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. This also means that sometimes top level
156 function will be used when top level binder is really meant.
158 As an example, consider the following Haskell snippet:
167 Here, \hs{foo} is a top level binder, whereas \hs{inc} is a
168 function (since it is bound to a lambda extraction, indicated by
169 the backslash) but is not a top level binder or function. Since
170 the type of \hs{foo} is a function type, namely \hs{Int -> Int},
171 it is also a top level function.
175 Although describing components and connections allows us to describe a lot of
176 hardware designs already, there is an obvious thing missing: choice. We
177 need some way to be able to choose between values based on another value.
178 In Haskell, choice is achieved by \hs{case} expressions, \hs{if}
179 expressions, pattern matching and guards.
181 An obvious way to add choice to our language without having to recognize
182 any of Haskell's syntax, would be to add a primivite \quote{\hs{if}}
183 function. This function would take three arguments: the condition, the
184 value to return when the condition is true and the value to return when
185 the condition is false.
187 This \hs{if} function would then essentially describe a multiplexer and
188 allows us to describe any architecture that uses multiplexers.
190 However, to be able to describe our hardware in a more convenient way, we
191 also want to translate Haskell's choice mechanisms. The easiest of these
192 are of course case expressions (and \hs{if} expressions, which can be very
193 directly translated to \hs{case} expressions). A \hs{case} expression can in turn
194 simply be translated to a conditional assignment, where the conditions use
195 equality comparisons against the constructors in the \hs{case} expressions.
197 In \in{example}[ex:Inv] two versions of an inverter are shown. The first
198 uses a simple \hs{case} expression, scrutinizing a boolean value. The
199 corresponding architecture has a comparator to determine which of the
200 constructors is on the \hs{in} input. There is a multiplexer to select the
201 output signal (which is just a conditional assignment in the generated
202 \VHDL). The two options for the output signals are just constants,
203 but these could have been more complex expressions (in which case also
204 both of them would be working in parallel, regardless of which output
205 would be chosen eventually). The \VHDL\ generated for (both versions of)
206 this inverter is shown in \in{example}[ex:InvVHDL].
208 If we would translate a Boolean to a bit value, we could of course remove
209 the comparator and directly feed 'in' into the multiplexer (or even use an
210 inverter instead of a multiplexer). However, we will try to make a
211 general translation, which works for all possible \hs{case} expressions.
212 Optimizations such as these are left for the \VHDL\ synthesizer, which
213 handles them very well.
216 \startframedtext[width=8cm,background=box,frame=no]
217 \startalignment[center]
218 {\tfa Arguments / results vs. inputs / outputs}
221 Due to the translation chosen for function application, there is a
222 very strong relation between arguments, results, inputs and outputs.
223 For clarity, the former two will always refer to the arguments and
224 results in the functional description (either Haskell or Core). The
225 latter two will refer to input and output ports in the generated
228 Even though these concepts seem to be nearly identical, when stateful
229 functions are introduces we will see arguments and results that will
230 not get translated into input and output ports, making this
231 distinction more important.
235 A slightly more complex (but very powerful) form of choice is pattern
236 matching. A function can be defined in multiple clauses, where each clause
237 specifies a pattern. When the arguments match the pattern, the
238 corresponding clause will be used.
240 \in{Example}[ex:Inv] also shows an inverter that uses pattern matching.
241 The architecture it describes is of course the
242 same one as the description with a case expression. The general interpretation
243 of pattern matching is also similar to that of \hs{case} expressions: generate
244 hardware for each of the clauses (like each of the clauses of a \hs{case}
245 expression) and connect them to the function output through (a number of
246 nested) multiplexers. These multiplexers are driven by comparators and
247 other logic, that check each pattern in turn.
249 In these examples we have seen only binary case expressions and pattern
250 matches (\ie, with two alternatives). In practice, case expressions can
251 choose between more than two values, resulting in a number of nested
254 \startbuffer[CaseInv]
261 \startbuffer[PatternInv]
267 \startuseMPgraphic{Inv}
268 save in, truecmp, falseout, trueout, out, cmp, mux;
271 newCircle.in(btex $in$ etex) "framed(false)";
272 newCircle.out(btex $out$ etex) "framed(false)";
274 newBox.truecmp(btex $True$ etex) "framed(false)";
275 newBox.trueout(btex $True$ etex) "framed(false)";
276 newBox.falseout(btex $False$ etex) "framed(false)";
279 newCircle.cmp(btex $==$ etex);
283 cmp.c = in.c + (3cm, 0cm);
284 truecmp.c = cmp.c + (-1cm, 1cm);
285 mux.sel = cmp.e + (1cm, -1cm);
286 falseout.c = mux.inpa - (2cm, 0cm);
287 trueout.c = mux.inpb - (2cm, 0cm);
288 out.c = mux.out + (2cm, 0cm);
290 % Draw objects and lines
291 drawObj(in, out, truecmp, trueout, falseout, cmp, mux);
295 nccurve(cmp.e)(mux.sel) "angleA(0)", "angleB(-90)";
296 ncline(falseout)(mux) "posB(inpa)";
297 ncline(trueout)(mux) "posB(inpb)";
298 ncline(mux)(out) "posA(out)";
301 \startbuffer[InvVHDL]
302 entity invComponent_0 is
303 port (\xzAMo2\ : in boolean;
304 \reszAMuzAMu2\ : out boolean;
305 clock : in std_logic;
306 resetn : in std_logic);
307 end entity invComponent_0;
310 architecture structural of invComponent_0 is
312 \reszAMuzAMu2\ <= false when \xzAMo2\ = true else
314 end architecture structural;
317 \placeexample[][ex:Inv]{Simple inverter.}{
318 % Use placesidebyside, since nesting combinations doesn't seem to work
319 % here. This does break centering, but well...
321 % Use 2*2 instead of 1*2 to insert some extra space (\placesidebyside
322 % places stuff very close together)
323 {\startcombination[2*2]
324 {\typebufferhs{CaseInv}}{Haskell description using a Case expression.}
326 {\typebufferhs{PatternInv}}{Haskell description using Pattern matching expression.}
329 % Use a 1*1 combination to add a caption
330 {\startcombination[1*1]
331 {\boxedgraphic{Inv}}{The architecture described by the Haskell descriptions.}
335 % \placeexample[][ex:Inv]{Simple inverter.}{
336 % \startcombination[2*2]
337 % {\typebufferhs{CaseInv}}{Haskell description using a Case expression.}
339 % {\typebufferhs{PatternInv}}{Haskell description using Pattern matching expression.}
340 % {\boxedgraphic{Inv}}{The architecture described by the Haskell description.}
343 \placeexample[][ex:InvVHDL]{\VHDL\ generated for (both versions of) \hs{inv} from \in{example}[ex:Inv]}
344 {\typebuffervhdl{InvVHDL}}
347 Translation of two most basic functional concepts has been
348 discussed: function application and choice. Before looking further
349 into less obvious concepts like higher-order expressions and
350 polymorphism, the possible types that can be used in hardware
351 descriptions will be discussed.
353 Some way is needed to translate every values used to its hardware
354 equivalents. In particular, this means a hardware equivalent for
355 every \emph{type} used in a hardware description is needed
357 Since most functional languages have a lot of standard types that
358 are hard to translate (integers without a fixed size, lists without
359 a static length, etc.), a number of \quote{built-in} types will be
360 defined first. These types are built-in in the sense that our
361 compiler will have a fixed VHDL type for these. User defined types,
362 on the other hand, will have their hardware type derived directly
363 from their Haskell declaration automatically, according to the rules
366 \todo{Introduce Haskell type syntax (type constructors, type application,
369 \subsection{Built-in types}
370 The language currently supports the following built-in types. Of these,
371 only the \hs{Bool} type is supported by Haskell out of the box (the
372 others are defined by the Cλash package, so they are user-defined types
373 from Haskell's point of view).
376 This is the most basic type available. It is mapped directly onto
377 the \type{std_logic} \small{VHDL} type. Mapping this to the
378 \type{bit} type might make more sense (since the Haskell version
379 only has two values), but using \type{std_logic} is more standard
380 (and allowed for some experimentation with don't care values)
382 \todo{Sidenote bit vs stdlogic}
384 \startdesc{\hs{Bool}}
385 This is the only built-in Haskell type supported and is translated
386 exactly like the Bit type (where a value of \hs{True} corresponds to a
387 value of \hs{High}). Supporting the Bool type is particularly
388 useful to support \hs{if ... then ... else ...} expressions, which
389 always have a \hs{Bool} value for the condition.
391 A \hs{Bool} is translated to a \type{std_logic}, just like \hs{Bit}.
393 \startdesc{\hs{SizedWord}, \hs{SizedInt}}
394 These are types to represent integers. A \hs{SizedWord} is unsigned,
395 while a \hs{SizedInt} is signed. These types are parameterized by a
396 length type, so you can define an unsigned word of 32 bits wide as
400 type Word32 = SizedWord D32
403 Here, a type synonym \hs{Word32} is defined that is equal to the
404 \hs{SizedWord} type constructor applied to the type \hs{D32}. \hs{D32}
405 is the \emph{type level representation} of the decimal number 32,
406 making the \hs{Word32} type a 32-bit unsigned word.
408 These types are translated to the \small{VHDL} \type{unsigned} and
409 \type{signed} respectively.
410 \todo{Sidenote on dependent typing?}
412 \startdesc{\hs{Vector}}
413 This is a vector type, that can contain elements of any other type and
414 has a fixed length. It has two type parameters: its
415 length and the type of the elements contained in it. By putting the
416 length parameter in the type, the length of a vector can be determined
417 at compile time, instead of only at runtime for conventional lists.
419 The \hs{Vector} type constructor takes two type arguments: the length
420 of the vector and the type of the elements contained in it. The state
421 type of an 8 element register bank would then for example be:
424 type RegisterState = Vector D8 Word32
427 Here, a type synonym \hs{RegisterState} is defined that is equal to
428 the \hs{Vector} type constructor applied to the types \hs{D8} (The type
429 level representation of the decimal number 8) and \hs{Word32} (The 32
430 bit word type as defined above). In other words, the
431 \hs{RegisterState} type is a vector of 8 32-bit words.
433 A fixed size vector is translated to a \small{VHDL} array type.
435 \startdesc{\hs{RangedWord}}
436 This is another type to describe integers, but unlike the previous
437 two it has no specific bitwidth, but an upper bound. This means that
438 its range is not limited to powers of two, but can be any number.
439 A \hs{RangedWord} only has an upper bound, its lower bound is
440 implicitly zero. There is a lot of added implementation complexity
441 when adding a lower bound and having just an upper bound was enough
442 for the primary purpose of this type: typesafely indexing vectors.
444 To define an index for the 8 element vector above, we would do:
447 type RegisterIndex = RangedWord D7
450 Here, a type synonym \hs{RegisterIndex} is defined that is equal to
451 the \hs{RangedWord} type constructor applied to the type \hs{D7}. In
452 other words, this defines an unsigned word with values from
453 {\definedfont[Serif*normalnum]0 to 7} (inclusive). This word can be be used to index the
454 8 element vector \hs{RegisterState} above.
456 This type is translated to the \type{unsigned} \small{VHDL} type.
459 The integer and vector built-in types are discussed in more detail
462 \subsection{User-defined types}
463 There are three ways to define new types in Haskell: algebraic
464 datatypes with the \hs{data} keyword, type synonyms with the \hs{type}
465 keyword and type renamings with the \hs{newtype} keyword. \GHC\
466 offers a few more advanced ways to introduce types (type families,
467 existential typing, \small{GADT}s, etc.) which are not standard
468 Haskell. These will be left outside the scope of this research.
470 Only an algebraic datatype declaration actually introduces a
471 completely new type, for which we provide the \VHDL\ translation
472 below. Type synonyms and renamings only define new names for
473 existing types (where synonyms are completely interchangeable and
474 renamings need explicit conversion). Therefore, these do not need
475 any particular \VHDL\ translation, a synonym or renamed type will
476 just use the same representation as the original type. The
477 distinction between a renaming and a synonym does no longer matter
478 in hardware and can be disregarded in the generated \VHDL.
480 For algebraic types, we can make the following distinction:
482 \startdesc{Product types}
483 A product type is an algebraic datatype with a single constructor with
484 two or more fields, denoted in practice like (a,b), (a,b,c), etc. This
485 is essentially a way to pack a few values together in a record-like
486 structure. In fact, the built-in tuple types are just algebraic product
487 types (and are thus supported in exactly the same way).
489 The \quote{product} in its name refers to the collection of values belonging
490 to this type. The collection for a product type is the Cartesian
491 product of the collections for the types of its fields.
493 These types are translated to \VHDL\ record types, with one field for
494 every field in the constructor. This translation applies to all single
495 constructor algebraic datatypes, including those with just one
496 field (which are technically not a product, but generate a VHDL
497 record for implementation simplicity).
499 \startdesc{Enumerated types}
500 \defref{enumerated types}
501 An enumerated type is an algebraic datatype with multiple constructors, but
502 none of them have fields. This is essentially a way to get an
503 enum-like type containing alternatives.
505 Note that Haskell's \hs{Bool} type is also defined as an
506 enumeration type, but we have a fixed translation for that.
508 These types are translated to \VHDL\ enumerations, with one value for
509 each constructor. This allows references to these constructors to be
510 translated to the corresponding enumeration value.
512 \startdesc{Sum types}
513 A sum type is an algebraic datatype with multiple constructors, where
514 the constructors have one or more fields. Technically, a type with
515 more than one field per constructor is a sum of products type, but
516 for our purposes this distinction does not really make a
517 difference, so this distinction is note made.
519 The \quote{sum} in its name refers again to the collection of values
520 belonging to this type. The collection for a sum type is the
521 union of the the collections for each of the constructors.
523 Sum types are currently not supported by the prototype, since there is
524 no obvious \VHDL\ alternative. They can easily be emulated, however, as
525 we will see from an example:
528 data Sum = A Bit Word | B Word
531 An obvious way to translate this would be to create an enumeration to
532 distinguish the constructors and then create a big record that
533 contains all the fields of all the constructors. This is the same
534 translation that would result from the following enumeration and
535 product type (using a tuple for clarity):
539 type Sum = (SumC, Bit, Word, Word)
542 Here, the \hs{SumC} type effectively signals which of the latter three
543 fields of the \hs{Sum} type are valid (the first two if \hs{A}, the
544 last one if \hs{B}), all the other ones have no useful value.
546 An obvious problem with this naive approach is the space usage: the
547 example above generates a fairly big \VHDL\ type. Since we can be
548 sure that the two \hs{Word}s in the \hs{Sum} type will never be valid
549 at the same time, this is a waste of space.
551 Obviously, duplication detection could be used to reuse a
552 particular field for another constructor, but this would only
553 partially solve the problem. If two fields would be, for
554 example, an array of 8 bits and an 8 bit unsiged word, these are
555 different types and could not be shared. However, in the final
556 hardware, both of these types would simply be 8 bit connections,
557 so we have a 100\% size increase by not sharing these.
560 Another interesting case is that of recursive types. In Haskell, an
561 algebraic datatype can be recursive: any of its field types can be (or
562 contain) the type being defined. The most well-known recursive type is
563 probably the list type, which is defined is:
566 data List t = Empty | Cons t (List t)
569 Note that \hs{Empty} is usually written as \hs{[]} and \hs{Cons} as
570 \hs{:}, but this would make the definition harder to read. This
571 immediately shows the problem with recursive types: what hardware type
574 If the naive approach for sum types described above would be used,
575 a record would be created where the first field is an enumeration
576 to distinguish \hs{Empty} from \hs{Cons}. Furthermore, two more
577 fields would be added: one with the (\VHDL\ equivalent of) type
578 \hs{t} (assuming this type is actually known at compile time, this
579 should not be a problem) and a second one with type \hs{List t}.
580 The latter one is of course a problem: this is exactly the type
581 that was to be translated in the first place.
583 The resulting \VHDL\ type will thus become infinitely deep. In
584 other words, there is no way to statically determine how long
585 (deep) the list will be (it could even be infinite).
587 In general, recursive types can never be properly translated: all
588 recursive types have a potentially infinite value (even though in
589 practice they will have a bounded value, there is no way for the
590 compiler to automatically determine an upper bound on its size).
592 \subsection{Partial application}
593 Now the translation of application, choice and types has been
594 discussed, a more complex concept can be considered: partial
595 applications. A \emph{partial application} is any application whose
596 (return) type is (again) a function type.
598 From this, it should be clear that the translation rules for full
599 application does not apply to a partial application: there are not
600 enough values for all the input ports in the resulting \VHDL.
601 \in{Example}[ex:Quadruple] shows an example use of partial application
602 and the corresponding architecture.
604 \startbuffer[Quadruple]
605 -- Multiply the input word by four.
606 quadruple :: Word -> Word
607 quadruple n = mul (mul n)
612 \startuseMPgraphic{Quadruple}
613 save in, two, mula, mulb, out;
616 newCircle.in(btex $n$ etex) "framed(false)";
617 newCircle.two(btex $2$ etex) "framed(false)";
618 newCircle.out(btex $out$ etex) "framed(false)";
621 newCircle.mula(btex $\times$ etex);
622 newCircle.mulb(btex $\times$ etex);
625 in.c = two.c + (0cm, 1cm);
626 mula.c = in.c + (2cm, 0cm);
627 mulb.c = mula.c + (2cm, 0cm);
628 out.c = mulb.c + (2cm, 0cm);
630 % Draw objects and lines
631 drawObj(in, two, mula, mulb, out);
633 nccurve(two)(mula) "angleA(0)", "angleB(45)";
634 nccurve(two)(mulb) "angleA(0)", "angleB(45)";
640 \placeexample[][ex:Quadruple]{Simple three port and.}
641 \startcombination[2*1]
642 {\typebufferhs{Quadruple}}{Haskell description using function applications.}
643 {\boxedgraphic{Quadruple}}{The architecture described by the Haskell description.}
646 Here, the definition of mul is a partial function application: it applies
647 the function \hs{(*) :: Word -> Word -> Word} to the value \hs{2 :: Word},
648 resulting in the expression \hs{(*) 2 :: Word -> Word}. Since this resulting
649 expression is again a function, hardware cannot be generated for it
650 directly. This is because the hardware to generate for \hs{mul}
651 depends completely on where and how it is used. In this example, it is
654 However, it is clear that the above hardware description actually
655 describes valid hardware. In general, any partial applied function
656 must eventually become completely applied, at which point hardware for
657 it can be generated using the rules for function application given in
658 \in{section}[sec:description:application]. It might mean that a
659 partial application is passed around quite a bit (even beyond function
660 boundaries), but eventually, the partial application will become
661 completely applied. An example of this principe is given in
662 \in{section}[sec:normalization:defunctionalization].
664 \section{Costless specialization}
665 Each (complete) function application in our description generates a
666 component instantiation, or a specific piece of hardware in the final
667 design. It is interesting to note that each application of a function
668 generates a \emph{separate} piece of hardware. In the final design, none
669 of the hardware is shared between applications, even when the applied
670 function is the same (of course, if a particular value, such as the result
671 of a function application, is used twice, it is not calculated twice).
673 This is distinctly different from normal program compilation: two separate
674 calls to the same function share the same machine code. Having more
675 machine code has implications for speed (due to less efficient caching)
676 and memory usage. For normal compilation, it is therefore important to
677 keep the amount of functions limited and maximize the code sharing
678 (though there is a tradeoff between speed and memory usage here).
680 When generating hardware, this is hardly an issue. Having more \quote{code
681 sharing} does reduce the amount of \small{VHDL} output (Since different
682 component instantiations still share the same component), but after
683 synthesis, the amount of hardware generated is not affected. This
684 means there is no tradeoff between speed and memory (or rather,
687 In particular, if we would duplicate all functions so that there is a
688 separate function for every application in the program (\eg, each function
689 is then only applied exactly once), there would be no increase in hardware
692 Because of this, a common optimization technique called
693 \emph{specialization} can be applied to hardware generation without any
694 performance or area cost (unlike for software).
696 \fxnote{Perhaps these next three sections are a bit too
697 implementation-oriented?}
699 \subsection{Specialization}
700 \defref{specialization}
701 Given some function that has a \emph{domain} $D$ (\eg, the set of
702 all possible arguments that the function could be applied to), we
703 create a specialized function with exactly the same behaviour, but
704 with a domain $D' \subset D$. This subset can be chosen in all
705 sorts of ways. Any subset is valid for the general definition of
706 specialization, but in practice only some of them provide useful
707 optimization opportunities.
709 Common subsets include limiting a polymorphic argument to a single type
710 (\ie, removing polymorphism) or limiting an argument to just a single
711 value (\ie, cross-function constant propagation, effectively removing
714 Since we limit the argument domain of the specialized function, its
715 definition can often be optimized further (since now more types or even
716 values of arguments are already known). By replacing any application of
717 the function that falls within the reduced domain by an application of
718 the specialized version, the code gets faster (but the code also gets
719 bigger, since we now have two versions instead of one). If we apply
720 this technique often enough, we can often replace all applications of a
721 function by specialized versions, allowing the original function to be
722 removed (in some cases, this can even give a net reduction of the code
723 compared to the non-specialized version).
725 Specialization is useful for our hardware descriptions for functions
726 that contain arguments that cannot be translated to hardware directly
727 (polymorphic or higher-order arguments, for example). If we can create
728 specialized functions that remove the argument, or make it translatable,
729 we can use specialization to make the original, untranslatable, function
732 \section{Higher order values}
733 What holds for partial application, can be easily generalized to any
734 higher-order expression. This includes partial applications, plain
735 variables (e.g., a binder referring to a top level function), lambda
736 expressions and more complex expressions with a function type (a \hs{case}
737 expression returning lambda's, for example).
739 Each of these values cannot be directly represented in hardware (just like
740 partial applications). Also, to make them representable, they need to be
741 applied: function variables and partial applications will then eventually
742 become complete applications, applied lambda expressions disappear by
743 applying β-reduction, etc.
745 So any higher-order value will be \quote{pushed down} towards its
746 application just like partial applications. Whenever a function boundary
747 needs to be crossed, the called function can be specialized.
749 \fxnote{This section needs improvement and an example}
751 \section{Polymorphism}
752 In Haskell, values can be \emph{polymorphic}: they can have multiple types. For
753 example, the function \hs{fst :: (a, b) -> a} is an example of a
754 polymorphic function: it works for tuples with any two element types. Haskell
755 type classes allow a function to work on a specific set of types, but the
756 general idea is the same. The opposite of this is a \emph{monomorphic}
757 value, which has a single, fixed, type.
759 % A type class is a collection of types for which some operations are
760 % defined. It is thus possible for a value to be polymorphic while having
761 % any number of \emph{class constraints}: the value is not defined for
762 % every type, but only for types in the type class. An example of this is
763 % the \hs{even :: (Integral a) => a -> Bool} function, which can map any
764 % value of a type that is member of the \hs{Integral} type class
766 When generating hardware, polymorphism cannot be easily translated. How
767 many wires will you lay down for a value that could have any type? When
768 type classes are involved, what hardware components will you lay down for
769 a class method (whose behaviour depends on the type of its arguments)?
770 Note that Cλash currently does not allow user-defined type classes,
771 but does partly support some of the built-in type classes (like \hs{Num}).
773 Fortunately, we can again use the principle of specialization: since every
774 function application generates a separate piece of hardware, we can know
775 the types of all arguments exactly. Provided that existential typing
776 (which is a \GHC\ extension) is not used typing, all of the
777 polymorphic types in a function must depend on the types of the
778 arguments (In other words, the only way to introduce a type variable
779 is in a lambda abstraction).
781 If a function is monomorphic, all values inside it are monomorphic as
782 well, so any function that is applied within the function can only be
783 applied to monomorphic values. The applied functions can then be
784 specialized to work just for these specific types, removing the
785 polymorphism from the applied functions as well.
787 \defref{entry function}The entry function must not have a
788 polymorphic type (otherwise no hardware interface could be generated
789 for the entry function).
791 By induction, this means that all functions that are (indirectly) called
792 by our top level function (meaning all functions that are translated in
793 the final hardware) become monomorphic.
796 A very important concept in hardware designs is \emph{state}. In a
797 stateless (or, \emph{combinational}) design, every output is directly and solely dependent on the
798 inputs. In a stateful design, the outputs can depend on the history of
799 inputs, or the \emph{state}. State is usually stored in \emph{registers},
800 which retain their value during a clockcycle, and are typically updated at
801 the start of every clockcycle. Since the updating of the state is tightly
802 coupled (synchronized) to the clock signal, these state updates are often
803 called \emph{synchronous} behaviour.
805 \todo{Sidenote? Registers can contain any (complex) type}
807 To make Cλash useful to describe more than simple combinational
808 designs, it needs to be able to describe state in some way.
810 \subsection{Approaches to state}
811 In Haskell, functions are always pure (except when using unsafe
812 functions with the \hs{IO} monad, which is not supported by Cλash). This
813 means that the output of a function solely depends on its inputs. If you
814 evaluate a given function with given inputs, it will always provide the
819 \startframedtext[width=8cm,background=box,frame=no]
820 \startalignment[center]
825 A function is said to be pure if it satisfies two conditions:
828 \item When a pure function is called with the same arguments twice, it should
829 return the same value in both cases.
830 \item When a pure function is called, it should have not
831 observable side-effects.
834 Purity is an important property in functional languages, since
835 it enables all kinds of mathematical reasoning and
836 optimizattions with pure functions, that are not guaranteed to
837 be correct for impure functions.
839 An example of a pure function is the square root function
840 \hs{sqrt}. An example of an impure function is the \hs{today}
841 function that returns the current date (which of course cannot
842 exist like this in Haskell).
846 This is a perfect match for a combinational circuit, where the output
847 also solely depends on the inputs. However, when state is involved, this
848 no longer holds. Of course this purity constraint cannot just be
849 removed from Haskell. But even when designing a completely new (hardware
850 description) language, it does not seem to be a good idea to
851 remove this purity. This would that all kinds of interesting properties of
852 the functional language get lost, and all kinds of transformations
853 and optimizations are no longer be meaning preserving.
855 So our functions must remain pure, meaning the current state has
856 to be present in the function's arguments in some way. There seem
857 to be two obvious ways to do this: adding the current state as an
858 argument, or including the full history of each argument.
860 \subsubsection{Stream arguments and results}
861 Including the entire history of each input (\eg, the value of that
862 input for each previous clockcycle) is an obvious way to make outputs
863 depend on all previous input. This is easily done by making every
864 input a list instead of a single value, containing all previous values
865 as well as the current value.
867 An obvious downside of this solution is that on each cycle, all the
868 previous cycles must be resimulated to obtain the current state. To do
869 this, it might be needed to have a recursive helper function as well,
870 which might be hard to be properly analyzed by the compiler.
872 A slight variation on this approach is one taken by some of the other
873 functional \small{HDL}s in the field: \todo{References to Lava,
874 ForSyDe, ...} Make functions operate on complete streams. This means
875 that a function is no longer called on every cycle, but just once. It
876 takes stream as inputs instead of values, where each stream contains
877 all the values for every clockcycle since system start. This is easily
878 modeled using an (infinite) list, with one element for each clock
879 cycle. Since the function is only evaluated once, its output must also
880 be a stream. Note that, since we are working with infinite lists and
881 still want to be able to simulate the system cycle-by-cycle, this
882 relies heavily on the lazy semantics of Haskell.
884 Since our inputs and outputs are streams, all other (intermediate)
885 values must be streams. All of our primitive operators (\eg, addition,
886 substraction, bitwise operations, etc.) must operate on streams as
887 well (note that changing a single-element operation to a stream
888 operation can done with \hs{map}, \hs{zipwith}, etc.).
890 This also means that primitive operations from an existing
891 language such as Haskell cannot be used directly (including some
892 syntax constructs, like \hs{case} expressions and \hs{if}
893 expressions). This mkes this approach well suited for use in
894 \small{EDSL}s, since those already impose these same
895 limitations. \refdef{EDSL}
897 Note that the concept of \emph{state} is no more than having some way
898 to communicate a value from one cycle to the next. By introducing a
899 \hs{delay} function, we can do exactly that: delay (each value in) a
900 stream so that we can "look into" the past. This \hs{delay} function
901 simply outputs a stream where each value is the same as the input
902 value, but shifted one cycle. This causes a \quote{gap} at the
903 beginning of the stream: what is the value of the delay output in the
904 first cycle? For this, the \hs{delay} function has a second input, of
905 which only a single value is used.
907 \in{Example}[ex:DelayAcc] shows a simple accumulator expressed in this
910 \startbuffer[DelayAcc]
911 acc :: Stream Word -> Stream Word
914 out = (delay out 0) + in
917 \startuseMPgraphic{DelayAcc}
918 save in, out, add, reg;
921 newCircle.in(btex $in$ etex) "framed(false)";
922 newCircle.out(btex $out$ etex) "framed(false)";
925 newReg.reg("") "dx(4mm)", "dy(6mm)", "reflect(true)";
926 newCircle.add(btex + etex);
929 add.c = in.c + (2cm, 0cm);
930 out.c = add.c + (2cm, 0cm);
931 reg.c = add.c + (0cm, 2cm);
933 % Draw objects and lines
934 drawObj(in, out, add, reg);
936 nccurve(add)(reg) "angleA(0)", "angleB(180)", "posB(d)";
937 nccurve(reg)(add) "angleA(180)", "angleB(-45)", "posA(out)";
943 \placeexample[][ex:DelayAcc]{Simple accumulator architecture.}
944 \startcombination[2*1]
945 {\typebufferhs{DelayAcc}}{Haskell description using streams.}
946 {\boxedgraphic{DelayAcc}}{The architecture described by the Haskell description.}
950 This notation can be confusing (especially due to the loop in the
951 definition of out), but is essentially easy to interpret. There is a
952 single call to delay, resulting in a circuit with a single register,
953 whose input is connected to \hs{out} (which is the output of the
954 adder), and its output is the expression \hs{delay out 0} (which is
955 connected to one of the adder inputs).
957 \subsubsection{Explicit state arguments and results}
958 A more explicit way to model state, is to simply add an extra argument
959 containing the current state value. This allows an output to depend on
960 both the inputs as well as the current state while keeping the
961 function pure (letting the result depend only on the arguments), since
962 the current state is now an argument.
964 In Haskell, this would look like
965 \in{example}[ex:ExplicitAcc]\footnote[notfinalsyntax]{This
966 example is not in the final Cλash syntax}. \todo{Referencing
967 notfinalsyntax from Introduction.tex doesn't work}
969 \startbuffer[ExplicitAcc]
970 -- input -> current state -> (new state, output)
971 acc :: Word -> Word -> (Word, Word)
978 \placeexample[][ex:ExplicitAcc]{Simple accumulator architecture.}
979 \startcombination[2*1]
980 {\typebufferhs{ExplicitAcc}}{Haskell description using explicit state arguments.}
981 % Picture is identical to the one we had just now.
982 {\boxedgraphic{DelayAcc}}{The architecture described by the Haskell description.}
985 This approach makes a function's state very explicit, which state
986 variables are used by a function can be completely determined from its
987 type signature (as opposed to the stream approach, where a function
988 looks the same from the outside, regardless of what state variables it
989 uses or whether it is stateful at all).
991 This approach to state has been one of the initial drives behind
992 this research. Unlike a stream based approach it is well suited
993 to completely use existing code and language features (like
994 \hs{if} and \hs{case} expressions) because it operates on normal
995 values. Because of these reasons, this is the approach chosen
996 for Cλash. It will be examined more closely below.
998 \subsection{Explicit state specification}
999 The concept of explicit state has been introduced with some
1000 examples above, but what are the implications of this approach?
1002 \subsubsection{Substates}
1003 Since a function's state is reflected directly in its type signature,
1004 if a function calls other stateful functions (\eg, has subcircuits), it
1005 has to somehow know the current state for these called functions. The
1006 only way to do this, is to put these \emph{substates} inside the
1007 caller's state. This means that a function's state is the sum of the
1008 states of all functions it calls, and its own state. This sum
1009 can be obtained using something simple like a tuple, or possibly
1010 custom algebraic types for clarity.
1012 This also means that the type of a function (at least the "state"
1013 part) is dependent on its own implementation and of the functions it
1016 This is the major downside of this approach: the separation between
1017 interface and implementation is limited. However, since Cλash is not
1018 very suitable for separate compilation (see
1019 \in{section}[sec:prototype:separate]) this is not a big problem in
1022 Additionally, when using a type synonym for the state type
1023 of each function, we can still provide explicit type signatures
1024 while keeping the state specification for a function near its
1028 \subsubsection{Which arguments and results are stateful?}
1029 \fxnote{This section should get some examples}
1030 We need some way to know which arguments should become input ports and
1031 which argument(s?) should become the current state (\eg, be bound to
1032 the register outputs). This does not hold just for the top
1033 level function, but also for any subfunction. Or could we perhaps
1034 deduce the statefulness of subfunctions by analyzing the flow of data
1035 in the calling functions?
1037 To explore this matter, the following observeration is interesting: we
1038 get completely correct behaviour when we put all state registers in
1039 the top level entity (or even outside of it). All of the state
1040 arguments and results on subfunctions are treated as normal input and
1041 output ports. Effectively, a stateful function results in a stateless
1042 hardware component that has one of its input ports connected to the
1043 output of a register and one of its output ports connected to the
1044 input of the same register.
1048 Of course, even though the hardware described like this has the
1049 correct behaviour, unless the layout tool does smart optimizations,
1050 there will be a lot of extra wire in the design (since registers will
1051 not be close to the component that uses them). Also, when working with
1052 the generated \small{VHDL} code, there will be a lot of extra ports
1053 just to pass on state values, which can get quite confusing.
1055 To fix this, we can simply \quote{push} the registers down into the
1056 subcircuits. When we see a register that is connected directly to a
1057 subcircuit, we remove the corresponding input and output port and put
1058 the register inside the subcircuit instead. This is slightly less
1059 trivial when looking at the Haskell code instead of the resulting
1060 circuit, but the idea is still the same.
1064 However, when applying this technique, we might push registers down
1065 too far. When you intend to store a result of a stateless subfunction
1066 in the caller's state and pass the current value of that state
1067 variable to that same function, the register might get pushed down too
1068 far. It is impossible to distinguish this case from similar code where
1069 the called function is in fact stateful. From this we can conclude
1070 that we have to either:
1072 \todo{Example of wrong downpushing}
1075 \item accept that the generated hardware might not be exactly what we
1076 intended, in some specific cases. In most cases, the hardware will be
1078 \item explicitly annotate state arguments and results in the input
1082 The first option causes (non-obvious) exceptions in the language
1083 intepretation. Also, automatically determining where registers should
1084 end up is easier to implement correctly with explicit annotations, so
1085 for these reasons we will look at how this annotations could work.
1087 \todo{Sidenote: one or more state arguments?}
1089 \subsection[sec:description:stateann]{Explicit state annotation}
1090 To make our stateful descriptions unambigious and easier to translate,
1091 we need some way for the developer to describe which arguments and
1092 results are intended to become stateful.
1094 Roughly, we have two ways to achieve this:
1096 \item Use some kind of annotation method or syntactic construction in
1097 the language to indicate exactly which argument and (part of the)
1098 result is stateful. This means that the annotation lives
1099 \quote{outside} of the function, it is completely invisible when
1100 looking at the function body.
1101 \item Use some kind of annotation on the type level, \ie\ give stateful
1102 arguments and stateful (parts of) results a different type. This has the
1103 potential to make this annotation visible inside the function as well,
1104 such that when looking at a value inside the function body you can
1105 tell if it is stateful by looking at its type. This could possibly make
1106 the translation process a lot easier, since less analysis of the
1107 program flow might be required.
1110 From these approaches, the type level \quote{annotations} have been
1111 implemented in Cλash. \in{Section}[sec:prototype:statetype] expands on
1112 the possible ways this could have been implemented.
1114 \todo{Note about conditions on state variables and checking them}
1116 \section[sec:recursion]{Recursion}
1117 An important concept in functional languages is recursion. In its most basic
1118 form, recursion is a definition that is described in terms of itself. A
1119 recursive function is thus a function that uses itself in its body. This
1120 usually requires multiple evaluations of this function, with changing
1121 arguments, until eventually an evaluation of the function no longer requires
1124 Given the notion that each function application will translate to a
1125 component instantiation, we are presented with a problem. A recursive
1126 function would translate to a component that contains itself. Or, more
1127 precisely, that contains an instance of itself. This instance would again
1128 contain an instance of itself, and again, into infinity. This is obviously a
1129 problem for generating hardware.
1131 This is expected for functions that describe infinite recursion. In that
1132 case, we cannot generate hardware that shows correct behaviour in a single
1133 cycle (at best, we could generate hardware that needs an infinite number of
1134 cycles to complete).
1137 \startframedtext[width=8cm,background=box,frame=no]
1138 \startalignment[center]
1139 {\tfa \hs{null}, \hs{head} and \hs{tail}}
1142 The functions \hs{null}, \hs{head} and \hs{tail} are common list
1143 functions in Haskell. The \hs{null} function simply checks if a list is
1144 empty. The \hs{head} function returns the first element of a list. The
1145 \hs{tail} function returns containing everything \emph{except} the first
1148 In Cλash, there are vector versions of these functions, which do exactly
1153 However, most recursive definitions will describe finite
1154 recursion. This is because the recursive call is done conditionally. There
1155 is usually a \hs{case} expression where at least one alternative does not contain
1156 the recursive call, which we call the "base case". If, for each call to the
1157 recursive function, we would be able to detect at compile time which
1158 alternative applies, we would be able to remove the \hs{case} expression and
1159 leave only the base case when it applies. This will ensure that expanding
1160 the recursive functions will terminate after a bounded number of expansions.
1162 This does imply the extra requirement that the base case is detectable at
1163 compile time. In particular, this means that the decision between the base
1164 case and the recursive case must not depend on runtime data.
1166 \subsection{List recursion}
1167 The most common deciding factor in recursion is the length of a list that is
1168 passed in as an argument. Since we represent lists as vectors that encode
1169 the length in the vector type, it seems easy to determine the base case. We
1170 can simply look at the argument type for this. However, it turns out that
1171 this is rather non-trivial to write down in Haskell already, not even
1172 looking at translation. As an example, we would like to write down something
1176 sum :: Vector n Word -> Word
1177 sum xs = case null xs of
1179 False -> head xs + sum (tail xs)
1182 However, the Haskell typechecker will now use the following reasoning.
1183 For simplicity, the element type of a vector is left out, all vectors
1184 are assumed to have the same element type. Below, we write conditions
1185 on type variables before the \hs{=>} operator. This is not completely
1186 valid Haskell syntax, but serves to illustrate the typechecker
1187 reasoning. Also note that a vector can never have a negative length,
1188 so \hs{Vector n} implicitly means \hs{(n >= 0) => Vector n}.
1190 \todo{This typechecker disregards the type signature}
1192 \item tail has the type \hs{(n > 0) => Vector n -> Vector (n - 1)}
1193 \item This means that xs must have the type \hs{(n > 0) => Vector n}
1194 \item This means that sum must have the type \hs{(n > 0) => Vector n -> a}
1195 (The type \hs{a} is can be anything at this stage, we will not try to finds
1196 its actual type in this example).
1197 \item sum is called with the result of tail as an argument, which has the
1198 type \hs{Vector n} (since \hs{(n > 0) => Vector (n - 1)} is the same as \hs{(n >= 0)
1199 => Vector n}, which is the same as just \hs{Vector n}).
1200 \item This means that sum must have the type \hs{Vector n -> a}
1201 \item This is a contradiction between the type deduced from the body of sum
1202 (the input vector must be non-empty) and the use of sum (the input vector
1203 could have any length).
1206 As you can see, using a simple \hs{case} expression at value level causes
1207 the type checker to always typecheck both alternatives, which cannot be
1208 done. The typechecker is unable to distinguish the two case
1209 alternatives (this is partly possible using \small{GADT}s, but that
1210 approach faced other problems \todo{ref christiaan?}).
1212 This is a fundamental problem, that would seem perfectly suited for a
1213 type class. Considering that we need to switch between to
1214 implementations of the sum function, based on the type of the
1215 argument, this sounds like the perfect problem to solve with a type
1216 class. However, this approach has its own problems (not the least of
1217 them that you need to define a new type class for every recursive
1218 function you want to define).
1220 \todo{This should reference Christiaan}
1222 \subsection{General recursion}
1223 Of course there are other forms of recursion, that do not depend on the
1224 length (and thus type) of a list. For example, simple recursion using a
1225 counter could be expressed, but only translated to hardware for a fixed
1226 number of iterations. Also, this would require extensive support for compile
1227 time simplification (constant propagation) and compile time evaluation
1228 (evaluation of constant comparisons), to ensure non-termination.
1229 Supporting general recursion will probably require strict conditions
1230 on the input descriptions. Even then, it will be hard (if not
1231 impossible) to really guarantee termination, since the user (or \GHC\
1232 desugarer) might use some obscure notation that results in a corner
1233 case of the simplifier that is not caught and thus non-termination.
1235 Evaluating all possible (and non-possible) ways to add recursion to
1236 our descriptions, it seems better to limit the scope of this research
1237 to exclude recursion. This allows for focusing on other interesting
1238 areas instead. By including (built-in) support for a number of
1239 higher-order functions like \hs{map} and \hs{fold}, we can still
1240 express most of the things we would use (list) recursion for.
1243 % vim: set sw=2 sts=2 expandtab: