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:CaseInv] a simple \hs{case} expression is shown,
197 scrutinizing a boolean value. The corresponding architecture has a
198 comparator to determine which of the constructors is on the \hs{in}
199 input. There is a multiplexer to select the output signal. The two options
200 for the output signals are just constants, but these could have been more
201 complex expressions (in which case also both of them would be working in
202 parallel, regardless of which output would be chosen eventually).
204 If we would translate a Boolean to a bit value, we could of course remove
205 the comparator and directly feed 'in' into the multiplexer (or even use an
206 inverter instead of a multiplexer). However, we will try to make a
207 general translation, which works for all possible \hs{case} expressions.
208 Optimizations such as these are left for the \VHDL\ synthesizer, which
209 handles them very well.
212 \startframedtext[width=8cm,background=box,frame=no]
213 \startalignment[center]
214 {\tfa Arguments / results vs. inputs / outputs}
217 Due to the translation chosen for function application, there is a
218 very strong relation between arguments, results, inputs and outputs.
219 For clarity, the former two will always refer to the arguments and
220 results in the functional description (either Haskell or Core). The
221 latter two will refer to input and output ports in the generated
224 Even though these concepts seem to be nearly identical, when stateful
225 functions are introduces we will see arguments and results that will
226 not get translated into input and output ports, making this
227 distinction more important.
231 A slightly more complex (but very powerful) form of choice is pattern
232 matching. A function can be defined in multiple clauses, where each clause
233 specifies a pattern. When the arguments match the pattern, the
234 corresponding clause will be used.
236 The architecture described by \in{example}[ex:PatternInv] is of course the
237 same one as the one in \in{example}[ex:CaseInv]. The general interpretation
238 of pattern matching is also similar to that of \hs{case} expressions: generate
239 hardware for each of the clauses (like each of the clauses of a \hs{case}
240 expression) and connect them to the function output through (a number of
241 nested) multiplexers. These multiplexers are driven by comparators and
242 other logic, that check each pattern in turn.
244 In these examples we have seen only binary case expressions and pattern
245 matches (\ie, with two alternatives). In practice, case expressions can
246 choose between more than two values, resulting in a number of nested
249 \startbuffer[CaseInv]
256 \startbuffer[PatternInv]
262 \startuseMPgraphic{Inv}
263 save in, truecmp, falseout, trueout, out, cmp, mux;
266 newCircle.in(btex $in$ etex) "framed(false)";
267 newCircle.out(btex $out$ etex) "framed(false)";
269 newBox.truecmp(btex $True$ etex) "framed(false)";
270 newBox.trueout(btex $True$ etex) "framed(false)";
271 newBox.falseout(btex $False$ etex) "framed(false)";
274 newCircle.cmp(btex $==$ etex);
278 cmp.c = in.c + (3cm, 0cm);
279 truecmp.c = cmp.c + (-1cm, 1cm);
280 mux.sel = cmp.e + (1cm, -1cm);
281 falseout.c = mux.inpa - (2cm, 0cm);
282 trueout.c = mux.inpb - (2cm, 0cm);
283 out.c = mux.out + (2cm, 0cm);
285 % Draw objects and lines
286 drawObj(in, out, truecmp, trueout, falseout, cmp, mux);
290 nccurve(cmp.e)(mux.sel) "angleA(0)", "angleB(-90)";
291 ncline(falseout)(mux) "posB(inpa)";
292 ncline(trueout)(mux) "posB(inpb)";
293 ncline(mux)(out) "posA(out)";
296 \startbuffer[InvVHDL]
297 entity invComponent_0 is
298 port (\xzAMo2\ : in boolean;
299 \reszAMuzAMu2\ : out boolean;
300 clock : in std_logic;
301 resetn : in std_logic);
302 end entity invComponent_0;
305 architecture structural of invComponent_0 is
307 \reszAMuzAMu2\ <= false when \xzAMo2\ = true else
309 end architecture structural;
312 \placeexample[][ex:Inv]{Simple inverter.}{
313 % Use placesidebyside, since nesting combinations doesn't seem to work
314 % here. This does break centering, but well...
316 % Use 2*2 instead of 1*2 to insert some extra space (\placesidebyside
317 % places stuff very close together)
318 {\startcombination[2*2]
319 {\typebufferhs{CaseInv}}{Haskell description using a Case expression.}
321 {\typebufferhs{PatternInv}}{Haskell description using Pattern matching expression.}
324 % Use a 1*1 combination to add a caption
325 {\startcombination[1*1]
326 {\boxedgraphic{Inv}}{The architecture described by the Haskell descriptions.}
330 % \placeexample[][ex:Inv]{Simple inverter.}{
331 % \startcombination[2*2]
332 % {\typebufferhs{CaseInv}}{Haskell description using a Case expression.}
334 % {\typebufferhs{PatternInv}}{Haskell description using Pattern matching expression.}
335 % {\boxedgraphic{Inv}}{The architecture described by the Haskell description.}
338 \placeexample[][ex:InvVHDL]{\VHDL\ generated for \hs{inv} from \in{example}[ex:Inv]}
339 {\typebuffervhdl{InvVHDL}}
342 Translation of two most basic functional concepts has been
343 discussed: function application and choice. Before looking further
344 into less obvious concepts like higher-order expressions and
345 polymorphism, the possible types that can be used in hardware
346 descriptions will be discussed.
348 Some way is needed to translate every values used to its hardware
349 equivalents. In particular, this means a hardware equivalent for
350 every \emph{type} used in a hardware description is needed
352 Since most functional languages have a lot of standard types that
353 are hard to translate (integers without a fixed size, lists without
354 a static length, etc.), a number of \quote{built-in} types will be
355 defined first. These types are built-in in the sense that our
356 compiler will have a fixed VHDL type for these. User defined types,
357 on the other hand, will have their hardware type derived directly
358 from their Haskell declaration automatically, according to the rules
361 \todo{Introduce Haskell type syntax (type constructors, type application,
364 \subsection{Built-in types}
365 The language currently supports the following built-in types. Of these,
366 only the \hs{Bool} type is supported by Haskell out of the box (the
367 others are defined by the Cλash package, so they are user-defined types
368 from Haskell's point of view).
371 This is the most basic type available. It is mapped directly onto
372 the \type{std_logic} \small{VHDL} type. Mapping this to the
373 \type{bit} type might make more sense (since the Haskell version
374 only has two values), but using \type{std_logic} is more standard
375 (and allowed for some experimentation with don't care values)
377 \todo{Sidenote bit vs stdlogic}
379 \startdesc{\hs{Bool}}
380 This is the only built-in Haskell type supported and is translated
381 exactly like the Bit type (where a value of \hs{True} corresponds to a
382 value of \hs{High}). Supporting the Bool type is particularly
383 useful to support \hs{if ... then ... else ...} expressions, which
384 always have a \hs{Bool} value for the condition.
386 A \hs{Bool} is translated to a \type{std_logic}, just like \hs{Bit}.
388 \startdesc{\hs{SizedWord}, \hs{SizedInt}}
389 These are types to represent integers. A \hs{SizedWord} is unsigned,
390 while a \hs{SizedInt} is signed. These types are parameterized by a
391 length type, so you can define an unsigned word of 32 bits wide as
395 type Word32 = SizedWord D32
398 Here, a type synonym \hs{Word32} is defined that is equal to the
399 \hs{SizedWord} type constructor applied to the type \hs{D32}. \hs{D32}
400 is the \emph{type level representation} of the decimal number 32,
401 making the \hs{Word32} type a 32-bit unsigned word.
403 These types are translated to the \small{VHDL} \type{unsigned} and
404 \type{signed} respectively.
405 \todo{Sidenote on dependent typing?}
407 \startdesc{\hs{Vector}}
408 This is a vector type, that can contain elements of any other type and
409 has a fixed length. It has two type parameters: its
410 length and the type of the elements contained in it. By putting the
411 length parameter in the type, the length of a vector can be determined
412 at compile time, instead of only at runtime for conventional lists.
414 The \hs{Vector} type constructor takes two type arguments: the length
415 of the vector and the type of the elements contained in it. The state
416 type of an 8 element register bank would then for example be:
419 type RegisterState = Vector D8 Word32
422 Here, a type synonym \hs{RegisterState} is defined that is equal to
423 the \hs{Vector} type constructor applied to the types \hs{D8} (The type
424 level representation of the decimal number 8) and \hs{Word32} (The 32
425 bit word type as defined above). In other words, the
426 \hs{RegisterState} type is a vector of 8 32-bit words.
428 A fixed size vector is translated to a \small{VHDL} array type.
430 \startdesc{\hs{RangedWord}}
431 This is another type to describe integers, but unlike the previous
432 two it has no specific bitwidth, but an upper bound. This means that
433 its range is not limited to powers of two, but can be any number.
434 A \hs{RangedWord} only has an upper bound, its lower bound is
435 implicitly zero. There is a lot of added implementation complexity
436 when adding a lower bound and having just an upper bound was enough
437 for the primary purpose of this type: typesafely indexing vectors.
439 To define an index for the 8 element vector above, we would do:
442 type RegisterIndex = RangedWord D7
445 Here, a type synonym \hs{RegisterIndex} is defined that is equal to
446 the \hs{RangedWord} type constructor applied to the type \hs{D7}. In
447 other words, this defines an unsigned word with values from
448 {\definedfont[Serif*normalnum]0 to 7} (inclusive). This word can be be used to index the
449 8 element vector \hs{RegisterState} above.
451 This type is translated to the \type{unsigned} \small{VHDL} type.
454 The integer and vector built-in types are discussed in more detail
457 \subsection{User-defined types}
458 There are three ways to define new types in Haskell: algebraic
459 datatypes with the \hs{data} keyword, type synonyms with the \hs{type}
460 keyword and type renamings with the \hs{newtype} keyword. \GHC\
461 offers a few more advanced ways to introduce types (type families,
462 existential typing, \small{GADT}s, etc.) which are not standard
463 Haskell. These will be left outside the scope of this research.
465 Only an algebraic datatype declaration actually introduces a
466 completely new type, for which we provide the \VHDL\ translation
467 below. Type synonyms and renamings only define new names for
468 existing types (where synonyms are completely interchangeable and
469 renamings need explicit conversion). Therefore, these do not need
470 any particular \VHDL\ translation, a synonym or renamed type will
471 just use the same representation as the original type. The
472 distinction between a renaming and a synonym does no longer matter
473 in hardware and can be disregarded in the generated \VHDL.
475 For algebraic types, we can make the following distinction:
477 \startdesc{Product types}
478 A product type is an algebraic datatype with a single constructor with
479 two or more fields, denoted in practice like (a,b), (a,b,c), etc. This
480 is essentially a way to pack a few values together in a record-like
481 structure. In fact, the built-in tuple types are just algebraic product
482 types (and are thus supported in exactly the same way).
484 The \quote{product} in its name refers to the collection of values belonging
485 to this type. The collection for a product type is the Cartesian
486 product of the collections for the types of its fields.
488 These types are translated to \VHDL\ record types, with one field for
489 every field in the constructor. This translation applies to all single
490 constructor algebraic datatypes, including those with just one
491 field (which are technically not a product, but generate a VHDL
492 record for implementation simplicity).
494 \startdesc{Enumerated types}
495 \defref{enumerated types}
496 An enumerated type is an algebraic datatype with multiple constructors, but
497 none of them have fields. This is essentially a way to get an
498 enum-like type containing alternatives.
500 Note that Haskell's \hs{Bool} type is also defined as an
501 enumeration type, but we have a fixed translation for that.
503 These types are translated to \VHDL\ enumerations, with one value for
504 each constructor. This allows references to these constructors to be
505 translated to the corresponding enumeration value.
507 \startdesc{Sum types}
508 A sum type is an algebraic datatype with multiple constructors, where
509 the constructors have one or more fields. Technically, a type with
510 more than one field per constructor is a sum of products type, but
511 for our purposes this distinction does not really make a
512 difference, so this distinction is note made.
514 The \quote{sum} in its name refers again to the collection of values
515 belonging to this type. The collection for a sum type is the
516 union of the the collections for each of the constructors.
518 Sum types are currently not supported by the prototype, since there is
519 no obvious \VHDL\ alternative. They can easily be emulated, however, as
520 we will see from an example:
523 data Sum = A Bit Word | B Word
526 An obvious way to translate this would be to create an enumeration to
527 distinguish the constructors and then create a big record that
528 contains all the fields of all the constructors. This is the same
529 translation that would result from the following enumeration and
530 product type (using a tuple for clarity):
534 type Sum = (SumC, Bit, Word, Word)
537 Here, the \hs{SumC} type effectively signals which of the latter three
538 fields of the \hs{Sum} type are valid (the first two if \hs{A}, the
539 last one if \hs{B}), all the other ones have no useful value.
541 An obvious problem with this naive approach is the space usage: the
542 example above generates a fairly big \VHDL\ type. Since we can be
543 sure that the two \hs{Word}s in the \hs{Sum} type will never be valid
544 at the same time, this is a waste of space.
546 Obviously, duplication detection could be used to reuse a
547 particular field for another constructor, but this would only
548 partially solve the problem. If two fields would be, for
549 example, an array of 8 bits and an 8 bit unsiged word, these are
550 different types and could not be shared. However, in the final
551 hardware, both of these types would simply be 8 bit connections,
552 so we have a 100\% size increase by not sharing these.
555 Another interesting case is that of recursive types. In Haskell, an
556 algebraic datatype can be recursive: any of its field types can be (or
557 contain) the type being defined. The most well-known recursive type is
558 probably the list type, which is defined is:
561 data List t = Empty | Cons t (List t)
564 Note that \hs{Empty} is usually written as \hs{[]} and \hs{Cons} as
565 \hs{:}, but this would make the definition harder to read. This
566 immediately shows the problem with recursive types: what hardware type
569 If the naive approach for sum types described above would be used,
570 a record would be created where the first field is an enumeration
571 to distinguish \hs{Empty} from \hs{Cons}. Furthermore, two more
572 fields would be added: one with the (\VHDL\ equivalent of) type
573 \hs{t} (assuming this type is actually known at compile time, this
574 should not be a problem) and a second one with type \hs{List t}.
575 The latter one is of course a problem: this is exactly the type
576 that was to be translated in the first place.
578 The resulting \VHDL\ type will thus become infinitely deep. In
579 other words, there is no way to statically determine how long
580 (deep) the list will be (it could even be infinite).
582 In general, recursive types can never be properly translated: all
583 recursive types have a potentially infinite value (even though in
584 practice they will have a bounded value, there is no way for the
585 compiler to automatically determine an upper bound on its size).
587 \subsection{Partial application}
588 Now the translation of application, choice and types has been
589 discussed, a more complex concept can be considered: partial
590 applications. A \emph{partial application} is any application whose
591 (return) type is (again) a function type.
593 From this, it should be clear that the translation rules for full
594 application does not apply to a partial application: there are not
595 enough values for all the input ports in the resulting \VHDL.
596 \in{Example}[ex:Quadruple] shows an example use of partial application
597 and the corresponding architecture.
599 \startbuffer[Quadruple]
600 -- Multiply the input word by four.
601 quadruple :: Word -> Word
602 quadruple n = mul (mul n)
607 \startuseMPgraphic{Quadruple}
608 save in, two, mula, mulb, out;
611 newCircle.in(btex $n$ etex) "framed(false)";
612 newCircle.two(btex $2$ etex) "framed(false)";
613 newCircle.out(btex $out$ etex) "framed(false)";
616 newCircle.mula(btex $\times$ etex);
617 newCircle.mulb(btex $\times$ etex);
620 in.c = two.c + (0cm, 1cm);
621 mula.c = in.c + (2cm, 0cm);
622 mulb.c = mula.c + (2cm, 0cm);
623 out.c = mulb.c + (2cm, 0cm);
625 % Draw objects and lines
626 drawObj(in, two, mula, mulb, out);
628 nccurve(two)(mula) "angleA(0)", "angleB(45)";
629 nccurve(two)(mulb) "angleA(0)", "angleB(45)";
635 \placeexample[][ex:Quadruple]{Simple three port and.}
636 \startcombination[2*1]
637 {\typebufferhs{Quadruple}}{Haskell description using function applications.}
638 {\boxedgraphic{Quadruple}}{The architecture described by the Haskell description.}
641 Here, the definition of mul is a partial function application: it applies
642 the function \hs{(*) :: Word -> Word -> Word} to the value \hs{2 :: Word},
643 resulting in the expression \hs{(*) 2 :: Word -> Word}. Since this resulting
644 expression is again a function, hardware cannot be generated for it
645 directly. This is because the hardware to generate for \hs{mul}
646 depends completely on where and how it is used. In this example, it is
649 However, it is clear that the above hardware description actually
650 describes valid hardware. In general, any partial applied function
651 must eventually become completely applied, at which point hardware for
652 it can be generated using the rules for function application given in
653 \in{section}[sec:description:application]. It might mean that a
654 partial application is passed around quite a bit (even beyond function
655 boundaries), but eventually, the partial application will become
656 completely applied. An example of this principe is given in
657 \in{section}[sec:normalization:defunctionalization].
659 \section{Costless specialization}
660 Each (complete) function application in our description generates a
661 component instantiation, or a specific piece of hardware in the final
662 design. It is interesting to note that each application of a function
663 generates a \emph{separate} piece of hardware. In the final design, none
664 of the hardware is shared between applications, even when the applied
665 function is the same (of course, if a particular value, such as the result
666 of a function application, is used twice, it is not calculated twice).
668 This is distinctly different from normal program compilation: two separate
669 calls to the same function share the same machine code. Having more
670 machine code has implications for speed (due to less efficient caching)
671 and memory usage. For normal compilation, it is therefore important to
672 keep the amount of functions limited and maximize the code sharing
673 (though there is a tradeoff between speed and memory usage here).
675 When generating hardware, this is hardly an issue. Having more \quote{code
676 sharing} does reduce the amount of \small{VHDL} output (Since different
677 component instantiations still share the same component), but after
678 synthesis, the amount of hardware generated is not affected. This
679 means there is no tradeoff between speed and memory (or rather,
682 In particular, if we would duplicate all functions so that there is a
683 separate function for every application in the program (\eg, each function
684 is then only applied exactly once), there would be no increase in hardware
687 Because of this, a common optimization technique called
688 \emph{specialization} can be applied to hardware generation without any
689 performance or area cost (unlike for software).
691 \fxnote{Perhaps these next three sections are a bit too
692 implementation-oriented?}
694 \subsection{Specialization}
695 \defref{specialization}
696 Given some function that has a \emph{domain} $D$ (\eg, the set of
697 all possible arguments that the function could be applied to), we
698 create a specialized function with exactly the same behaviour, but
699 with a domain $D' \subset D$. This subset can be chosen in all
700 sorts of ways. Any subset is valid for the general definition of
701 specialization, but in practice only some of them provide useful
702 optimization opportunities.
704 Common subsets include limiting a polymorphic argument to a single type
705 (\ie, removing polymorphism) or limiting an argument to just a single
706 value (\ie, cross-function constant propagation, effectively removing
709 Since we limit the argument domain of the specialized function, its
710 definition can often be optimized further (since now more types or even
711 values of arguments are already known). By replacing any application of
712 the function that falls within the reduced domain by an application of
713 the specialized version, the code gets faster (but the code also gets
714 bigger, since we now have two versions instead of one). If we apply
715 this technique often enough, we can often replace all applications of a
716 function by specialized versions, allowing the original function to be
717 removed (in some cases, this can even give a net reduction of the code
718 compared to the non-specialized version).
720 Specialization is useful for our hardware descriptions for functions
721 that contain arguments that cannot be translated to hardware directly
722 (polymorphic or higher-order arguments, for example). If we can create
723 specialized functions that remove the argument, or make it translatable,
724 we can use specialization to make the original, untranslatable, function
727 \section{Higher order values}
728 What holds for partial application, can be easily generalized to any
729 higher-order expression. This includes partial applications, plain
730 variables (e.g., a binder referring to a top level function), lambda
731 expressions and more complex expressions with a function type (a \hs{case}
732 expression returning lambda's, for example).
734 Each of these values cannot be directly represented in hardware (just like
735 partial applications). Also, to make them representable, they need to be
736 applied: function variables and partial applications will then eventually
737 become complete applications, applied lambda expressions disappear by
738 applying β-reduction, etc.
740 So any higher-order value will be \quote{pushed down} towards its
741 application just like partial applications. Whenever a function boundary
742 needs to be crossed, the called function can be specialized.
744 \fxnote{This section needs improvement and an example}
746 \section{Polymorphism}
747 In Haskell, values can be \emph{polymorphic}: they can have multiple types. For
748 example, the function \hs{fst :: (a, b) -> a} is an example of a
749 polymorphic function: it works for tuples with any two element types. Haskell
750 type classes allow a function to work on a specific set of types, but the
751 general idea is the same. The opposite of this is a \emph{monomorphic}
752 value, which has a single, fixed, type.
754 % A type class is a collection of types for which some operations are
755 % defined. It is thus possible for a value to be polymorphic while having
756 % any number of \emph{class constraints}: the value is not defined for
757 % every type, but only for types in the type class. An example of this is
758 % the \hs{even :: (Integral a) => a -> Bool} function, which can map any
759 % value of a type that is member of the \hs{Integral} type class
761 When generating hardware, polymorphism cannot be easily translated. How
762 many wires will you lay down for a value that could have any type? When
763 type classes are involved, what hardware components will you lay down for
764 a class method (whose behaviour depends on the type of its arguments)?
765 Note that Cλash currently does not allow user-defined type classes,
766 but does partly support some of the built-in type classes (like \hs{Num}).
768 Fortunately, we can again use the principle of specialization: since every
769 function application generates a separate piece of hardware, we can know
770 the types of all arguments exactly. Provided that existential typing
771 (which is a \GHC\ extension) is not used typing, all of the
772 polymorphic types in a function must depend on the types of the
773 arguments (In other words, the only way to introduce a type variable
774 is in a lambda abstraction).
776 If a function is monomorphic, all values inside it are monomorphic as
777 well, so any function that is applied within the function can only be
778 applied to monomorphic values. The applied functions can then be
779 specialized to work just for these specific types, removing the
780 polymorphism from the applied functions as well.
782 \defref{entry function}The entry function must not have a
783 polymorphic type (otherwise no hardware interface could be generated
784 for the entry function).
786 By induction, this means that all functions that are (indirectly) called
787 by our top level function (meaning all functions that are translated in
788 the final hardware) become monomorphic.
791 A very important concept in hardware designs is \emph{state}. In a
792 stateless (or, \emph{combinational}) design, every output is directly and solely dependent on the
793 inputs. In a stateful design, the outputs can depend on the history of
794 inputs, or the \emph{state}. State is usually stored in \emph{registers},
795 which retain their value during a clockcycle, and are typically updated at
796 the start of every clockcycle. Since the updating of the state is tightly
797 coupled (synchronized) to the clock signal, these state updates are often
798 called \emph{synchronous} behaviour.
800 \todo{Sidenote? Registers can contain any (complex) type}
802 To make Cλash useful to describe more than simple combinational
803 designs, it needs to be able to describe state in some way.
805 \subsection{Approaches to state}
806 In Haskell, functions are always pure (except when using unsafe
807 functions with the \hs{IO} monad, which is not supported by Cλash). This
808 means that the output of a function solely depends on its inputs. If you
809 evaluate a given function with given inputs, it will always provide the
814 \startframedtext[width=8cm,background=box,frame=no]
815 \startalignment[center]
820 A function is said to be pure if it satisfies two conditions:
823 \item When a pure function is called with the same arguments twice, it should
824 return the same value in both cases.
825 \item When a pure function is called, it should have not
826 observable side-effects.
829 Purity is an important property in functional languages, since
830 it enables all kinds of mathematical reasoning and
831 optimizattions with pure functions, that are not guaranteed to
832 be correct for impure functions.
834 An example of a pure function is the square root function
835 \hs{sqrt}. An example of an impure function is the \hs{today}
836 function that returns the current date (which of course cannot
837 exist like this in Haskell).
841 This is a perfect match for a combinational circuit, where the output
842 also solely depends on the inputs. However, when state is involved, this
843 no longer holds. Of course this purity constraint cannot just be
844 removed from Haskell. But even when designing a completely new (hardware
845 description) language, it does not seem to be a good idea to
846 remove this purity. This would that all kinds of interesting properties of
847 the functional language get lost, and all kinds of transformations
848 and optimizations are no longer be meaning preserving.
850 So our functions must remain pure, meaning the current state has
851 to be present in the function's arguments in some way. There seem
852 to be two obvious ways to do this: adding the current state as an
853 argument, or including the full history of each argument.
855 \subsubsection{Stream arguments and results}
856 Including the entire history of each input (\eg, the value of that
857 input for each previous clockcycle) is an obvious way to make outputs
858 depend on all previous input. This is easily done by making every
859 input a list instead of a single value, containing all previous values
860 as well as the current value.
862 An obvious downside of this solution is that on each cycle, all the
863 previous cycles must be resimulated to obtain the current state. To do
864 this, it might be needed to have a recursive helper function as well,
865 which might be hard to be properly analyzed by the compiler.
867 A slight variation on this approach is one taken by some of the other
868 functional \small{HDL}s in the field: \todo{References to Lava,
869 ForSyDe, ...} Make functions operate on complete streams. This means
870 that a function is no longer called on every cycle, but just once. It
871 takes stream as inputs instead of values, where each stream contains
872 all the values for every clockcycle since system start. This is easily
873 modeled using an (infinite) list, with one element for each clock
874 cycle. Since the function is only evaluated once, its output must also
875 be a stream. Note that, since we are working with infinite lists and
876 still want to be able to simulate the system cycle-by-cycle, this
877 relies heavily on the lazy semantics of Haskell.
879 Since our inputs and outputs are streams, all other (intermediate)
880 values must be streams. All of our primitive operators (\eg, addition,
881 substraction, bitwise operations, etc.) must operate on streams as
882 well (note that changing a single-element operation to a stream
883 operation can done with \hs{map}, \hs{zipwith}, etc.).
885 This also means that primitive operations from an existing
886 language such as Haskell cannot be used directly (including some
887 syntax constructs, like \hs{case} expressions and \hs{if}
888 expressions). This mkes this approach well suited for use in
889 \small{EDSL}s, since those already impose these same
890 limitations. \refdef{EDSL}
892 Note that the concept of \emph{state} is no more than having some way
893 to communicate a value from one cycle to the next. By introducing a
894 \hs{delay} function, we can do exactly that: delay (each value in) a
895 stream so that we can "look into" the past. This \hs{delay} function
896 simply outputs a stream where each value is the same as the input
897 value, but shifted one cycle. This causes a \quote{gap} at the
898 beginning of the stream: what is the value of the delay output in the
899 first cycle? For this, the \hs{delay} function has a second input, of
900 which only a single value is used.
902 \in{Example}[ex:DelayAcc] shows a simple accumulator expressed in this
905 \startbuffer[DelayAcc]
906 acc :: Stream Word -> Stream Word
909 out = (delay out 0) + in
912 \startuseMPgraphic{DelayAcc}
913 save in, out, add, reg;
916 newCircle.in(btex $in$ etex) "framed(false)";
917 newCircle.out(btex $out$ etex) "framed(false)";
920 newReg.reg("") "dx(4mm)", "dy(6mm)", "reflect(true)";
921 newCircle.add(btex + etex);
924 add.c = in.c + (2cm, 0cm);
925 out.c = add.c + (2cm, 0cm);
926 reg.c = add.c + (0cm, 2cm);
928 % Draw objects and lines
929 drawObj(in, out, add, reg);
931 nccurve(add)(reg) "angleA(0)", "angleB(180)", "posB(d)";
932 nccurve(reg)(add) "angleA(180)", "angleB(-45)", "posA(out)";
938 \placeexample[][ex:DelayAcc]{Simple accumulator architecture.}
939 \startcombination[2*1]
940 {\typebufferhs{DelayAcc}}{Haskell description using streams.}
941 {\boxedgraphic{DelayAcc}}{The architecture described by the Haskell description.}
945 This notation can be confusing (especially due to the loop in the
946 definition of out), but is essentially easy to interpret. There is a
947 single call to delay, resulting in a circuit with a single register,
948 whose input is connected to \hs{out} (which is the output of the
949 adder), and its output is the expression \hs{delay out 0} (which is
950 connected to one of the adder inputs).
952 \subsubsection{Explicit state arguments and results}
953 A more explicit way to model state, is to simply add an extra argument
954 containing the current state value. This allows an output to depend on
955 both the inputs as well as the current state while keeping the
956 function pure (letting the result depend only on the arguments), since
957 the current state is now an argument.
959 In Haskell, this would look like
960 \in{example}[ex:ExplicitAcc]\footnote[notfinalsyntax]{This
961 example is not in the final Cλash syntax}. \todo{Referencing
962 notfinalsyntax from Introduction.tex doesn't work}
964 \startbuffer[ExplicitAcc]
965 -- input -> current state -> (new state, output)
966 acc :: Word -> Word -> (Word, Word)
973 \placeexample[][ex:ExplicitAcc]{Simple accumulator architecture.}
974 \startcombination[2*1]
975 {\typebufferhs{ExplicitAcc}}{Haskell description using explicit state arguments.}
976 % Picture is identical to the one we had just now.
977 {\boxedgraphic{DelayAcc}}{The architecture described by the Haskell description.}
980 This approach makes a function's state very explicit, which state
981 variables are used by a function can be completely determined from its
982 type signature (as opposed to the stream approach, where a function
983 looks the same from the outside, regardless of what state variables it
984 uses or whether it is stateful at all).
986 This approach to state has been one of the initial drives behind
987 this research. Unlike a stream based approach it is well suited
988 to completely use existing code and language features (like
989 \hs{if} and \hs{case} expressions) because it operates on normal
990 values. Because of these reasons, this is the approach chosen
991 for Cλash. It will be examined more closely below.
993 \subsection{Explicit state specification}
994 The concept of explicit state has been introduced with some
995 examples above, but what are the implications of this approach?
997 \subsubsection{Substates}
998 Since a function's state is reflected directly in its type signature,
999 if a function calls other stateful functions (\eg, has subcircuits), it
1000 has to somehow know the current state for these called functions. The
1001 only way to do this, is to put these \emph{substates} inside the
1002 caller's state. This means that a function's state is the sum of the
1003 states of all functions it calls, and its own state. This sum
1004 can be obtained using something simple like a tuple, or possibly
1005 custom algebraic types for clarity.
1007 This also means that the type of a function (at least the "state"
1008 part) is dependent on its own implementation and of the functions it
1011 This is the major downside of this approach: the separation between
1012 interface and implementation is limited. However, since Cλash is not
1013 very suitable for separate compilation (see
1014 \in{section}[sec:prototype:separate]) this is not a big problem in
1017 Additionally, when using a type synonym for the state type
1018 of each function, we can still provide explicit type signatures
1019 while keeping the state specification for a function near its
1023 \subsubsection{Which arguments and results are stateful?}
1024 \fxnote{This section should get some examples}
1025 We need some way to know which arguments should become input ports and
1026 which argument(s?) should become the current state (\eg, be bound to
1027 the register outputs). This does not hold just for the top
1028 level function, but also for any subfunction. Or could we perhaps
1029 deduce the statefulness of subfunctions by analyzing the flow of data
1030 in the calling functions?
1032 To explore this matter, the following observeration is interesting: we
1033 get completely correct behaviour when we put all state registers in
1034 the top level entity (or even outside of it). All of the state
1035 arguments and results on subfunctions are treated as normal input and
1036 output ports. Effectively, a stateful function results in a stateless
1037 hardware component that has one of its input ports connected to the
1038 output of a register and one of its output ports connected to the
1039 input of the same register.
1043 Of course, even though the hardware described like this has the
1044 correct behaviour, unless the layout tool does smart optimizations,
1045 there will be a lot of extra wire in the design (since registers will
1046 not be close to the component that uses them). Also, when working with
1047 the generated \small{VHDL} code, there will be a lot of extra ports
1048 just to pass on state values, which can get quite confusing.
1050 To fix this, we can simply \quote{push} the registers down into the
1051 subcircuits. When we see a register that is connected directly to a
1052 subcircuit, we remove the corresponding input and output port and put
1053 the register inside the subcircuit instead. This is slightly less
1054 trivial when looking at the Haskell code instead of the resulting
1055 circuit, but the idea is still the same.
1059 However, when applying this technique, we might push registers down
1060 too far. When you intend to store a result of a stateless subfunction
1061 in the caller's state and pass the current value of that state
1062 variable to that same function, the register might get pushed down too
1063 far. It is impossible to distinguish this case from similar code where
1064 the called function is in fact stateful. From this we can conclude
1065 that we have to either:
1067 \todo{Example of wrong downpushing}
1070 \item accept that the generated hardware might not be exactly what we
1071 intended, in some specific cases. In most cases, the hardware will be
1073 \item explicitly annotate state arguments and results in the input
1077 The first option causes (non-obvious) exceptions in the language
1078 intepretation. Also, automatically determining where registers should
1079 end up is easier to implement correctly with explicit annotations, so
1080 for these reasons we will look at how this annotations could work.
1082 \todo{Sidenote: one or more state arguments?}
1084 \subsection[sec:description:stateann]{Explicit state annotation}
1085 To make our stateful descriptions unambigious and easier to translate,
1086 we need some way for the developer to describe which arguments and
1087 results are intended to become stateful.
1089 Roughly, we have two ways to achieve this:
1091 \item Use some kind of annotation method or syntactic construction in
1092 the language to indicate exactly which argument and (part of the)
1093 result is stateful. This means that the annotation lives
1094 \quote{outside} of the function, it is completely invisible when
1095 looking at the function body.
1096 \item Use some kind of annotation on the type level, \ie\ give stateful
1097 arguments and stateful (parts of) results a different type. This has the
1098 potential to make this annotation visible inside the function as well,
1099 such that when looking at a value inside the function body you can
1100 tell if it is stateful by looking at its type. This could possibly make
1101 the translation process a lot easier, since less analysis of the
1102 program flow might be required.
1105 From these approaches, the type level \quote{annotations} have been
1106 implemented in Cλash. \in{Section}[sec:prototype:statetype] expands on
1107 the possible ways this could have been implemented.
1109 \todo{Note about conditions on state variables and checking them}
1111 \section[sec:recursion]{Recursion}
1112 An important concept in functional languages is recursion. In its most basic
1113 form, recursion is a definition that is described in terms of itself. A
1114 recursive function is thus a function that uses itself in its body. This
1115 usually requires multiple evaluations of this function, with changing
1116 arguments, until eventually an evaluation of the function no longer requires
1119 Given the notion that each function application will translate to a
1120 component instantiation, we are presented with a problem. A recursive
1121 function would translate to a component that contains itself. Or, more
1122 precisely, that contains an instance of itself. This instance would again
1123 contain an instance of itself, and again, into infinity. This is obviously a
1124 problem for generating hardware.
1126 This is expected for functions that describe infinite recursion. In that
1127 case, we cannot generate hardware that shows correct behaviour in a single
1128 cycle (at best, we could generate hardware that needs an infinite number of
1129 cycles to complete).
1132 \startframedtext[width=8cm,background=box,frame=no]
1133 \startalignment[center]
1134 {\tfa \hs{null}, \hs{head} and \hs{tail}}
1137 The functions \hs{null}, \hs{head} and \hs{tail} are common list
1138 functions in Haskell. The \hs{null} function simply checks if a list is
1139 empty. The \hs{head} function returns the first element of a list. The
1140 \hs{tail} function returns containing everything \emph{except} the first
1143 In Cλash, there are vector versions of these functions, which do exactly
1148 However, most recursive definitions will describe finite
1149 recursion. This is because the recursive call is done conditionally. There
1150 is usually a \hs{case} expression where at least one alternative does not contain
1151 the recursive call, which we call the "base case". If, for each call to the
1152 recursive function, we would be able to detect at compile time which
1153 alternative applies, we would be able to remove the \hs{case} expression and
1154 leave only the base case when it applies. This will ensure that expanding
1155 the recursive functions will terminate after a bounded number of expansions.
1157 This does imply the extra requirement that the base case is detectable at
1158 compile time. In particular, this means that the decision between the base
1159 case and the recursive case must not depend on runtime data.
1161 \subsection{List recursion}
1162 The most common deciding factor in recursion is the length of a list that is
1163 passed in as an argument. Since we represent lists as vectors that encode
1164 the length in the vector type, it seems easy to determine the base case. We
1165 can simply look at the argument type for this. However, it turns out that
1166 this is rather non-trivial to write down in Haskell already, not even
1167 looking at translation. As an example, we would like to write down something
1171 sum :: Vector n Word -> Word
1172 sum xs = case null xs of
1174 False -> head xs + sum (tail xs)
1177 However, the Haskell typechecker will now use the following reasoning.
1178 For simplicity, the element type of a vector is left out, all vectors
1179 are assumed to have the same element type. Below, we write conditions
1180 on type variables before the \hs{=>} operator. This is not completely
1181 valid Haskell syntax, but serves to illustrate the typechecker
1182 reasoning. Also note that a vector can never have a negative length,
1183 so \hs{Vector n} implicitly means \hs{(n >= 0) => Vector n}.
1185 \todo{This typechecker disregards the type signature}
1187 \item tail has the type \hs{(n > 0) => Vector n -> Vector (n - 1)}
1188 \item This means that xs must have the type \hs{(n > 0) => Vector n}
1189 \item This means that sum must have the type \hs{(n > 0) => Vector n -> a}
1190 (The type \hs{a} is can be anything at this stage, we will not try to finds
1191 its actual type in this example).
1192 \item sum is called with the result of tail as an argument, which has the
1193 type \hs{Vector n} (since \hs{(n > 0) => Vector (n - 1)} is the same as \hs{(n >= 0)
1194 => Vector n}, which is the same as just \hs{Vector n}).
1195 \item This means that sum must have the type \hs{Vector n -> a}
1196 \item This is a contradiction between the type deduced from the body of sum
1197 (the input vector must be non-empty) and the use of sum (the input vector
1198 could have any length).
1201 As you can see, using a simple \hs{case} expression at value level causes
1202 the type checker to always typecheck both alternatives, which cannot be
1203 done. The typechecker is unable to distinguish the two case
1204 alternatives (this is partly possible using \small{GADT}s, but that
1205 approach faced other problems \todo{ref christiaan?}).
1207 This is a fundamental problem, that would seem perfectly suited for a
1208 type class. Considering that we need to switch between to
1209 implementations of the sum function, based on the type of the
1210 argument, this sounds like the perfect problem to solve with a type
1211 class. However, this approach has its own problems (not the least of
1212 them that you need to define a new type class for every recursive
1213 function you want to define).
1215 \todo{This should reference Christiaan}
1217 \subsection{General recursion}
1218 Of course there are other forms of recursion, that do not depend on the
1219 length (and thus type) of a list. For example, simple recursion using a
1220 counter could be expressed, but only translated to hardware for a fixed
1221 number of iterations. Also, this would require extensive support for compile
1222 time simplification (constant propagation) and compile time evaluation
1223 (evaluation of constant comparisons), to ensure non-termination.
1224 Supporting general recursion will probably require strict conditions
1225 on the input descriptions. Even then, it will be hard (if not
1226 impossible) to really guarantee termination, since the user (or \GHC\
1227 desugarer) might use some obscure notation that results in a corner
1228 case of the simplifier that is not caught and thus non-termination.
1230 Evaluating all possible (and non-possible) ways to add recursion to
1231 our descriptions, it seems better to limit the scope of this research
1232 to exclude recursion. This allows for focusing on other interesting
1233 areas instead. By including (built-in) support for a number of
1234 higher-order functions like \hs{map} and \hs{fold}, we can still
1235 express most of the things we would use (list) recursion for.
1238 % vim: set sw=2 sts=2 expandtab: