument,
146 which might be limiting.
148 \subsubsection{Example}
149 As an example of the used approach, a simple averaging circuit, that lets
150 the accumulation of the inputs be done by a subcomponent.
153 newtype State s = State s
155 type AccumState = State Bit
156 accum :: Word -> AccumState -> (AccumState, Word)
157 accum i (State s) = (State (s + i), s + i)
159 type AvgState = (AccumState, Word)
160 avg :: Word -> AvgState -> (AvgState, Word)
161 avg i (State s) = (State s', o)
164 -- Pass our input through the accumulator, which outputs a sum
165 (accums', sum) = accum i accums
166 -- Increment the count (which will be our new state)
168 -- Compute the average
170 s' = (accums', count')
173 And the normalized, core-like versions:
176 accum i spacked = res
178 s = case spacked of (State s) -> s
186 s = case spacked of (State s) -> s
187 accums = case s of (accums, \_) -> accums
188 count = case s of (\_, count) -> count
189 accumres = accum i accums
190 accums' = case accumres of (accums', \_) -> accums'
191 sum = case accumres of (\_, sum) -> sum
194 s' = (accums', count')
201 As noted above, any component of a function's state that is a substate,
202 \eg passed on as the state of another function, should have no influence
203 on the hardware generated for the calling function. Any state-specific
204 \small{VHDL} for this component can be generated entirely within the called
205 function. So,we can completely leave out substates from any function.
207 From this observation, we might think to remove the substates from a
208 function's states alltogether, and leave only the state components which
209 are actual states of the current function. While doing this would not
210 remove any information needed to generate \small{VHDL} from the function, it would
211 cause the function definition to become invalid (since we won't have any
212 substate to pass to the functions anymore). We could solve the syntactic
213 problems by passing \type{undefined} for state variables, but that would
214 still break the code on the semantic level (\ie, the function would no
215 longer be semantically equivalent to the original input).
217 To keep the function definition correct until the very end of the process,
218 we will not deal with (sub)states until we get to the \small{VHDL} generation.
219 Here, we are translating from Core to \small{VHDL}, and we can simply not generate
220 \small{VHDL} for substates, effectively removing the substate components
223 There are a few important points when ignore substates.
225 First, we have to have some definition of "substate". Since any state
226 argument or return value that represents state must be of the \type{State}
227 type, we can simply look at its type. However, we must be careful to
228 ignore only {\em substates}, and not a function's own state.
230 In the example above, this means we should remove \type{accums'} from
231 \type{s'}, but not throw away \type{s'} entirely. We should, however,
232 remove \type{s'} from the output port of the function, since the state
233 will be handled by a \small{VHDL} procedure within the function.
235 When looking at substates, these can appear in two places: As part of an
236 argument and as part of a return value. As noted above, these substates
237 can only be used in very specific ways.
239 \desc{State variables can appear as an argument.} When generating \small{VHDL}, we
240 completely ignore the argument and generate no input port for it.
242 \desc{State variables can be extracted from other state variables.} When
243 extracting a state variable from another state variable, this always means
244 we're extracting a substate, which we can ignore. So, we simply generate no
245 \small{VHDL} for any extraction operation that has a state variable as a result.
247 \desc{State variables can be passed to functions.} When passing a
248 state variable to a function, this always means we're passing a substate
249 to a subcomponent. The entire argument can simply be ingored in the
252 \desc{State variables can be returned from functions.} When returning a
253 state variable from a function (probably as a part of an algebraic
254 datatype), this always mean we're returning a substate from a
255 subcomponent. The entire state variable should be ignored in the resulting
256 port map. The type binder of the binder that the function call is bound
257 to should not include the state type either.
259 \startdesc{State variables can be inserted into other variables.} When inserting
260 a state variable into another variable (usually by constructing that new
261 variable using its constructor), we can identify two cases:
264 \item The state is inserted into another state variable. In this case,
265 the inserted state is a substate, and can be safely left out of the
266 constructed variable.
267 \item The state is inserted into a non-state variable. This happens when
268 building up the return value of a function, where you put state and
269 retsult variables together in an algebraic type (usually a tuple). In
270 this case, we should leave the state variable out as well, since we
271 don't want it to be included as an output port.
274 So, in both cases, we can simply leave out the state variable from the
275 resulting value. In the latter case, however, we should generate a state
276 proc instead, which assigns the state variable to the input state variable
280 \desc{State variables can appear as (part of) a function result.} When
281 generating \small{VHDL}, we can completely ignore any part of a function result
282 that has a state type. If the entire result is a state type, this will
283 mean the entity will not have an output port. Otherwise, the state
284 elements will be removed from the type of the output port.
287 Now, we know how to handle each use of a state variable separately. If we
288 look at the whole, we can conclude the following:
291 \item A state unpack operation should not generate any \small{VHDL}. The binder
292 to which the unpacked state is bound should still be declared, this signal
293 will become the register and will hold the current state.
294 \item A state pack operation should not generate any \small{VHDL}. The binder th
295 which the packed state is bound should not be declared. The binder that is
296 packed is the signal that will hold the new state.
297 \item Any values of a State type should not be translated to \small{VHDL}. In
298 particular, State elements should be removed from tuples (and other
299 datatypes) and arguments with a state type should not generate ports.
300 \item To make the state actually work, a simple \small{VHDL} proc should be
301 generated. This proc updates the state at every clockcycle, by assigning
302 the new state to the current state. This will be recognized by synthesis
303 tools as a register specification.
307 When applying these rules to the example program (in normal form), we will
308 get the following result. All the parts that don't generate any value are
309 crossed out, leaving some very boring assignments here and there.
313 avg i --spacked-- = res
315 s = --case spacked of (State s) -> s--
316 --accums = case s of (accums, \_) -> accums--
317 count = case s of (--\_,-- count) -> count
318 accumres = accum i --accums--
319 --accums' = case accumres of (accums', \_) -> accums'--
320 sum = case accumres of (--\_,-- sum) -> sum
323 s' = (--accums',-- count')
324 --spacked' = State s'--
325 res = (--spacked',-- o)
328 When we would really leave out the crossed out parts, we get a slightly
329 weird program: There is a variable \type{s} which has no value, and there
330 is a variable \type{s'} that is never used. Together, these two will form
331 the state proc of the function. \type{s} contains the "current" state,
332 \type{s'} is assigned the "next" state. So, at the end of each clock
333 cycle, \type{s'} should be assigned to \type{s}.
335 Note that the definition of \type{s'} is not removed, even though one
336 might think it as having a state type. Since the state type has a single
337 argument constructor \type{State}, some type that should be the resulting
338 state should always be explicitly packed with the State constructor,
339 allowing us to remove the packed version, but still generate \small{VHDL} for the
340 unpacked version (of course with any substates removed).
342 As you can see, the definition of \type{s'} is still present, since it
343 does not have a state type (The State constructor. The \type{accums'} substate has been removed,
344 leaving us just with the state of \type{avg} itself.
345 \subsection{Initial state}
346 How to specify the initial state? Cannot be done inside a hardware
347 function, since the initial state is its own state argument for the first
348 call (unless you add an explicit, synchronous reset port).
350 External init state is natural for simulation.
352 External init state works for hardware generation as well.
354 Implementation issues: state splitting, linking input to output state,
355 checking usage constraints on state variables.
357 \section[sec:recursion]{Recursion}
358 An import concept in functional languages is recursion. In it's most basic
359 form, recursion is a function that is defined in terms of itself. This
360 usually requires multiple evaluations of this function, with changing
361 arguments, until eventually an evaluation of the function no longer requires
364 Recursion in a hardware description is a bit of a funny thing. Usually,
365 recursion is associated with a lot of nondeterminism, stack overflows, but
366 also flexibility and expressive power.
368 Given the notion that each function application will translate to a
369 component instantiation, we are presented with a problem. A recursive
370 function would translate to a component that contains itself. Or, more
371 precisely, that contains an instance of itself. This instance would again
372 contain an instance of itself, and again, into infinity. This is obviously a
373 problem for generating hardware.
375 This is expected for functions that describe infinite recursion. In that
376 case, we can't generate hardware that shows correct behaviour in a single
377 cycle (at best, we could generate hardware that needs an infinite number of
380 However, most recursive hardware descriptions will describe finite
381 recursion. This is because the recursive call is done conditionally. There
382 is usually a case statement where at least one alternative does not contain
383 the recursive call, which we call the "base case". If, for each call to the
384 recursive function, we would be able to detect which alternative applies,
385 we would be able to remove the case expression and leave only the base case
386 when it applies. This will ensure that expanding the recursive functions
387 will terminate after a bounded number of expansions.
389 This does imply the extra requirement that the base case is detectable at
390 compile time. In particular, this means that the decision between the base
391 case and the recursive case must not depend on runtime data.
393 \subsection{List recursion}
394 The most common deciding factor in recursion is the length of a list that is
395 passed in as an argument. Since we represent lists as vectors that encode
396 the length in the vector type, it seems easy to determine the base case. We
397 can simply look at the argument type for this. However, it turns out that
398 this is rather non-trivial to write down in Haskell in the first place. As
399 an example, we would like to write down something like this:
402 sum :: Vector n Word -> Word
403 sum xs = case null xs of
405 False -> head xs + sum (tail xs)
408 However, the typechecker will now use the following reasoning (element type
409 of the vector is left out):
412 \item tail has the type \hs{(n > 0) => Vector n -> Vector (n - 1)}
413 \item This means that xs must have the type \hs{(n > 0) => Vector n}
414 \item This means that sum must have the type \hs{(n > 0) => Vector n -> a}
415 \item sum is called with the result of tail as an argument, which has the
416 type \hs{Vector n} (since \hs{(n > 0) => n - 1 == m}).
417 \item This means that sum must have the type \hs{Vector n -> a}
418 \item This is a contradiction between the type deduced from the body of sum
419 (the input vector must be non-empty) and the use of sum (the input vector
420 could have any length).
423 As you can see, using a simple case at value level causes the type checker
424 to always typecheck both alternatives, which can't be done! This is a
425 fundamental problem, that would seem perfectly suited for a type class.
426 Considering that we need to switch between to implementations of the sum
427 function, based on the type of the argument, this sounds like the perfect
428 problem to solve with a type class. However, this approach has its own
429 problems (not the least of them that you need to define a new typeclass for
430 every recursive function you want to define).
432 Another approach tried involved using GADTs to be able to do pattern
433 matching on empty / non empty lists. While this worked partially, it also
434 created problems with more complex expressions.
436 TODO: How much detail should there be here? I can probably refer to
439 Evaluating all possible (and non-possible) ways to add recursion to our
440 descriptions, it seems better to leave out list recursion alltogether. This
441 allows us to focus on other interesting areas instead. By including
442 (builtin) support for a number of higher order functions like map and fold,
443 we can still express most of the things we would use list recursion for.
445 \subsection{General recursion}
446 Of course there are other forms of recursion, that do not depend on the
447 length (and thus type) of a list. For example, simple recursion using a
448 counter could be expressed, but only translated to hardware for a fixed
449 number of iterations. Also, this would require extensive support for compile
450 time simplification (constant propagation) and compile time evaluation
451 (evaluation constant comparisons), to ensure non-termination. Even then, it
452 is hard to really guarantee termination, since the user (or GHC desugarer)
453 might use some obscure notation that results in a corner case of the
454 simplifier that is not caught and thus non-termination.
456 Due to these complications, we leave other forms of recursion as