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 To translate Haskell to hardware, every Haskell construct needs a
10 translation to \VHDL. There are often multiple valid translations
11 possible. When faced with choices, the most obvious choice has been
12 chosen wherever possible. In a lot of cases, when a programmer looks
13 at a functional hardware description it is completely clear what
14 hardware is described. We want our translator to generate exactly that
15 hardware whenever possible, to make working with Cλash as intuitive as
18 In this chapter we describe how to interpret a Haskell program from a
19 hardware perspective. We provide a description of each Haskell language
20 element that needs translation, to provide a clear picture of what is
23 \section[sec:description:application]{Function application}
24 \todo{Sidenote: Inputs vs arguments}
25 The basic syntactic elements of a functional program are functions and
26 function application. These have a single obvious \small{VHDL} translation: Each
27 function becomes a hardware component, where each argument is an input port
28 and the result value is the (single) output port. This output port can have
29 a complex type (such as a tuple), so having just a single output port does
30 not pose a limitation.
32 Each function application in turn becomes component instantiation. Here, the
33 result of each argument expression is assigned to a signal, which is mapped
34 to the corresponding input port. The output port of the function is also
35 mapped to a signal, which is used as the result of the application.
37 \in{Example}[ex:And3] shows a simple program using only function
38 application and the corresponding architecture.
41 -- | A simple function that returns
42 -- conjunction of three bits
43 and3 :: Bit -> Bit -> Bit -> Bit
44 and3 a b c = and (and a b) c
47 \todo{Mirror this image vertically}
48 \startuseMPgraphic{And3}
49 save a, b, c, anda, andb, out;
52 newCircle.a(btex $a$ etex) "framed(false)";
53 newCircle.b(btex $b$ etex) "framed(false)";
54 newCircle.c(btex $c$ etex) "framed(false)";
55 newCircle.out(btex $out$ etex) "framed(false)";
58 newCircle.anda(btex $and$ etex);
59 newCircle.andb(btex $and$ etex);
62 b.c = a.c + (0cm, 1cm);
63 c.c = b.c + (0cm, 1cm);
64 anda.c = midpoint(a.c, b.c) + (2cm, 0cm);
65 andb.c = midpoint(b.c, c.c) + (4cm, 0cm);
67 out.c = andb.c + (2cm, 0cm);
69 % Draw objects and lines
70 drawObj(a, b, c, anda, andb, out);
72 ncarc(a)(anda) "arcangle(-10)";
79 \placeexample[here][ex:And3]{Simple three input and gate.}
80 \startcombination[2*1]
81 {\typebufferhs{And3}}{Haskell description using function applications.}
82 {\boxedgraphic{And3}}{The architecture described by the Haskell description.}
85 \todo{This section should also mention hierarchy, top level functions and
89 Although describing components and connections allows us to describe a lot of
90 hardware designs already, there is an obvious thing missing: Choice. We
91 need some way to be able to choose between values based on another value.
92 In Haskell, choice is achieved by \hs{case} expressions, \hs{if}
93 expressions, pattern matching and guards.
95 An obvious way to add choice to our language without having to recognize
96 any of Haskell's syntax, would be to add a primivite \quote{\hs{if}}
97 function. This function would take three arguments: The condition, the
98 value to return when the condition is true and the value to return when
99 the condition is false.
101 This \hs{if} function would then essentially describe a multiplexer and
102 allows us to describe any architecture that uses multiplexers. \fxnote{Are
103 there other mechanisms of choice in hardware?}
105 However, to be able to describe our hardware in a more convenient way, we
106 also want to translate Haskell's choice mechanisms. The easiest of these
107 are of course case expressions (and \hs{if} expressions, which can be very
108 directly translated to \hs{case} expressions). A \hs{case} expression can in turn
109 simply be translated to a conditional assignment, where the conditions use
110 equality comparisons against the constructors in the \hs{case} expressions.
112 \todo{Assignment vs multiplexers}
114 In \in{example}[ex:CaseInv] a simple \hs{case} expression is shown,
115 scrutinizing a boolean value. The corresponding architecture has a
116 comparator to determine which of the constructors is on the \hs{in}
117 input. There is a multiplexer to select the output signal. The two options
118 for the output signals are just constants, but these could have been more
119 complex expressions (in which case also both of them would be working in
120 parallel, regardless of which output would be chosen eventually).
122 If we would translate a Boolean to a bit value, we could of course remove
123 the comparator and directly feed 'in' into the multiplex (or even use an
124 inverter instead of a multiplexer). However, we will try to make a
125 general translation, which works for all possible \hs{case} expressions.
126 Optimizations such as these are left for the \VHDL synthesizer, which
127 handles them very well.
129 \todo{Be more explicit about >2 alternatives}
131 \startbuffer[CaseInv]
138 \startuseMPgraphic{CaseInv}
139 save in, truecmp, falseout, trueout, out, cmp, mux;
142 newCircle.in(btex $in$ etex) "framed(false)";
143 newCircle.out(btex $out$ etex) "framed(false)";
145 newBox.truecmp(btex $True$ etex) "framed(false)";
146 newBox.trueout(btex $True$ etex) "framed(false)";
147 newBox.falseout(btex $False$ etex) "framed(false)";
150 newCircle.cmp(btex $==$ etex);
154 cmp.c = in.c + (3cm, 0cm);
155 truecmp.c = cmp.c + (-1cm, 1cm);
156 mux.sel = cmp.e + (1cm, -1cm);
157 falseout.c = mux.inpa - (2cm, 0cm);
158 trueout.c = mux.inpb - (2cm, 0cm);
159 out.c = mux.out + (2cm, 0cm);
161 % Draw objects and lines
162 drawObj(in, out, truecmp, trueout, falseout, cmp, mux);
166 nccurve(cmp.e)(mux.sel) "angleA(0)", "angleB(-90)";
167 ncline(falseout)(mux) "posB(inpa)";
168 ncline(trueout)(mux) "posB(inpb)";
169 ncline(mux)(out) "posA(out)";
172 \placeexample[here][ex:CaseInv]{Simple inverter.}
173 \startcombination[2*1]
174 {\typebufferhs{CaseInv}}{Haskell description using a Case expression.}
175 {\boxedgraphic{CaseInv}}{The architecture described by the Haskell description.}
178 A slightly more complex (but very powerful) form of choice is pattern
179 matching. A function can be defined in multiple clauses, where each clause
180 specifies a pattern. When the arguments match the pattern, the
181 corresponding clause will be used.
183 \startbuffer[PatternInv]
189 \placeexample[here][ex:PatternInv]{Simple inverter using pattern matching.
190 Describes the same architecture as \in{example}[ex:CaseInv].}
191 {\typebufferhs{PatternInv}}
193 The architecture described by \in{example}[ex:PatternInv] is of course the
194 same one as the one in \in{example}[ex:CaseInv]. The general interpretation
195 of pattern matching is also similar to that of \hs{case} expressions: Generate
196 hardware for each of the clauses (like each of the clauses of a \hs{case}
197 expression) and connect them to the function output through (a number of
198 nested) multiplexers. These multiplexers are driven by comparators and
199 other logic, that check each pattern in turn.
202 We've seen the two most basic functional concepts translated to hardware:
203 Function application and choice. Before we look further into less obvious
204 concepts like higher order expressions and polymorphism, we will have a
205 look at the types of the values we use in our descriptions.
207 When working with values in our descriptions, we'll also need to provide
208 some way to translate the values used to hardware equivalents. In
209 particular, this means having to come up with a hardware equivalent for
210 every \emph{type} used in our program.
212 Since most functional languages have a lot of standard types that are
213 hard to translate (integers without a fixed size, lists without a static
214 length, etc.), we will start out by defining a number of \quote{builtin}
215 types ourselves. These types are builtin in the sense that our compiler
216 will have a fixed VHDL type for these. User defined types, on the other
217 hand, will have their hardware type derived directly from their Haskell
218 declaration automatically, according to the rules we sketch here.
220 \todo{Introduce Haskell type syntax (type constructors, type application,
223 \subsection{Builtin types}
224 The language currently supports the following builtin types. Of these,
225 only the \hs{Bool} type is supported by Haskell out of the box (the
226 others are defined by the Cλash package, so they are user-defined types
227 from Haskell's point of view).
230 This is the most basic type available. It is mapped directly onto
231 the \type{std_logic} \small{VHDL} type. Mapping this to the
232 \type{bit} type might make more sense (since the Haskell version
233 only has two values), but using \type{std_logic} is more standard
234 (and allowed for some experimentation with don't care values)
236 \todo{Sidenote bit vs stdlogic}
238 \startdesc{\hs{Bool}}
239 This is the only builtin Haskell type supported and is translated
240 exactly like the Bit type (where a value of \hs{True} corresponds to a
241 value of \hs{High}). Supporting the Bool type is particularly
242 useful to support \hs{if ... then ... else ...} expressions, which
243 always have a \hs{Bool} value for the condition.
245 A \hs{Bool} is translated to a \type{std_logic}, just like \hs{Bit}.
247 \startdesc{\hs{SizedWord}, \hs{SizedInt}}
248 These are types to represent integers. A \hs{SizedWord} is unsigned,
249 while a \hs{SizedInt} is signed. These types are parameterized by a
250 length type, so you can define an unsigned word of 32 bits wide as
254 type Word32 = SizedWord D32
257 Here, a type synonym \hs{Word32} is defined that is equal to the
258 \hs{SizedWord} type constructor applied to the type \hs{D32}. \hs{D32}
259 is the \emph{type level representation} of the decimal number 32,
260 making the \hs{Word32} type a 32-bit unsigned word.
262 These types are translated to the \small{VHDL} \type{unsigned} and
263 \type{signed} respectively.
264 \todo{Sidenote on dependent typing?}
266 \startdesc{\hs{Vector}}
267 This is a vector type, that can contain elements of any other type and
268 has a fixed length. It has two type parameters: Its
269 length and the type of the elements contained in it. By putting the
270 length parameter in the type, the length of a vector can be determined
271 at compile time, instead of only at runtime for conventional lists.
273 The \hs{Vector} type constructor takes two type arguments: The length
274 of the vector and the type of the elements contained in it. The state
275 type of an 8 element register bank would then for example be:
278 type RegisterState = Vector D8 Word32
281 Here, a type synonym \hs{RegisterState} is defined that is equal to
282 the \hs{Vector} type constructor applied to the types \hs{D8} (The type
283 level representation of the decimal number 8) and \hs{Word32} (The 32
284 bit word type as defined above). In other words, the
285 \hs{RegisterState} type is a vector of 8 32-bit words.
287 A fixed size vector is translated to a \small{VHDL} array type.
289 \startdesc{\hs{RangedWord}}
290 This is another type to describe integers, but unlike the previous
291 two it has no specific bitwidth, but an upper bound. This means that
292 its range is not limited to powers of two, but can be any number.
293 A \hs{RangedWord} only has an upper bound, its lower bound is
294 implicitly zero. There is a lot of added implementation complexity
295 when adding a lower bound and having just an upper bound was enough
296 for the primary purpose of this type: Typesafely indexing vectors.
298 To define an index for the 8 element vector above, we would do:
301 type RegisterIndex = RangedWord D7
304 Here, a type synonym \hs{RegisterIndex} is defined that is equal to
305 the \hs{RangedWord} type constructor applied to the type \hs{D7}. In
306 other words, this defines an unsigned word with values from 0 to 7
307 (inclusive). This word can be be used to index the 8 element vector
308 \hs{RegisterState} above.
310 This type is translated to the \type{unsigned} \small{VHDL} type.
312 \fxnote{There should be a reference to Christiaan's work here.}
314 \subsection{User-defined types}
315 There are three ways to define new types in Haskell: Algebraic
316 datatypes with the \hs{data} keyword, type synonyms with the \hs{type}
317 keyword and type renamings with the \hs{newtype} keyword. \GHC
318 offers a few more advanced ways to introduce types (type families,
319 existential typing, \small{GADT}s, etc.) which are not standard
320 Haskell. These will be left outside the scope of this research.
322 Only an algebraic datatype declaration actually introduces a
323 completely new type, for which we provide the \VHDL translation
324 below. Type synonyms and renamings only define new names for
325 existing types (where synonyms are completely interchangeable and
326 renamings need explicit conversion). Therefore, these don't need
327 any particular \VHDL translation, a synonym or renamed type will
328 just use the same representation as the original type. The
329 distinction between a renaming and a synonym does no longer matter
330 in hardware and can be disregarded in the generated \VHDL.
332 For algebraic types, we can make the following distinction:
334 \startdesc{Product types}
335 A product type is an algebraic datatype with a single constructor with
336 two or more fields, denoted in practice like (a,b), (a,b,c), etc. This
337 is essentially a way to pack a few values together in a record-like
338 structure. In fact, the builtin tuple types are just algebraic product
339 types (and are thus supported in exactly the same way).
341 The \quote{product} in its name refers to the collection of values belonging
342 to this type. The collection for a product type is the cartesian
343 product of the collections for the types of its fields.
345 These types are translated to \VHDL record types, with one field for
346 every field in the constructor. This translation applies to all single
347 constructor algebraic datatypes, including those with just one
348 field (which are technically not a product, but generate a VHDL
349 record for implementation simplicity).
351 \startdesc{Enumerated types}
352 \defref{enumerated types}
353 An enumerated type is an algebraic datatype with multiple constructors, but
354 none of them have fields. This is essentially a way to get an
355 enum-like type containing alternatives.
357 Note that Haskell's \hs{Bool} type is also defined as an
358 enumeration type, but we have a fixed translation for that.
360 These types are translated to \VHDL enumerations, with one value for
361 each constructor. This allows references to these constructors to be
362 translated to the corresponding enumeration value.
364 \startdesc{Sum types}
365 A sum type is an algebraic datatype with multiple constructors, where
366 the constructors have one or more fields. Technically, a type with
367 more than one field per constructor is a sum of products type, but
368 for our purposes this distinction does not really make a difference,
369 so we'll leave it out.
371 The \quote{sum} in its name refers again to the collection of values
372 belonging to this type. The collection for a sum type is the
373 union of the the collections for each of the constructors.
375 Sum types are currently not supported by the prototype, since there is
376 no obvious \VHDL alternative. They can easily be emulated, however, as
377 we will see from an example:
380 data Sum = A Bit Word | B Word
383 An obvious way to translate this would be to create an enumeration to
384 distinguish the constructors and then create a big record that
385 contains all the fields of all the constructors. This is the same
386 translation that would result from the following enumeration and
387 product type (using a tuple for clarity):
391 type Sum = (SumC, Bit, Word, Word)
394 Here, the \hs{SumC} type effectively signals which of the latter three
395 fields of the \hs{Sum} type are valid (the first two if \hs{A}, the
396 last one if \hs{B}), all the other ones have no useful value.
398 An obvious problem with this naive approach is the space usage: The
399 example above generates a fairly big \VHDL type. Since we can be
400 sure that the two \hs{Word}s in the \hs{Sum} type will never be valid
401 at the same time, this is a waste of space.
403 Obviously, we could do some duplication detection here to reuse a
404 particular field for another constructor, but this would only
405 partially solve the problem. If I would, for example, have an array of
406 8 bits and an 8 bit unsiged word, these are different types and could
407 not be shared. However, in the final hardware, both of these types
408 would simply be 8 bit connections, so we have a 100\% size increase by
412 Another interesting case is that of recursive types. In Haskell, an
413 algebraic datatype can be recursive: Any of its field types can be (or
414 contain) the type being defined. The most well-known recursive type is
415 probably the list type, which is defined is:
418 data List t = Empty | Cons t (List t)
421 Note that \hs{Empty} is usually written as \hs{[]} and \hs{Cons} as
422 \hs{:}, but this would make the definition harder to read. This
423 immediately shows the problem with recursive types: What hardware type
426 If we would use the naive approach for sum types we described above, we
427 would create a record where the first field is an enumeration to
428 distinguish \hs{Empty} from \hs{Cons}. Furthermore, we would add two
429 more fields: One with the (\VHDL equivalent of) type \hs{t} (assuming we
430 actually know what type this is at compile time, this should not be a
431 problem) and a second one with type \hs{List t}. The latter one is of
432 course a problem: This is exactly the type we were trying to translate
435 Our \VHDL type will thus become infinitely deep. In other words, there
436 is no way to statically determine how long (deep) the list will be
437 (it could even be infinite).
439 In general, we can say we can never properly translate recursive types:
440 All recursive types have a potentially infinite value (even though in
441 practice they will have a bounded value, there is no way for the
442 compiler to automatically determine an upper bound on its size).
444 \subsection{Partial application}
445 Now we've seen how to to translate application, choice and types, we will
446 get to a more complex concept: Partial applications. A \emph{partial
447 application} is any application whose (return) type is (again) a function
450 From this, we can see that the translation rules for full application do not
451 apply to a partial application. \in{Example}[ex:Quadruple] shows an example
452 use of partial application and the corresponding architecture.
454 \startbuffer[Quadruple]
455 -- | Multiply the input word by four.
456 quadruple :: Word -> Word
457 quadruple n = mul (mul n)
462 \startuseMPgraphic{Quadruple}
463 save in, two, mula, mulb, out;
466 newCircle.in(btex $n$ etex) "framed(false)";
467 newCircle.two(btex $2$ etex) "framed(false)";
468 newCircle.out(btex $out$ etex) "framed(false)";
471 newCircle.mula(btex $\times$ etex);
472 newCircle.mulb(btex $\times$ etex);
475 in.c = two.c + (0cm, 1cm);
476 mula.c = in.c + (2cm, 0cm);
477 mulb.c = mula.c + (2cm, 0cm);
478 out.c = mulb.c + (2cm, 0cm);
480 % Draw objects and lines
481 drawObj(in, two, mula, mulb, out);
483 nccurve(two)(mula) "angleA(0)", "angleB(45)";
484 nccurve(two)(mulb) "angleA(0)", "angleB(45)";
490 \placeexample[here][ex:Quadruple]{Simple three port and.}
491 \startcombination[2*1]
492 {\typebufferhs{Quadruple}}{Haskell description using function applications.}
493 {\boxedgraphic{Quadruple}}{The architecture described by the Haskell description.}
496 Here, the definition of mul is a partial function application: It applies
497 the function \hs{(*) :: Word -> Word -> Word} to the value \hs{2 :: Word},
498 resulting in the expression \hs{(*) 2 :: Word -> Word}. Since this resulting
499 expression is again a function, we can't generate hardware for it directly.
500 This is because the hardware to generate for \hs{mul} depends completely on
501 where and how it is used. In this example, it is even used twice.
503 However, it is clear that the above hardware description actually describes
504 valid hardware. In general, we can see that any partial applied function
505 must eventually become completely applied, at which point we can generate
506 hardware for it using the rules for function application given in
507 \in{section}[sec:description:application]. It might mean that a partial
508 application is passed around quite a bit (even beyond function boundaries),
509 but eventually, the partial application will become completely applied.
510 \todo{Provide a step-by-step example of how this works}
512 \section{Costless specialization}
513 Each (complete) function application in our description generates a
514 component instantiation, or a specific piece of hardware in the final
515 design. It is interesting to note that each application of a function
516 generates a \emph{separate} piece of hardware. In the final design, none
517 of the hardware is shared between applications, even when the applied
518 function is the same (of course, if a particular value, such as the result
519 of a function application, is used twice, it is not calculated twice).
521 This is distinctly different from normal program compilation: Two separate
522 calls to the same function share the same machine code. Having more
523 machine code has implications for speed (due to less efficient caching)
524 and memory usage. For normal compilation, it is therefore important to
525 keep the amount of functions limited and maximize the code sharing
526 (though there is a tradeoff between speed and memory usage here).
528 When generating hardware, this is hardly an issue. Having more \quote{code
529 sharing} does reduce the amount of \small{VHDL} output (Since different
530 component instantiations still share the same component), but after
531 synthesis, the amount of hardware generated is not affected. This
532 means there is no tradeoff between speed and memory (or rather,
535 In particular, if we would duplicate all functions so that there is a
536 separate function for every application in the program (\eg, each function
537 is then only applied exactly once), there would be no increase in hardware
540 Because of this, a common optimization technique called
541 \emph{specialization} can be applied to hardware generation without any
542 performance or area cost (unlike for software).
544 \fxnote{Perhaps these next three sections are a bit too
545 implementation-oriented?}
547 \subsection{Specialization}
548 \defref{specialization}
549 Given some function that has a \emph{domain} $D$ (\eg, the set of
550 all possible arguments that the function could be applied to), we
551 create a specialized function with exactly the same behaviour, but
552 with a domain $D' \subset D$. This subset can be chosen in all
553 sorts of ways. Any subset is valid for the general definition of
554 specialization, but in practice only some of them provide useful
555 optimization opportunities.
557 Common subsets include limiting a polymorphic argument to a single type
558 (\ie, removing polymorphism) or limiting an argument to just a single
559 value (\ie, cross-function constant propagation, effectively removing
562 Since we limit the argument domain of the specialized function, its
563 definition can often be optimized further (since now more types or even
564 values of arguments are already known). By replacing any application of
565 the function that falls within the reduced domain by an application of
566 the specialized version, the code gets faster (but the code also gets
567 bigger, since we now have two versions instead of one). If we apply
568 this technique often enough, we can often replace all applications of a
569 function by specialized versions, allowing the original function to be
570 removed (in some cases, this can even give a net reduction of the code
571 compared to the non-specialized version).
573 Specialization is useful for our hardware descriptions for functions
574 that contain arguments that cannot be translated to hardware directly
575 (polymorphic or higher order arguments, for example). If we can create
576 specialized functions that remove the argument, or make it translatable,
577 we can use specialization to make the original, untranslatable, function
580 \section{Higher order values}
581 What holds for partial application, can be easily generalized to any
582 higher order expression. This includes partial applications, plain
583 variables (e.g., a binder referring to a top level function), lambda
584 expressions and more complex expressions with a function type (a \hs{case}
585 expression returning lambda's, for example).
587 Each of these values cannot be directly represented in hardware (just like
588 partial applications). Also, to make them representable, they need to be
589 applied: function variables and partial applications will then eventually
590 become complete applications, applied lambda expressions disappear by
591 applying β-reduction, etc.
593 So any higher order value will be \quote{pushed down} towards its
594 application just like partial applications. Whenever a function boundary
595 needs to be crossed, the called function can be specialized.
597 \fxnote{This section needs improvement and an example}
599 \section{Polymorphism}
600 In Haskell, values can be \emph{polymorphic}: They can have multiple types. For
601 example, the function \hs{fst :: (a, b) -> a} is an example of a
602 polymorphic function: It works for tuples with any two element types. Haskell
603 typeclasses allow a function to work on a specific set of types, but the
604 general idea is the same. The opposite of this is a \emph{monomorphic}
605 value, which has a single, fixed, type.
607 % A type class is a collection of types for which some operations are
608 % defined. It is thus possible for a value to be polymorphic while having
609 % any number of \emph{class constraints}: The value is not defined for
610 % every type, but only for types in the type class. An example of this is
611 % the \hs{even :: (Integral a) => a -> Bool} function, which can map any
612 % value of a type that is member of the \hs{Integral} type class
614 When generating hardware, polymorphism can't be easily translated. How
615 many wires will you lay down for a value that could have any type? When
616 type classes are involved, what hardware components will you lay down for
617 a class method (whose behaviour depends on the type of its arguments)?
618 Note that Cλash currently does not allow user-defined typeclasses,
619 but does partly support some of the builtin typeclasses (like \hs{Num}).
621 Fortunately, we can again use the principle of specialization: Since every
622 function application generates a separate piece of hardware, we can know
623 the types of all arguments exactly. Provided that we don't use existential
624 typing, all of the polymorphic types in a function must depend on the
625 types of the arguments (In other words, the only way to introduce a type
626 variable is in a lambda abstraction).
628 If a function is monomorphic, all values inside it are monomorphic as
629 well, so any function that is applied within the function can only be
630 applied to monomorphic values. The applied functions can then be
631 specialized to work just for these specific types, removing the
632 polymorphism from the applied functions as well.
634 \defref{top level function}Our top level function must not have a polymorphic type (otherwise we
635 wouldn't know the hardware interface to our top level function).
637 By induction, this means that all functions that are (indirectly) called
638 by our top level function (meaning all functions that are translated in
639 the final hardware) become monomorphic.
642 A very important concept in hardware designs is \emph{state}. In a
643 stateless (or, \emph{combinatoric}) design, every output is directly and solely dependent on the
644 inputs. In a stateful design, the outputs can depend on the history of
645 inputs, or the \emph{state}. State is usually stored in \emph{registers},
646 which retain their value during a clockcycle, and are typically updated at
647 the start of every clockcycle. Since the updating of the state is tightly
648 coupled (synchronized) to the clock signal, these state updates are often
649 called \emph{synchronous} behaviour.
651 \todo{Sidenote? Registers can contain any (complex) type}
653 To make our hardware description language useful to describe more than
654 simple combinatoric designs, we'll need to be able to describe state in
657 \subsection{Approaches to state}
658 In Haskell, functions are always pure (except when using unsafe
659 functions with the \hs{IO} monad, which is not supported by Cλash). This
660 means that the output of a function solely depends on its inputs. If you
661 evaluate a given function with given inputs, it will always provide the
666 This is a perfect match for a combinatoric circuit, where the output
667 also solely depends on the inputs. However, when state is involved, this
668 no longer holds. Since we're in charge of our own language (or at least
669 let's pretend we aren't using Haskell and we are), we could remove this
670 purity constraint and allow a function to return different values
671 depending on the cycle in which it is evaluated (or rather, the current
672 state). However, this means that all kinds of interesting properties of
673 our functional language get lost, and all kinds of transformations and
674 optimizations might no longer be meaning preserving.
676 Provided that we want to keep the function pure, the current state has
677 to be present in the function's arguments in some way. There seem to be
678 two obvious ways to do this: Adding the current state as an argument, or
679 including the full history of each argument.
681 \subsubsection{Stream arguments and results}
682 Including the entire history of each input (\eg, the value of that
683 input for each previous clockcycle) is an obvious way to make outputs
684 depend on all previous input. This is easily done by making every
685 input a list instead of a single value, containing all previous values
686 as well as the current value.
688 An obvious downside of this solution is that on each cycle, all the
689 previous cycles must be resimulated to obtain the current state. To do
690 this, it might be needed to have a recursive helper function as well,
691 wich might be hard to be properly analyzed by the compiler.
693 A slight variation on this approach is one taken by some of the other
694 functional \small{HDL}s in the field: \todo{References to Lava,
695 ForSyDe, ...} Make functions operate on complete streams. This means
696 that a function is no longer called on every cycle, but just once. It
697 takes stream as inputs instead of values, where each stream contains
698 all the values for every clockcycle since system start. This is easily
699 modeled using an (infinite) list, with one element for each clock
700 cycle. Since the function is only evaluated once, its output must also
701 be a stream. Note that, since we are working with infinite lists and
702 still want to be able to simulate the system cycle-by-cycle, this
703 relies heavily on the lazy semantics of Haskell.
705 Since our inputs and outputs are streams, all other (intermediate)
706 values must be streams. All of our primitive operators (\eg, addition,
707 substraction, bitwise operations, etc.) must operate on streams as
708 well (note that changing a single-element operation to a stream
709 operation can done with \hs{map}, \hs{zipwith}, etc.).
711 Note that the concept of \emph{state} is no more than having some way
712 to communicate a value from one cycle to the next. By introducing a
713 \hs{delay} function, we can do exactly that: Delay (each value in) a
714 stream so that we can "look into" the past. This \hs{delay} function
715 simply outputs a stream where each value is the same as the input
716 value, but shifted one cycle. This causes a \quote{gap} at the
717 beginning of the stream: What is the value of the delay output in the
718 first cycle? For this, the \hs{delay} function has a second input
719 (which is a value, not a stream!).
721 \in{Example}[ex:DelayAcc] shows a simple accumulator expressed in this
724 \startbuffer[DelayAcc]
725 acc :: Stream Word -> Stream Word
728 out = (delay out 0) + in
731 \startuseMPgraphic{DelayAcc}
732 save in, out, add, reg;
735 newCircle.in(btex $in$ etex) "framed(false)";
736 newCircle.out(btex $out$ etex) "framed(false)";
739 newReg.reg("") "dx(4mm)", "dy(6mm)", "reflect(true)";
740 newCircle.add(btex + etex);
743 add.c = in.c + (2cm, 0cm);
744 out.c = add.c + (2cm, 0cm);
745 reg.c = add.c + (0cm, 2cm);
747 % Draw objects and lines
748 drawObj(in, out, add, reg);
750 nccurve(add)(reg) "angleA(0)", "angleB(180)", "posB(d)";
751 nccurve(reg)(add) "angleA(180)", "angleB(-45)", "posA(out)";
757 \placeexample[here][ex:DelayAcc]{Simple accumulator architecture.}
758 \startcombination[2*1]
759 {\typebufferhs{DelayAcc}}{Haskell description using streams.}
760 {\boxedgraphic{DelayAcc}}{The architecture described by the Haskell description.}
764 This notation can be confusing (especially due to the loop in the
765 definition of out), but is essentially easy to interpret. There is a
766 single call to delay, resulting in a circuit with a single register,
767 whose input is connected to \hs{out} (which is the output of the
768 adder), and it's output is the expression \hs{delay out 0} (which is
769 connected to one of the adder inputs).
771 This notation has a number of downsides, amongst which are limited
772 readability and ambiguity in the interpretation. \todo{Reference
773 Christiaan, who has done further investigation}
775 \subsubsection{Explicit state arguments and results}
776 A more explicit way to model state, is to simply add an extra argument
777 containing the current state value. This allows an output to depend on
778 both the inputs as well as the current state while keeping the
779 function pure (letting the result depend only on the arguments), since
780 the current state is now an argument.
782 In Haskell, this would look like
783 \in{example}[ex:ExplicitAcc]\footnote[notfinalsyntax]{This example is not in the final
784 Cλash syntax}. \todo{Referencing notfinalsyntax from Introduction.tex doesn't
787 \startbuffer[ExplicitAcc]
788 -- input -> current state -> (new state, output)
789 acc :: Word -> Word -> (Word, Word)
796 \placeexample[here][ex:ExplicitAcc]{Simple accumulator architecture.}
797 \startcombination[2*1]
798 {\typebufferhs{ExplicitAcc}}{Haskell description using explicit state arguments.}
799 % Picture is identical to the one we had just now.
800 {\boxedgraphic{DelayAcc}}{The architecture described by the Haskell description.}
803 This approach makes a function's state very explicit, which state
804 variables are used by a function can be completely determined from its
805 type signature (as opposed to the stream approach, where a function
806 looks the same from the outside, regardless of what state variables it
807 uses or whether it's stateful at all).
809 This approach is the one chosen for Cλash and will be examined more
812 \subsection{Explicit state specification}
813 We've seen the concept of explicit state in a simple example below, but
814 what are the implications of this approach?
816 \subsubsection{Substates}
817 Since a function's state is reflected directly in its type signature,
818 if a function calls other stateful functions (\eg, has subcircuits), it
819 has to somehow know the current state for these called functions. The
820 only way to do this, is to put these \emph{substates} inside the
821 caller's state. This means that a function's state is the sum of the
822 states of all functions it calls, and its own state. This sum
823 can be obtained using something simple like a tuple, or possibly
824 custom algebraic types for clarity.
826 This also means that the type of a function (at least the "state"
827 part) is dependent on its own implementation and of the functions it
830 This is the major downside of this approach: The separation between
831 interface and implementation is limited. However, since Cλash is not
832 very suitable for separate compilation (see
833 \in{section}[sec:prototype:separate]) this is not a big problem in
836 Additionally, when using a type synonym for the state type
837 of each function, we can still provide explicit type signatures
838 while keeping the state specification for a function near its
842 \subsubsection{Which arguments and results are stateful?}
843 \fxnote{This section should get some examples}
844 We need some way to know which arguments should become input ports and
845 which argument(s?) should become the current state (\eg, be bound to
846 the register outputs). This does not hold just for the top
847 level function, but also for any subfunction. Or could we perhaps
848 deduce the statefulness of subfunctions by analyzing the flow of data
849 in the calling functions?
851 To explore this matter, the following observeration is interesting: We
852 get completely correct behaviour when we put all state registers in
853 the top level entity (or even outside of it). All of the state
854 arguments and results on subfunctions are treated as normal input and
855 output ports. Effectively, a stateful function results in a stateless
856 hardware component that has one of its input ports connected to the
857 output of a register and one of its output ports connected to the
858 input of the same register.
862 Of course, even though the hardware described like this has the
863 correct behaviour, unless the layout tool does smart optimizations,
864 there will be a lot of extra wire in the design (since registers will
865 not be close to the component that uses them). Also, when working with
866 the generated \small{VHDL} code, there will be a lot of extra ports
867 just to pass on state values, which can get quite confusing.
869 To fix this, we can simply \quote{push} the registers down into the
870 subcircuits. When we see a register that is connected directly to a
871 subcircuit, we remove the corresponding input and output port and put
872 the register inside the subcircuit instead. This is slightly less
873 trivial when looking at the Haskell code instead of the resulting
874 circuit, but the idea is still the same.
878 However, when applying this technique, we might push registers down
879 too far. When you intend to store a result of a stateless subfunction
880 in the caller's state and pass the current value of that state
881 variable to that same function, the register might get pushed down too
882 far. It is impossible to distinguish this case from similar code where
883 the called function is in fact stateful. From this we can conclude
884 that we have to either:
886 \todo{Example of wrong downpushing}
889 \item accept that the generated hardware might not be exactly what we
890 intended, in some specific cases. In most cases, the hardware will be
892 \item explicitely annotate state arguments and results in the input
896 The first option causes (non-obvious) exceptions in the language
897 intepretation. Also, automatically determining where registers should
898 end up is easier to implement correctly with explicit annotations, so
899 for these reasons we will look at how this annotations could work.
901 \todo{Sidenote: One or more state arguments?}
903 \subsection[sec:description:stateann]{Explicit state annotation}
904 To make our stateful descriptions unambigious and easier to translate,
905 we need some way for the developer to describe which arguments and
906 results are intended to become stateful.
908 Roughly, we have two ways to achieve this:
910 \item Use some kind of annotation method or syntactic construction in
911 the language to indicate exactly which argument and (part of the)
912 result is stateful. This means that the annotation lives
913 \quote{outside} of the function, it is completely invisible when
914 looking at the function body.
915 \item Use some kind of annotation on the type level, \ie give stateful
916 arguments and stateful (parts of) results a different type. This has the
917 potential to make this annotation visible inside the function as well,
918 such that when looking at a value inside the function body you can
919 tell if it's stateful by looking at its type. This could possibly make
920 the translation process a lot easier, since less analysis of the
921 program flow might be required.
924 From these approaches, the type level \quote{annotations} have been
925 implemented in Cλash. \in{Section}[sec:prototype:statetype] expands on
926 the possible ways this could have been implemented.
928 \todo{Note about conditions on state variables and checking them}
930 \section[sec:recursion]{Recursion}
931 An import concept in functional languages is recursion. In it's most basic
932 form, recursion is a definition that is described in terms of itself. A
933 recursive function is thus a function that uses itself in its body. This
934 usually requires multiple evaluations of this function, with changing
935 arguments, until eventually an evaluation of the function no longer requires
938 Given the notion that each function application will translate to a
939 component instantiation, we are presented with a problem. A recursive
940 function would translate to a component that contains itself. Or, more
941 precisely, that contains an instance of itself. This instance would again
942 contain an instance of itself, and again, into infinity. This is obviously a
943 problem for generating hardware.
945 This is expected for functions that describe infinite recursion. In that
946 case, we can't generate hardware that shows correct behaviour in a single
947 cycle (at best, we could generate hardware that needs an infinite number of
950 However, most recursive definitions will describe finite
951 recursion. This is because the recursive call is done conditionally. There
952 is usually a \hs{case} expression where at least one alternative does not contain
953 the recursive call, which we call the "base case". If, for each call to the
954 recursive function, we would be able to detect at compile time which
955 alternative applies, we would be able to remove the \hs{case} expression and
956 leave only the base case when it applies. This will ensure that expanding
957 the recursive functions will terminate after a bounded number of expansions.
959 This does imply the extra requirement that the base case is detectable at
960 compile time. In particular, this means that the decision between the base
961 case and the recursive case must not depend on runtime data.
963 \subsection{List recursion}
964 The most common deciding factor in recursion is the length of a list that is
965 passed in as an argument. Since we represent lists as vectors that encode
966 the length in the vector type, it seems easy to determine the base case. We
967 can simply look at the argument type for this. However, it turns out that
968 this is rather non-trivial to write down in Haskell already, not even
969 looking at translation. As an example, we would like to write down something
973 sum :: Vector n Word -> Word
974 sum xs = case null xs of
976 False -> head xs + sum (tail xs)
979 However, the Haskell typechecker will now use the following reasoning.
980 For simplicity, the element type of a vector is left out, all vectors
981 are assumed to have the same element type. Below, we write conditions
982 on type variables before the \hs{=>} operator. This is not completely
983 valid Haskell syntax, but serves to illustrate the typechecker
984 reasoning. Also note that a vector can never have a negative length,
985 so \hs{Vector n} implicitly means \hs{(n >= 0) => Vector n}.
987 \todo{This typechecker disregards the type signature}
989 \item tail has the type \hs{(n > 0) => Vector n -> Vector (n - 1)}
990 \item This means that xs must have the type \hs{(n > 0) => Vector n}
991 \item This means that sum must have the type \hs{(n > 0) => Vector n -> a}
992 \item sum is called with the result of tail as an argument, which has the
993 type \hs{Vector n} (since \hs{(n > 0) => Vector (n - 1)} is the same as \hs{(n >= 0)
994 => Vector n}, which is the same as just \hs{Vector n}).
995 \item This means that sum must have the type \hs{Vector n -> a}
996 \item This is a contradiction between the type deduced from the body of sum
997 (the input vector must be non-empty) and the use of sum (the input vector
998 could have any length).
1001 As you can see, using a simple \hs{case} expression at value level causes
1002 the type checker to always typecheck both alternatives, which can't be
1003 done! The typechecker is unable to distinguish the two case
1004 alternatives (this is partly possible using \small{GADT}s, but that
1005 approach faced other problems \todo{ref christiaan?}).
1007 This is a fundamental problem, that would seem perfectly suited for a
1008 type class. Considering that we need to switch between to
1009 implementations of the sum function, based on the type of the
1010 argument, this sounds like the perfect problem to solve with a type
1011 class. However, this approach has its own problems (not the least of
1012 them that you need to define a new typeclass for every recursive
1013 function you want to define).
1015 \todo{This should reference Christiaan}
1017 \subsection{General recursion}
1018 Of course there are other forms of recursion, that do not depend on the
1019 length (and thus type) of a list. For example, simple recursion using a
1020 counter could be expressed, but only translated to hardware for a fixed
1021 number of iterations. Also, this would require extensive support for compile
1022 time simplification (constant propagation) and compile time evaluation
1023 (evaluation of constant comparisons), to ensure non-termination. Even then, it
1024 is hard to really guarantee termination, since the user (or GHC desugarer)
1025 might use some obscure notation that results in a corner case of the
1026 simplifier that is not caught and thus non-termination.
1028 Evaluating all possible (and non-possible) ways to add recursion to
1029 our descriptions, it seems better to limit the scope of this research
1030 to exclude recursion. This allows for focusing on other interesting
1031 areas instead. By including (builtin) support for a number of higher
1032 order functions like \hs{map} and \hs{fold}, we can still express most
1033 of the things we would use (list) recursion for.
1036 % vim: set sw=2 sts=2 expandtab: