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 \placeexample[][ex:And3]{Simple three input and gate.}
112 \startcombination[2*1]
113 {\typebufferhs{And3}}{Haskell description using function applications.}
114 {\boxedgraphic{And3}}{The architecture described by the Haskell description.}
118 \defref{top level binder}
119 \defref{top level function}
120 \startframedtext[width=8cm,background=box,frame=no]
121 \startalignment[center]
122 {\tfa Top level binders and functions}
125 A top level binder is any binder (variable) that is declared in
126 the \quote{global} scope of a Haskell program (as opposed to a
127 binder that is bound inside a function.
129 In Haskell, there is no sharp distinction between a variable and a
130 function: a function is just a variable (binder) with a function
131 type. This means that a top level function is just any top level
132 binder with a function type.
134 As an example, consider the following Haskell snippet:
143 Here, \hs{foo} is a top level binder, whereas \hs{inc} is a
144 function (since it is bound to a lambda extraction, indicated by
145 the backslash) but is not a top level binder or function. Since
146 the type of \hs{foo} is a function type, namely \hs{Int -> Int},
147 it is also a top level function.
151 Although describing components and connections allows us to describe a lot of
152 hardware designs already, there is an obvious thing missing: choice. We
153 need some way to be able to choose between values based on another value.
154 In Haskell, choice is achieved by \hs{case} expressions, \hs{if}
155 expressions, pattern matching and guards.
157 An obvious way to add choice to our language without having to recognize
158 any of Haskell's syntax, would be to add a primivite \quote{\hs{if}}
159 function. This function would take three arguments: the condition, the
160 value to return when the condition is true and the value to return when
161 the condition is false.
163 This \hs{if} function would then essentially describe a multiplexer and
164 allows us to describe any architecture that uses multiplexers.
166 However, to be able to describe our hardware in a more convenient way, we
167 also want to translate Haskell's choice mechanisms. The easiest of these
168 are of course case expressions (and \hs{if} expressions, which can be very
169 directly translated to \hs{case} expressions). A \hs{case} expression can in turn
170 simply be translated to a conditional assignment, where the conditions use
171 equality comparisons against the constructors in the \hs{case} expressions.
173 In \in{example}[ex:CaseInv] a simple \hs{case} expression is shown,
174 scrutinizing a boolean value. The corresponding architecture has a
175 comparator to determine which of the constructors is on the \hs{in}
176 input. There is a multiplexer to select the output signal. The two options
177 for the output signals are just constants, but these could have been more
178 complex expressions (in which case also both of them would be working in
179 parallel, regardless of which output would be chosen eventually).
181 If we would translate a Boolean to a bit value, we could of course remove
182 the comparator and directly feed 'in' into the multiplexer (or even use an
183 inverter instead of a multiplexer). However, we will try to make a
184 general translation, which works for all possible \hs{case} expressions.
185 Optimizations such as these are left for the \VHDL\ synthesizer, which
186 handles them very well.
189 \startframedtext[width=8cm,background=box,frame=no]
190 \startalignment[center]
191 {\tfa Arguments / results vs. inputs / outputs}
194 Due to the translation chosen for function application, there is a
195 very strong relation between arguments, results, inputs and outputs.
196 For clarity, the former two will always refer to the arguments and
197 results in the functional description (either Haskell or Core). The
198 latter two will refer to input and output ports in the generated
201 Even though these concepts seem to be nearly identical, when stateful
202 functions are introduces we will see arguments and results that will
203 not get translated into input and output ports, making this
204 distinction more important.
208 A slightly more complex (but very powerful) form of choice is pattern
209 matching. A function can be defined in multiple clauses, where each clause
210 specifies a pattern. When the arguments match the pattern, the
211 corresponding clause will be used.
213 The architecture described by \in{example}[ex:PatternInv] is of course the
214 same one as the one in \in{example}[ex:CaseInv]. The general interpretation
215 of pattern matching is also similar to that of \hs{case} expressions: generate
216 hardware for each of the clauses (like each of the clauses of a \hs{case}
217 expression) and connect them to the function output through (a number of
218 nested) multiplexers. These multiplexers are driven by comparators and
219 other logic, that check each pattern in turn.
221 In these examples we have seen only binary case expressions and pattern
222 matches (\ie, with two alternatives). In practice, case expressions can
223 choose between more than two values, resulting in a number of nested
226 \startbuffer[CaseInv]
233 \startuseMPgraphic{CaseInv}
234 save in, truecmp, falseout, trueout, out, cmp, mux;
237 newCircle.in(btex $in$ etex) "framed(false)";
238 newCircle.out(btex $out$ etex) "framed(false)";
240 newBox.truecmp(btex $True$ etex) "framed(false)";
241 newBox.trueout(btex $True$ etex) "framed(false)";
242 newBox.falseout(btex $False$ etex) "framed(false)";
245 newCircle.cmp(btex $==$ etex);
249 cmp.c = in.c + (3cm, 0cm);
250 truecmp.c = cmp.c + (-1cm, 1cm);
251 mux.sel = cmp.e + (1cm, -1cm);
252 falseout.c = mux.inpa - (2cm, 0cm);
253 trueout.c = mux.inpb - (2cm, 0cm);
254 out.c = mux.out + (2cm, 0cm);
256 % Draw objects and lines
257 drawObj(in, out, truecmp, trueout, falseout, cmp, mux);
261 nccurve(cmp.e)(mux.sel) "angleA(0)", "angleB(-90)";
262 ncline(falseout)(mux) "posB(inpa)";
263 ncline(trueout)(mux) "posB(inpb)";
264 ncline(mux)(out) "posA(out)";
267 \placeexample[][ex:CaseInv]{Simple inverter.}
268 \startcombination[2*1]
269 {\typebufferhs{CaseInv}}{Haskell description using a Case expression.}
270 {\boxedgraphic{CaseInv}}{The architecture described by the Haskell description.}
273 \startbuffer[PatternInv]
279 \placeexample[][ex:PatternInv]{Simple inverter using pattern matching.
280 Describes the same architecture as \in{example}[ex:CaseInv].}
281 {\typebufferhs{PatternInv}}
284 Translation of two most basic functional concepts has been
285 discussed: function application and choice. Before looking further
286 into less obvious concepts like higher-order expressions and
287 polymorphism, the possible types that can be used in hardware
288 descriptions will be discussed.
290 Some way is needed to translate every values used to its hardware
291 equivalents. In particular, this means a hardware equivalent for
292 every \emph{type} used in a hardware description is needed
294 Since most functional languages have a lot of standard types that
295 are hard to translate (integers without a fixed size, lists without
296 a static length, etc.), a number of \quote{built-in} types will be
297 defined first. These types are built-in in the sense that our
298 compiler will have a fixed VHDL type for these. User defined types,
299 on the other hand, will have their hardware type derived directly
300 from their Haskell declaration automatically, according to the rules
303 \todo{Introduce Haskell type syntax (type constructors, type application,
306 \subsection{Built-in types}
307 The language currently supports the following built-in types. Of these,
308 only the \hs{Bool} type is supported by Haskell out of the box (the
309 others are defined by the Cλash package, so they are user-defined types
310 from Haskell's point of view).
313 This is the most basic type available. It is mapped directly onto
314 the \type{std_logic} \small{VHDL} type. Mapping this to the
315 \type{bit} type might make more sense (since the Haskell version
316 only has two values), but using \type{std_logic} is more standard
317 (and allowed for some experimentation with don't care values)
319 \todo{Sidenote bit vs stdlogic}
321 \startdesc{\hs{Bool}}
322 This is the only built-in Haskell type supported and is translated
323 exactly like the Bit type (where a value of \hs{True} corresponds to a
324 value of \hs{High}). Supporting the Bool type is particularly
325 useful to support \hs{if ... then ... else ...} expressions, which
326 always have a \hs{Bool} value for the condition.
328 A \hs{Bool} is translated to a \type{std_logic}, just like \hs{Bit}.
330 \startdesc{\hs{SizedWord}, \hs{SizedInt}}
331 These are types to represent integers. A \hs{SizedWord} is unsigned,
332 while a \hs{SizedInt} is signed. These types are parameterized by a
333 length type, so you can define an unsigned word of 32 bits wide as
337 type Word32 = SizedWord D32
340 Here, a type synonym \hs{Word32} is defined that is equal to the
341 \hs{SizedWord} type constructor applied to the type \hs{D32}. \hs{D32}
342 is the \emph{type level representation} of the decimal number 32,
343 making the \hs{Word32} type a 32-bit unsigned word.
345 These types are translated to the \small{VHDL} \type{unsigned} and
346 \type{signed} respectively.
347 \todo{Sidenote on dependent typing?}
349 \startdesc{\hs{Vector}}
350 This is a vector type, that can contain elements of any other type and
351 has a fixed length. It has two type parameters: its
352 length and the type of the elements contained in it. By putting the
353 length parameter in the type, the length of a vector can be determined
354 at compile time, instead of only at runtime for conventional lists.
356 The \hs{Vector} type constructor takes two type arguments: the length
357 of the vector and the type of the elements contained in it. The state
358 type of an 8 element register bank would then for example be:
361 type RegisterState = Vector D8 Word32
364 Here, a type synonym \hs{RegisterState} is defined that is equal to
365 the \hs{Vector} type constructor applied to the types \hs{D8} (The type
366 level representation of the decimal number 8) and \hs{Word32} (The 32
367 bit word type as defined above). In other words, the
368 \hs{RegisterState} type is a vector of 8 32-bit words.
370 A fixed size vector is translated to a \small{VHDL} array type.
372 \startdesc{\hs{RangedWord}}
373 This is another type to describe integers, but unlike the previous
374 two it has no specific bitwidth, but an upper bound. This means that
375 its range is not limited to powers of two, but can be any number.
376 A \hs{RangedWord} only has an upper bound, its lower bound is
377 implicitly zero. There is a lot of added implementation complexity
378 when adding a lower bound and having just an upper bound was enough
379 for the primary purpose of this type: typesafely indexing vectors.
381 To define an index for the 8 element vector above, we would do:
384 type RegisterIndex = RangedWord D7
387 Here, a type synonym \hs{RegisterIndex} is defined that is equal to
388 the \hs{RangedWord} type constructor applied to the type \hs{D7}. In
389 other words, this defines an unsigned word with values from
390 {\definedfont[Serif*normalnum]0 to 7} (inclusive). This word can be be used to index the
391 8 element vector \hs{RegisterState} above.
393 This type is translated to the \type{unsigned} \small{VHDL} type.
396 The integer and vector built-in types are discussed in more detail
399 \subsection{User-defined types}
400 There are three ways to define new types in Haskell: algebraic
401 datatypes with the \hs{data} keyword, type synonyms with the \hs{type}
402 keyword and type renamings with the \hs{newtype} keyword. \GHC\
403 offers a few more advanced ways to introduce types (type families,
404 existential typing, \small{GADT}s, etc.) which are not standard
405 Haskell. These will be left outside the scope of this research.
407 Only an algebraic datatype declaration actually introduces a
408 completely new type, for which we provide the \VHDL\ translation
409 below. Type synonyms and renamings only define new names for
410 existing types (where synonyms are completely interchangeable and
411 renamings need explicit conversion). Therefore, these do not need
412 any particular \VHDL\ translation, a synonym or renamed type will
413 just use the same representation as the original type. The
414 distinction between a renaming and a synonym does no longer matter
415 in hardware and can be disregarded in the generated \VHDL.
417 For algebraic types, we can make the following distinction:
419 \startdesc{Product types}
420 A product type is an algebraic datatype with a single constructor with
421 two or more fields, denoted in practice like (a,b), (a,b,c), etc. This
422 is essentially a way to pack a few values together in a record-like
423 structure. In fact, the built-in tuple types are just algebraic product
424 types (and are thus supported in exactly the same way).
426 The \quote{product} in its name refers to the collection of values belonging
427 to this type. The collection for a product type is the Cartesian
428 product of the collections for the types of its fields.
430 These types are translated to \VHDL\ record types, with one field for
431 every field in the constructor. This translation applies to all single
432 constructor algebraic datatypes, including those with just one
433 field (which are technically not a product, but generate a VHDL
434 record for implementation simplicity).
436 \startdesc{Enumerated types}
437 \defref{enumerated types}
438 An enumerated type is an algebraic datatype with multiple constructors, but
439 none of them have fields. This is essentially a way to get an
440 enum-like type containing alternatives.
442 Note that Haskell's \hs{Bool} type is also defined as an
443 enumeration type, but we have a fixed translation for that.
445 These types are translated to \VHDL\ enumerations, with one value for
446 each constructor. This allows references to these constructors to be
447 translated to the corresponding enumeration value.
449 \startdesc{Sum types}
450 A sum type is an algebraic datatype with multiple constructors, where
451 the constructors have one or more fields. Technically, a type with
452 more than one field per constructor is a sum of products type, but
453 for our purposes this distinction does not really make a
454 difference, so this distinction is note made.
456 The \quote{sum} in its name refers again to the collection of values
457 belonging to this type. The collection for a sum type is the
458 union of the the collections for each of the constructors.
460 Sum types are currently not supported by the prototype, since there is
461 no obvious \VHDL\ alternative. They can easily be emulated, however, as
462 we will see from an example:
465 data Sum = A Bit Word | B Word
468 An obvious way to translate this would be to create an enumeration to
469 distinguish the constructors and then create a big record that
470 contains all the fields of all the constructors. This is the same
471 translation that would result from the following enumeration and
472 product type (using a tuple for clarity):
476 type Sum = (SumC, Bit, Word, Word)
479 Here, the \hs{SumC} type effectively signals which of the latter three
480 fields of the \hs{Sum} type are valid (the first two if \hs{A}, the
481 last one if \hs{B}), all the other ones have no useful value.
483 An obvious problem with this naive approach is the space usage: the
484 example above generates a fairly big \VHDL\ type. Since we can be
485 sure that the two \hs{Word}s in the \hs{Sum} type will never be valid
486 at the same time, this is a waste of space.
488 Obviously, duplication detection could be used to reuse a
489 particular field for another constructor, but this would only
490 partially solve the problem. If two fields would be, for
491 example, an array of 8 bits and an 8 bit unsiged word, these are
492 different types and could not be shared. However, in the final
493 hardware, both of these types would simply be 8 bit connections,
494 so we have a 100\% size increase by not sharing these.
497 Another interesting case is that of recursive types. In Haskell, an
498 algebraic datatype can be recursive: any of its field types can be (or
499 contain) the type being defined. The most well-known recursive type is
500 probably the list type, which is defined is:
503 data List t = Empty | Cons t (List t)
506 Note that \hs{Empty} is usually written as \hs{[]} and \hs{Cons} as
507 \hs{:}, but this would make the definition harder to read. This
508 immediately shows the problem with recursive types: what hardware type
511 If the naive approach for sum types described above would be used,
512 a record would be created where the first field is an enumeration
513 to distinguish \hs{Empty} from \hs{Cons}. Furthermore, two more
514 fields would be added: one with the (\VHDL\ equivalent of) type
515 \hs{t} (assuming this type is actually known at compile time, this
516 should not be a problem) and a second one with type \hs{List t}.
517 The latter one is of course a problem: this is exactly the type
518 that was to be translated in the first place.
520 The resulting \VHDL\ type will thus become infinitely deep. In
521 other words, there is no way to statically determine how long
522 (deep) the list will be (it could even be infinite).
524 In general, recursive types can never be properly translated: all
525 recursive types have a potentially infinite value (even though in
526 practice they will have a bounded value, there is no way for the
527 compiler to automatically determine an upper bound on its size).
529 \subsection{Partial application}
530 Now the translation of application, choice and types has been
531 discussed, a more complex concept can be considered: partial
532 applications. A \emph{partial application} is any application whose
533 (return) type is (again) a function type.
535 From this, it should be clear that the translation rules for full
536 application does not apply to a partial application: there are not
537 enough values for all the input ports in the resulting \VHDL.
538 \in{Example}[ex:Quadruple] shows an example use of partial application
539 and the corresponding architecture.
541 \startbuffer[Quadruple]
542 -- Multiply the input word by four.
543 quadruple :: Word -> Word
544 quadruple n = mul (mul n)
549 \startuseMPgraphic{Quadruple}
550 save in, two, mula, mulb, out;
553 newCircle.in(btex $n$ etex) "framed(false)";
554 newCircle.two(btex $2$ etex) "framed(false)";
555 newCircle.out(btex $out$ etex) "framed(false)";
558 newCircle.mula(btex $\times$ etex);
559 newCircle.mulb(btex $\times$ etex);
562 in.c = two.c + (0cm, 1cm);
563 mula.c = in.c + (2cm, 0cm);
564 mulb.c = mula.c + (2cm, 0cm);
565 out.c = mulb.c + (2cm, 0cm);
567 % Draw objects and lines
568 drawObj(in, two, mula, mulb, out);
570 nccurve(two)(mula) "angleA(0)", "angleB(45)";
571 nccurve(two)(mulb) "angleA(0)", "angleB(45)";
577 \placeexample[][ex:Quadruple]{Simple three port and.}
578 \startcombination[2*1]
579 {\typebufferhs{Quadruple}}{Haskell description using function applications.}
580 {\boxedgraphic{Quadruple}}{The architecture described by the Haskell description.}
583 Here, the definition of mul is a partial function application: it applies
584 the function \hs{(*) :: Word -> Word -> Word} to the value \hs{2 :: Word},
585 resulting in the expression \hs{(*) 2 :: Word -> Word}. Since this resulting
586 expression is again a function, hardware cannot be generated for it
587 directly. This is because the hardware to generate for \hs{mul}
588 depends completely on where and how it is used. In this example, it is
591 However, it is clear that the above hardware description actually
592 describes valid hardware. In general, any partial applied function
593 must eventually become completely applied, at which point hardware for
594 it can be generated using the rules for function application given in
595 \in{section}[sec:description:application]. It might mean that a
596 partial application is passed around quite a bit (even beyond function
597 boundaries), but eventually, the partial application will become
598 completely applied. An example of this principe is given in
599 \in{section}[sec:normalization:defunctionalization].
601 \section{Costless specialization}
602 Each (complete) function application in our description generates a
603 component instantiation, or a specific piece of hardware in the final
604 design. It is interesting to note that each application of a function
605 generates a \emph{separate} piece of hardware. In the final design, none
606 of the hardware is shared between applications, even when the applied
607 function is the same (of course, if a particular value, such as the result
608 of a function application, is used twice, it is not calculated twice).
610 This is distinctly different from normal program compilation: two separate
611 calls to the same function share the same machine code. Having more
612 machine code has implications for speed (due to less efficient caching)
613 and memory usage. For normal compilation, it is therefore important to
614 keep the amount of functions limited and maximize the code sharing
615 (though there is a tradeoff between speed and memory usage here).
617 When generating hardware, this is hardly an issue. Having more \quote{code
618 sharing} does reduce the amount of \small{VHDL} output (Since different
619 component instantiations still share the same component), but after
620 synthesis, the amount of hardware generated is not affected. This
621 means there is no tradeoff between speed and memory (or rather,
624 In particular, if we would duplicate all functions so that there is a
625 separate function for every application in the program (\eg, each function
626 is then only applied exactly once), there would be no increase in hardware
629 Because of this, a common optimization technique called
630 \emph{specialization} can be applied to hardware generation without any
631 performance or area cost (unlike for software).
633 \fxnote{Perhaps these next three sections are a bit too
634 implementation-oriented?}
636 \subsection{Specialization}
637 \defref{specialization}
638 Given some function that has a \emph{domain} $D$ (\eg, the set of
639 all possible arguments that the function could be applied to), we
640 create a specialized function with exactly the same behaviour, but
641 with a domain $D' \subset D$. This subset can be chosen in all
642 sorts of ways. Any subset is valid for the general definition of
643 specialization, but in practice only some of them provide useful
644 optimization opportunities.
646 Common subsets include limiting a polymorphic argument to a single type
647 (\ie, removing polymorphism) or limiting an argument to just a single
648 value (\ie, cross-function constant propagation, effectively removing
651 Since we limit the argument domain of the specialized function, its
652 definition can often be optimized further (since now more types or even
653 values of arguments are already known). By replacing any application of
654 the function that falls within the reduced domain by an application of
655 the specialized version, the code gets faster (but the code also gets
656 bigger, since we now have two versions instead of one). If we apply
657 this technique often enough, we can often replace all applications of a
658 function by specialized versions, allowing the original function to be
659 removed (in some cases, this can even give a net reduction of the code
660 compared to the non-specialized version).
662 Specialization is useful for our hardware descriptions for functions
663 that contain arguments that cannot be translated to hardware directly
664 (polymorphic or higher-order arguments, for example). If we can create
665 specialized functions that remove the argument, or make it translatable,
666 we can use specialization to make the original, untranslatable, function
669 \section{Higher order values}
670 What holds for partial application, can be easily generalized to any
671 higher-order expression. This includes partial applications, plain
672 variables (e.g., a binder referring to a top level function), lambda
673 expressions and more complex expressions with a function type (a \hs{case}
674 expression returning lambda's, for example).
676 Each of these values cannot be directly represented in hardware (just like
677 partial applications). Also, to make them representable, they need to be
678 applied: function variables and partial applications will then eventually
679 become complete applications, applied lambda expressions disappear by
680 applying β-reduction, etc.
682 So any higher-order value will be \quote{pushed down} towards its
683 application just like partial applications. Whenever a function boundary
684 needs to be crossed, the called function can be specialized.
686 \fxnote{This section needs improvement and an example}
688 \section{Polymorphism}
689 In Haskell, values can be \emph{polymorphic}: they can have multiple types. For
690 example, the function \hs{fst :: (a, b) -> a} is an example of a
691 polymorphic function: it works for tuples with any two element types. Haskell
692 type classes allow a function to work on a specific set of types, but the
693 general idea is the same. The opposite of this is a \emph{monomorphic}
694 value, which has a single, fixed, type.
696 % A type class is a collection of types for which some operations are
697 % defined. It is thus possible for a value to be polymorphic while having
698 % any number of \emph{class constraints}: the value is not defined for
699 % every type, but only for types in the type class. An example of this is
700 % the \hs{even :: (Integral a) => a -> Bool} function, which can map any
701 % value of a type that is member of the \hs{Integral} type class
703 When generating hardware, polymorphism cannot be easily translated. How
704 many wires will you lay down for a value that could have any type? When
705 type classes are involved, what hardware components will you lay down for
706 a class method (whose behaviour depends on the type of its arguments)?
707 Note that Cλash currently does not allow user-defined type classes,
708 but does partly support some of the built-in type classes (like \hs{Num}).
710 Fortunately, we can again use the principle of specialization: since every
711 function application generates a separate piece of hardware, we can know
712 the types of all arguments exactly. Provided that existential typing
713 (which is a \GHC\ extension) is not used typing, all of the
714 polymorphic types in a function must depend on the types of the
715 arguments (In other words, the only way to introduce a type variable
716 is in a lambda abstraction).
718 If a function is monomorphic, all values inside it are monomorphic as
719 well, so any function that is applied within the function can only be
720 applied to monomorphic values. The applied functions can then be
721 specialized to work just for these specific types, removing the
722 polymorphism from the applied functions as well.
724 \defref{entry function}The entry function must not have a
725 polymorphic type (otherwise no hardware interface could be generated
726 for the entry function).
728 By induction, this means that all functions that are (indirectly) called
729 by our top level function (meaning all functions that are translated in
730 the final hardware) become monomorphic.
733 A very important concept in hardware designs is \emph{state}. In a
734 stateless (or, \emph{combinational}) design, every output is directly and solely dependent on the
735 inputs. In a stateful design, the outputs can depend on the history of
736 inputs, or the \emph{state}. State is usually stored in \emph{registers},
737 which retain their value during a clockcycle, and are typically updated at
738 the start of every clockcycle. Since the updating of the state is tightly
739 coupled (synchronized) to the clock signal, these state updates are often
740 called \emph{synchronous} behaviour.
742 \todo{Sidenote? Registers can contain any (complex) type}
744 To make Cλash useful to describe more than simple combinational
745 designs, it needs to be able to describe state in some way.
747 \subsection{Approaches to state}
748 In Haskell, functions are always pure (except when using unsafe
749 functions with the \hs{IO} monad, which is not supported by Cλash). This
750 means that the output of a function solely depends on its inputs. If you
751 evaluate a given function with given inputs, it will always provide the
756 \startframedtext[width=8cm,background=box,frame=no]
757 \startalignment[center]
762 A function is said to be pure if it satisfies two conditions:
765 \item When a pure function is called with the same arguments twice, it should
766 return the same value in both cases.
767 \item When a pure function is called, it should have not
768 observable side-effects.
771 Purity is an important property in functional languages, since
772 it enables all kinds of mathematical reasoning and
773 optimizattions with pure functions, that are not guaranteed to
774 be correct for impure functions.
776 An example of a pure function is the square root function
777 \hs{sqrt}. An example of an impure function is the \hs{today}
778 function that returns the current date (which of course cannot
779 exist like this in Haskell).
783 This is a perfect match for a combinational circuit, where the output
784 also solely depends on the inputs. However, when state is involved, this
785 no longer holds. Of course this purity constraint cannot just be
786 removed from Haskell. But even when designing a completely new (hardware
787 description) language, it does not seem to be a good idea to
788 remove this purity. This would that all kinds of interesting properties of
789 the functional language get lost, and all kinds of transformations
790 and optimizations are no longer be meaning preserving.
792 So our functions must remain pure, meaning the current state has
793 to be present in the function's arguments in some way. There seem
794 to be two obvious ways to do this: adding the current state as an
795 argument, or including the full history of each argument.
797 \subsubsection{Stream arguments and results}
798 Including the entire history of each input (\eg, the value of that
799 input for each previous clockcycle) is an obvious way to make outputs
800 depend on all previous input. This is easily done by making every
801 input a list instead of a single value, containing all previous values
802 as well as the current value.
804 An obvious downside of this solution is that on each cycle, all the
805 previous cycles must be resimulated to obtain the current state. To do
806 this, it might be needed to have a recursive helper function as well,
807 which might be hard to be properly analyzed by the compiler.
809 A slight variation on this approach is one taken by some of the other
810 functional \small{HDL}s in the field: \todo{References to Lava,
811 ForSyDe, ...} Make functions operate on complete streams. This means
812 that a function is no longer called on every cycle, but just once. It
813 takes stream as inputs instead of values, where each stream contains
814 all the values for every clockcycle since system start. This is easily
815 modeled using an (infinite) list, with one element for each clock
816 cycle. Since the function is only evaluated once, its output must also
817 be a stream. Note that, since we are working with infinite lists and
818 still want to be able to simulate the system cycle-by-cycle, this
819 relies heavily on the lazy semantics of Haskell.
821 Since our inputs and outputs are streams, all other (intermediate)
822 values must be streams. All of our primitive operators (\eg, addition,
823 substraction, bitwise operations, etc.) must operate on streams as
824 well (note that changing a single-element operation to a stream
825 operation can done with \hs{map}, \hs{zipwith}, etc.).
827 This also means that primitive operations from an existing
828 language such as Haskell cannot be used directly (including some
829 syntax constructs, like \hs{case} expressions and \hs{if}
830 expressions). This mkes this approach well suited for use in
831 \small{EDSL}s, since those already impose these same
832 limitations. \refdef{EDSL}
834 Note that the concept of \emph{state} is no more than having some way
835 to communicate a value from one cycle to the next. By introducing a
836 \hs{delay} function, we can do exactly that: delay (each value in) a
837 stream so that we can "look into" the past. This \hs{delay} function
838 simply outputs a stream where each value is the same as the input
839 value, but shifted one cycle. This causes a \quote{gap} at the
840 beginning of the stream: what is the value of the delay output in the
841 first cycle? For this, the \hs{delay} function has a second input, of
842 which only a single value is used.
844 \in{Example}[ex:DelayAcc] shows a simple accumulator expressed in this
847 \startbuffer[DelayAcc]
848 acc :: Stream Word -> Stream Word
851 out = (delay out 0) + in
854 \startuseMPgraphic{DelayAcc}
855 save in, out, add, reg;
858 newCircle.in(btex $in$ etex) "framed(false)";
859 newCircle.out(btex $out$ etex) "framed(false)";
862 newReg.reg("") "dx(4mm)", "dy(6mm)", "reflect(true)";
863 newCircle.add(btex + etex);
866 add.c = in.c + (2cm, 0cm);
867 out.c = add.c + (2cm, 0cm);
868 reg.c = add.c + (0cm, 2cm);
870 % Draw objects and lines
871 drawObj(in, out, add, reg);
873 nccurve(add)(reg) "angleA(0)", "angleB(180)", "posB(d)";
874 nccurve(reg)(add) "angleA(180)", "angleB(-45)", "posA(out)";
880 \placeexample[][ex:DelayAcc]{Simple accumulator architecture.}
881 \startcombination[2*1]
882 {\typebufferhs{DelayAcc}}{Haskell description using streams.}
883 {\boxedgraphic{DelayAcc}}{The architecture described by the Haskell description.}
887 This notation can be confusing (especially due to the loop in the
888 definition of out), but is essentially easy to interpret. There is a
889 single call to delay, resulting in a circuit with a single register,
890 whose input is connected to \hs{out} (which is the output of the
891 adder), and its output is the expression \hs{delay out 0} (which is
892 connected to one of the adder inputs).
894 \subsubsection{Explicit state arguments and results}
895 A more explicit way to model state, is to simply add an extra argument
896 containing the current state value. This allows an output to depend on
897 both the inputs as well as the current state while keeping the
898 function pure (letting the result depend only on the arguments), since
899 the current state is now an argument.
901 In Haskell, this would look like
902 \in{example}[ex:ExplicitAcc]\footnote[notfinalsyntax]{This
903 example is not in the final Cλash syntax}. \todo{Referencing
904 notfinalsyntax from Introduction.tex doesn't work}
906 \startbuffer[ExplicitAcc]
907 -- input -> current state -> (new state, output)
908 acc :: Word -> Word -> (Word, Word)
915 \placeexample[][ex:ExplicitAcc]{Simple accumulator architecture.}
916 \startcombination[2*1]
917 {\typebufferhs{ExplicitAcc}}{Haskell description using explicit state arguments.}
918 % Picture is identical to the one we had just now.
919 {\boxedgraphic{DelayAcc}}{The architecture described by the Haskell description.}
922 This approach makes a function's state very explicit, which state
923 variables are used by a function can be completely determined from its
924 type signature (as opposed to the stream approach, where a function
925 looks the same from the outside, regardless of what state variables it
926 uses or whether it is stateful at all).
928 This approach to state has been one of the initial drives behind
929 this research. Unlike a stream based approach it is well suited
930 to completely use existing code and language features (like
931 \hs{if} and \hs{case} expressions) because it operates on normal
932 values. Because of these reasons, this is the approach chosen
933 for Cλash. It will be examined more closely below.
935 \subsection{Explicit state specification}
936 The concept of explicit state has been introduced with some
937 examples above, but what are the implications of this approach?
939 \subsubsection{Substates}
940 Since a function's state is reflected directly in its type signature,
941 if a function calls other stateful functions (\eg, has subcircuits), it
942 has to somehow know the current state for these called functions. The
943 only way to do this, is to put these \emph{substates} inside the
944 caller's state. This means that a function's state is the sum of the
945 states of all functions it calls, and its own state. This sum
946 can be obtained using something simple like a tuple, or possibly
947 custom algebraic types for clarity.
949 This also means that the type of a function (at least the "state"
950 part) is dependent on its own implementation and of the functions it
953 This is the major downside of this approach: the separation between
954 interface and implementation is limited. However, since Cλash is not
955 very suitable for separate compilation (see
956 \in{section}[sec:prototype:separate]) this is not a big problem in
959 Additionally, when using a type synonym for the state type
960 of each function, we can still provide explicit type signatures
961 while keeping the state specification for a function near its
965 \subsubsection{Which arguments and results are stateful?}
966 \fxnote{This section should get some examples}
967 We need some way to know which arguments should become input ports and
968 which argument(s?) should become the current state (\eg, be bound to
969 the register outputs). This does not hold just for the top
970 level function, but also for any subfunction. Or could we perhaps
971 deduce the statefulness of subfunctions by analyzing the flow of data
972 in the calling functions?
974 To explore this matter, the following observeration is interesting: we
975 get completely correct behaviour when we put all state registers in
976 the top level entity (or even outside of it). All of the state
977 arguments and results on subfunctions are treated as normal input and
978 output ports. Effectively, a stateful function results in a stateless
979 hardware component that has one of its input ports connected to the
980 output of a register and one of its output ports connected to the
981 input of the same register.
985 Of course, even though the hardware described like this has the
986 correct behaviour, unless the layout tool does smart optimizations,
987 there will be a lot of extra wire in the design (since registers will
988 not be close to the component that uses them). Also, when working with
989 the generated \small{VHDL} code, there will be a lot of extra ports
990 just to pass on state values, which can get quite confusing.
992 To fix this, we can simply \quote{push} the registers down into the
993 subcircuits. When we see a register that is connected directly to a
994 subcircuit, we remove the corresponding input and output port and put
995 the register inside the subcircuit instead. This is slightly less
996 trivial when looking at the Haskell code instead of the resulting
997 circuit, but the idea is still the same.
1001 However, when applying this technique, we might push registers down
1002 too far. When you intend to store a result of a stateless subfunction
1003 in the caller's state and pass the current value of that state
1004 variable to that same function, the register might get pushed down too
1005 far. It is impossible to distinguish this case from similar code where
1006 the called function is in fact stateful. From this we can conclude
1007 that we have to either:
1009 \todo{Example of wrong downpushing}
1012 \item accept that the generated hardware might not be exactly what we
1013 intended, in some specific cases. In most cases, the hardware will be
1015 \item explicitly annotate state arguments and results in the input
1019 The first option causes (non-obvious) exceptions in the language
1020 intepretation. Also, automatically determining where registers should
1021 end up is easier to implement correctly with explicit annotations, so
1022 for these reasons we will look at how this annotations could work.
1024 \todo{Sidenote: one or more state arguments?}
1026 \subsection[sec:description:stateann]{Explicit state annotation}
1027 To make our stateful descriptions unambigious and easier to translate,
1028 we need some way for the developer to describe which arguments and
1029 results are intended to become stateful.
1031 Roughly, we have two ways to achieve this:
1033 \item Use some kind of annotation method or syntactic construction in
1034 the language to indicate exactly which argument and (part of the)
1035 result is stateful. This means that the annotation lives
1036 \quote{outside} of the function, it is completely invisible when
1037 looking at the function body.
1038 \item Use some kind of annotation on the type level, \ie\ give stateful
1039 arguments and stateful (parts of) results a different type. This has the
1040 potential to make this annotation visible inside the function as well,
1041 such that when looking at a value inside the function body you can
1042 tell if it is stateful by looking at its type. This could possibly make
1043 the translation process a lot easier, since less analysis of the
1044 program flow might be required.
1047 From these approaches, the type level \quote{annotations} have been
1048 implemented in Cλash. \in{Section}[sec:prototype:statetype] expands on
1049 the possible ways this could have been implemented.
1051 \todo{Note about conditions on state variables and checking them}
1053 \section[sec:recursion]{Recursion}
1054 An important concept in functional languages is recursion. In its most basic
1055 form, recursion is a definition that is described in terms of itself. A
1056 recursive function is thus a function that uses itself in its body. This
1057 usually requires multiple evaluations of this function, with changing
1058 arguments, until eventually an evaluation of the function no longer requires
1061 Given the notion that each function application will translate to a
1062 component instantiation, we are presented with a problem. A recursive
1063 function would translate to a component that contains itself. Or, more
1064 precisely, that contains an instance of itself. This instance would again
1065 contain an instance of itself, and again, into infinity. This is obviously a
1066 problem for generating hardware.
1068 This is expected for functions that describe infinite recursion. In that
1069 case, we cannot generate hardware that shows correct behaviour in a single
1070 cycle (at best, we could generate hardware that needs an infinite number of
1071 cycles to complete).
1074 \startframedtext[width=8cm,background=box,frame=no]
1075 \startalignment[center]
1076 {\tfa \hs{null}, \hs{head} and \hs{tail}}
1079 The functions \hs{null}, \hs{head} and \hs{tail} are common list
1080 functions in Haskell. The \hs{null} function simply checks if a list is
1081 empty. The \hs{head} function returns the first element of a list. The
1082 \hs{tail} function returns containing everything \emph{except} the first
1085 In Cλash, there are vector versions of these functions, which do exactly
1090 However, most recursive definitions will describe finite
1091 recursion. This is because the recursive call is done conditionally. There
1092 is usually a \hs{case} expression where at least one alternative does not contain
1093 the recursive call, which we call the "base case". If, for each call to the
1094 recursive function, we would be able to detect at compile time which
1095 alternative applies, we would be able to remove the \hs{case} expression and
1096 leave only the base case when it applies. This will ensure that expanding
1097 the recursive functions will terminate after a bounded number of expansions.
1099 This does imply the extra requirement that the base case is detectable at
1100 compile time. In particular, this means that the decision between the base
1101 case and the recursive case must not depend on runtime data.
1103 \subsection{List recursion}
1104 The most common deciding factor in recursion is the length of a list that is
1105 passed in as an argument. Since we represent lists as vectors that encode
1106 the length in the vector type, it seems easy to determine the base case. We
1107 can simply look at the argument type for this. However, it turns out that
1108 this is rather non-trivial to write down in Haskell already, not even
1109 looking at translation. As an example, we would like to write down something
1113 sum :: Vector n Word -> Word
1114 sum xs = case null xs of
1116 False -> head xs + sum (tail xs)
1119 However, the Haskell typechecker will now use the following reasoning.
1120 For simplicity, the element type of a vector is left out, all vectors
1121 are assumed to have the same element type. Below, we write conditions
1122 on type variables before the \hs{=>} operator. This is not completely
1123 valid Haskell syntax, but serves to illustrate the typechecker
1124 reasoning. Also note that a vector can never have a negative length,
1125 so \hs{Vector n} implicitly means \hs{(n >= 0) => Vector n}.
1127 \todo{This typechecker disregards the type signature}
1129 \item tail has the type \hs{(n > 0) => Vector n -> Vector (n - 1)}
1130 \item This means that xs must have the type \hs{(n > 0) => Vector n}
1131 \item This means that sum must have the type \hs{(n > 0) => Vector n -> a}
1132 (The type \hs{a} is can be anything at this stage, we will not try to finds
1133 its actual type in this example).
1134 \item sum is called with the result of tail as an argument, which has the
1135 type \hs{Vector n} (since \hs{(n > 0) => Vector (n - 1)} is the same as \hs{(n >= 0)
1136 => Vector n}, which is the same as just \hs{Vector n}).
1137 \item This means that sum must have the type \hs{Vector n -> a}
1138 \item This is a contradiction between the type deduced from the body of sum
1139 (the input vector must be non-empty) and the use of sum (the input vector
1140 could have any length).
1143 As you can see, using a simple \hs{case} expression at value level causes
1144 the type checker to always typecheck both alternatives, which cannot be
1145 done. The typechecker is unable to distinguish the two case
1146 alternatives (this is partly possible using \small{GADT}s, but that
1147 approach faced other problems \todo{ref christiaan?}).
1149 This is a fundamental problem, that would seem perfectly suited for a
1150 type class. Considering that we need to switch between to
1151 implementations of the sum function, based on the type of the
1152 argument, this sounds like the perfect problem to solve with a type
1153 class. However, this approach has its own problems (not the least of
1154 them that you need to define a new type class for every recursive
1155 function you want to define).
1157 \todo{This should reference Christiaan}
1159 \subsection{General recursion}
1160 Of course there are other forms of recursion, that do not depend on the
1161 length (and thus type) of a list. For example, simple recursion using a
1162 counter could be expressed, but only translated to hardware for a fixed
1163 number of iterations. Also, this would require extensive support for compile
1164 time simplification (constant propagation) and compile time evaluation
1165 (evaluation of constant comparisons), to ensure non-termination.
1166 Supporting general recursion will probably require strict conditions
1167 on the input descriptions. Even then, it will be hard (if not
1168 impossible) to really guarantee termination, since the user (or \GHC\
1169 desugarer) might use some obscure notation that results in a corner
1170 case of the simplifier that is not caught and thus non-termination.
1172 Evaluating all possible (and non-possible) ways to add recursion to
1173 our descriptions, it seems better to limit the scope of this research
1174 to exclude recursion. This allows for focusing on other interesting
1175 areas instead. By including (built-in) support for a number of
1176 higher-order functions like \hs{map} and \hs{fold}, we can still
1177 express most of the things we would use (list) recursion for.
1180 % vim: set sw=2 sts=2 expandtab:
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