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 \todo{Shortshort introduction to Cλash (Bit, Word, and, not, etc.)}
10 To translate Haskell to hardware, every Haskell construct needs a
11 translation to \VHDL. There are often multiple valid translations
12 possible. When faced with choices, the most obvious choice has been
13 chosen wherever possible. In a lot of cases, when a programmer looks
14 at a functional hardware description it is completely clear what
15 hardware is described. We want our translator to generate exactly that
16 hardware whenever possible, to make working with Cλash as intuitive as
20 \defref{top level binder}
21 \defref{top level function}
22 \startframedtext[width=8cm,background=box,frame=no]
23 \startalignment[center]
24 {\tfa Top level binders and functions}
27 A top level binder is any binder (variable) that is declared in
28 the \quote{global} scope of a Haskell program (as opposed to a
29 binder that is bound inside a function.
31 In Haskell, there is no sharp distinction between a variable and a
32 function: a function is just a variable (binder) with a function
33 type. This means that a top level function is just any top level
34 binder with a function type.
36 As an example, consider the following Haskell snippet:
45 Here, \hs{foo} is a top level binder, whereas \hs{inc} is a
46 function (since it is bound to a lambda extraction, indicated by
47 the backslash) but is not a top level binder or function. Since
48 the type of \hs{foo} is a function type, namely \hs{Int -> Int},
49 it is also a top level function.
52 In this chapter we describe how to interpret a Haskell program from a
53 hardware perspective. We provide a description of each Haskell language
54 element that needs translation, to provide a clear picture of what is
58 \section[sec:description:application]{Function application}
59 The basic syntactic elements of a functional program are functions and
60 function application. These have a single obvious \small{VHDL}
61 translation: each top level function becomes a hardware component, where each
62 argument is an input port and the result value is the (single) output
63 port. This output port can have a complex type (such as a tuple), so
64 having just a single output port does not pose a limitation.
66 Each function application in turn becomes component instantiation. Here, the
67 result of each argument expression is assigned to a signal, which is mapped
68 to the corresponding input port. The output port of the function is also
69 mapped to a signal, which is used as the result of the application.
71 Since every top level function generates its own component, the
72 hierarchy of of function calls is reflected in the final \VHDL\ output
73 as well, creating a hierarchical \VHDL\ description of the hardware.
74 This separation in different components makes the resulting \VHDL\
75 output easier to read and debug.
77 \in{Example}[ex:And3] shows a simple program using only function
78 application and the corresponding architecture.
81 -- | A simple function that returns
82 -- conjunction of three bits
83 and3 :: Bit -> Bit -> Bit -> Bit
84 and3 a b c = and (and a b) c
87 \startuseMPgraphic{And3}
88 save a, b, c, anda, andb, out;
91 newCircle.a(btex $a$ etex) "framed(false)";
92 newCircle.b(btex $b$ etex) "framed(false)";
93 newCircle.c(btex $c$ etex) "framed(false)";
94 newCircle.out(btex $out$ etex) "framed(false)";
97 newCircle.anda(btex $and$ etex);
98 newCircle.andb(btex $and$ etex);
101 b.c = a.c + (0cm, 1cm);
102 c.c = b.c + (0cm, 1cm);
103 anda.c = midpoint(a.c, b.c) + (2cm, 0cm);
104 andb.c = midpoint(b.c, c.c) + (4cm, 0cm);
106 out.c = andb.c + (2cm, 0cm);
108 % Draw objects and lines
109 drawObj(a, b, c, anda, andb, out);
111 ncarc(a)(anda) "arcangle(-10)";
118 \placeexample[here][ex:And3]{Simple three input and gate.}
119 \startcombination[2*1]
120 {\typebufferhs{And3}}{Haskell description using function applications.}
121 {\boxedgraphic{And3}}{The architecture described by the Haskell description.}
125 Although describing components and connections allows us to describe a lot of
126 hardware designs already, there is an obvious thing missing: choice. We
127 need some way to be able to choose between values based on another value.
128 In Haskell, choice is achieved by \hs{case} expressions, \hs{if}
129 expressions, pattern matching and guards.
131 An obvious way to add choice to our language without having to recognize
132 any of Haskell's syntax, would be to add a primivite \quote{\hs{if}}
133 function. This function would take three arguments: the condition, the
134 value to return when the condition is true and the value to return when
135 the condition is false.
137 This \hs{if} function would then essentially describe a multiplexer and
138 allows us to describe any architecture that uses multiplexers.
140 However, to be able to describe our hardware in a more convenient way, we
141 also want to translate Haskell's choice mechanisms. The easiest of these
142 are of course case expressions (and \hs{if} expressions, which can be very
143 directly translated to \hs{case} expressions). A \hs{case} expression can in turn
144 simply be translated to a conditional assignment, where the conditions use
145 equality comparisons against the constructors in the \hs{case} expressions.
148 \defref{substitution notation}
149 \startframedtext[width=8cm,background=box,frame=no]
150 \startalignment[center]
151 {\tfa Arguments / results vs. inputs / outputs}
154 Due to the translation chosen for function application, there is a
155 very strong relation between arguments, results, inputs and outputs.
156 For clarity, the former two will always refer to the arguments and
157 results in the functional description (either Haskell or Core). The
158 latter two will refer to input and output ports in the generated
161 Even though these concepts seem to be nearly identical, when stateful
162 functions are introduces we will see arguments and results that will
163 not get translated into input and output ports, making this
164 distinction more important.
168 In \in{example}[ex:CaseInv] a simple \hs{case} expression is shown,
169 scrutinizing a boolean value. The corresponding architecture has a
170 comparator to determine which of the constructors is on the \hs{in}
171 input. There is a multiplexer to select the output signal. The two options
172 for the output signals are just constants, but these could have been more
173 complex expressions (in which case also both of them would be working in
174 parallel, regardless of which output would be chosen eventually).
176 If we would translate a Boolean to a bit value, we could of course remove
177 the comparator and directly feed 'in' into the multiplexer (or even use an
178 inverter instead of a multiplexer). However, we will try to make a
179 general translation, which works for all possible \hs{case} expressions.
180 Optimizations such as these are left for the \VHDL\ synthesizer, which
181 handles them very well.
183 \startbuffer[CaseInv]
190 \startuseMPgraphic{CaseInv}
191 save in, truecmp, falseout, trueout, out, cmp, mux;
194 newCircle.in(btex $in$ etex) "framed(false)";
195 newCircle.out(btex $out$ etex) "framed(false)";
197 newBox.truecmp(btex $True$ etex) "framed(false)";
198 newBox.trueout(btex $True$ etex) "framed(false)";
199 newBox.falseout(btex $False$ etex) "framed(false)";
202 newCircle.cmp(btex $==$ etex);
206 cmp.c = in.c + (3cm, 0cm);
207 truecmp.c = cmp.c + (-1cm, 1cm);
208 mux.sel = cmp.e + (1cm, -1cm);
209 falseout.c = mux.inpa - (2cm, 0cm);
210 trueout.c = mux.inpb - (2cm, 0cm);
211 out.c = mux.out + (2cm, 0cm);
213 % Draw objects and lines
214 drawObj(in, out, truecmp, trueout, falseout, cmp, mux);
218 nccurve(cmp.e)(mux.sel) "angleA(0)", "angleB(-90)";
219 ncline(falseout)(mux) "posB(inpa)";
220 ncline(trueout)(mux) "posB(inpb)";
221 ncline(mux)(out) "posA(out)";
224 \placeexample[here][ex:CaseInv]{Simple inverter.}
225 \startcombination[2*1]
226 {\typebufferhs{CaseInv}}{Haskell description using a Case expression.}
227 {\boxedgraphic{CaseInv}}{The architecture described by the Haskell description.}
230 A slightly more complex (but very powerful) form of choice is pattern
231 matching. A function can be defined in multiple clauses, where each clause
232 specifies a pattern. When the arguments match the pattern, the
233 corresponding clause will be used.
235 \startbuffer[PatternInv]
241 \placeexample[here][ex:PatternInv]{Simple inverter using pattern matching.
242 Describes the same architecture as \in{example}[ex:CaseInv].}
243 {\typebufferhs{PatternInv}}
245 The architecture described by \in{example}[ex:PatternInv] is of course the
246 same one as the one in \in{example}[ex:CaseInv]. The general interpretation
247 of pattern matching is also similar to that of \hs{case} expressions: generate
248 hardware for each of the clauses (like each of the clauses of a \hs{case}
249 expression) and connect them to the function output through (a number of
250 nested) multiplexers. These multiplexers are driven by comparators and
251 other logic, that check each pattern in turn.
253 In these examples we have seen only binary case expressions and pattern
254 matches (\ie, with two alternatives). In practice, case expressions can
255 choose between more than two values, resulting in a number of nested
259 Translation of two most basic functional concepts has been
260 discussed: function application and choice. Before looking further
261 into less obvious concepts like higher-order expressions and
262 polymorphism, the possible types that can be used in hardware
263 descriptions will be discussed.
265 Some way is needed to translate every values used to its hardware
266 equivalents. In particular, this means a hardware equivalent for
267 every \emph{type} used in a hardware description is needed
269 Since most functional languages have a lot of standard types that
270 are hard to translate (integers without a fixed size, lists without
271 a static length, etc.), a number of \quote{built-in} types will be
272 defined first. These types are built-in in the sense that our
273 compiler will have a fixed VHDL type for these. User defined types,
274 on the other hand, will have their hardware type derived directly
275 from their Haskell declaration automatically, according to the rules
278 \todo{Introduce Haskell type syntax (type constructors, type application,
281 \subsection{Built-in types}
282 The language currently supports the following built-in types. Of these,
283 only the \hs{Bool} type is supported by Haskell out of the box (the
284 others are defined by the Cλash package, so they are user-defined types
285 from Haskell's point of view).
288 This is the most basic type available. It is mapped directly onto
289 the \type{std_logic} \small{VHDL} type. Mapping this to the
290 \type{bit} type might make more sense (since the Haskell version
291 only has two values), but using \type{std_logic} is more standard
292 (and allowed for some experimentation with don't care values)
294 \todo{Sidenote bit vs stdlogic}
296 \startdesc{\hs{Bool}}
297 This is the only built-in Haskell type supported and is translated
298 exactly like the Bit type (where a value of \hs{True} corresponds to a
299 value of \hs{High}). Supporting the Bool type is particularly
300 useful to support \hs{if ... then ... else ...} expressions, which
301 always have a \hs{Bool} value for the condition.
303 A \hs{Bool} is translated to a \type{std_logic}, just like \hs{Bit}.
305 \startdesc{\hs{SizedWord}, \hs{SizedInt}}
306 These are types to represent integers. A \hs{SizedWord} is unsigned,
307 while a \hs{SizedInt} is signed. These types are parameterized by a
308 length type, so you can define an unsigned word of 32 bits wide as
312 type Word32 = SizedWord D32
315 Here, a type synonym \hs{Word32} is defined that is equal to the
316 \hs{SizedWord} type constructor applied to the type \hs{D32}. \hs{D32}
317 is the \emph{type level representation} of the decimal number 32,
318 making the \hs{Word32} type a 32-bit unsigned word.
320 These types are translated to the \small{VHDL} \type{unsigned} and
321 \type{signed} respectively.
322 \todo{Sidenote on dependent typing?}
324 \startdesc{\hs{Vector}}
325 This is a vector type, that can contain elements of any other type and
326 has a fixed length. It has two type parameters: its
327 length and the type of the elements contained in it. By putting the
328 length parameter in the type, the length of a vector can be determined
329 at compile time, instead of only at runtime for conventional lists.
331 The \hs{Vector} type constructor takes two type arguments: the length
332 of the vector and the type of the elements contained in it. The state
333 type of an 8 element register bank would then for example be:
336 type RegisterState = Vector D8 Word32
339 Here, a type synonym \hs{RegisterState} is defined that is equal to
340 the \hs{Vector} type constructor applied to the types \hs{D8} (The type
341 level representation of the decimal number 8) and \hs{Word32} (The 32
342 bit word type as defined above). In other words, the
343 \hs{RegisterState} type is a vector of 8 32-bit words.
345 A fixed size vector is translated to a \small{VHDL} array type.
347 \startdesc{\hs{RangedWord}}
348 This is another type to describe integers, but unlike the previous
349 two it has no specific bitwidth, but an upper bound. This means that
350 its range is not limited to powers of two, but can be any number.
351 A \hs{RangedWord} only has an upper bound, its lower bound is
352 implicitly zero. There is a lot of added implementation complexity
353 when adding a lower bound and having just an upper bound was enough
354 for the primary purpose of this type: typesafely indexing vectors.
356 To define an index for the 8 element vector above, we would do:
359 type RegisterIndex = RangedWord D7
362 Here, a type synonym \hs{RegisterIndex} is defined that is equal to
363 the \hs{RangedWord} type constructor applied to the type \hs{D7}. In
364 other words, this defines an unsigned word with values from
365 {\definedfont[Serif*normalnum]0 to 7} (inclusive). This word can be be used to index the
366 8 element vector \hs{RegisterState} above.
368 This type is translated to the \type{unsigned} \small{VHDL} type.
371 The integer and vector built-in types are discussed in more detail
374 \subsection{User-defined types}
375 There are three ways to define new types in Haskell: algebraic
376 datatypes with the \hs{data} keyword, type synonyms with the \hs{type}
377 keyword and type renamings with the \hs{newtype} keyword. \GHC\
378 offers a few more advanced ways to introduce types (type families,
379 existential typing, \small{GADT}s, etc.) which are not standard
380 Haskell. These will be left outside the scope of this research.
382 Only an algebraic datatype declaration actually introduces a
383 completely new type, for which we provide the \VHDL\ translation
384 below. Type synonyms and renamings only define new names for
385 existing types (where synonyms are completely interchangeable and
386 renamings need explicit conversion). Therefore, these do not need
387 any particular \VHDL\ translation, a synonym or renamed type will
388 just use the same representation as the original type. The
389 distinction between a renaming and a synonym does no longer matter
390 in hardware and can be disregarded in the generated \VHDL.
392 For algebraic types, we can make the following distinction:
394 \startdesc{Product types}
395 A product type is an algebraic datatype with a single constructor with
396 two or more fields, denoted in practice like (a,b), (a,b,c), etc. This
397 is essentially a way to pack a few values together in a record-like
398 structure. In fact, the built-in tuple types are just algebraic product
399 types (and are thus supported in exactly the same way).
401 The \quote{product} in its name refers to the collection of values belonging
402 to this type. The collection for a product type is the Cartesian
403 product of the collections for the types of its fields.
405 These types are translated to \VHDL\ record types, with one field for
406 every field in the constructor. This translation applies to all single
407 constructor algebraic datatypes, including those with just one
408 field (which are technically not a product, but generate a VHDL
409 record for implementation simplicity).
411 \startdesc{Enumerated types}
412 \defref{enumerated types}
413 An enumerated type is an algebraic datatype with multiple constructors, but
414 none of them have fields. This is essentially a way to get an
415 enum-like type containing alternatives.
417 Note that Haskell's \hs{Bool} type is also defined as an
418 enumeration type, but we have a fixed translation for that.
420 These types are translated to \VHDL\ enumerations, with one value for
421 each constructor. This allows references to these constructors to be
422 translated to the corresponding enumeration value.
424 \startdesc{Sum types}
425 A sum type is an algebraic datatype with multiple constructors, where
426 the constructors have one or more fields. Technically, a type with
427 more than one field per constructor is a sum of products type, but
428 for our purposes this distinction does not really make a
429 difference, so this distinction is note made.
431 The \quote{sum} in its name refers again to the collection of values
432 belonging to this type. The collection for a sum type is the
433 union of the the collections for each of the constructors.
435 Sum types are currently not supported by the prototype, since there is
436 no obvious \VHDL\ alternative. They can easily be emulated, however, as
437 we will see from an example:
440 data Sum = A Bit Word | B Word
443 An obvious way to translate this would be to create an enumeration to
444 distinguish the constructors and then create a big record that
445 contains all the fields of all the constructors. This is the same
446 translation that would result from the following enumeration and
447 product type (using a tuple for clarity):
451 type Sum = (SumC, Bit, Word, Word)
454 Here, the \hs{SumC} type effectively signals which of the latter three
455 fields of the \hs{Sum} type are valid (the first two if \hs{A}, the
456 last one if \hs{B}), all the other ones have no useful value.
458 An obvious problem with this naive approach is the space usage: the
459 example above generates a fairly big \VHDL\ type. Since we can be
460 sure that the two \hs{Word}s in the \hs{Sum} type will never be valid
461 at the same time, this is a waste of space.
463 Obviously, duplication detection could be used to reuse a
464 particular field for another constructor, but this would only
465 partially solve the problem. If two fields would be, for
466 example, an array of 8 bits and an 8 bit unsiged word, these are
467 different types and could not be shared. However, in the final
468 hardware, both of these types would simply be 8 bit connections,
469 so we have a 100\% size increase by not sharing these.
472 Another interesting case is that of recursive types. In Haskell, an
473 algebraic datatype can be recursive: any of its field types can be (or
474 contain) the type being defined. The most well-known recursive type is
475 probably the list type, which is defined is:
478 data List t = Empty | Cons t (List t)
481 Note that \hs{Empty} is usually written as \hs{[]} and \hs{Cons} as
482 \hs{:}, but this would make the definition harder to read. This
483 immediately shows the problem with recursive types: what hardware type
486 If the naive approach for sum types described above would be used,
487 a record would be created where the first field is an enumeration
488 to distinguish \hs{Empty} from \hs{Cons}. Furthermore, two more
489 fields would be added: one with the (\VHDL\ equivalent of) type
490 \hs{t} (assuming this type is actually known at compile time, this
491 should not be a problem) and a second one with type \hs{List t}.
492 The latter one is of course a problem: this is exactly the type
493 that was to be translated in the first place.
495 The resulting \VHDL\ type will thus become infinitely deep. In
496 other words, there is no way to statically determine how long
497 (deep) the list will be (it could even be infinite).
499 In general, recursive types can never be properly translated: all
500 recursive types have a potentially infinite value (even though in
501 practice they will have a bounded value, there is no way for the
502 compiler to automatically determine an upper bound on its size).
504 \subsection{Partial application}
505 Now the translation of application, choice and types has been
506 discussed, a more complex concept can be considered: partial
507 applications. A \emph{partial application} is any application whose
508 (return) type is (again) a function type.
510 From this, it should be clear that the translation rules for full
511 application does not apply to a partial application: there are not
512 enough values for all the input ports in the resulting \VHDL.
513 \in{Example}[ex:Quadruple] shows an example use of partial application
514 and the corresponding architecture.
516 \startbuffer[Quadruple]
517 -- | Multiply the input word by four.
518 quadruple :: Word -> Word
519 quadruple n = mul (mul n)
524 \startuseMPgraphic{Quadruple}
525 save in, two, mula, mulb, out;
528 newCircle.in(btex $n$ etex) "framed(false)";
529 newCircle.two(btex $2$ etex) "framed(false)";
530 newCircle.out(btex $out$ etex) "framed(false)";
533 newCircle.mula(btex $\times$ etex);
534 newCircle.mulb(btex $\times$ etex);
537 in.c = two.c + (0cm, 1cm);
538 mula.c = in.c + (2cm, 0cm);
539 mulb.c = mula.c + (2cm, 0cm);
540 out.c = mulb.c + (2cm, 0cm);
542 % Draw objects and lines
543 drawObj(in, two, mula, mulb, out);
545 nccurve(two)(mula) "angleA(0)", "angleB(45)";
546 nccurve(two)(mulb) "angleA(0)", "angleB(45)";
552 \placeexample[here][ex:Quadruple]{Simple three port and.}
553 \startcombination[2*1]
554 {\typebufferhs{Quadruple}}{Haskell description using function applications.}
555 {\boxedgraphic{Quadruple}}{The architecture described by the Haskell description.}
558 Here, the definition of mul is a partial function application: it applies
559 the function \hs{(*) :: Word -> Word -> Word} to the value \hs{2 :: Word},
560 resulting in the expression \hs{(*) 2 :: Word -> Word}. Since this resulting
561 expression is again a function, hardware cannot be generated for it
562 directly. This is because the hardware to generate for \hs{mul}
563 depends completely on where and how it is used. In this example, it is
566 However, it is clear that the above hardware description actually
567 describes valid hardware. In general, any partial applied function
568 must eventually become completely applied, at which point hardware for
569 it can be generated using the rules for function application given in
570 \in{section}[sec:description:application]. It might mean that a
571 partial application is passed around quite a bit (even beyond function
572 boundaries), but eventually, the partial application will become
573 completely applied. An example of this principe is given in
574 \in{section}[sec:normalization:defunctionalization].
576 \section{Costless specialization}
577 Each (complete) function application in our description generates a
578 component instantiation, or a specific piece of hardware in the final
579 design. It is interesting to note that each application of a function
580 generates a \emph{separate} piece of hardware. In the final design, none
581 of the hardware is shared between applications, even when the applied
582 function is the same (of course, if a particular value, such as the result
583 of a function application, is used twice, it is not calculated twice).
585 This is distinctly different from normal program compilation: two separate
586 calls to the same function share the same machine code. Having more
587 machine code has implications for speed (due to less efficient caching)
588 and memory usage. For normal compilation, it is therefore important to
589 keep the amount of functions limited and maximize the code sharing
590 (though there is a tradeoff between speed and memory usage here).
592 When generating hardware, this is hardly an issue. Having more \quote{code
593 sharing} does reduce the amount of \small{VHDL} output (Since different
594 component instantiations still share the same component), but after
595 synthesis, the amount of hardware generated is not affected. This
596 means there is no tradeoff between speed and memory (or rather,
599 In particular, if we would duplicate all functions so that there is a
600 separate function for every application in the program (\eg, each function
601 is then only applied exactly once), there would be no increase in hardware
604 Because of this, a common optimization technique called
605 \emph{specialization} can be applied to hardware generation without any
606 performance or area cost (unlike for software).
608 \fxnote{Perhaps these next three sections are a bit too
609 implementation-oriented?}
611 \subsection{Specialization}
612 \defref{specialization}
613 Given some function that has a \emph{domain} $D$ (\eg, the set of
614 all possible arguments that the function could be applied to), we
615 create a specialized function with exactly the same behaviour, but
616 with a domain $D' \subset D$. This subset can be chosen in all
617 sorts of ways. Any subset is valid for the general definition of
618 specialization, but in practice only some of them provide useful
619 optimization opportunities.
621 Common subsets include limiting a polymorphic argument to a single type
622 (\ie, removing polymorphism) or limiting an argument to just a single
623 value (\ie, cross-function constant propagation, effectively removing
626 Since we limit the argument domain of the specialized function, its
627 definition can often be optimized further (since now more types or even
628 values of arguments are already known). By replacing any application of
629 the function that falls within the reduced domain by an application of
630 the specialized version, the code gets faster (but the code also gets
631 bigger, since we now have two versions instead of one). If we apply
632 this technique often enough, we can often replace all applications of a
633 function by specialized versions, allowing the original function to be
634 removed (in some cases, this can even give a net reduction of the code
635 compared to the non-specialized version).
637 Specialization is useful for our hardware descriptions for functions
638 that contain arguments that cannot be translated to hardware directly
639 (polymorphic or higher-order arguments, for example). If we can create
640 specialized functions that remove the argument, or make it translatable,
641 we can use specialization to make the original, untranslatable, function
644 \section{Higher order values}
645 What holds for partial application, can be easily generalized to any
646 higher-order expression. This includes partial applications, plain
647 variables (e.g., a binder referring to a top level function), lambda
648 expressions and more complex expressions with a function type (a \hs{case}
649 expression returning lambda's, for example).
651 Each of these values cannot be directly represented in hardware (just like
652 partial applications). Also, to make them representable, they need to be
653 applied: function variables and partial applications will then eventually
654 become complete applications, applied lambda expressions disappear by
655 applying β-reduction, etc.
657 So any higher-order value will be \quote{pushed down} towards its
658 application just like partial applications. Whenever a function boundary
659 needs to be crossed, the called function can be specialized.
661 \fxnote{This section needs improvement and an example}
663 \section{Polymorphism}
664 In Haskell, values can be \emph{polymorphic}: they can have multiple types. For
665 example, the function \hs{fst :: (a, b) -> a} is an example of a
666 polymorphic function: it works for tuples with any two element types. Haskell
667 type classes allow a function to work on a specific set of types, but the
668 general idea is the same. The opposite of this is a \emph{monomorphic}
669 value, which has a single, fixed, type.
671 % A type class is a collection of types for which some operations are
672 % defined. It is thus possible for a value to be polymorphic while having
673 % any number of \emph{class constraints}: the value is not defined for
674 % every type, but only for types in the type class. An example of this is
675 % the \hs{even :: (Integral a) => a -> Bool} function, which can map any
676 % value of a type that is member of the \hs{Integral} type class
678 When generating hardware, polymorphism cannot be easily translated. How
679 many wires will you lay down for a value that could have any type? When
680 type classes are involved, what hardware components will you lay down for
681 a class method (whose behaviour depends on the type of its arguments)?
682 Note that Cλash currently does not allow user-defined type classes,
683 but does partly support some of the built-in type classes (like \hs{Num}).
685 Fortunately, we can again use the principle of specialization: since every
686 function application generates a separate piece of hardware, we can know
687 the types of all arguments exactly. Provided that existential typing
688 (which is a \GHC\ extension) is not used typing, all of the
689 polymorphic types in a function must depend on the types of the
690 arguments (In other words, the only way to introduce a type variable
691 is in a lambda abstraction).
693 If a function is monomorphic, all values inside it are monomorphic as
694 well, so any function that is applied within the function can only be
695 applied to monomorphic values. The applied functions can then be
696 specialized to work just for these specific types, removing the
697 polymorphism from the applied functions as well.
699 \defref{entry function}The entry function must not have a
700 polymorphic type (otherwise no hardware interface could be generated
701 for the entry function).
703 By induction, this means that all functions that are (indirectly) called
704 by our top level function (meaning all functions that are translated in
705 the final hardware) become monomorphic.
708 A very important concept in hardware designs is \emph{state}. In a
709 stateless (or, \emph{combinational}) design, every output is directly and solely dependent on the
710 inputs. In a stateful design, the outputs can depend on the history of
711 inputs, or the \emph{state}. State is usually stored in \emph{registers},
712 which retain their value during a clockcycle, and are typically updated at
713 the start of every clockcycle. Since the updating of the state is tightly
714 coupled (synchronized) to the clock signal, these state updates are often
715 called \emph{synchronous} behaviour.
717 \todo{Sidenote? Registers can contain any (complex) type}
719 To make Cλash useful to describe more than simple combinational
720 designs, it needs to be able to describe state in some way.
722 \subsection{Approaches to state}
723 In Haskell, functions are always pure (except when using unsafe
724 functions with the \hs{IO} monad, which is not supported by Cλash). This
725 means that the output of a function solely depends on its inputs. If you
726 evaluate a given function with given inputs, it will always provide the
731 \startframedtext[width=8cm,background=box,frame=no]
732 \startalignment[center]
737 A function is said to be pure if it satisfies two conditions:
740 \item When a pure function is called with the same arguments twice, it should
741 return the same value in both cases.
742 \item When a pure function is called, it should have not
743 observable side-effects.
746 Purity is an important property in functional languages, since
747 it enables all kinds of mathematical reasoning and
748 optimizattions with pure functions, that are not guaranteed to
749 be correct for impure functions.
751 An example of a pure function is the square root function
752 \hs{sqrt}. An example of an impure function is the \hs{today}
753 function that returns the current date (which of course cannot
754 exist like this in Haskell).
758 This is a perfect match for a combinational circuit, where the output
759 also solely depends on the inputs. However, when state is involved, this
760 no longer holds. Of course this purity constraint cannot just be
761 removed from Haskell. But even when designing a completely new (hardware
762 description) language, it does not seem to be a good idea to
763 remove this purity. This would that all kinds of interesting properties of
764 the functional language get lost, and all kinds of transformations
765 and optimizations are no longer be meaning preserving.
767 So our functions must remain pure, meaning the current state has
768 to be present in the function's arguments in some way. There seem
769 to be two obvious ways to do this: adding the current state as an
770 argument, or including the full history of each argument.
772 \subsubsection{Stream arguments and results}
773 Including the entire history of each input (\eg, the value of that
774 input for each previous clockcycle) is an obvious way to make outputs
775 depend on all previous input. This is easily done by making every
776 input a list instead of a single value, containing all previous values
777 as well as the current value.
779 An obvious downside of this solution is that on each cycle, all the
780 previous cycles must be resimulated to obtain the current state. To do
781 this, it might be needed to have a recursive helper function as well,
782 which might be hard to be properly analyzed by the compiler.
784 A slight variation on this approach is one taken by some of the other
785 functional \small{HDL}s in the field: \todo{References to Lava,
786 ForSyDe, ...} Make functions operate on complete streams. This means
787 that a function is no longer called on every cycle, but just once. It
788 takes stream as inputs instead of values, where each stream contains
789 all the values for every clockcycle since system start. This is easily
790 modeled using an (infinite) list, with one element for each clock
791 cycle. Since the function is only evaluated once, its output must also
792 be a stream. Note that, since we are working with infinite lists and
793 still want to be able to simulate the system cycle-by-cycle, this
794 relies heavily on the lazy semantics of Haskell.
796 Since our inputs and outputs are streams, all other (intermediate)
797 values must be streams. All of our primitive operators (\eg, addition,
798 substraction, bitwise operations, etc.) must operate on streams as
799 well (note that changing a single-element operation to a stream
800 operation can done with \hs{map}, \hs{zipwith}, etc.).
802 This also means that primitive operations from an existing
803 language such as Haskell cannot be used directly (including some
804 syntax constructs, like \hs{case} expressions and \hs{if}
805 expressions). This mkes this approach well suited for use in
806 \small{EDSL}s, since those already impose these same
807 limitations. \refdef{EDSL}
809 Note that the concept of \emph{state} is no more than having some way
810 to communicate a value from one cycle to the next. By introducing a
811 \hs{delay} function, we can do exactly that: delay (each value in) a
812 stream so that we can "look into" the past. This \hs{delay} function
813 simply outputs a stream where each value is the same as the input
814 value, but shifted one cycle. This causes a \quote{gap} at the
815 beginning of the stream: what is the value of the delay output in the
816 first cycle? For this, the \hs{delay} function has a second input, of
817 which only a single value is used.
819 \in{Example}[ex:DelayAcc] shows a simple accumulator expressed in this
822 \startbuffer[DelayAcc]
823 acc :: Stream Word -> Stream Word
826 out = (delay out 0) + in
829 \startuseMPgraphic{DelayAcc}
830 save in, out, add, reg;
833 newCircle.in(btex $in$ etex) "framed(false)";
834 newCircle.out(btex $out$ etex) "framed(false)";
837 newReg.reg("") "dx(4mm)", "dy(6mm)", "reflect(true)";
838 newCircle.add(btex + etex);
841 add.c = in.c + (2cm, 0cm);
842 out.c = add.c + (2cm, 0cm);
843 reg.c = add.c + (0cm, 2cm);
845 % Draw objects and lines
846 drawObj(in, out, add, reg);
848 nccurve(add)(reg) "angleA(0)", "angleB(180)", "posB(d)";
849 nccurve(reg)(add) "angleA(180)", "angleB(-45)", "posA(out)";
855 \placeexample[here][ex:DelayAcc]{Simple accumulator architecture.}
856 \startcombination[2*1]
857 {\typebufferhs{DelayAcc}}{Haskell description using streams.}
858 {\boxedgraphic{DelayAcc}}{The architecture described by the Haskell description.}
862 This notation can be confusing (especially due to the loop in the
863 definition of out), but is essentially easy to interpret. There is a
864 single call to delay, resulting in a circuit with a single register,
865 whose input is connected to \hs{out} (which is the output of the
866 adder), and its output is the expression \hs{delay out 0} (which is
867 connected to one of the adder inputs).
869 \subsubsection{Explicit state arguments and results}
870 A more explicit way to model state, is to simply add an extra argument
871 containing the current state value. This allows an output to depend on
872 both the inputs as well as the current state while keeping the
873 function pure (letting the result depend only on the arguments), since
874 the current state is now an argument.
876 In Haskell, this would look like
877 \in{example}[ex:ExplicitAcc]\footnote[notfinalsyntax]{This
878 example is not in the final Cλash syntax}. \todo{Referencing
879 notfinalsyntax from Introduction.tex doesn't work}
881 \startbuffer[ExplicitAcc]
882 -- input -> current state -> (new state, output)
883 acc :: Word -> Word -> (Word, Word)
890 \placeexample[here][ex:ExplicitAcc]{Simple accumulator architecture.}
891 \startcombination[2*1]
892 {\typebufferhs{ExplicitAcc}}{Haskell description using explicit state arguments.}
893 % Picture is identical to the one we had just now.
894 {\boxedgraphic{DelayAcc}}{The architecture described by the Haskell description.}
897 This approach makes a function's state very explicit, which state
898 variables are used by a function can be completely determined from its
899 type signature (as opposed to the stream approach, where a function
900 looks the same from the outside, regardless of what state variables it
901 uses or whether it is stateful at all).
903 This approach to state has been one of the initial drives behind
904 this research. Unlike a stream based approach it is well suited
905 to completely use existing code and language features (like
906 \hs{if} and \hs{case} expressions) because it operates on normal
907 values. Because of these reasons, this is the approach chosen
908 for Cλash. It will be examined more closely below.
910 \subsection{Explicit state specification}
911 The concept of explicit state has been introduced with some
912 examples above, but what are the implications of this approach?
914 \subsubsection{Substates}
915 Since a function's state is reflected directly in its type signature,
916 if a function calls other stateful functions (\eg, has subcircuits), it
917 has to somehow know the current state for these called functions. The
918 only way to do this, is to put these \emph{substates} inside the
919 caller's state. This means that a function's state is the sum of the
920 states of all functions it calls, and its own state. This sum
921 can be obtained using something simple like a tuple, or possibly
922 custom algebraic types for clarity.
924 This also means that the type of a function (at least the "state"
925 part) is dependent on its own implementation and of the functions it
928 This is the major downside of this approach: the separation between
929 interface and implementation is limited. However, since Cλash is not
930 very suitable for separate compilation (see
931 \in{section}[sec:prototype:separate]) this is not a big problem in
934 Additionally, when using a type synonym for the state type
935 of each function, we can still provide explicit type signatures
936 while keeping the state specification for a function near its
940 \subsubsection{Which arguments and results are stateful?}
941 \fxnote{This section should get some examples}
942 We need some way to know which arguments should become input ports and
943 which argument(s?) should become the current state (\eg, be bound to
944 the register outputs). This does not hold just for the top
945 level function, but also for any subfunction. Or could we perhaps
946 deduce the statefulness of subfunctions by analyzing the flow of data
947 in the calling functions?
949 To explore this matter, the following observeration is interesting: we
950 get completely correct behaviour when we put all state registers in
951 the top level entity (or even outside of it). All of the state
952 arguments and results on subfunctions are treated as normal input and
953 output ports. Effectively, a stateful function results in a stateless
954 hardware component that has one of its input ports connected to the
955 output of a register and one of its output ports connected to the
956 input of the same register.
960 Of course, even though the hardware described like this has the
961 correct behaviour, unless the layout tool does smart optimizations,
962 there will be a lot of extra wire in the design (since registers will
963 not be close to the component that uses them). Also, when working with
964 the generated \small{VHDL} code, there will be a lot of extra ports
965 just to pass on state values, which can get quite confusing.
967 To fix this, we can simply \quote{push} the registers down into the
968 subcircuits. When we see a register that is connected directly to a
969 subcircuit, we remove the corresponding input and output port and put
970 the register inside the subcircuit instead. This is slightly less
971 trivial when looking at the Haskell code instead of the resulting
972 circuit, but the idea is still the same.
976 However, when applying this technique, we might push registers down
977 too far. When you intend to store a result of a stateless subfunction
978 in the caller's state and pass the current value of that state
979 variable to that same function, the register might get pushed down too
980 far. It is impossible to distinguish this case from similar code where
981 the called function is in fact stateful. From this we can conclude
982 that we have to either:
984 \todo{Example of wrong downpushing}
987 \item accept that the generated hardware might not be exactly what we
988 intended, in some specific cases. In most cases, the hardware will be
990 \item explicitly annotate state arguments and results in the input
994 The first option causes (non-obvious) exceptions in the language
995 intepretation. Also, automatically determining where registers should
996 end up is easier to implement correctly with explicit annotations, so
997 for these reasons we will look at how this annotations could work.
999 \todo{Sidenote: one or more state arguments?}
1001 \subsection[sec:description:stateann]{Explicit state annotation}
1002 To make our stateful descriptions unambigious and easier to translate,
1003 we need some way for the developer to describe which arguments and
1004 results are intended to become stateful.
1006 Roughly, we have two ways to achieve this:
1008 \item Use some kind of annotation method or syntactic construction in
1009 the language to indicate exactly which argument and (part of the)
1010 result is stateful. This means that the annotation lives
1011 \quote{outside} of the function, it is completely invisible when
1012 looking at the function body.
1013 \item Use some kind of annotation on the type level, \ie\ give stateful
1014 arguments and stateful (parts of) results a different type. This has the
1015 potential to make this annotation visible inside the function as well,
1016 such that when looking at a value inside the function body you can
1017 tell if it is stateful by looking at its type. This could possibly make
1018 the translation process a lot easier, since less analysis of the
1019 program flow might be required.
1022 From these approaches, the type level \quote{annotations} have been
1023 implemented in Cλash. \in{Section}[sec:prototype:statetype] expands on
1024 the possible ways this could have been implemented.
1026 \todo{Note about conditions on state variables and checking them}
1028 \section[sec:recursion]{Recursion}
1029 An important concept in functional languages is recursion. In its most basic
1030 form, recursion is a definition that is described in terms of itself. A
1031 recursive function is thus a function that uses itself in its body. This
1032 usually requires multiple evaluations of this function, with changing
1033 arguments, until eventually an evaluation of the function no longer requires
1036 Given the notion that each function application will translate to a
1037 component instantiation, we are presented with a problem. A recursive
1038 function would translate to a component that contains itself. Or, more
1039 precisely, that contains an instance of itself. This instance would again
1040 contain an instance of itself, and again, into infinity. This is obviously a
1041 problem for generating hardware.
1043 This is expected for functions that describe infinite recursion. In that
1044 case, we cannot generate hardware that shows correct behaviour in a single
1045 cycle (at best, we could generate hardware that needs an infinite number of
1046 cycles to complete).
1048 However, most recursive definitions will describe finite
1049 recursion. This is because the recursive call is done conditionally. There
1050 is usually a \hs{case} expression where at least one alternative does not contain
1051 the recursive call, which we call the "base case". If, for each call to the
1052 recursive function, we would be able to detect at compile time which
1053 alternative applies, we would be able to remove the \hs{case} expression and
1054 leave only the base case when it applies. This will ensure that expanding
1055 the recursive functions will terminate after a bounded number of expansions.
1057 This does imply the extra requirement that the base case is detectable at
1058 compile time. In particular, this means that the decision between the base
1059 case and the recursive case must not depend on runtime data.
1061 \subsection{List recursion}
1062 The most common deciding factor in recursion is the length of a list that is
1063 passed in as an argument. Since we represent lists as vectors that encode
1064 the length in the vector type, it seems easy to determine the base case. We
1065 can simply look at the argument type for this. However, it turns out that
1066 this is rather non-trivial to write down in Haskell already, not even
1067 looking at translation. As an example, we would like to write down something
1071 sum :: Vector n Word -> Word
1072 sum xs = case null xs of
1074 False -> head xs + sum (tail xs)
1077 However, the Haskell typechecker will now use the following reasoning.
1078 For simplicity, the element type of a vector is left out, all vectors
1079 are assumed to have the same element type. Below, we write conditions
1080 on type variables before the \hs{=>} operator. This is not completely
1081 valid Haskell syntax, but serves to illustrate the typechecker
1082 reasoning. Also note that a vector can never have a negative length,
1083 so \hs{Vector n} implicitly means \hs{(n >= 0) => Vector n}.
1085 \todo{This typechecker disregards the type signature}
1087 \item tail has the type \hs{(n > 0) => Vector n -> Vector (n - 1)}
1088 \item This means that xs must have the type \hs{(n > 0) => Vector n}
1089 \item This means that sum must have the type \hs{(n > 0) => Vector n -> a}
1090 (The type \hs{a} is can be anything at this stage, we will not try to finds
1091 its actual type in this example).
1092 \item sum is called with the result of tail as an argument, which has the
1093 type \hs{Vector n} (since \hs{(n > 0) => Vector (n - 1)} is the same as \hs{(n >= 0)
1094 => Vector n}, which is the same as just \hs{Vector n}).
1095 \item This means that sum must have the type \hs{Vector n -> a}
1096 \item This is a contradiction between the type deduced from the body of sum
1097 (the input vector must be non-empty) and the use of sum (the input vector
1098 could have any length).
1101 As you can see, using a simple \hs{case} expression at value level causes
1102 the type checker to always typecheck both alternatives, which cannot be
1103 done. The typechecker is unable to distinguish the two case
1104 alternatives (this is partly possible using \small{GADT}s, but that
1105 approach faced other problems \todo{ref christiaan?}).
1107 This is a fundamental problem, that would seem perfectly suited for a
1108 type class. Considering that we need to switch between to
1109 implementations of the sum function, based on the type of the
1110 argument, this sounds like the perfect problem to solve with a type
1111 class. However, this approach has its own problems (not the least of
1112 them that you need to define a new type class for every recursive
1113 function you want to define).
1115 \todo{This should reference Christiaan}
1117 \subsection{General recursion}
1118 Of course there are other forms of recursion, that do not depend on the
1119 length (and thus type) of a list. For example, simple recursion using a
1120 counter could be expressed, but only translated to hardware for a fixed
1121 number of iterations. Also, this would require extensive support for compile
1122 time simplification (constant propagation) and compile time evaluation
1123 (evaluation of constant comparisons), to ensure non-termination.
1124 Supporting general recursion will probably require strict conditions
1125 on the input descriptions. Even then, it will be hard (if not
1126 impossible) to really guarantee termination, since the user (or \GHC\
1127 desugarer) might use some obscure notation that results in a corner
1128 case of the simplifier that is not caught and thus non-termination.
1130 Evaluating all possible (and non-possible) ways to add recursion to
1131 our descriptions, it seems better to limit the scope of this research
1132 to exclude recursion. This allows for focusing on other interesting
1133 areas instead. By including (built-in) support for a number of
1134 higher-order functions like \hs{map} and \hs{fold}, we can still
1135 express most of the things we would use (list) recursion for.
1138 % vim: set sw=2 sts=2 expandtab: