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 \defref{enumerated types}
346 An enumerated type is an algebraic datatype with multiple constructors, but
347 none of them have fields. This is essentially a way to get an
348 enum-like type containing alternatives.
350 Note that Haskell's \hs{Bool} type is also defined as an
351 enumeration type, but we have a fixed translation for that.
353 These types are translated to \VHDL enumerations, with one value for
354 each constructor. This allows references to these constructors to be
355 translated to the corresponding enumeration value.
357 \startdesc{Sum types}
358 A sum type is an algebraic datatype with multiple constructors, where
359 the constructors have one or more fields. Technically, a type with
360 more than one field per constructor is a sum of products type, but
361 for our purposes this distinction does not really make a difference,
362 so we'll leave it out.
364 Sum types are currently not supported by the prototype, since there is
365 no obvious \VHDL alternative. They can easily be emulated, however, as
366 we will see from an example:
369 data Sum = A Bit Word | B Word
372 An obvious way to translate this would be to create an enumeration to
373 distinguish the constructors and then create a big record that
374 contains all the fields of all the constructors. This is the same
375 translation that would result from the following enumeration and
376 product type (using a tuple for clarity):
380 type Sum = (SumC, Bit, Word, Word)
383 Here, the \hs{SumC} type effectively signals which of the latter three
384 fields of the \hs{Sum} type are valid (the first two if \hs{A}, the
385 last one if \hs{B}), all the other ones have no useful value.
387 An obvious problem with this naive approach is the space usage: The
388 example above generates a fairly big \VHDL type. However, we can be
389 sure that the two \hs{Word}s in the \hs{Sum} type will never be valid
390 at the same time, so this is a waste of space.
392 Obviously, we could do some duplication detection here to reuse a
393 particular field for another constructor, but this would only
394 partially solve the problem. If I would, for example, have an array of
395 8 bits and a 8 bit unsiged word, these are different types and could
396 not be shared. However, in the final hardware, both of these types
397 would simply be 8 bit connections, so we have a 100\% size increase by
401 Another interesting case is that of recursive types. In Haskell, an
402 algebraic datatype can be recursive: Any of its field types can be (or
403 contain) the type being defined. The most well-known recursive type is
404 probably the list type, which is defined is:
407 data List t = Empty | Cons t (List t)
410 Note that \hs{Empty} is usually written as \hs{[]} and \hs{Cons} as
411 \hs{:}, but this would make the definition harder to read. This
412 immediately shows the problem with recursive types: What hardware type
415 If we would use the naive approach for sum types we described above, we
416 would create a record where the first field is an enumeration to
417 distinguish \hs{Empty} from \hs{Cons}. Furthermore, we would add two
418 more fields: One with the (\VHDL equivalent of) type \hs{t} (assuming we
419 actually know what type this is at compile time, this should not be a
420 problem) and a second one with type \hs{List t}. The latter one is of
421 course a problem: This is exactly the type we were trying to translate
424 Our \VHDL type will thus become infinitely deep. In other words, there
425 is no way to statically determine how long (deep) the list will be
426 (it could even be infinite).
428 In general, we can say we can never properly translate recursive types:
429 All recursive types have a potentially infinite value (even though in
430 practice they will have a bounded value, there is no way for the
431 compiler to determine an upper bound on its size).
433 \subsection{Partial application}
434 Now we've seen how to to translate application, choice and types, we will
435 get to a more complex concept: Partial applications. A \emph{partial
436 application} is any application whose (return) type is (again) a function
439 From this, we can see that the translation rules for full application do not
440 apply to a partial application. \in{Example}[ex:Quadruple] shows an example
441 use of partial application and the corresponding architecture.
443 \startbuffer[Quadruple]
444 -- | Multiply the input word by four.
445 quadruple :: Word -> Word
446 quadruple n = mul (mul n)
451 \startuseMPgraphic{Quadruple}
452 save in, two, mula, mulb, out;
455 newCircle.in(btex $n$ etex) "framed(false)";
456 newCircle.two(btex $2$ etex) "framed(false)";
457 newCircle.out(btex $out$ etex) "framed(false)";
460 newCircle.mula(btex $\times$ etex);
461 newCircle.mulb(btex $\times$ etex);
464 in.c = two.c + (0cm, 1cm);
465 mula.c = in.c + (2cm, 0cm);
466 mulb.c = mula.c + (2cm, 0cm);
467 out.c = mulb.c + (2cm, 0cm);
469 % Draw objects and lines
470 drawObj(in, two, mula, mulb, out);
472 nccurve(two)(mula) "angleA(0)", "angleB(45)";
473 nccurve(two)(mulb) "angleA(0)", "angleB(45)";
479 \placeexample[here][ex:Quadruple]{Simple three port and.}
480 \startcombination[2*1]
481 {\typebufferhs{Quadruple}}{Haskell description using function applications.}
482 {\boxedgraphic{Quadruple}}{The architecture described by the Haskell description.}
485 Here, the definition of mul is a partial function application: It applies
486 the function \hs{(*) :: Word -> Word -> Word} to the value \hs{2 :: Word},
487 resulting in the expression \hs{(*) 2 :: Word -> Word}. Since this resulting
488 expression is again a function, we can't generate hardware for it directly.
489 This is because the hardware to generate for \hs{mul} depends completely on
490 where and how it is used. In this example, it is even used twice.
492 However, it is clear that the above hardware description actually describes
493 valid hardware. In general, we can see that any partial applied function
494 must eventually become completely applied, at which point we can generate
495 hardware for it using the rules for function application given in
496 \in{section}[sec:description:application]. It might mean that a partial
497 application is passed around quite a bit (even beyond function boundaries),
498 but eventually, the partial application will become completely applied.
499 \todo{Provide a step-by-step example of how this works}
501 \section{Costless specialization}
502 Each (complete) function application in our description generates a
503 component instantiation, or a specific piece of hardware in the final
504 design. It is interesting to note that each application of a function
505 generates a \emph{separate} piece of hardware. In the final design, none
506 of the hardware is shared between applications, even when the applied
507 function is the same (of course, if a particular value, such as the result
508 of a function application, is used twice, it is not calculated twice).
510 This is distinctly different from normal program compilation: Two separate
511 calls to the same function share the same machine code. Having more
512 machine code has implications for speed (due to less efficient caching)
513 and memory usage. For normal compilation, it is therefore important to
514 keep the amount of functions limited and maximize the code sharing.
516 When generating hardware, this is hardly an issue. Having more \quote{code
517 sharing} does reduce the amount of \small{VHDL} output (Since different
518 component instantiations still share the same component), but after
519 synthesis, the amount of hardware generated is not affected.
521 In particular, if we would duplicate all functions so that there is a
522 separate function for every application in the program (\eg, each function
523 is then only applied exactly once), there would be no increase in hardware
526 Because of this, a common optimization technique called
527 \emph{specialization} can be applied to hardware generation without any
528 performance or area cost (unlike for software).
530 \fxnote{Perhaps these next three sections are a bit too
531 implementation-oriented?}
533 \subsection{Specialization}
534 \defref{specialization}
535 Given some function that has a \emph{domain} $D$ (\eg, the set of all
536 possible arguments that could be applied), we create a specialized
537 function with exactly the same behaviour, but with a domain $D' \subset
538 D$. This subset can be chosen in all sorts of ways. Any subset is valid
539 for the general definition of specialization, but in practice only some
540 of them provide useful optimization opportunities.
542 Common subsets include limiting a polymorphic argument to a single type
543 (\ie, removing polymorphism) or limiting an argument to just a single
544 value (\ie, cross-function constant propagation, effectively removing
547 Since we limit the argument domain of the specialized function, its
548 definition can often be optimized further (since now more types or even
549 values of arguments are already known). By replacing any application of
550 the function that falls within the reduced domain by an application of
551 the specialized version, the code gets faster (but the code also gets
552 bigger, since we now have two versions instead of one). If we apply
553 this technique often enough, we can often replace all applications of a
554 function by specialized versions, allowing the original function to be
555 removed (in some cases, this can even give a net reduction of the code
556 compared to the non-specialized version).
558 Specialization is useful for our hardware descriptions for functions
559 that contain arguments that cannot be translated to hardware directly
560 (polymorphic or higher order arguments, for example). If we can create
561 specialized functions that remove the argument, or make it translatable,
562 we can use specialization to make the original, untranslatable, function
565 \section{Higher order values}
566 What holds for partial application, can be easily generalized to any
567 higher order expression. This includes partial applications, plain
568 variables (e.g., a binder referring to a top level function), lambda
569 expressions and more complex expressions with a function type (a \hs{case}
570 expression returning lambda's, for example).
572 Each of these values cannot be directly represented in hardware (just like
573 partial applications). Also, to make them representable, they need to be
574 applied: function variables and partial applications will then eventually
575 become complete applications, applied lambda expressions disappear by
576 applying β-reduction, etc.
578 So any higher order value will be \quote{pushed down} towards its
579 application just like partial applications. Whenever a function boundary
580 needs to be crossed, the called function can be specialized.
582 \fxnote{This section needs improvement and an example}
584 \section{Polymorphism}
585 In Haskell, values can be \emph{polymorphic}: They can have multiple types. For
586 example, the function \hs{fst :: (a, b) -> a} is an example of a
587 polymorphic function: It works for tuples with any two element types. Haskell
588 typeclasses allow a function to work on a specific set of types, but the
589 general idea is the same. The opposite of this is a \emph{monomorphic}
590 value, which has a single, fixed, type.
592 % A type class is a collection of types for which some operations are
593 % defined. It is thus possible for a value to be polymorphic while having
594 % any number of \emph{class constraints}: The value is not defined for
595 % every type, but only for types in the type class. An example of this is
596 % the \hs{even :: (Integral a) => a -> Bool} function, which can map any
597 % value of a type that is member of the \hs{Integral} type class
599 When generating hardware, polymorphism can't be easily translated. How
600 many wires will you lay down for a value that could have any type? When
601 type classes are involved, what hardware components will you lay down for
602 a class method (whose behaviour depends on the type of its arguments)?
603 Note that the language currently does not allow user-defined typeclasses,
604 but does support partly some of the builtin typeclasses (like \hs{Num}).
606 Fortunately, we can again use the principle of specialization: Since every
607 function application generates a separate piece of hardware, we can know
608 the types of all arguments exactly. Provided that we don't use existential
609 typing, all of the polymorphic types in a function must depend on the
610 types of the arguments (In other words, the only way to introduce a type
611 variable is in a lambda abstraction).
613 If a function is monomorphic, all values inside it are monomorphic as
614 well, so any function that is applied within the function can only be
615 applied to monomorphic values. The applied functions can then be
616 specialized to work just for these specific types, removing the
617 polymorphism from the applied functions as well.
619 Our top level function must not have a polymorphic type (otherwise we
620 wouldn't know the hardware interface to our top level function).
622 By induction, this means that all functions that are (indirectly) called
623 by our top level function (meaning all functions that are translated in
624 the final hardware) become monomorphic.
627 A very important concept in hardware designs is \emph{state}. In a
628 stateless (or, \emph{combinatoric}) design, every output is directly and solely dependent on the
629 inputs. In a stateful design, the outputs can depend on the history of
630 inputs, or the \emph{state}. State is usually stored in \emph{registers},
631 which retain their value during a clockcycle, and are typically updated at
632 the start of every clockcycle. Since the updating of the state is tightly
633 coupled (synchronized) to the clock signal, these state updates are often
634 called \emph{synchronous} behaviour.
636 \todo{Sidenote? Registers can contain any (complex) type}
638 To make our hardware description language useful to describe more than
639 simple combinatoric designs, we'll need to be able to describe state in
642 \subsection{Approaches to state}
643 In Haskell, functions are always pure (except when using unsafe
644 functions with the \hs{IO} monad, which is not supported by Cλash). This
645 means that the output of a function solely depends on its inputs. If you
646 evaluate a given function with given inputs, it will always provide the
651 This is a perfect match for a combinatoric circuit, where the output
652 also soley depends on the inputs. However, when state is involved, this
653 no longer holds. Since we're in charge of our own language (or at least
654 let's pretend we aren't using Haskell and we are), we could remove this
655 purity constraint and allow a function to return different values
656 depending on the cycle in which it is evaluated (or rather, the current
657 state). However, this means that all kinds of interesting properties of
658 our functional language get lost, and all kinds of transformations and
659 optimizations might no longer be meaning preserving.
661 Provided that we want to keep the function pure, the current state has
662 to be present in the function's arguments in some way. There seem to be
663 two obvious ways to do this: Adding the current state as an argument, or
664 including the full history of each argument.
666 \subsubsection{Stream arguments and results}
667 Including the entire history of each input (\eg, the value of that
668 input for each previous clockcycle) is an obvious way to make outputs
669 depend on all previous input. This is easily done by making every
670 input a list instead of a single value, containing all previous values
671 as well as the current value.
673 An obvious downside of this solution is that on each cycle, all the
674 previous cycles must be resimulated to obtain the current state. To do
675 this, it might be needed to have a recursive helper function as well,
676 wich might be hard to be properly analyzed by the compiler.
678 A slight variation on this approach is one taken by some of the other
679 functional \small{HDL}s in the field: \todo{References to Lava,
680 ForSyDe, ...} Make functions operate on complete streams. This means
681 that a function is no longer called on every cycle, but just once. It
682 takes stream as inputs instead of values, where each stream contains
683 all the values for every clockcycle since system start. This is easily
684 modeled using an (infinite) list, with one element for each clock
685 cycle. Since the function is only evaluated once, its output must also
686 be a stream. Note that, since we are working with infinite lists and
687 still want to be able to simulate the system cycle-by-cycle, this
688 relies heavily on the lazy semantics of Haskell.
690 Since our inputs and outputs are streams, all other (intermediate)
691 values must be streams. All of our primitive operators (\eg, addition,
692 substraction, bitwise operations, etc.) must operate on streams as
693 well (note that changing a single-element operation to a stream
694 operation can done with \hs{map}, \hs{zipwith}, etc.).
696 Note that the concept of \emph{state} is no more than having some way
697 to communicate a value from one cycle to the next. By introducing a
698 \hs{delay} function, we can do exactly that: Delay (each value in) a
699 stream so that we can "look into" the past. This \hs{delay} function
700 simply outputs a stream where each value is the same as the input
701 value, but shifted one cycle. This causes a \quote{gap} at the
702 beginning of the stream: What is the value of the delay output in the
703 first cycle? For this, the \hs{delay} function has a second input
704 (which is a value, not a stream!).
706 \in{Example}[ex:DelayAcc] shows a simple accumulator expressed in this
709 \startbuffer[DelayAcc]
710 acc :: Stream Word -> Stream Word
713 out = (delay out 0) + in
716 \startuseMPgraphic{DelayAcc}
717 save in, out, add, reg;
720 newCircle.in(btex $in$ etex) "framed(false)";
721 newCircle.out(btex $out$ etex) "framed(false)";
724 newReg.reg("") "dx(4mm)", "dy(6mm)", "reflect(true)";
725 newCircle.add(btex + etex);
728 add.c = in.c + (2cm, 0cm);
729 out.c = add.c + (2cm, 0cm);
730 reg.c = add.c + (0cm, 2cm);
732 % Draw objects and lines
733 drawObj(in, out, add, reg);
735 nccurve(add)(reg) "angleA(0)", "angleB(180)", "posB(d)";
736 nccurve(reg)(add) "angleA(180)", "angleB(-45)", "posA(out)";
742 \placeexample[here][ex:DelayAcc]{Simple accumulator architecture.}
743 \startcombination[2*1]
744 {\typebufferhs{DelayAcc}}{Haskell description using streams.}
745 {\boxedgraphic{DelayAcc}}{The architecture described by the Haskell description.}
749 This notation can be confusing (especially due to the loop in the
750 definition of out), but is essentially easy to interpret. There is a
751 single call to delay, resulting in a circuit with a single register,
752 whose input is connected to \hs{out} (which is the output of the
753 adder), and it's output is the expression \hs{delay out 0} (which is
754 connected to one of the adder inputs).
756 This notation has a number of downsides, amongst which are limited
757 readability and ambiguity in the interpretation. \note{Reference
758 Christiaan, who has done further investigation}
760 \subsubsection{Explicit state arguments and results}
761 A more explicit way to model state, is to simply add an extra argument
762 containing the current state value. This allows an output to depend on
763 both the inputs as well as the current state while keeping the
764 function pure (letting the result depend only on the arguments), since
765 the current state is now an argument.
767 In Haskell, this would look like
768 \in{example}[ex:ExplicitAcc]\footnote[notfinalsyntax]{Note that this example is not in the final
769 Cλash syntax}. \todo{Referencing notfinalsyntax from Introduction.tex doesn't
772 \startbuffer[ExplicitAcc]
773 -- input -> current state -> (new state, output)
774 acc :: Word -> Word -> (Word, Word)
781 \placeexample[here][ex:ExplicitAcc]{Simple accumulator architecture.}
782 \startcombination[2*1]
783 {\typebufferhs{ExplicitAcc}}{Haskell description using explicit state arguments.}
784 % Picture is identical to the one we had just now.
785 {\boxedgraphic{DelayAcc}}{The architecture described by the Haskell description.}
788 This approach makes a function's state very explicit, which state
789 variables are used by a function can be completely determined from its
790 type signature (as opposed to the stream approach, where a function
791 looks the same from the outside, regardless of what state variables it
792 uses or whether it's stateful at all).
794 This approach is the one chosen for Cλash and will be examined more
797 \subsection{Explicit state specification}
798 We've seen the concept of explicit state in a simple example below, but
799 what are the implications of this approach?
801 \subsubsection{Substates}
802 Since a function's state is reflected directly in its type signature,
803 if a function calls other stateful functions (\eg, has subcircuits), it
804 has to somehow know the current state for these called functions. The
805 only way to do this, is to put these \emph{substates} inside the
806 caller's state. This means that a function's state is the sum of the
807 states of all functions it calls, and its own state. This sum
808 can be obtained using something simple like a tuple, or possibly
809 custom algebraic types for clarity.
811 This also means that the type of a function (at least the "state"
812 part) is dependent on its own implementation and of the functions it
815 This is the major downside of this approach: The separation between
816 interface and implementation is limited. However, since Cλash is not
817 very suitable for separate compilation (see
818 \in{section}[sec:prototype:separate]) this is not a big problem in
821 Additionally, when using a type synonym for the state type
822 of each function, we can still provide explicit type signatures
823 while keeping the state specification for a function near its
827 \subsubsection{Which arguments and results are stateful?}
828 \fxnote{This section should get some examples}
829 We need some way to know which arguments should become input ports and
830 which argument(s?) should become the current state (\eg, be bound to
831 the register outputs). This does not hold just for the top
832 level function, but also for any subfunction. Or could we perhaps
833 deduce the statefulness of subfunctions by analyzing the flow of data
834 in the calling functions?
836 To explore this matter, the following observeration is interesting: We
837 get completely correct behaviour when we put all state registers in
838 the top level entity (or even outside of it). All of the state
839 arguments and results on subfunctions are treated as normal input and
840 output ports. Effectively, a stateful function results in a stateless
841 hardware component that has one of its input ports connected to the
842 output of a register and one of its output ports connected to the
843 input of the same register.
847 Of course, even though the hardware described like this has the
848 correct behaviour, unless the layout tool does smart optimizations,
849 there will be a lot of extra wire in the design (since registers will
850 not be close to the component that uses them). Also, when working with
851 the generated \small{VHDL} code, there will be a lot of extra ports
852 just to pass on state values, which can get quite confusing.
854 To fix this, we can simply \quote{push} the registers down into the
855 subcircuits. When we see a register that is connected directly to a
856 subcircuit, we remove the corresponding input and output port and put
857 the register inside the subcircuit instead. This is slightly less
858 trivial when looking at the Haskell code instead of the resulting
859 circuit, but the idea is still the same.
863 However, when applying this technique, we might push registers down
864 too far. When you intend to store a result of a stateless subfunction
865 in the caller's state and pass the current value of that state
866 variable to that same function, the register might get pushed down too
867 far. It is impossible to distinguish this case from similar code where
868 the called function is in fact stateful. From this we can conclude
869 that we have to either:
871 \todo{Example of wrong downpushing}
874 \item accept that the generated hardware might not be exactly what we
875 intended, in some specific cases. In most cases, the hardware will be
877 \item explicitely annotate state arguments and results in the input
881 The first option causes (non-obvious) exceptions in the language
882 intepretation. Also, automatically determining where registers should
883 end up is easier to implement correctly with explicit annotations, so
884 for these reasons we will look at how this annotations could work.
886 \todo{Sidenote: One or more state arguments?}
888 \subsection[sec:description:stateann]{Explicit state annotation}
889 To make our stateful descriptions unambigious and easier to translate,
890 we need some way for the developer to describe which arguments and
891 results are intended to become stateful.
893 Roughly, we have two ways to achieve this:
895 \item Use some kind of annotation method or syntactic construction in
896 the language to indicate exactly which argument and (part of the)
897 result is stateful. This means that the annotation lives
898 \quote{outside} of the function, it is completely invisible when
899 looking at the function body.
900 \item Use some kind of annotation on the type level, \ie give stateful
901 arguments and stateful (parts of) results a different type. This has the
902 potential to make this annotation visible inside the function as well,
903 such that when looking at a value inside the function body you can
904 tell if it's stateful by looking at its type. This could possibly make
905 the translation process a lot easier, since less analysis of the
906 program flow might be required.
909 From these approaches, the type level \quote{annotations} have been
910 implemented in Cλash. \in{Section}[sec:prototype:statetype] expands on
911 the possible ways this could have been implemented.
913 \todo{Note about conditions on state variables and checking them}
915 \section[sec:recursion]{Recursion}
916 An import concept in functional languages is recursion. In it's most basic
917 form, recursion is a definition that is defined in terms of itself. A
918 recursive function is thus a function that uses itself in its body. This
919 usually requires multiple evaluations of this function, with changing
920 arguments, until eventually an evaluation of the function no longer requires
923 Recursion in a hardware description is a bit of a funny thing. Usually,
924 recursion is associated with a lot of nondeterminism, stack overflows, but
925 also flexibility and expressive power.
927 Given the notion that each function application will translate to a
928 component instantiation, we are presented with a problem. A recursive
929 function would translate to a component that contains itself. Or, more
930 precisely, that contains an instance of itself. This instance would again
931 contain an instance of itself, and again, into infinity. This is obviously a
932 problem for generating hardware.
934 This is expected for functions that describe infinite recursion. In that
935 case, we can't generate hardware that shows correct behaviour in a single
936 cycle (at best, we could generate hardware that needs an infinite number of
939 However, most recursive hardware descriptions will describe finite
940 recursion. This is because the recursive call is done conditionally. There
941 is usually a \hs{case} expression where at least one alternative does not contain
942 the recursive call, which we call the "base case". If, for each call to the
943 recursive function, we would be able to detect at compile time which
944 alternative applies, we would be able to remove the \hs{case} expression and
945 leave only the base case when it applies. This will ensure that expanding
946 the recursive functions will terminate after a bounded number of expansions.
948 This does imply the extra requirement that the base case is detectable at
949 compile time. In particular, this means that the decision between the base
950 case and the recursive case must not depend on runtime data.
952 \subsection{List recursion}
953 The most common deciding factor in recursion is the length of a list that is
954 passed in as an argument. Since we represent lists as vectors that encode
955 the length in the vector type, it seems easy to determine the base case. We
956 can simply look at the argument type for this. However, it turns out that
957 this is rather non-trivial to write down in Haskell already, not even
958 looking at translation. As an example, we would like to write down something
962 sum :: Vector n Word -> Word
963 sum xs = case null xs of
965 False -> head xs + sum (tail xs)
968 However, the Haskell typechecker will now use the following reasoning (element
969 type of the vector is left out). Below, we write conditions on type
970 variables before the \hs{=>} operator. This is not completely valid Haskell
971 syntax, but serves to illustrate the typechecker reasoning. Also note that a
972 vector can never have a negative length, so \hs{Vector n} implicitly means
973 \hs{(n >= 0) => Vector n}.
975 \todo{This typechecker disregards the type signature}
977 \item tail has the type \hs{(n > 0) => Vector n -> Vector (n - 1)}
978 \item This means that xs must have the type \hs{(n > 0) => Vector n}
979 \item This means that sum must have the type \hs{(n > 0) => Vector n -> a}
980 \item sum is called with the result of tail as an argument, which has the
981 type \hs{Vector n} (since \hs{(n > 0) => Vector (n - 1)} is the same as \hs{(n >= 0)
982 => Vector n}, which is the same as just \hs{Vector n}).
983 \item This means that sum must have the type \hs{Vector n -> a}
984 \item This is a contradiction between the type deduced from the body of sum
985 (the input vector must be non-empty) and the use of sum (the input vector
986 could have any length).
989 As you can see, using a simple \hs{case} expression at value level causes
990 the type checker to always typecheck both alternatives, which can't be done!
991 This is a fundamental problem, that would seem perfectly suited for a type
992 class. Considering that we need to switch between to implementations of the
993 sum function, based on the type of the argument, this sounds like the
994 perfect problem to solve with a type class. However, this approach has its
995 own problems (not the least of them that you need to define a new typeclass
996 for every recursive function you want to define).
998 Another approach tried involved using GADTs to be able to do pattern
999 matching on empty / non empty lists. While this worked partially, it also
1000 created problems with more complex expressions.
1002 \note{This should reference Christiaan}
1004 Evaluating all possible (and non-possible) ways to add recursion to our
1005 descriptions, it seems better to leave out list recursion alltogether. This
1006 allows us to focus on other interesting areas instead. By including
1007 (builtin) support for a number of higher order functions like map and fold,
1008 we can still express most of the things we would use list recursion for.
1010 \todo{Expand on this decision a bit}
1012 \subsection{General recursion}
1013 Of course there are other forms of recursion, that do not depend on the
1014 length (and thus type) of a list. For example, simple recursion using a
1015 counter could be expressed, but only translated to hardware for a fixed
1016 number of iterations. Also, this would require extensive support for compile
1017 time simplification (constant propagation) and compile time evaluation
1018 (evaluation of constant comparisons), to ensure non-termination. Even then, it
1019 is hard to really guarantee termination, since the user (or GHC desugarer)
1020 might use some obscure notation that results in a corner case of the
1021 simplifier that is not caught and thus non-termination.
1023 Due to these complications and limited time available, we leave other forms
1024 of recursion as future work as well.
1026 % vim: set sw=2 sts=2 expandtab: