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. The prototype
5 implementation will be discussed in \in{chapter}[chap:prototype].
7 \todo{Shortshort introduction to Cλash (Bit, Word, and, not, etc.)}
9 When translating Haskell to hardware, we need to make choices about what kind
10 of hardware to generate for which Haskell constructs. When faced with
11 choices, we've tried to stick with the most obvious choice wherever
12 possible. In a lot of cases, when you look at a hardware description it is
13 comletely clear what hardware is described. We want our translator to
14 generate exactly that hardware whenever possible, to minimize the amount of
15 surprise for people working with it.
17 In this chapter we try to describe how we interpret a Haskell program from a
18 hardware perspective. We provide a description of each Haskell language
19 element that needs translation, to provide a clear picture of what is
22 \section[sec:description:application]{Function application}
23 \todo{Sidenote: Inputs vs arguments}
24 The basic syntactic element of a functional program are functions and
25 function application. These have a single obvious \small{VHDL} translation: Each
26 function becomes a hardware component, where each argument is an input port
27 and the result value is the (single) output port. This output port can have
28 a complex type (such as a tuple), so having just a single output port does
29 not pose a limitation.
31 Each function application in turn becomes component instantiation. Here, the
32 result of each argument expression is assigned to a signal, which is mapped
33 to the corresponding input port. The output port of the function is also
34 mapped to a signal, which is used as the result of the application.
36 \in{Example}[ex:And3] shows a simple program using only function
37 application and the corresponding architecture.
40 -- | A simple function that returns
41 -- conjunction of three bits
42 and3 :: Bit -> Bit -> Bit -> Bit
43 and3 a b c = and (and a b) c
46 \todo{Mirror this image vertically}
47 \startuseMPgraphic{And3}
48 save a, b, c, anda, andb, out;
51 newCircle.a(btex $a$ etex) "framed(false)";
52 newCircle.b(btex $b$ etex) "framed(false)";
53 newCircle.c(btex $c$ etex) "framed(false)";
54 newCircle.out(btex $out$ etex) "framed(false)";
57 newCircle.anda(btex $and$ etex);
58 newCircle.andb(btex $and$ etex);
61 b.c = a.c + (0cm, 1cm);
62 c.c = b.c + (0cm, 1cm);
63 anda.c = midpoint(a.c, b.c) + (2cm, 0cm);
64 andb.c = midpoint(b.c, c.c) + (4cm, 0cm);
66 out.c = andb.c + (2cm, 0cm);
68 % Draw objects and lines
69 drawObj(a, b, c, anda, andb, out);
71 ncarc(a)(anda) "arcangle(-10)";
78 \placeexample[here][ex:And3]{Simple three input and gate.}
79 \startcombination[2*1]
80 {\typebufferhs{And3}}{Haskell description using function applications.}
81 {\boxedgraphic{And3}}{The architecture described by the Haskell description.}
84 \note{This section should also mention hierarchy, top level functions and
88 Although describing components and connections allows us to describe a lot of
89 hardware designs already, there is an obvious thing missing: Choice. We
90 need some way to be able to choose between values based on another value.
91 In Haskell, choice is achieved by \hs{case} expressions, \hs{if} expressions or
94 An obvious way to add choice to our language without having to recognize
95 any of Haskell's syntax, would be to add a primivite \quote{\hs{if}}
96 function. This function would take three arguments: The condition, the
97 value to return when the condition is true and the value to return when
98 the condition is false.
100 This \hs{if} function would then essentially describe a multiplexer and
101 allows us to describe any architecture that uses multiplexers. \fxnote{Are
102 there other mechanisms of choice in hardware?}
104 However, to be able to describe our hardware in a more convenient way, we
105 also want to translate Haskell's choice mechanisms. The easiest of these
106 are of course case expressions (and \hs{if} expressions, which can be very
107 directly translated to \hs{case} expressions). A \hs{case} expression can in turn
108 simply be translated to a conditional assignment, where the conditions use
109 equality comparisons against the constructors in the \hs{case} expressions.
111 \todo{Assignment vs multiplexers}
113 In \in{example}[ex:CaseInv] a simple \hs{case} expression is shown,
114 scrutinizing a boolean value. The corresponding architecture has a
115 comparator to determine which of the constructors is on the \hs{in}
116 input. There is a multiplexer to select the output signal. The two options
117 for the output signals are just constants, but these could have been more
118 complex expressions (in which case also both of them would be working in
119 parallel, regardless of which output would be chosen eventually).
121 If we would translate a Boolean to a bit value, we could of course remove
122 the comparator and directly feed 'in' into the multiplex (or even use an
123 inverter instead of a multiplexer). However, we will try to make a
124 general translation, which works for all possible \hs{case} expressions.
125 Optimizations such as these are left for the \VHDL synthesizer, which
126 handles them very well.
128 \todo{Be more explicit about >2 alternatives}
130 \startbuffer[CaseInv]
136 \startuseMPgraphic{CaseInv}
137 save in, truecmp, falseout, trueout, out, cmp, mux;
140 newCircle.in(btex $in$ etex) "framed(false)";
141 newCircle.out(btex $out$ etex) "framed(false)";
143 newBox.truecmp(btex $True$ etex) "framed(false)";
144 newBox.trueout(btex $True$ etex) "framed(false)";
145 newBox.falseout(btex $False$ etex) "framed(false)";
148 newCircle.cmp(btex $==$ etex);
152 cmp.c = in.c + (3cm, 0cm);
153 truecmp.c = cmp.c + (-1cm, 1cm);
154 mux.sel = cmp.e + (1cm, -1cm);
155 falseout.c = mux.inpa - (2cm, 0cm);
156 trueout.c = mux.inpb - (2cm, 0cm);
157 out.c = mux.out + (2cm, 0cm);
159 % Draw objects and lines
160 drawObj(in, out, truecmp, trueout, falseout, cmp, mux);
164 nccurve(cmp.e)(mux.sel) "angleA(0)", "angleB(-90)";
165 ncline(falseout)(mux) "posB(inpa)";
166 ncline(trueout)(mux) "posB(inpb)";
167 ncline(mux)(out) "posA(out)";
170 \placeexample[here][ex:CaseInv]{Simple inverter.}
171 \startcombination[2*1]
172 {\typebufferhs{CaseInv}}{Haskell description using a Case expression.}
173 {\boxedgraphic{CaseInv}}{The architecture described by the Haskell description.}
176 A slightly more complex (but very powerful) form of choice is pattern
177 matching. A function can be defined in multiple clauses, where each clause
178 specifies a pattern. When the arguments match the pattern, the
179 corresponding clause will be used.
181 \startbuffer[PatternInv]
187 \placeexample[here][ex:PatternInv]{Simple inverter using pattern matching.
188 Describes the same architecture as \in{example}[ex:CaseInv].}
189 {\typebufferhs{CaseInv}}
191 The architecture described by \in{example}[ex:PatternInv] is of course the
192 same one as the one in \in{example}[ex:CaseInv]. The general interpretation
193 of pattern matching is also similar to that of \hs{case} expressions: Generate
194 hardware for each of the clauses (like each of the clauses of a \hs{case}
195 expression) and connect them to the function output through (a number of
196 nested) multiplexers. These multiplexers are driven by comparators and
197 other logic, that check each pattern in turn.
200 We've seen the two most basic functional concepts translated to hardware:
201 Function application and choice. Before we look further into less obvious
202 concepts like higher order expressions and polymorphism, we will have a
203 look at the types of the values we use in our descriptions.
205 When working with values in our descriptions, we'll also need to provide
206 some way to translate the values used to hardware equivalents. In
207 particular, this means having to come up with a hardware equivalent for
208 every \emph{type} used in our program.
210 Since most functional languages have a lot of standard types that are
211 hard to translate (integers without a fixed size, lists without a static
212 length, etc.), we will start out by defining a number of \quote{builtin}
213 types ourselves. These types are builtin in the sense that our compiler
214 will have a fixed VHDL type for these. User defined types, on the other
215 hand, will have their hardware type derived directly from their Haskell
216 declaration automatically, according to the rules we sketch here.
218 \todo{Introduce Haskell type syntax (type constructors, type application,
221 \subsection{Builtin types}
222 The language currently supports the following builtin types. Of these,
223 only the \hs{Bool} type is supported by Haskell out of the box (the
224 others are defined by the Cλash package, so they are user-defined types
225 from Haskell's point of view).
228 This is the most basic type available. It is mapped directly onto
229 the \type{std_logic} \small{VHDL} type. Mapping this to the
230 \type{bit} type might make more sense (since the Haskell version
231 only has two values), but using \type{std_logic} is more standard
232 (and allowed for some experimentation with don't care values)
234 \todo{Sidenote bit vs stdlogic}
236 \startdesc{\hs{Bool}}
237 This is the only builtin Haskell type supported and is translated
238 exactly like the Bit type (where a value of \hs{True} corresponds to a
239 value of \hs{High}). Supporting the Bool type is particularly
240 useful to support \hs{if ... then ... else ...} expressions, which
241 always have a \hs{Bool} value for the condition.
243 A \hs{Bool} is translated to a \type{std_logic}, just like \hs{Bit}.
245 \startdesc{\hs{SizedWord}, \hs{SizedInt}}
246 These are types to represent integers. A \hs{SizedWord} is unsigned,
247 while a \hs{SizedInt} is signed. These types are parameterized by a
248 length type, so you can define an unsigned word of 32 bits wide as
252 type Word32 = SizedWord D32
255 Here, a type synonym \hs{Word32} is defined that is equal to the
256 \hs{SizedWord} type constructor applied to the type \hs{D32}. \hs{D32}
257 is the \emph{type level representation} of the decimal number 32,
258 making the \hs{Word32} type a 32-bit unsigned word.
260 These types are translated to the \small{VHDL} \type{unsigned} and
261 \type{signed} respectively.
262 \todo{Sidenote on dependent typing?}
264 \startdesc{\hs{Vector}}
265 This is a vector type, that can contain elements of any other type and
266 has a fixed length. It has two type parameters: Its
267 length and the type of the elements contained in it. By putting the
268 length parameter in the type, the length of a vector can be determined
269 at compile time, instead of only at runtime for conventional lists.
271 The \hs{Vector} type constructor takes two type arguments: The length
272 of the vector and the type of the elements contained in it. The state
273 type of an 8 element register bank would then for example be:
276 type RegisterState = Vector D8 Word32
279 Here, a type synonym \hs{RegisterState} is defined that is equal to
280 the \hs{Vector} type constructor applied to the types \hs{D8} (The type
281 level representation of the decimal number 8) and \hs{Word32} (The 32
282 bit word type as defined above). In other words, the
283 \hs{RegisterState} type is a vector of 8 32-bit words.
285 A fixed size vector is translated to a \small{VHDL} array type.
287 \startdesc{\hs{RangedWord}}
288 This is another type to describe integers, but unlike the previous
289 two it has no specific bitwidth, but an upper bound. This means that
290 its range is not limited to powers of two, but can be any number.
291 A \hs{RangedWord} only has an upper bound, its lower bound is
292 implicitly zero. There is a lot of added implementation complexity
293 when adding a lower bound and having just an upper bound was enough
294 for the primary purpose of this type: Typesafely indexing vectors.
296 To define an index for the 8 element vector above, we would do:
299 type Register = RangedWord D7
302 Here, a type synonym \hs{RegisterIndex} is defined that is equal to
303 the \hs{RangedWord} type constructor applied to the type \hs{D7}. In
304 other words, this defines an unsigned word with values from 0 to 7
305 (inclusive). This word can be be used to index the 8 element vector
306 \hs{RegisterState} above.
308 This type is translated to the \type{unsigned} \small{VHDL} type.
310 \fxnote{There should be a reference to Christiaan's work here.}
312 \subsection{User-defined types}
313 There are three ways to define new types in Haskell: Algebraic
314 datatypes with the \hs{data} keyword, type synonyms with the \hs{type}
315 keyword and type renamings with the \hs{newtype} keyword. This
316 explicitly excludes more advanced type creation from \GHC extensions
317 such as type families, existential typing, \small{GADT}s, etc.
319 The first of these actually introduces a new type, for which we provide
320 the \VHDL translation below. The latter two only define new names for
321 existing types (where synonyms are completely interchangeable and
322 renamings need explicit conversion). Therefore, these don't need any
323 particular \VHDL translation, a synonym or renamed type will just use
324 the same representation as the equivalent type.
326 For algebraic types, we can make the following distinction:
328 \startdesc{Product types}
329 A product type is an algebraic datatype with a single constructor with
330 two or more fields, denoted in practice like (a,b), (a,b,c), etc. This
331 is essentially a way to pack a few values together in a record-like
332 structure. In fact, the builtin tuple types are just algebraic product
333 types (and are thus supported in exactly the same way).
335 The "product" in its name refers to the collection of values belonging
336 to this type. The collection for a product type is the cartesian
337 product of the collections for the types of its fields.
339 These types are translated to \VHDL, record types, with one field for
340 every field in the constructor. This translation applies to all single
341 constructor algebraic datatypes, including those with no fields (unit
342 types) and just one field (which are technically not a product).
344 \startdesc{Enumerated types}
345 An enumerated type is an algebraic datatype with multiple constructors, but
346 none of them have fields. This is essentially a way to get an
347 enum-like type containing alternatives.
349 Note that Haskell's \hs{Bool} type is also defined as an
350 enumeration type, but we have a fixed translation for that.
352 These types are translated to \VHDL enumerations, with one value for
353 each constructor. This allows references to these constructors to be
354 translated to the corresponding enumeration value.
356 \startdesc{Sum types}
357 A sum type is an algebraic datatype with multiple constructors, where
358 the constructors have one or more fields. Technically, a type with
359 more than one field per constructor is a sum of products type, but
360 for our purposes this distinction does not really make a difference,
361 so we'll leave it out.
363 Sum types are currently not supported by the prototype, since there is
364 no obvious \VHDL alternative. They can easily be emulated, however, as
365 we will see from an example:
368 data Sum = A Bit Word | B Word
371 An obvious way to translate this would be to create an enumeration to
372 distinguish the constructors and then create a big record that
373 contains all the fields of all the constructors. This is the same
374 translation that would result from the following enumeration and
375 product type (using a tuple for clarity):
379 type Sum = (SumC, Bit, Word, Word)
382 Here, the \hs{SumC} type effectively signals which of the latter three
383 fields of the \hs{Sum} type are valid (the first two if \hs{A}, the
384 last one if \hs{B}), all the other ones have no useful value.
386 An obvious problem with this naive approach is the space usage: The
387 example above generates a fairly big \VHDL type. However, we can be
388 sure that the two \hs{Word}s in the \hs{Sum} type will never be valid
389 at the same time, so this is a waste of space.
391 Obviously, we could do some duplication detection here to reuse a
392 particular field for another constructor, but this would only
393 partially solve the problem. If I would, for example, have an array of
394 8 bits and a 8 bit unsiged word, these are different types and could
395 not be shared. However, in the final hardware, both of these types
396 would simply be 8 bit connections, so we have a 100\% size increase by
400 Another interesting case is that of recursive types. In Haskell, an
401 algebraic datatype can be recursive: Any of its field types can be (or
402 contain) the type being defined. The most well-known recursive type is
403 probably the list type, which is defined is:
406 data List t = Empty | Cons t (List t)
409 Note that \hs{Empty} is usually written as \hs{[]} and \hs{Cons} as
410 \hs{:}, but this would make the definition harder to read. This
411 immediately shows the problem with recursive types: What hardware type
414 If we would use the naive approach for sum types we described above, we
415 would create a record where the first field is an enumeration to
416 distinguish \hs{Empty} from \hs{Cons}. Furthermore, we would add two
417 more fields: One with the (\VHDL equivalent of) type \hs{t} (assuming we
418 actually know what type this is at compile time, this should not be a
419 problem) and a second one with type \hs{List t}. The latter one is of
420 course a problem: This is exactly the type we were trying to translate
423 Our \VHDL type will thus become infinitely deep. In other words, there
424 is no way to statically determine how long (deep) the list will be
425 (it could even be infinite).
427 In general, we can say we can never properly translate recursive types:
428 All recursive types have a potentially infinite value (even though in
429 practice they will have a bounded value, there is no way for the
430 compiler to determine an upper bound on its size).
432 \subsection{Partial application}
433 Now we've seen how to to translate application, choice and types, we will
434 get to a more complex concept: Partial applications. A \emph{partial
435 application} is any application whose (return) type is (again) a function
438 From this, we can see that the translation rules for full application do not
439 apply to a partial application. \in{Example}[ex:Quadruple] shows an example
440 use of partial application and the corresponding architecture.
442 \startbuffer[Quadruple]
443 -- | Multiply the input word by four.
444 quadruple :: Word -> Word
445 quadruple n = mul (mul n)
450 \startuseMPgraphic{Quadruple}
451 save in, two, mula, mulb, out;
454 newCircle.in(btex $n$ etex) "framed(false)";
455 newCircle.two(btex $2$ etex) "framed(false)";
456 newCircle.out(btex $out$ etex) "framed(false)";
459 newCircle.mula(btex $\times$ etex);
460 newCircle.mulb(btex $\times$ etex);
463 in.c = two.c + (0cm, 1cm);
464 mula.c = in.c + (2cm, 0cm);
465 mulb.c = mula.c + (2cm, 0cm);
466 out.c = mulb.c + (2cm, 0cm);
468 % Draw objects and lines
469 drawObj(in, two, mula, mulb, out);
471 nccurve(two)(mula) "angleA(0)", "angleB(45)";
472 nccurve(two)(mulb) "angleA(0)", "angleB(45)";
478 \placeexample[here][ex:Quadruple]{Simple three port and.}
479 \startcombination[2*1]
480 {\typebufferhs{Quadruple}}{Haskell description using function applications.}
481 {\boxedgraphic{Quadruple}}{The architecture described by the Haskell description.}
484 Here, the definition of mul is a partial function application: It applies
485 the function \hs{(*) :: Word -> Word -> Word} to the value \hs{2 :: Word},
486 resulting in the expression \hs{(*) 2 :: Word -> Word}. Since this resulting
487 expression is again a function, we can't generate hardware for it directly.
488 This is because the hardware to generate for \hs{mul} depends completely on
489 where and how it is used. In this example, it is even used twice.
491 However, it is clear that the above hardware description actually describes
492 valid hardware. In general, we can see that any partial applied function
493 must eventually become completely applied, at which point we can generate
494 hardware for it using the rules for function application given in
495 \in{section}[sec:description:application]. It might mean that a partial
496 application is passed around quite a bit (even beyond function boundaries),
497 but eventually, the partial application will become completely applied.
498 \todo{Provide a step-by-step example of how this works}
500 \section{Costless specialization}
501 Each (complete) function application in our description generates a
502 component instantiation, or a specific piece of hardware in the final
503 design. It is interesting to note that each application of a function
504 generates a \emph{separate} piece of hardware. In the final design, none
505 of the hardware is shared between applications, even when the applied
506 function is the same (of course, if a particular value, such as the result
507 of a function application, is used twice, it is not calculated twice).
509 This is distinctly different from normal program compilation: Two separate
510 calls to the same function share the same machine code. Having more
511 machine code has implications for speed (due to less efficient caching)
512 and memory usage. For normal compilation, it is therefore important to
513 keep the amount of functions limited and maximize the code sharing.
515 When generating hardware, this is hardly an issue. Having more \quote{code
516 sharing} does reduce the amount of \small{VHDL} output (Since different
517 component instantiations still share the same component), but after
518 synthesis, the amount of hardware generated is not affected.
520 In particular, if we would duplicate all functions so that there is a
521 separate function for every application in the program (\eg, each function
522 is then only applied exactly once), there would be no increase in hardware
525 Because of this, a common optimization technique called
526 \emph{specialization} can be applied to hardware generation without any
527 performance or area cost (unlike for software).
529 \fxnote{Perhaps these next three sections are a bit too
530 implementation-oriented?}
532 \subsection{Specialization}
533 \defref{specialization}
534 Given some function that has a \emph{domain} $D$ (\eg, the set of all
535 possible arguments that could be applied), we create a specialized
536 function with exactly the same behaviour, but with a domain $D' \subset
537 D$. This subset can be chosen in all sorts of ways. Any subset is valid
538 for the general definition of specialization, but in practice only some
539 of them provide useful optimization opportunities.
541 Common subsets include limiting a polymorphic argument to a single type
542 (\ie, removing polymorphism) or limiting an argument to just a single
543 value (\ie, cross-function constant propagation, effectively removing
546 Since we limit the argument domain of the specialized function, its
547 definition can often be optimized further (since now more types or even
548 values of arguments are already known). By replacing any application of
549 the function that falls within the reduced domain by an application of
550 the specialized version, the code gets faster (but the code also gets
551 bigger, since we now have two versions instead of one). If we apply
552 this technique often enough, we can often replace all applications of a
553 function by specialized versions, allowing the original function to be
554 removed (in some cases, this can even give a net reduction of the code
555 compared to the non-specialized version).
557 Specialization is useful for our hardware descriptions for functions
558 that contain arguments that cannot be translated to hardware directly
559 (polymorphic or higher order arguments, for example). If we can create
560 specialized functions that remove the argument, or make it translatable,
561 we can use specialization to make the original, untranslatable, function
564 \section{Higher order values}
565 What holds for partial application, can be easily generalized to any
566 higher order expression. This includes partial applications, plain
567 variables (e.g., a binder referring to a top level function), lambda
568 expressions and more complex expressions with a function type (a \hs{case}
569 expression returning lambda's, for example).
571 Each of these values cannot be directly represented in hardware (just like
572 partial applications). Also, to make them representable, they need to be
573 applied: function variables and partial applications will then eventually
574 become complete applications, applied lambda expressions disappear by
575 applying β-reduction, etc.
577 So any higher order value will be \quote{pushed down} towards its
578 application just like partial applications. Whenever a function boundary
579 needs to be crossed, the called function can be specialized.
581 \fxnote{This section needs improvement and an example}
583 \section{Polymorphism}
584 In Haskell, values can be \emph{polymorphic}: They can have multiple types. For
585 example, the function \hs{fst :: (a, b) -> a} is an example of a
586 polymorphic function: It works for tuples with any two element types. Haskell
587 typeclasses allow a function to work on a specific set of types, but the
588 general idea is the same. The opposite of this is a \emph{monomorphic}
589 value, which has a single, fixed, type.
591 % A type class is a collection of types for which some operations are
592 % defined. It is thus possible for a value to be polymorphic while having
593 % any number of \emph{class constraints}: The value is not defined for
594 % every type, but only for types in the type class. An example of this is
595 % the \hs{even :: (Integral a) => a -> Bool} function, which can map any
596 % value of a type that is member of the \hs{Integral} type class
598 When generating hardware, polymorphism can't be easily translated. How
599 many wires will you lay down for a value that could have any type? When
600 type classes are involved, what hardware components will you lay down for
601 a class method (whose behaviour depends on the type of its arguments)?
602 Note that the language currently does not allow user-defined typeclasses,
603 but does support partly some of the builtin typeclasses (like \hs{Num}).
605 Fortunately, we can again use the principle of specialization: Since every
606 function application generates a separate piece of hardware, we can know
607 the types of all arguments exactly. Provided that we don't use existential
608 typing, all of the polymorphic types in a function must depend on the
609 types of the arguments (In other words, the only way to introduce a type
610 variable is in a lambda abstraction).
612 If a function is monomorphic, all values inside it are monomorphic as
613 well, so any function that is applied within the function can only be
614 applied to monomorphic values. The applied functions can then be
615 specialized to work just for these specific types, removing the
616 polymorphism from the applied functions as well.
618 Our top level function must not have a polymorphic type (otherwise we
619 wouldn't know the hardware interface to our top level function).
621 By induction, this means that all functions that are (indirectly) called
622 by our top level function (meaning all functions that are translated in
623 the final hardware) become monomorphic.
626 A very important concept in hardware designs is \emph{state}. In a
627 stateless (or, \emph{combinatoric}) design, every output is directly and solely dependent on the
628 inputs. In a stateful design, the outputs can depend on the history of
629 inputs, or the \emph{state}. State is usually stored in \emph{registers},
630 which retain their value during a clockcycle, and are typically updated at
631 the start of every clockcycle. Since the updating of the state is tightly
632 coupled (synchronized) to the clock signal, these state updates are often
633 called \emph{synchronous} behaviour.
635 \todo{Sidenote? Registers can contain any (complex) type}
637 To make our hardware description language useful to describe more than
638 simple combinatoric designs, we'll need to be able to describe state in
641 \subsection{Approaches to state}
642 In Haskell, functions are always pure (except when using unsafe
643 functions with the \hs{IO} monad, which is not supported by Cλash). This
644 means that the output of a function solely depends on its inputs. If you
645 evaluate a given function with given inputs, it will always provide the
650 This is a perfect match for a combinatoric circuit, where the output
651 also soley depends on the inputs. However, when state is involved, this
652 no longer holds. Since we're in charge of our own language (or at least
653 let's pretend we aren't using Haskell and we are), we could remove this
654 purity constraint and allow a function to return different values
655 depending on the cycle in which it is evaluated (or rather, the current
656 state). However, this means that all kinds of interesting properties of
657 our functional language get lost, and all kinds of transformations and
658 optimizations might no longer be meaning preserving.
660 Provided that we want to keep the function pure, the current state has
661 to be present in the function's arguments in some way. There seem to be
662 two obvious ways to do this: Adding the current state as an argument, or
663 including the full history of each argument.
665 \subsubsection{Stream arguments and results}
666 Including the entire history of each input (\eg, the value of that
667 input for each previous clockcycle) is an obvious way to make outputs
668 depend on all previous input. This is easily done by making every
669 input a list instead of a single value, containing all previous values
670 as well as the current value.
672 An obvious downside of this solution is that on each cycle, all the
673 previous cycles must be resimulated to obtain the current state. To do
674 this, it might be needed to have a recursive helper function as well,
675 wich might be hard to be properly analyzed by the compiler.
677 A slight variation on this approach is one taken by some of the other
678 functional \small{HDL}s in the field: \todo{References to Lava,
679 ForSyDe, ...} Make functions operate on complete streams. This means
680 that a function is no longer called on every cycle, but just once. It
681 takes stream as inputs instead of values, where each stream contains
682 all the values for every clockcycle since system start. This is easily
683 modeled using an (infinite) list, with one element for each clock
684 cycle. Since the function is only evaluated once, its output must also
685 be a stream. Note that, since we are working with infinite lists and
686 still want to be able to simulate the system cycle-by-cycle, this
687 relies heavily on the lazy semantics of Haskell.
689 Since our inputs and outputs are streams, all other (intermediate)
690 values must be streams. All of our primitive operators (\eg, addition,
691 substraction, bitwise operations, etc.) must operate on streams as
692 well (note that changing a single-element operation to a stream
693 operation can done with \hs{map}, \hs{zipwith}, etc.).
695 Note that the concept of \emph{state} is no more than having some way
696 to communicate a value from one cycle to the next. By introducing a
697 \hs{delay} function, we can do exactly that: Delay (each value in) a
698 stream so that we can "look into" the past. This \hs{delay} function
699 simply outputs a stream where each value is the same as the input
700 value, but shifted one cycle. This causes a \quote{gap} at the
701 beginning of the stream: What is the value of the delay output in the
702 first cycle? For this, the \hs{delay} function has a second input
703 (which is a value, not a stream!).
705 \in{Example}[ex:DelayAcc] shows a simple accumulator expressed in this
708 \startbuffer[DelayAcc]
709 acc :: Stream Word -> Stream Word
712 out = (delay out 0) + in
715 \startuseMPgraphic{DelayAcc}
716 save in, out, add, reg;
719 newCircle.in(btex $in$ etex) "framed(false)";
720 newCircle.out(btex $out$ etex) "framed(false)";
723 newReg.reg("") "dx(4mm)", "dy(6mm)", "reflect(true)";
724 newCircle.add(btex + etex);
727 add.c = in.c + (2cm, 0cm);
728 out.c = add.c + (2cm, 0cm);
729 reg.c = add.c + (0cm, 2cm);
731 % Draw objects and lines
732 drawObj(in, out, add, reg);
734 nccurve(add)(reg) "angleA(0)", "angleB(180)", "posB(d)";
735 nccurve(reg)(add) "angleA(180)", "angleB(-45)", "posA(out)";
741 \placeexample[here][ex:DelayAcc]{Simple accumulator architecture.}
742 \startcombination[2*1]
743 {\typebufferhs{DelayAcc}}{Haskell description using streams.}
744 {\boxedgraphic{DelayAcc}}{The architecture described by the Haskell description.}
748 This notation can be confusing (especially due to the loop in the
749 definition of out), but is essentially easy to interpret. There is a
750 single call to delay, resulting in a circuit with a single register,
751 whose input is connected to \hs{out} (which is the output of the
752 adder), and it's output is the expression \hs{delay out 0} (which is
753 connected to one of the adder inputs).
755 This notation has a number of downsides, amongst which are limited
756 readability and ambiguity in the interpretation. \note{Reference
757 Christiaan, who has done further investigation}
759 \subsubsection{Explicit state arguments and results}
760 A more explicit way to model state, is to simply add an extra argument
761 containing the current state value. This allows an output to depend on
762 both the inputs as well as the current state while keeping the
763 function pure (letting the result depend only on the arguments), since
764 the current state is now an argument.
766 In Haskell, this would look like
767 \in{example}[ex:ExplicitAcc]\footnote[notfinalsyntax]{Note that this example is not in the final
768 Cλash syntax}. \todo{Referencing notfinalsyntax from Introduction.tex doesn't
771 \startbuffer[ExplicitAcc]
772 -- input -> current state -> (new state, output)
773 acc :: Word -> Word -> (Word, Word)
780 \placeexample[here][ex:ExplicitAcc]{Simple accumulator architecture.}
781 \startcombination[2*1]
782 {\typebufferhs{ExplicitAcc}}{Haskell description using explicit state arguments.}
783 % Picture is identical to the one we had just now.
784 {\boxedgraphic{DelayAcc}}{The architecture described by the Haskell description.}
787 This approach makes a function's state very explicit, which state
788 variables are used by a function can be completely determined from its
789 type signature (as opposed to the stream approach, where a function
790 looks the same from the outside, regardless of what state variables it
791 uses (or whether it's stateful at all).
793 This approach is the one chosen for Cλash and will be examined more
796 \subsection{Explicit state specification}
797 We've seen the concept of explicit state in a simple example below, but
798 what are the implications of this approach?
800 \subsubsection{Substates}
801 Since a function's state is reflected directly in its type signature,
802 if a function calls other stateful functions (\eg, has subcircuits), it
803 has to somehow know the current state for these called functions. The
804 only way to do this, is to put these \emph{substates} inside the
805 caller's state. This means that a function's state is the sum of the
806 states of all functions it calls, and its own state.
808 This also means that the type of a function (at least the "state"
809 part) is dependent on its own implementation and of the functions it
812 This is the major downside of this approach: The separation between
813 interface and implementation is limited. However, since Cλash is not
814 very suitable for separate compilation (see
815 \in{section}[sec:prototype:separate]) this is not a big problem in
818 Additionally, when using a type synonym for the state type
819 of each function, we can still provide explicit type signatures
820 while keeping the state specification for a function near its
824 \subsubsection{Which arguments and results are stateful?}
825 \fxnote{This section should get some examples}
826 We need some way to know which arguments should become input ports and
827 which argument(s?) should become the current state (\eg, be bound to
828 the register outputs). This does not hold just for the top
829 level function, but also for any subfunction. Or could we perhaps
830 deduce the statefulness of subfunctions by analyzing the flow of data
831 in the calling functions?
833 To explore this matter, the following observeration is interesting: We
834 get completely correct behaviour when we put all state registers in
835 the top level entity (or even outside of it). All of the state
836 arguments and results on subfunctions are treated as normal input and
837 output ports. Effectively, a stateful function results in a stateless
838 hardware component that has one of its input ports connected to the
839 output of a register and one of its output ports connected to the
840 input of the same register.
844 Of course, even though the hardware described like this has the
845 correct behaviour, unless the layout tool does smart optimizations,
846 there will be a lot of extra wire in the design (since registers will
847 not be close to the component that uses them). Also, when working with
848 the generated \small{VHDL} code, there will be a lot of extra ports
849 just to pass on state values, which can get quite confusing.
851 To fix this, we can simply \quote{push} the registers down into the
852 subcircuits. When we see a register that is connected directly to a
853 subcircuit, we remove the corresponding input and output port and put
854 the register inside the subcircuit instead. This is slightly less
855 trivial when looking at the Haskell code instead of the resulting
856 circuit, but the idea is still the same.
860 However, when applying this technique, we might push registers down
861 too far. When you intend to store a result of a stateless subfunction
862 in the caller's state and pass the current value of that state
863 variable to that same function, the register might get pushed down too
864 far. It is impossible to distinguish this case from similar code where
865 the called function is in fact stateful. From this we can conclude
866 that we have to either:
868 \todo{Example of wrong downpushing}
871 \item accept that the generated hardware might not be exactly what we
872 intended, in some specific cases. In most cases, the hardware will be
874 \item explicitely annotate state arguments and results in the input
878 The first option causes (non-obvious) exceptions in the language
879 intepretation. Also, automatically determining where registers should
880 end up is easier to implement correctly with explicit annotations, so
881 for these reasons we will look at how this annotations could work.
883 \todo{Sidenote: One or more state arguments?}
885 \subsection{Explicit state annotation}
886 To make our stateful descriptions unambigious and easier to translate,
887 we need some way for the developer to describe which arguments and
888 results are intended to become stateful.
890 Roughly, we have two ways to achieve this:
892 \item Use some kind of annotation method or syntactic construction in
893 the language to indicate exactly which argument and (part of the)
894 result is stateful. This means that the annotation lives
895 \quote{outside} of the function, it is completely invisible when
896 looking at the function body.
897 \item Use some kind of annotation on the type level, \ie give stateful
898 arguments and stateful (parts of) results a different type. This has the
899 potential to make this annotation visible inside the function as well,
900 such that when looking at a value inside the function body you can
901 tell if it's stateful by looking at its type. This could possibly make
902 the translation process a lot easier, since less analysis of the
903 program flow might be required.
906 From these approaches, the type level \quote{annotations} have been
907 implemented in Cλash. \in{Section}[sec:prototype:statetype] expands on
908 the possible ways this could have been implemented.
910 \todo{Note about conditions on state variables and checking them}
912 \section[sec:recursion]{Recursion}
913 An import concept in functional languages is recursion. In it's most basic
914 form, recursion is a definition that is defined in terms of itself. A
915 recursive function is thus a function that uses itself in its body. This
916 usually requires multiple evaluations of this function, with changing
917 arguments, until eventually an evaluation of the function no longer requires
920 Recursion in a hardware description is a bit of a funny thing. Usually,
921 recursion is associated with a lot of nondeterminism, stack overflows, but
922 also flexibility and expressive power.
924 Given the notion that each function application will translate to a
925 component instantiation, we are presented with a problem. A recursive
926 function would translate to a component that contains itself. Or, more
927 precisely, that contains an instance of itself. This instance would again
928 contain an instance of itself, and again, into infinity. This is obviously a
929 problem for generating hardware.
931 This is expected for functions that describe infinite recursion. In that
932 case, we can't generate hardware that shows correct behaviour in a single
933 cycle (at best, we could generate hardware that needs an infinite number of
936 However, most recursive hardware descriptions will describe finite
937 recursion. This is because the recursive call is done conditionally. There
938 is usually a \hs{case} expression where at least one alternative does not contain
939 the recursive call, which we call the "base case". If, for each call to the
940 recursive function, we would be able to detect at compile time which
941 alternative applies, we would be able to remove the \hs{case} expression and
942 leave only the base case when it applies. This will ensure that expanding
943 the recursive functions will terminate after a bounded number of expansions.
945 This does imply the extra requirement that the base case is detectable at
946 compile time. In particular, this means that the decision between the base
947 case and the recursive case must not depend on runtime data.
949 \subsection{List recursion}
950 The most common deciding factor in recursion is the length of a list that is
951 passed in as an argument. Since we represent lists as vectors that encode
952 the length in the vector type, it seems easy to determine the base case. We
953 can simply look at the argument type for this. However, it turns out that
954 this is rather non-trivial to write down in Haskell already, not even
955 looking at translation. As an example, we would like to write down something
959 sum :: Vector n Word -> Word
960 sum xs = case null xs of
962 False -> head xs + sum (tail xs)
965 However, the Haskell typechecker will now use the following reasoning (element
966 type of the vector is left out). Below, we write conditions on type
967 variables before the \hs{=>} operator. This is not completely valid Haskell
968 syntax, but serves to illustrate the typechecker reasoning. Also note that a
969 vector can never have a negative length, so \hs{Vector n} implicitly means
970 \hs{(n >= 0) => Vector n}.
972 \todo{This typechecker disregards the type signature}
974 \item tail has the type \hs{(n > 0) => Vector n -> Vector (n - 1)}
975 \item This means that xs must have the type \hs{(n > 0) => Vector n}
976 \item This means that sum must have the type \hs{(n > 0) => Vector n -> a}
977 \item sum is called with the result of tail as an argument, which has the
978 type \hs{Vector n} (since \hs{(n > 0) => Vector (n - 1)} is the same as \hs{(n >= 0)
979 => Vector n}, which is the same as just \hs{Vector n}).
980 \item This means that sum must have the type \hs{Vector n -> a}
981 \item This is a contradiction between the type deduced from the body of sum
982 (the input vector must be non-empty) and the use of sum (the input vector
983 could have any length).
986 As you can see, using a simple \hs{case} expression at value level causes
987 the type checker to always typecheck both alternatives, which can't be done!
988 This is a fundamental problem, that would seem perfectly suited for a type
989 class. Considering that we need to switch between to implementations of the
990 sum function, based on the type of the argument, this sounds like the
991 perfect problem to solve with a type class. However, this approach has its
992 own problems (not the least of them that you need to define a new typeclass
993 for every recursive function you want to define).
995 Another approach tried involved using GADTs to be able to do pattern
996 matching on empty / non empty lists. While this worked partially, it also
997 created problems with more complex expressions.
999 \note{This should reference Christiaan}
1001 Evaluating all possible (and non-possible) ways to add recursion to our
1002 descriptions, it seems better to leave out list recursion alltogether. This
1003 allows us to focus on other interesting areas instead. By including
1004 (builtin) support for a number of higher order functions like map and fold,
1005 we can still express most of the things we would use list recursion for.
1007 \todo{Expand on this decision a bit}
1009 \subsection{General recursion}
1010 Of course there are other forms of recursion, that do not depend on the
1011 length (and thus type) of a list. For example, simple recursion using a
1012 counter could be expressed, but only translated to hardware for a fixed
1013 number of iterations. Also, this would require extensive support for compile
1014 time simplification (constant propagation) and compile time evaluation
1015 (evaluation of constant comparisons), to ensure non-termination. Even then, it
1016 is hard to really guarantee termination, since the user (or GHC desugarer)
1017 might use some obscure notation that results in a corner case of the
1018 simplifier that is not caught and thus non-termination.
1020 Due to these complications and limited time available, we leave other forms
1021 of recursion as future work as well.
1023 % vim: set sw=2 sts=2 expandtab: