1 \chapter[chap:description]{Hardware description}
2 In this chapter an overview will be provided of the hardware
3 description language that was created and the issues that have arisen
4 in the process. The focus will be on the issues of the language, not
5 the implementation. The prototype implementation will be discussed in
6 \in{chapter}[chap:prototype].
8 To translate Haskell to hardware, every Haskell construct needs a
9 translation to \VHDL. There are often multiple valid translations
10 possible. When faced with choices, the most obvious choice has been
11 chosen wherever possible. In a lot of cases, when a programmer looks
12 at a functional hardware description it is completely clear what
13 hardware is described. We want our translator to generate exactly that
14 hardware whenever possible, to make working with Cλash as intuitive as
18 \defref{reading examples}
19 \startframedtext[width=9cm,background=box,frame=no]
20 \startalignment[center]
21 {\tfa Reading the examples}
24 In this thesis, a lot of functional code will be presented. Part of this
25 will be valid Cλash code, but others will just be small Haskell or Core
26 snippets to illustrate a concept.
28 In these examples, some functions and types will be used, without
29 properly defining every one of them. These functions (like \hs{and},
30 \hs{not}, \hs{add}, \hs{+}, etc.) and types (like \hs{Bit}, \hs{Word},
31 \hs{Bool}, etc.) are usually pretty self-explanatory.
33 The special type \hs{[t]} means \quote{list of \hs{t}'s}, where \hs{t}
34 can be any other type.
36 Of particular note is the use of the \hs{::} operator. It is used in
37 Haskell to explicitly declare the type of function or let binding. In
38 these examples and the text, we will occasionally use this operator to
39 show the type of arbitrary expressions, even where this would not
40 normally be valid. Just reading the \hs{::} operator as \quote{and also
41 note that \emph{this} expression has \emph{this} type} should work out.
45 In this chapter we describe how to interpret a Haskell program from a
46 hardware perspective. We provide a description of each Haskell language
47 element that needs translation, to provide a clear picture of what is
51 \section[sec:description:application]{Function application}
52 The basic syntactic elements of a functional program are functions and
53 function application. These have a single obvious \small{VHDL}
54 translation: each top level function becomes a hardware component, where each
55 argument is an input port and the result value is the (single) output
56 port. This output port can have a complex type (such as a tuple), so
57 having just a single output port does not pose a limitation.
59 Each function application in turn becomes component instantiation. Here, the
60 result of each argument expression is assigned to a signal, which is mapped
61 to the corresponding input port. The output port of the function is also
62 mapped to a signal, which is used as the result of the application.
64 Since every top level function generates its own component, the
65 hierarchy of of function calls is reflected in the final \VHDL\ output
66 as well, creating a hierarchical \VHDL\ description of the hardware.
67 This separation in different components makes the resulting \VHDL\
68 output easier to read and debug.
70 \in{Example}[ex:And3] shows a simple program using only function
71 application and the corresponding architecture.
74 -- A simple function that returns
75 -- conjunction of three bits
76 and3 :: Bit -> Bit -> Bit -> Bit
77 and3 a b c = and (and a b) c
80 \startuseMPgraphic{And3}
81 save a, b, c, anda, andb, out;
84 newCircle.a(btex $a$ etex) "framed(false)";
85 newCircle.b(btex $b$ etex) "framed(false)";
86 newCircle.c(btex $c$ etex) "framed(false)";
87 newCircle.out(btex $out$ etex) "framed(false)";
90 newCircle.anda(btex $and$ etex);
91 newCircle.andb(btex $and$ etex);
94 b.c = a.c + (0cm, 1cm);
95 c.c = b.c + (0cm, 1cm);
96 anda.c = midpoint(a.c, b.c) + (2cm, 0cm);
97 andb.c = midpoint(b.c, c.c) + (4cm, 0cm);
99 out.c = andb.c + (2cm, 0cm);
101 % Draw objects and lines
102 drawObj(a, b, c, anda, andb, out);
104 ncarc(a)(anda) "arcangle(-10)";
111 \startbuffer[And3VHDL]
112 entity and3Component_0 is
113 port (\azMyG2\ : in std_logic;
114 \bzMyI2\ : in std_logic;
115 \czMyK2\ : in std_logic;
116 \foozMySzMyS2\ : out std_logic;
117 clock : in std_logic;
118 resetn : in std_logic);
119 end entity and3Component_0;
122 architecture structural of and3Component_0 is
123 signal \argzMyMzMyM2\ : std_logic;
125 \argzMyMzMyM2\ <= \azMyG2\ and \bzMyI2\;
127 \foozMySzMyS2\ <= \argzMyMzMyM2\ and \czMyK2\;
128 end architecture structural;
131 \placeexample[][ex:And3]{Simple three input and gate.}
132 \startcombination[2*1]
133 {\typebufferhs{And3}}{Haskell description using function applications.}
134 {\boxedgraphic{And3}}{The architecture described by the Haskell description.}
137 \placeexample[][ex:And3VHDL]{\VHDL\ generated for \hs{and3} from \in{example}[ex:And3]}
138 {\typebuffervhdl{And3VHDL}}
141 \defref{top level binder}
142 \defref{top level function}
143 \startframedtext[width=8cm,background=box,frame=no]
144 \startalignment[center]
145 {\tfa Top level binders and functions}
148 A top level binder is any binder (variable) that is declared in
149 the \quote{global} scope of a Haskell program (as opposed to a
150 binder that is bound inside a function.
152 In Haskell, there is no sharp distinction between a variable and a
153 function: a function is just a variable (binder) with a function
154 type. This means that a top level function is just any top level
155 binder with a function type.
157 As an example, consider the following Haskell snippet:
166 Here, \hs{foo} is a top level binder, whereas \hs{inc} is a
167 function (since it is bound to a lambda extraction, indicated by
168 the backslash) but is not a top level binder or function. Since
169 the type of \hs{foo} is a function type, namely \hs{Int -> Int},
170 it is also a top level function.
174 Although describing components and connections allows us to describe a lot of
175 hardware designs already, there is an obvious thing missing: choice. We
176 need some way to be able to choose between values based on another value.
177 In Haskell, choice is achieved by \hs{case} expressions, \hs{if}
178 expressions, pattern matching and guards.
180 An obvious way to add choice to our language without having to recognize
181 any of Haskell's syntax, would be to add a primivite \quote{\hs{if}}
182 function. This function would take three arguments: the condition, the
183 value to return when the condition is true and the value to return when
184 the condition is false.
186 This \hs{if} function would then essentially describe a multiplexer and
187 allows us to describe any architecture that uses multiplexers.
189 However, to be able to describe our hardware in a more convenient way, we
190 also want to translate Haskell's choice mechanisms. The easiest of these
191 are of course case expressions (and \hs{if} expressions, which can be very
192 directly translated to \hs{case} expressions). A \hs{case} expression can in turn
193 simply be translated to a conditional assignment, where the conditions use
194 equality comparisons against the constructors in the \hs{case} expressions.
196 In \in{example}[ex:Inv] two versions of an inverter are shown. The first
197 uses a simple \hs{case} expression, scrutinizing a boolean value. The
198 corresponding architecture has a comparator to determine which of the
199 constructors is on the \hs{in} input. There is a multiplexer to select the
200 output signal (which is just a conditional assignment in the generated
201 \VHDL). The two options for the output signals are just constants,
202 but these could have been more complex expressions (in which case also
203 both of them would be working in parallel, regardless of which output
204 would be chosen eventually). The \VHDL\ generated for (both versions of)
205 this inverter is shown in \in{example}[ex:InvVHDL].
207 If we would translate a Boolean to a bit value, we could of course remove
208 the comparator and directly feed 'in' into the multiplexer (or even use an
209 inverter instead of a multiplexer). However, we will try to make a
210 general translation, which works for all possible \hs{case} expressions.
211 Optimizations such as these are left for the \VHDL\ synthesizer, which
212 handles them very well.
215 \startframedtext[width=8cm,background=box,frame=no]
216 \startalignment[center]
217 {\tfa Arguments / results vs. inputs / outputs}
220 Due to the translation chosen for function application, there is a
221 very strong relation between arguments, results, inputs and outputs.
222 For clarity, the former two will always refer to the arguments and
223 results in the functional description (either Haskell or Core). The
224 latter two will refer to input and output ports in the generated
227 Even though these concepts seem to be nearly identical, when stateful
228 functions are introduces we will see arguments and results that will
229 not get translated into input and output ports, making this
230 distinction more important.
234 A slightly more complex (but very powerful) form of choice is pattern
235 matching. A function can be defined in multiple clauses, where each clause
236 specifies a pattern. When the arguments match the pattern, the
237 corresponding clause will be used.
239 \in{Example}[ex:Inv] also shows an inverter that uses pattern matching.
240 The architecture it describes is of course the
241 same one as the description with a case expression. The general interpretation
242 of pattern matching is also similar to that of \hs{case} expressions: generate
243 hardware for each of the clauses (like each of the clauses of a \hs{case}
244 expression) and connect them to the function output through (a number of
245 nested) multiplexers. These multiplexers are driven by comparators and
246 other logic, that check each pattern in turn.
248 In these examples we have seen only binary case expressions and pattern
249 matches (\ie, with two alternatives). In practice, case expressions can
250 choose between more than two values, resulting in a number of nested
253 \startbuffer[CaseInv]
260 \startbuffer[PatternInv]
266 \startuseMPgraphic{Inv}
267 save in, truecmp, falseout, trueout, out, cmp, mux;
270 newCircle.in(btex $in$ etex) "framed(false)";
271 newCircle.out(btex $out$ etex) "framed(false)";
273 newBox.truecmp(btex $True$ etex) "framed(false)";
274 newBox.trueout(btex $True$ etex) "framed(false)";
275 newBox.falseout(btex $False$ etex) "framed(false)";
278 newCircle.cmp(btex $==$ etex);
282 cmp.c = in.c + (3cm, 0cm);
283 truecmp.c = cmp.c + (-1cm, 1cm);
284 mux.sel = cmp.e + (1cm, -1cm);
285 falseout.c = mux.inpa - (2cm, 0cm);
286 trueout.c = mux.inpb - (2cm, 0cm);
287 out.c = mux.out + (2cm, 0cm);
289 % Draw objects and lines
290 drawObj(in, out, truecmp, trueout, falseout, cmp, mux);
294 nccurve(cmp.e)(mux.sel) "angleA(0)", "angleB(-90)";
295 ncline(falseout)(mux) "posB(inpa)";
296 ncline(trueout)(mux) "posB(inpb)";
297 ncline(mux)(out) "posA(out)";
300 \startbuffer[InvVHDL]
301 entity invComponent_0 is
302 port (\xzAMo2\ : in boolean;
303 \reszAMuzAMu2\ : out boolean;
304 clock : in std_logic;
305 resetn : in std_logic);
306 end entity invComponent_0;
309 architecture structural of invComponent_0 is
311 \reszAMuzAMu2\ <= false when \xzAMo2\ = true else
313 end architecture structural;
316 \placeexample[][ex:Inv]{Simple inverter.}{
317 % Use placesidebyside, since nesting combinations doesn't seem to work
318 % here. This does break centering, but well...
320 % Use 2*2 instead of 1*2 to insert some extra space (\placesidebyside
321 % places stuff very close together)
322 {\startcombination[2*2]
323 {\typebufferhs{CaseInv}}{Haskell description using a Case expression.}
325 {\typebufferhs{PatternInv}}{Haskell description using Pattern matching expression.}
328 % Use a 1*1 combination to add a caption
329 {\startcombination[1*1]
330 {\boxedgraphic{Inv}}{The architecture described by the Haskell descriptions.}
334 % \placeexample[][ex:Inv]{Simple inverter.}{
335 % \startcombination[2*2]
336 % {\typebufferhs{CaseInv}}{Haskell description using a Case expression.}
338 % {\typebufferhs{PatternInv}}{Haskell description using Pattern matching expression.}
339 % {\boxedgraphic{Inv}}{The architecture described by the Haskell description.}
342 \placeexample[][ex:InvVHDL]{\VHDL\ generated for (both versions of) \hs{inv} from \in{example}[ex:Inv]}
343 {\typebuffervhdl{InvVHDL}}
346 Translation of two most basic functional concepts has been
347 discussed: function application and choice. Before looking further
348 into less obvious concepts like higher-order expressions and
349 polymorphism, the possible types that can be used in hardware
350 descriptions will be discussed.
352 Some way is needed to translate every values used to its hardware
353 equivalents. In particular, this means a hardware equivalent for
354 every \emph{type} used in a hardware description is needed
356 Since most functional languages have a lot of standard types that
357 are hard to translate (integers without a fixed size, lists without
358 a static length, etc.), a number of \quote{built-in} types will be
359 defined first. These types are built-in in the sense that our
360 compiler will have a fixed VHDL type for these. User defined types,
361 on the other hand, will have their hardware type derived directly
362 from their Haskell declaration automatically, according to the rules
365 \todo{Introduce Haskell type syntax (type constructors, type application,
368 \subsection{Built-in types}
369 The language currently supports the following built-in types. Of these,
370 only the \hs{Bool} type is supported by Haskell out of the box (the
371 others are defined by the Cλash package, so they are user-defined types
372 from Haskell's point of view).
375 This is the most basic type available. It is mapped directly onto
376 the \type{std_logic} \small{VHDL} type. Mapping this to the
377 \type{bit} type might make more sense (since the Haskell version
378 only has two values), but using \type{std_logic} is more standard
379 (and allowed for some experimentation with don't care values)
381 \todo{Sidenote bit vs stdlogic}
383 \startdesc{\hs{Bool}}
384 This is the only built-in Haskell type supported and is translated
385 exactly like the Bit type (where a value of \hs{True} corresponds to a
386 value of \hs{High}). Supporting the Bool type is particularly
387 useful to support \hs{if ... then ... else ...} expressions, which
388 always have a \hs{Bool} value for the condition.
390 A \hs{Bool} is translated to a \type{std_logic}, just like \hs{Bit}.
392 \startdesc{\hs{SizedWord}, \hs{SizedInt}}
393 These are types to represent integers. A \hs{SizedWord} is unsigned,
394 while a \hs{SizedInt} is signed. These types are parameterized by a
395 length type, so you can define an unsigned word of 32 bits wide as
399 type Word32 = SizedWord D32
402 Here, a type synonym \hs{Word32} is defined that is equal to the
403 \hs{SizedWord} type constructor applied to the type \hs{D32}. \hs{D32}
404 is the \emph{type level representation} of the decimal number 32,
405 making the \hs{Word32} type a 32-bit unsigned word.
407 These types are translated to the \small{VHDL} \type{unsigned} and
408 \type{signed} respectively.
409 \todo{Sidenote on dependent typing?}
411 \startdesc{\hs{Vector}}
412 This is a vector type, that can contain elements of any other type and
413 has a fixed length. It has two type parameters: its
414 length and the type of the elements contained in it. By putting the
415 length parameter in the type, the length of a vector can be determined
416 at compile time, instead of only at runtime for conventional lists.
418 The \hs{Vector} type constructor takes two type arguments: the length
419 of the vector and the type of the elements contained in it. The state
420 type of an 8 element register bank would then for example be:
423 type RegisterState = Vector D8 Word32
426 Here, a type synonym \hs{RegisterState} is defined that is equal to
427 the \hs{Vector} type constructor applied to the types \hs{D8} (The type
428 level representation of the decimal number 8) and \hs{Word32} (The 32
429 bit word type as defined above). In other words, the
430 \hs{RegisterState} type is a vector of 8 32-bit words.
432 A fixed size vector is translated to a \small{VHDL} array type.
434 \startdesc{\hs{RangedWord}}
435 This is another type to describe integers, but unlike the previous
436 two it has no specific bitwidth, but an upper bound. This means that
437 its range is not limited to powers of two, but can be any number.
438 A \hs{RangedWord} only has an upper bound, its lower bound is
439 implicitly zero. There is a lot of added implementation complexity
440 when adding a lower bound and having just an upper bound was enough
441 for the primary purpose of this type: typesafely indexing vectors.
443 To define an index for the 8 element vector above, we would do:
446 type RegisterIndex = RangedWord D7
449 Here, a type synonym \hs{RegisterIndex} is defined that is equal to
450 the \hs{RangedWord} type constructor applied to the type \hs{D7}. In
451 other words, this defines an unsigned word with values from
452 {\definedfont[Serif*normalnum]0 to 7} (inclusive). This word can be be used to index the
453 8 element vector \hs{RegisterState} above.
455 This type is translated to the \type{unsigned} \small{VHDL} type.
458 The integer and vector built-in types are discussed in more detail
461 \subsection{User-defined types}
462 There are three ways to define new types in Haskell: algebraic
463 datatypes with the \hs{data} keyword, type synonyms with the \hs{type}
464 keyword and type renamings with the \hs{newtype} keyword. \GHC\
465 offers a few more advanced ways to introduce types (type families,
466 existential typing, \small{GADT}s, etc.) which are not standard
467 Haskell. These will be left outside the scope of this research.
469 Only an algebraic datatype declaration actually introduces a
470 completely new type, for which we provide the \VHDL\ translation
471 below. Type synonyms and renamings only define new names for
472 existing types (where synonyms are completely interchangeable and
473 renamings need explicit conversion). Therefore, these do not need
474 any particular \VHDL\ translation, a synonym or renamed type will
475 just use the same representation as the original type. The
476 distinction between a renaming and a synonym does no longer matter
477 in hardware and can be disregarded in the generated \VHDL.
479 For algebraic types, we can make the following distinction:
481 \startdesc{Product types}
482 A product type is an algebraic datatype with a single constructor with
483 two or more fields, denoted in practice like (a,b), (a,b,c), etc. This
484 is essentially a way to pack a few values together in a record-like
485 structure. In fact, the built-in tuple types are just algebraic product
486 types (and are thus supported in exactly the same way).
488 The \quote{product} in its name refers to the collection of values belonging
489 to this type. The collection for a product type is the Cartesian
490 product of the collections for the types of its fields.
492 These types are translated to \VHDL\ record types, with one field for
493 every field in the constructor. This translation applies to all single
494 constructor algebraic datatypes, including those with just one
495 field (which are technically not a product, but generate a VHDL
496 record for implementation simplicity).
498 \startdesc{Enumerated types}
499 \defref{enumerated types}
500 An enumerated type is an algebraic datatype with multiple constructors, but
501 none of them have fields. This is essentially a way to get an
502 enum-like type containing alternatives.
504 Note that Haskell's \hs{Bool} type is also defined as an
505 enumeration type, but we have a fixed translation for that.
507 These types are translated to \VHDL\ enumerations, with one value for
508 each constructor. This allows references to these constructors to be
509 translated to the corresponding enumeration value.
511 \startdesc{Sum types}
512 A sum type is an algebraic datatype with multiple constructors, where
513 the constructors have one or more fields. Technically, a type with
514 more than one field per constructor is a sum of products type, but
515 for our purposes this distinction does not really make a
516 difference, so this distinction is note made.
518 The \quote{sum} in its name refers again to the collection of values
519 belonging to this type. The collection for a sum type is the
520 union of the the collections for each of the constructors.
522 Sum types are currently not supported by the prototype, since there is
523 no obvious \VHDL\ alternative. They can easily be emulated, however, as
524 we will see from an example:
527 data Sum = A Bit Word | B Word
530 An obvious way to translate this would be to create an enumeration to
531 distinguish the constructors and then create a big record that
532 contains all the fields of all the constructors. This is the same
533 translation that would result from the following enumeration and
534 product type (using a tuple for clarity):
538 type Sum = (SumC, Bit, Word, Word)
541 Here, the \hs{SumC} type effectively signals which of the latter three
542 fields of the \hs{Sum} type are valid (the first two if \hs{A}, the
543 last one if \hs{B}), all the other ones have no useful value.
545 An obvious problem with this naive approach is the space usage: the
546 example above generates a fairly big \VHDL\ type. Since we can be
547 sure that the two \hs{Word}s in the \hs{Sum} type will never be valid
548 at the same time, this is a waste of space.
550 Obviously, duplication detection could be used to reuse a
551 particular field for another constructor, but this would only
552 partially solve the problem. If two fields would be, for
553 example, an array of 8 bits and an 8 bit unsiged word, these are
554 different types and could not be shared. However, in the final
555 hardware, both of these types would simply be 8 bit connections,
556 so we have a 100\% size increase by not sharing these.
559 Another interesting case is that of recursive types. In Haskell, an
560 algebraic datatype can be recursive: any of its field types can be (or
561 contain) the type being defined. The most well-known recursive type is
562 probably the list type, which is defined is:
565 data List t = Empty | Cons t (List t)
568 Note that \hs{Empty} is usually written as \hs{[]} and \hs{Cons} as
569 \hs{:}, but this would make the definition harder to read. This
570 immediately shows the problem with recursive types: what hardware type
573 If the naive approach for sum types described above would be used,
574 a record would be created where the first field is an enumeration
575 to distinguish \hs{Empty} from \hs{Cons}. Furthermore, two more
576 fields would be added: one with the (\VHDL\ equivalent of) type
577 \hs{t} (assuming this type is actually known at compile time, this
578 should not be a problem) and a second one with type \hs{List t}.
579 The latter one is of course a problem: this is exactly the type
580 that was to be translated in the first place.
582 The resulting \VHDL\ type will thus become infinitely deep. In
583 other words, there is no way to statically determine how long
584 (deep) the list will be (it could even be infinite).
586 In general, recursive types can never be properly translated: all
587 recursive types have a potentially infinite value (even though in
588 practice they will have a bounded value, there is no way for the
589 compiler to automatically determine an upper bound on its size).
591 \subsection{Partial application}
592 Now the translation of application, choice and types has been
593 discussed, a more complex concept can be considered: partial
594 applications. A \emph{partial application} is any application whose
595 (return) type is (again) a function type.
597 From this, it should be clear that the translation rules for full
598 application does not apply to a partial application: there are not
599 enough values for all the input ports in the resulting \VHDL.
600 \in{Example}[ex:Quadruple] shows an example use of partial application
601 and the corresponding architecture.
603 \startbuffer[Quadruple]
604 -- Multiply the input word by four.
605 quadruple :: Word -> Word
606 quadruple n = mul (mul n)
611 \startuseMPgraphic{Quadruple}
612 save in, two, mula, mulb, out;
615 newCircle.in(btex $n$ etex) "framed(false)";
616 newCircle.two(btex $2$ etex) "framed(false)";
617 newCircle.out(btex $out$ etex) "framed(false)";
620 newCircle.mula(btex $\times$ etex);
621 newCircle.mulb(btex $\times$ etex);
624 in.c = two.c + (0cm, 1cm);
625 mula.c = in.c + (2cm, 0cm);
626 mulb.c = mula.c + (2cm, 0cm);
627 out.c = mulb.c + (2cm, 0cm);
629 % Draw objects and lines
630 drawObj(in, two, mula, mulb, out);
632 nccurve(two)(mula) "angleA(0)", "angleB(45)";
633 nccurve(two)(mulb) "angleA(0)", "angleB(45)";
639 \placeexample[][ex:Quadruple]{Simple three port and.}
640 \startcombination[2*1]
641 {\typebufferhs{Quadruple}}{Haskell description using function applications.}
642 {\boxedgraphic{Quadruple}}{The architecture described by the Haskell description.}
645 Here, the definition of mul is a partial function application: it applies
646 the function \hs{(*) :: Word -> Word -> Word} to the value \hs{2 :: Word},
647 resulting in the expression \hs{(*) 2 :: Word -> Word}. Since this resulting
648 expression is again a function, hardware cannot be generated for it
649 directly. This is because the hardware to generate for \hs{mul}
650 depends completely on where and how it is used. In this example, it is
653 However, it is clear that the above hardware description actually
654 describes valid hardware. In general, any partial applied function
655 must eventually become completely applied, at which point hardware for
656 it can be generated using the rules for function application given in
657 \in{section}[sec:description:application]. It might mean that a
658 partial application is passed around quite a bit (even beyond function
659 boundaries), but eventually, the partial application will become
660 completely applied. An example of this principe is given in
661 \in{section}[sec:normalization:defunctionalization].
663 \section{Costless specialization}
664 Each (complete) function application in our description generates a
665 component instantiation, or a specific piece of hardware in the final
666 design. It is interesting to note that each application of a function
667 generates a \emph{separate} piece of hardware. In the final design, none
668 of the hardware is shared between applications, even when the applied
669 function is the same (of course, if a particular value, such as the result
670 of a function application, is used twice, it is not calculated twice).
672 This is distinctly different from normal program compilation: two separate
673 calls to the same function share the same machine code. Having more
674 machine code has implications for speed (due to less efficient caching)
675 and memory usage. For normal compilation, it is therefore important to
676 keep the amount of functions limited and maximize the code sharing
677 (though there is a tradeoff between speed and memory usage here).
679 When generating hardware, this is hardly an issue. Having more \quote{code
680 sharing} does reduce the amount of \small{VHDL} output (Since different
681 component instantiations still share the same component), but after
682 synthesis, the amount of hardware generated is not affected. This
683 means there is no tradeoff between speed and memory (or rather,
686 In particular, if we would duplicate all functions so that there is a
687 separate function for every application in the program (\eg, each function
688 is then only applied exactly once), there would be no increase in hardware
691 Because of this, a common optimization technique called
692 \emph{specialization} can be applied to hardware generation without any
693 performance or area cost (unlike for software).
695 \fxnote{Perhaps these next three sections are a bit too
696 implementation-oriented?}
698 \subsection{Specialization}
699 \defref{specialization}
700 Given some function that has a \emph{domain} $D$ (\eg, the set of
701 all possible arguments that the function could be applied to), we
702 create a specialized function with exactly the same behaviour, but
703 with a domain $D' \subset D$. This subset can be chosen in all
704 sorts of ways. Any subset is valid for the general definition of
705 specialization, but in practice only some of them provide useful
706 optimization opportunities.
708 Common subsets include limiting a polymorphic argument to a single type
709 (\ie, removing polymorphism) or limiting an argument to just a single
710 value (\ie, cross-function constant propagation, effectively removing
713 Since we limit the argument domain of the specialized function, its
714 definition can often be optimized further (since now more types or even
715 values of arguments are already known). By replacing any application of
716 the function that falls within the reduced domain by an application of
717 the specialized version, the code gets faster (but the code also gets
718 bigger, since we now have two versions instead of one). If we apply
719 this technique often enough, we can often replace all applications of a
720 function by specialized versions, allowing the original function to be
721 removed (in some cases, this can even give a net reduction of the code
722 compared to the non-specialized version).
724 Specialization is useful for our hardware descriptions for functions
725 that contain arguments that cannot be translated to hardware directly
726 (polymorphic or higher-order arguments, for example). If we can create
727 specialized functions that remove the argument, or make it translatable,
728 we can use specialization to make the original, untranslatable, function
731 \section{Higher order values}
732 What holds for partial application, can be easily generalized to any
733 higher-order expression. This includes partial applications, plain
734 variables (e.g., a binder referring to a top level function), lambda
735 expressions and more complex expressions with a function type (a \hs{case}
736 expression returning lambda's, for example).
738 Each of these values cannot be directly represented in hardware (just like
739 partial applications). Also, to make them representable, they need to be
740 applied: function variables and partial applications will then eventually
741 become complete applications, applied lambda expressions disappear by
742 applying β-reduction, etc.
744 So any higher-order value will be \quote{pushed down} towards its
745 application just like partial applications. Whenever a function boundary
746 needs to be crossed, the called function can be specialized.
748 \fxnote{This section needs improvement and an example}
750 \section{Polymorphism}
751 In Haskell, values can be \emph{polymorphic}: they can have multiple types. For
752 example, the function \hs{fst :: (a, b) -> a} is an example of a
753 polymorphic function: it works for tuples with any two element types. Haskell
754 type classes allow a function to work on a specific set of types, but the
755 general idea is the same. The opposite of this is a \emph{monomorphic}
756 value, which has a single, fixed, type.
758 % A type class is a collection of types for which some operations are
759 % defined. It is thus possible for a value to be polymorphic while having
760 % any number of \emph{class constraints}: the value is not defined for
761 % every type, but only for types in the type class. An example of this is
762 % the \hs{even :: (Integral a) => a -> Bool} function, which can map any
763 % value of a type that is member of the \hs{Integral} type class
765 When generating hardware, polymorphism cannot be easily translated. How
766 many wires will you lay down for a value that could have any type? When
767 type classes are involved, what hardware components will you lay down for
768 a class method (whose behaviour depends on the type of its arguments)?
769 Note that Cλash currently does not allow user-defined type classes,
770 but does partly support some of the built-in type classes (like \hs{Num}).
772 Fortunately, we can again use the principle of specialization: since every
773 function application generates a separate piece of hardware, we can know
774 the types of all arguments exactly. Provided that existential typing
775 (which is a \GHC\ extension) is not used typing, all of the
776 polymorphic types in a function must depend on the types of the
777 arguments (In other words, the only way to introduce a type variable
778 is in a lambda abstraction).
780 If a function is monomorphic, all values inside it are monomorphic as
781 well, so any function that is applied within the function can only be
782 applied to monomorphic values. The applied functions can then be
783 specialized to work just for these specific types, removing the
784 polymorphism from the applied functions as well.
786 \defref{entry function}The entry function must not have a
787 polymorphic type (otherwise no hardware interface could be generated
788 for the entry function).
790 By induction, this means that all functions that are (indirectly) called
791 by our top level function (meaning all functions that are translated in
792 the final hardware) become monomorphic.
795 A very important concept in hardware designs is \emph{state}. In a
796 stateless (or, \emph{combinational}) design, every output is directly and solely dependent on the
797 inputs. In a stateful design, the outputs can depend on the history of
798 inputs, or the \emph{state}. State is usually stored in \emph{registers},
799 which retain their value during a clockcycle, and are typically updated at
800 the start of every clockcycle. Since the updating of the state is tightly
801 coupled (synchronized) to the clock signal, these state updates are often
802 called \emph{synchronous} behaviour.
804 \todo{Sidenote? Registers can contain any (complex) type}
806 To make Cλash useful to describe more than simple combinational
807 designs, it needs to be able to describe state in some way.
809 \subsection{Approaches to state}
810 In Haskell, functions are always pure (except when using unsafe
811 functions with the \hs{IO} monad, which is not supported by Cλash). This
812 means that the output of a function solely depends on its inputs. If you
813 evaluate a given function with given inputs, it will always provide the
818 \startframedtext[width=8cm,background=box,frame=no]
819 \startalignment[center]
824 A function is said to be pure if it satisfies two conditions:
827 \item When a pure function is called with the same arguments twice, it should
828 return the same value in both cases.
829 \item When a pure function is called, it should have not
830 observable side-effects.
833 Purity is an important property in functional languages, since
834 it enables all kinds of mathematical reasoning and
835 optimizattions with pure functions, that are not guaranteed to
836 be correct for impure functions.
838 An example of a pure function is the square root function
839 \hs{sqrt}. An example of an impure function is the \hs{today}
840 function that returns the current date (which of course cannot
841 exist like this in Haskell).
845 This is a perfect match for a combinational circuit, where the output
846 also solely depends on the inputs. However, when state is involved, this
847 no longer holds. Of course this purity constraint cannot just be
848 removed from Haskell. But even when designing a completely new (hardware
849 description) language, it does not seem to be a good idea to
850 remove this purity. This would that all kinds of interesting properties of
851 the functional language get lost, and all kinds of transformations
852 and optimizations are no longer be meaning preserving.
854 So our functions must remain pure, meaning the current state has
855 to be present in the function's arguments in some way. There seem
856 to be two obvious ways to do this: adding the current state as an
857 argument, or including the full history of each argument.
859 \subsubsection{Stream arguments and results}
860 Including the entire history of each input (\eg, the value of that
861 input for each previous clockcycle) is an obvious way to make outputs
862 depend on all previous input. This is easily done by making every
863 input a list instead of a single value, containing all previous values
864 as well as the current value.
866 An obvious downside of this solution is that on each cycle, all the
867 previous cycles must be resimulated to obtain the current state. To do
868 this, it might be needed to have a recursive helper function as well,
869 which might be hard to be properly analyzed by the compiler.
871 A slight variation on this approach is one taken by some of the other
872 functional \small{HDL}s in the field: \todo{References to Lava,
873 ForSyDe, ...} Make functions operate on complete streams. This means
874 that a function is no longer called on every cycle, but just once. It
875 takes stream as inputs instead of values, where each stream contains
876 all the values for every clockcycle since system start. This is easily
877 modeled using an (infinite) list, with one element for each clock
878 cycle. Since the function is only evaluated once, its output must also
879 be a stream. Note that, since we are working with infinite lists and
880 still want to be able to simulate the system cycle-by-cycle, this
881 relies heavily on the lazy semantics of Haskell.
883 Since our inputs and outputs are streams, all other (intermediate)
884 values must be streams. All of our primitive operators (\eg, addition,
885 substraction, bitwise operations, etc.) must operate on streams as
886 well (note that changing a single-element operation to a stream
887 operation can done with \hs{map}, \hs{zipwith}, etc.).
889 This also means that primitive operations from an existing
890 language such as Haskell cannot be used directly (including some
891 syntax constructs, like \hs{case} expressions and \hs{if}
892 expressions). This mkes this approach well suited for use in
893 \small{EDSL}s, since those already impose these same
894 limitations. \refdef{EDSL}
896 Note that the concept of \emph{state} is no more than having some way
897 to communicate a value from one cycle to the next. By introducing a
898 \hs{delay} function, we can do exactly that: delay (each value in) a
899 stream so that we can "look into" the past. This \hs{delay} function
900 simply outputs a stream where each value is the same as the input
901 value, but shifted one cycle. This causes a \quote{gap} at the
902 beginning of the stream: what is the value of the delay output in the
903 first cycle? For this, the \hs{delay} function has a second input, of
904 which only a single value is used.
906 \in{Example}[ex:DelayAcc] shows a simple accumulator expressed in this
909 \startbuffer[DelayAcc]
910 acc :: Stream Word -> Stream Word
913 out = (delay out 0) + in
916 \startuseMPgraphic{DelayAcc}
917 save in, out, add, reg;
920 newCircle.in(btex $in$ etex) "framed(false)";
921 newCircle.out(btex $out$ etex) "framed(false)";
924 newReg.reg("") "dx(4mm)", "dy(6mm)", "reflect(true)";
925 newCircle.add(btex + etex);
928 add.c = in.c + (2cm, 0cm);
929 out.c = add.c + (2cm, 0cm);
930 reg.c = add.c + (0cm, 2cm);
932 % Draw objects and lines
933 drawObj(in, out, add, reg);
935 nccurve(add)(reg) "angleA(0)", "angleB(180)", "posB(d)";
936 nccurve(reg)(add) "angleA(180)", "angleB(-45)", "posA(out)";
942 \placeexample[][ex:DelayAcc]{Simple accumulator architecture.}
943 \startcombination[2*1]
944 {\typebufferhs{DelayAcc}}{Haskell description using streams.}
945 {\boxedgraphic{DelayAcc}}{The architecture described by the Haskell description.}
949 This notation can be confusing (especially due to the loop in the
950 definition of out), but is essentially easy to interpret. There is a
951 single call to delay, resulting in a circuit with a single register,
952 whose input is connected to \hs{out} (which is the output of the
953 adder), and its output is the expression \hs{delay out 0} (which is
954 connected to one of the adder inputs).
956 \subsubsection{Explicit state arguments and results}
957 A more explicit way to model state, is to simply add an extra argument
958 containing the current state value. This allows an output to depend on
959 both the inputs as well as the current state while keeping the
960 function pure (letting the result depend only on the arguments), since
961 the current state is now an argument.
963 In Haskell, this would look like
964 \in{example}[ex:ExplicitAcc]\footnote[notfinalsyntax]{This
965 example is not in the final Cλash syntax}. \todo{Referencing
966 notfinalsyntax from Introduction.tex doesn't work}
968 \startbuffer[ExplicitAcc]
969 -- input -> current state -> (new state, output)
970 acc :: Word -> Word -> (Word, Word)
977 \placeexample[][ex:ExplicitAcc]{Simple accumulator architecture.}
978 \startcombination[2*1]
979 {\typebufferhs{ExplicitAcc}}{Haskell description using explicit state arguments.}
980 % Picture is identical to the one we had just now.
981 {\boxedgraphic{DelayAcc}}{The architecture described by the Haskell description.}
984 This approach makes a function's state very explicit, which state
985 variables are used by a function can be completely determined from its
986 type signature (as opposed to the stream approach, where a function
987 looks the same from the outside, regardless of what state variables it
988 uses or whether it is stateful at all).
990 This approach to state has been one of the initial drives behind
991 this research. Unlike a stream based approach it is well suited
992 to completely use existing code and language features (like
993 \hs{if} and \hs{case} expressions) because it operates on normal
994 values. Because of these reasons, this is the approach chosen
995 for Cλash. It will be examined more closely below.
997 \subsection{Explicit state specification}
998 The concept of explicit state has been introduced with some
999 examples above, but what are the implications of this approach?
1001 \subsubsection{Substates}
1002 Since a function's state is reflected directly in its type signature,
1003 if a function calls other stateful functions (\eg, has subcircuits), it
1004 has to somehow know the current state for these called functions. The
1005 only way to do this, is to put these \emph{substates} inside the
1006 caller's state. This means that a function's state is the sum of the
1007 states of all functions it calls, and its own state. This sum
1008 can be obtained using something simple like a tuple, or possibly
1009 custom algebraic types for clarity.
1011 This also means that the type of a function (at least the "state"
1012 part) is dependent on its own implementation and of the functions it
1015 This is the major downside of this approach: the separation between
1016 interface and implementation is limited. However, since Cλash is not
1017 very suitable for separate compilation (see
1018 \in{section}[sec:prototype:separate]) this is not a big problem in
1021 Additionally, when using a type synonym for the state type
1022 of each function, we can still provide explicit type signatures
1023 while keeping the state specification for a function near its
1027 \subsubsection{Which arguments and results are stateful?}
1028 \fxnote{This section should get some examples}
1029 We need some way to know which arguments should become input ports and
1030 which argument(s?) should become the current state (\eg, be bound to
1031 the register outputs). This does not hold just for the top
1032 level function, but also for any subfunction. Or could we perhaps
1033 deduce the statefulness of subfunctions by analyzing the flow of data
1034 in the calling functions?
1036 To explore this matter, the following observeration is interesting: we
1037 get completely correct behaviour when we put all state registers in
1038 the top level entity (or even outside of it). All of the state
1039 arguments and results on subfunctions are treated as normal input and
1040 output ports. Effectively, a stateful function results in a stateless
1041 hardware component that has one of its input ports connected to the
1042 output of a register and one of its output ports connected to the
1043 input of the same register.
1047 Of course, even though the hardware described like this has the
1048 correct behaviour, unless the layout tool does smart optimizations,
1049 there will be a lot of extra wire in the design (since registers will
1050 not be close to the component that uses them). Also, when working with
1051 the generated \small{VHDL} code, there will be a lot of extra ports
1052 just to pass on state values, which can get quite confusing.
1054 To fix this, we can simply \quote{push} the registers down into the
1055 subcircuits. When we see a register that is connected directly to a
1056 subcircuit, we remove the corresponding input and output port and put
1057 the register inside the subcircuit instead. This is slightly less
1058 trivial when looking at the Haskell code instead of the resulting
1059 circuit, but the idea is still the same.
1063 However, when applying this technique, we might push registers down
1064 too far. When you intend to store a result of a stateless subfunction
1065 in the caller's state and pass the current value of that state
1066 variable to that same function, the register might get pushed down too
1067 far. It is impossible to distinguish this case from similar code where
1068 the called function is in fact stateful. From this we can conclude
1069 that we have to either:
1071 \todo{Example of wrong downpushing}
1074 \item accept that the generated hardware might not be exactly what we
1075 intended, in some specific cases. In most cases, the hardware will be
1077 \item explicitly annotate state arguments and results in the input
1081 The first option causes (non-obvious) exceptions in the language
1082 intepretation. Also, automatically determining where registers should
1083 end up is easier to implement correctly with explicit annotations, so
1084 for these reasons we will look at how this annotations could work.
1086 \todo{Sidenote: one or more state arguments?}
1088 \subsection[sec:description:stateann]{Explicit state annotation}
1089 To make our stateful descriptions unambigious and easier to translate,
1090 we need some way for the developer to describe which arguments and
1091 results are intended to become stateful.
1093 Roughly, we have two ways to achieve this:
1095 \item Use some kind of annotation method or syntactic construction in
1096 the language to indicate exactly which argument and (part of the)
1097 result is stateful. This means that the annotation lives
1098 \quote{outside} of the function, it is completely invisible when
1099 looking at the function body.
1100 \item Use some kind of annotation on the type level, \ie\ give stateful
1101 arguments and stateful (parts of) results a different type. This has the
1102 potential to make this annotation visible inside the function as well,
1103 such that when looking at a value inside the function body you can
1104 tell if it is stateful by looking at its type. This could possibly make
1105 the translation process a lot easier, since less analysis of the
1106 program flow might be required.
1109 From these approaches, the type level \quote{annotations} have been
1110 implemented in Cλash. \in{Section}[sec:prototype:statetype] expands on
1111 the possible ways this could have been implemented.
1113 \todo{Note about conditions on state variables and checking them}
1115 \section[sec:recursion]{Recursion}
1116 An important concept in functional languages is recursion. In its most basic
1117 form, recursion is a definition that is described in terms of itself. A
1118 recursive function is thus a function that uses itself in its body. This
1119 usually requires multiple evaluations of this function, with changing
1120 arguments, until eventually an evaluation of the function no longer requires
1123 Given the notion that each function application will translate to a
1124 component instantiation, we are presented with a problem. A recursive
1125 function would translate to a component that contains itself. Or, more
1126 precisely, that contains an instance of itself. This instance would again
1127 contain an instance of itself, and again, into infinity. This is obviously a
1128 problem for generating hardware.
1130 This is expected for functions that describe infinite recursion. In that
1131 case, we cannot generate hardware that shows correct behaviour in a single
1132 cycle (at best, we could generate hardware that needs an infinite number of
1133 cycles to complete).
1136 \startframedtext[width=8cm,background=box,frame=no]
1137 \startalignment[center]
1138 {\tfa \hs{null}, \hs{head} and \hs{tail}}
1141 The functions \hs{null}, \hs{head} and \hs{tail} are common list
1142 functions in Haskell. The \hs{null} function simply checks if a list is
1143 empty. The \hs{head} function returns the first element of a list. The
1144 \hs{tail} function returns containing everything \emph{except} the first
1147 In Cλash, there are vector versions of these functions, which do exactly
1152 However, most recursive definitions will describe finite
1153 recursion. This is because the recursive call is done conditionally. There
1154 is usually a \hs{case} expression where at least one alternative does not contain
1155 the recursive call, which we call the "base case". If, for each call to the
1156 recursive function, we would be able to detect at compile time which
1157 alternative applies, we would be able to remove the \hs{case} expression and
1158 leave only the base case when it applies. This will ensure that expanding
1159 the recursive functions will terminate after a bounded number of expansions.
1161 This does imply the extra requirement that the base case is detectable at
1162 compile time. In particular, this means that the decision between the base
1163 case and the recursive case must not depend on runtime data.
1165 \subsection{List recursion}
1166 The most common deciding factor in recursion is the length of a list that is
1167 passed in as an argument. Since we represent lists as vectors that encode
1168 the length in the vector type, it seems easy to determine the base case. We
1169 can simply look at the argument type for this. However, it turns out that
1170 this is rather non-trivial to write down in Haskell already, not even
1171 looking at translation. As an example, we would like to write down something
1175 sum :: Vector n Word -> Word
1176 sum xs = case null xs of
1178 False -> head xs + sum (tail xs)
1181 However, the Haskell typechecker will now use the following reasoning.
1182 For simplicity, the element type of a vector is left out, all vectors
1183 are assumed to have the same element type. Below, we write conditions
1184 on type variables before the \hs{=>} operator. This is not completely
1185 valid Haskell syntax, but serves to illustrate the typechecker
1186 reasoning. Also note that a vector can never have a negative length,
1187 so \hs{Vector n} implicitly means \hs{(n >= 0) => Vector n}.
1189 \todo{This typechecker disregards the type signature}
1191 \item tail has the type \hs{(n > 0) => Vector n -> Vector (n - 1)}
1192 \item This means that xs must have the type \hs{(n > 0) => Vector n}
1193 \item This means that sum must have the type \hs{(n > 0) => Vector n -> a}
1194 (The type \hs{a} is can be anything at this stage, we will not try to finds
1195 its actual type in this example).
1196 \item sum is called with the result of tail as an argument, which has the
1197 type \hs{Vector n} (since \hs{(n > 0) => Vector (n - 1)} is the same as \hs{(n >= 0)
1198 => Vector n}, which is the same as just \hs{Vector n}).
1199 \item This means that sum must have the type \hs{Vector n -> a}
1200 \item This is a contradiction between the type deduced from the body of sum
1201 (the input vector must be non-empty) and the use of sum (the input vector
1202 could have any length).
1205 As you can see, using a simple \hs{case} expression at value level causes
1206 the type checker to always typecheck both alternatives, which cannot be
1207 done. The typechecker is unable to distinguish the two case
1208 alternatives (this is partly possible using \small{GADT}s, but that
1209 approach faced other problems \todo{ref christiaan?}).
1211 This is a fundamental problem, that would seem perfectly suited for a
1212 type class. Considering that we need to switch between to
1213 implementations of the sum function, based on the type of the
1214 argument, this sounds like the perfect problem to solve with a type
1215 class. However, this approach has its own problems (not the least of
1216 them that you need to define a new type class for every recursive
1217 function you want to define).
1219 \todo{This should reference Christiaan}
1221 \subsection{General recursion}
1222 Of course there are other forms of recursion, that do not depend on the
1223 length (and thus type) of a list. For example, simple recursion using a
1224 counter could be expressed, but only translated to hardware for a fixed
1225 number of iterations. Also, this would require extensive support for compile
1226 time simplification (constant propagation) and compile time evaluation
1227 (evaluation of constant comparisons), to ensure non-termination.
1228 Supporting general recursion will probably require strict conditions
1229 on the input descriptions. Even then, it will be hard (if not
1230 impossible) to really guarantee termination, since the user (or \GHC\
1231 desugarer) might use some obscure notation that results in a corner
1232 case of the simplifier that is not caught and thus non-termination.
1234 Evaluating all possible (and non-possible) ways to add recursion to
1235 our descriptions, it seems better to limit the scope of this research
1236 to exclude recursion. This allows for focusing on other interesting
1237 areas instead. By including (built-in) support for a number of
1238 higher-order functions like \hs{map} and \hs{fold}, we can still
1239 express most of the things we would use (list) recursion for.
1242 % vim: set sw=2 sts=2 expandtab: