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 \todo{Sidenote: Inputs vs arguments}
60 The basic syntactic elements of a functional program are functions and
61 function application. These have a single obvious \small{VHDL}
62 translation: Each top level function becomes a hardware component, where each
63 argument is an input port and the result value is the (single) output
64 port. This output port can have a complex type (such as a tuple), so
65 having just a single output port does not pose a limitation.
67 Each function application in turn becomes component instantiation. Here, the
68 result of each argument expression is assigned to a signal, which is mapped
69 to the corresponding input port. The output port of the function is also
70 mapped to a signal, which is used as the result of the application.
72 Since every top level function generates its own component, the
73 hierarchy of of function calls is reflected in the final \VHDL\ output
74 as well, creating a hierarchical \VHDL\ description of the hardware.
75 This separation in different components makes the resulting \VHDL\
76 output easier to read and debug.
78 \in{Example}[ex:And3] shows a simple program using only function
79 application and the corresponding architecture.
82 -- | A simple function that returns
83 -- conjunction of three bits
84 and3 :: Bit -> Bit -> Bit -> Bit
85 and3 a b c = and (and a b) c
88 \startuseMPgraphic{And3}
89 save a, b, c, anda, andb, out;
92 newCircle.a(btex $a$ etex) "framed(false)";
93 newCircle.b(btex $b$ etex) "framed(false)";
94 newCircle.c(btex $c$ etex) "framed(false)";
95 newCircle.out(btex $out$ etex) "framed(false)";
98 newCircle.anda(btex $and$ etex);
99 newCircle.andb(btex $and$ etex);
102 b.c = a.c + (0cm, 1cm);
103 c.c = b.c + (0cm, 1cm);
104 anda.c = midpoint(a.c, b.c) + (2cm, 0cm);
105 andb.c = midpoint(b.c, c.c) + (4cm, 0cm);
107 out.c = andb.c + (2cm, 0cm);
109 % Draw objects and lines
110 drawObj(a, b, c, anda, andb, out);
112 ncarc(a)(anda) "arcangle(-10)";
119 \placeexample[here][ex:And3]{Simple three input and gate.}
120 \startcombination[2*1]
121 {\typebufferhs{And3}}{Haskell description using function applications.}
122 {\boxedgraphic{And3}}{The architecture described by the Haskell description.}
126 Although describing components and connections allows us to describe a lot of
127 hardware designs already, there is an obvious thing missing: Choice. We
128 need some way to be able to choose between values based on another value.
129 In Haskell, choice is achieved by \hs{case} expressions, \hs{if}
130 expressions, pattern matching and guards.
132 An obvious way to add choice to our language without having to recognize
133 any of Haskell's syntax, would be to add a primivite \quote{\hs{if}}
134 function. This function would take three arguments: The condition, the
135 value to return when the condition is true and the value to return when
136 the condition is false.
138 This \hs{if} function would then essentially describe a multiplexer and
139 allows us to describe any architecture that uses multiplexers. \fxnote{Are
140 there other mechanisms of choice in hardware?}
142 However, to be able to describe our hardware in a more convenient way, we
143 also want to translate Haskell's choice mechanisms. The easiest of these
144 are of course case expressions (and \hs{if} expressions, which can be very
145 directly translated to \hs{case} expressions). A \hs{case} expression can in turn
146 simply be translated to a conditional assignment, where the conditions use
147 equality comparisons against the constructors in the \hs{case} expressions.
149 \todo{Assignment vs multiplexers}
151 In \in{example}[ex:CaseInv] a simple \hs{case} expression is shown,
152 scrutinizing a boolean value. The corresponding architecture has a
153 comparator to determine which of the constructors is on the \hs{in}
154 input. There is a multiplexer to select the output signal. The two options
155 for the output signals are just constants, but these could have been more
156 complex expressions (in which case also both of them would be working in
157 parallel, regardless of which output would be chosen eventually).
159 If we would translate a Boolean to a bit value, we could of course remove
160 the comparator and directly feed 'in' into the multiplexer (or even use an
161 inverter instead of a multiplexer). However, we will try to make a
162 general translation, which works for all possible \hs{case} expressions.
163 Optimizations such as these are left for the \VHDL\ synthesizer, which
164 handles them very well.
166 \todo{Be more explicit about >2 alternatives}
168 \startbuffer[CaseInv]
175 \startuseMPgraphic{CaseInv}
176 save in, truecmp, falseout, trueout, out, cmp, mux;
179 newCircle.in(btex $in$ etex) "framed(false)";
180 newCircle.out(btex $out$ etex) "framed(false)";
182 newBox.truecmp(btex $True$ etex) "framed(false)";
183 newBox.trueout(btex $True$ etex) "framed(false)";
184 newBox.falseout(btex $False$ etex) "framed(false)";
187 newCircle.cmp(btex $==$ etex);
191 cmp.c = in.c + (3cm, 0cm);
192 truecmp.c = cmp.c + (-1cm, 1cm);
193 mux.sel = cmp.e + (1cm, -1cm);
194 falseout.c = mux.inpa - (2cm, 0cm);
195 trueout.c = mux.inpb - (2cm, 0cm);
196 out.c = mux.out + (2cm, 0cm);
198 % Draw objects and lines
199 drawObj(in, out, truecmp, trueout, falseout, cmp, mux);
203 nccurve(cmp.e)(mux.sel) "angleA(0)", "angleB(-90)";
204 ncline(falseout)(mux) "posB(inpa)";
205 ncline(trueout)(mux) "posB(inpb)";
206 ncline(mux)(out) "posA(out)";
209 \placeexample[here][ex:CaseInv]{Simple inverter.}
210 \startcombination[2*1]
211 {\typebufferhs{CaseInv}}{Haskell description using a Case expression.}
212 {\boxedgraphic{CaseInv}}{The architecture described by the Haskell description.}
215 A slightly more complex (but very powerful) form of choice is pattern
216 matching. A function can be defined in multiple clauses, where each clause
217 specifies a pattern. When the arguments match the pattern, the
218 corresponding clause will be used.
220 \startbuffer[PatternInv]
226 \placeexample[here][ex:PatternInv]{Simple inverter using pattern matching.
227 Describes the same architecture as \in{example}[ex:CaseInv].}
228 {\typebufferhs{PatternInv}}
230 The architecture described by \in{example}[ex:PatternInv] is of course the
231 same one as the one in \in{example}[ex:CaseInv]. The general interpretation
232 of pattern matching is also similar to that of \hs{case} expressions: Generate
233 hardware for each of the clauses (like each of the clauses of a \hs{case}
234 expression) and connect them to the function output through (a number of
235 nested) multiplexers. These multiplexers are driven by comparators and
236 other logic, that check each pattern in turn.
239 Translation of two most basic functional concepts has been
240 discussed: Function application and choice. Before looking further
241 into less obvious concepts like higher-order expressions and
242 polymorphism, the possible types that can be used in hardware
243 descriptions will be discussed.
245 Some way is needed to translate every values used to its hardware
246 equivalents. In particular, this means a hardware equivalent for
247 every \emph{type} used in a hardware description is needed
249 Since most functional languages have a lot of standard types that
250 are hard to translate (integers without a fixed size, lists without
251 a static length, etc.), a number of \quote{built-in} types will be
252 defined first. These types are built-in in the sense that our
253 compiler will have a fixed VHDL type for these. User defined types,
254 on the other hand, will have their hardware type derived directly
255 from their Haskell declaration automatically, according to the rules
258 \todo{Introduce Haskell type syntax (type constructors, type application,
261 \subsection{Built-in types}
262 The language currently supports the following built-in types. Of these,
263 only the \hs{Bool} type is supported by Haskell out of the box (the
264 others are defined by the Cλash package, so they are user-defined types
265 from Haskell's point of view).
268 This is the most basic type available. It is mapped directly onto
269 the \type{std_logic} \small{VHDL} type. Mapping this to the
270 \type{bit} type might make more sense (since the Haskell version
271 only has two values), but using \type{std_logic} is more standard
272 (and allowed for some experimentation with don't care values)
274 \todo{Sidenote bit vs stdlogic}
276 \startdesc{\hs{Bool}}
277 This is the only built-in Haskell type supported and is translated
278 exactly like the Bit type (where a value of \hs{True} corresponds to a
279 value of \hs{High}). Supporting the Bool type is particularly
280 useful to support \hs{if ... then ... else ...} expressions, which
281 always have a \hs{Bool} value for the condition.
283 A \hs{Bool} is translated to a \type{std_logic}, just like \hs{Bit}.
285 \startdesc{\hs{SizedWord}, \hs{SizedInt}}
286 These are types to represent integers. A \hs{SizedWord} is unsigned,
287 while a \hs{SizedInt} is signed. These types are parameterized by a
288 length type, so you can define an unsigned word of 32 bits wide as
292 type Word32 = SizedWord D32
295 Here, a type synonym \hs{Word32} is defined that is equal to the
296 \hs{SizedWord} type constructor applied to the type \hs{D32}. \hs{D32}
297 is the \emph{type level representation} of the decimal number 32,
298 making the \hs{Word32} type a 32-bit unsigned word.
300 These types are translated to the \small{VHDL} \type{unsigned} and
301 \type{signed} respectively.
302 \todo{Sidenote on dependent typing?}
304 \startdesc{\hs{Vector}}
305 This is a vector type, that can contain elements of any other type and
306 has a fixed length. It has two type parameters: Its
307 length and the type of the elements contained in it. By putting the
308 length parameter in the type, the length of a vector can be determined
309 at compile time, instead of only at runtime for conventional lists.
311 The \hs{Vector} type constructor takes two type arguments: The length
312 of the vector and the type of the elements contained in it. The state
313 type of an 8 element register bank would then for example be:
316 type RegisterState = Vector D8 Word32
319 Here, a type synonym \hs{RegisterState} is defined that is equal to
320 the \hs{Vector} type constructor applied to the types \hs{D8} (The type
321 level representation of the decimal number 8) and \hs{Word32} (The 32
322 bit word type as defined above). In other words, the
323 \hs{RegisterState} type is a vector of 8 32-bit words.
325 A fixed size vector is translated to a \small{VHDL} array type.
327 \startdesc{\hs{RangedWord}}
328 This is another type to describe integers, but unlike the previous
329 two it has no specific bitwidth, but an upper bound. This means that
330 its range is not limited to powers of two, but can be any number.
331 A \hs{RangedWord} only has an upper bound, its lower bound is
332 implicitly zero. There is a lot of added implementation complexity
333 when adding a lower bound and having just an upper bound was enough
334 for the primary purpose of this type: Typesafely indexing vectors.
336 To define an index for the 8 element vector above, we would do:
339 type RegisterIndex = RangedWord D7
342 Here, a type synonym \hs{RegisterIndex} is defined that is equal to
343 the \hs{RangedWord} type constructor applied to the type \hs{D7}. In
344 other words, this defines an unsigned word with values from
345 {\definedfont[Serif*normalnum]0 to 7} (inclusive). This word can be be used to index the
346 8 element vector \hs{RegisterState} above.
348 This type is translated to the \type{unsigned} \small{VHDL} type.
351 The integer and vector built-in types are discussed in more detail
354 \subsection{User-defined types}
355 There are three ways to define new types in Haskell: Algebraic
356 datatypes with the \hs{data} keyword, type synonyms with the \hs{type}
357 keyword and type renamings with the \hs{newtype} keyword. \GHC\
358 offers a few more advanced ways to introduce types (type families,
359 existential typing, \small{GADT}s, etc.) which are not standard
360 Haskell. These will be left outside the scope of this research.
362 Only an algebraic datatype declaration actually introduces a
363 completely new type, for which we provide the \VHDL\ translation
364 below. Type synonyms and renamings only define new names for
365 existing types (where synonyms are completely interchangeable and
366 renamings need explicit conversion). Therefore, these do not need
367 any particular \VHDL\ translation, a synonym or renamed type will
368 just use the same representation as the original type. The
369 distinction between a renaming and a synonym does no longer matter
370 in hardware and can be disregarded in the generated \VHDL.
372 For algebraic types, we can make the following distinction:
374 \startdesc{Product types}
375 A product type is an algebraic datatype with a single constructor with
376 two or more fields, denoted in practice like (a,b), (a,b,c), etc. This
377 is essentially a way to pack a few values together in a record-like
378 structure. In fact, the built-in tuple types are just algebraic product
379 types (and are thus supported in exactly the same way).
381 The \quote{product} in its name refers to the collection of values belonging
382 to this type. The collection for a product type is the Cartesian
383 product of the collections for the types of its fields.
385 These types are translated to \VHDL\ record types, with one field for
386 every field in the constructor. This translation applies to all single
387 constructor algebraic datatypes, including those with just one
388 field (which are technically not a product, but generate a VHDL
389 record for implementation simplicity).
391 \startdesc{Enumerated types}
392 \defref{enumerated types}
393 An enumerated type is an algebraic datatype with multiple constructors, but
394 none of them have fields. This is essentially a way to get an
395 enum-like type containing alternatives.
397 Note that Haskell's \hs{Bool} type is also defined as an
398 enumeration type, but we have a fixed translation for that.
400 These types are translated to \VHDL\ enumerations, with one value for
401 each constructor. This allows references to these constructors to be
402 translated to the corresponding enumeration value.
404 \startdesc{Sum types}
405 A sum type is an algebraic datatype with multiple constructors, where
406 the constructors have one or more fields. Technically, a type with
407 more than one field per constructor is a sum of products type, but
408 for our purposes this distinction does not really make a
409 difference, so this distinction is note made.
411 The \quote{sum} in its name refers again to the collection of values
412 belonging to this type. The collection for a sum type is the
413 union of the the collections for each of the constructors.
415 Sum types are currently not supported by the prototype, since there is
416 no obvious \VHDL\ alternative. They can easily be emulated, however, as
417 we will see from an example:
420 data Sum = A Bit Word | B Word
423 An obvious way to translate this would be to create an enumeration to
424 distinguish the constructors and then create a big record that
425 contains all the fields of all the constructors. This is the same
426 translation that would result from the following enumeration and
427 product type (using a tuple for clarity):
431 type Sum = (SumC, Bit, Word, Word)
434 Here, the \hs{SumC} type effectively signals which of the latter three
435 fields of the \hs{Sum} type are valid (the first two if \hs{A}, the
436 last one if \hs{B}), all the other ones have no useful value.
438 An obvious problem with this naive approach is the space usage: The
439 example above generates a fairly big \VHDL\ type. Since we can be
440 sure that the two \hs{Word}s in the \hs{Sum} type will never be valid
441 at the same time, this is a waste of space.
443 Obviously, duplication detection could be used to reuse a
444 particular field for another constructor, but this would only
445 partially solve the problem. If two fields would be, for
446 example, an array of 8 bits and an 8 bit unsiged word, these are
447 different types and could not be shared. However, in the final
448 hardware, both of these types would simply be 8 bit connections,
449 so we have a 100\% size increase by not sharing these.
452 Another interesting case is that of recursive types. In Haskell, an
453 algebraic datatype can be recursive: Any of its field types can be (or
454 contain) the type being defined. The most well-known recursive type is
455 probably the list type, which is defined is:
458 data List t = Empty | Cons t (List t)
461 Note that \hs{Empty} is usually written as \hs{[]} and \hs{Cons} as
462 \hs{:}, but this would make the definition harder to read. This
463 immediately shows the problem with recursive types: What hardware type
466 If the naive approach for sum types described above would be used,
467 a record would be created where the first field is an enumeration
468 to distinguish \hs{Empty} from \hs{Cons}. Furthermore, two more
469 fields would be added: One with the (\VHDL\ equivalent of) type
470 \hs{t} (assuming this type is actually known at compile time, this
471 should not be a problem) and a second one with type \hs{List t}.
472 The latter one is of course a problem: This is exactly the type
473 that was to be translated in the first place.
475 The resulting \VHDL\ type will thus become infinitely deep. In
476 other words, there is no way to statically determine how long
477 (deep) the list will be (it could even be infinite).
479 In general, recursive types can never be properly translated: All
480 recursive types have a potentially infinite value (even though in
481 practice they will have a bounded value, there is no way for the
482 compiler to automatically determine an upper bound on its size).
484 \subsection{Partial application}
485 Now the translation of application, choice and types has been
486 discussed, a more complex concept can be considered: Partial
487 applications. A \emph{partial application} is any application whose
488 (return) type is (again) a function type.
490 From this, it should be clear that the translation rules for full
491 application does not apply to a partial application: There are not
492 enough values for all the input ports in the resulting \VHDL.
493 \in{Example}[ex:Quadruple] shows an example use of partial application
494 and the corresponding architecture.
496 \startbuffer[Quadruple]
497 -- | Multiply the input word by four.
498 quadruple :: Word -> Word
499 quadruple n = mul (mul n)
504 \startuseMPgraphic{Quadruple}
505 save in, two, mula, mulb, out;
508 newCircle.in(btex $n$ etex) "framed(false)";
509 newCircle.two(btex $2$ etex) "framed(false)";
510 newCircle.out(btex $out$ etex) "framed(false)";
513 newCircle.mula(btex $\times$ etex);
514 newCircle.mulb(btex $\times$ etex);
517 in.c = two.c + (0cm, 1cm);
518 mula.c = in.c + (2cm, 0cm);
519 mulb.c = mula.c + (2cm, 0cm);
520 out.c = mulb.c + (2cm, 0cm);
522 % Draw objects and lines
523 drawObj(in, two, mula, mulb, out);
525 nccurve(two)(mula) "angleA(0)", "angleB(45)";
526 nccurve(two)(mulb) "angleA(0)", "angleB(45)";
532 \placeexample[here][ex:Quadruple]{Simple three port and.}
533 \startcombination[2*1]
534 {\typebufferhs{Quadruple}}{Haskell description using function applications.}
535 {\boxedgraphic{Quadruple}}{The architecture described by the Haskell description.}
538 Here, the definition of mul is a partial function application: It applies
539 the function \hs{(*) :: Word -> Word -> Word} to the value \hs{2 :: Word},
540 resulting in the expression \hs{(*) 2 :: Word -> Word}. Since this resulting
541 expression is again a function, hardware cannot be generated for it
542 directly. This is because the hardware to generate for \hs{mul}
543 depends completely on where and how it is used. In this example, it is
546 However, it is clear that the above hardware description actually
547 describes valid hardware. In general, any partial applied function
548 must eventually become completely applied, at which point hardware for
549 it can be generated using the rules for function application given in
550 \in{section}[sec:description:application]. It might mean that a
551 partial application is passed around quite a bit (even beyond function
552 boundaries), but eventually, the partial application will become
553 completely applied. An example of this principe is given in
554 \in{section}[sec:normalization:defunctionalization].
556 \section{Costless specialization}
557 Each (complete) function application in our description generates a
558 component instantiation, or a specific piece of hardware in the final
559 design. It is interesting to note that each application of a function
560 generates a \emph{separate} piece of hardware. In the final design, none
561 of the hardware is shared between applications, even when the applied
562 function is the same (of course, if a particular value, such as the result
563 of a function application, is used twice, it is not calculated twice).
565 This is distinctly different from normal program compilation: Two separate
566 calls to the same function share the same machine code. Having more
567 machine code has implications for speed (due to less efficient caching)
568 and memory usage. For normal compilation, it is therefore important to
569 keep the amount of functions limited and maximize the code sharing
570 (though there is a tradeoff between speed and memory usage here).
572 When generating hardware, this is hardly an issue. Having more \quote{code
573 sharing} does reduce the amount of \small{VHDL} output (Since different
574 component instantiations still share the same component), but after
575 synthesis, the amount of hardware generated is not affected. This
576 means there is no tradeoff between speed and memory (or rather,
579 In particular, if we would duplicate all functions so that there is a
580 separate function for every application in the program (\eg, each function
581 is then only applied exactly once), there would be no increase in hardware
584 Because of this, a common optimization technique called
585 \emph{specialization} can be applied to hardware generation without any
586 performance or area cost (unlike for software).
588 \fxnote{Perhaps these next three sections are a bit too
589 implementation-oriented?}
591 \subsection{Specialization}
592 \defref{specialization}
593 Given some function that has a \emph{domain} $D$ (\eg, the set of
594 all possible arguments that the function could be applied to), we
595 create a specialized function with exactly the same behaviour, but
596 with a domain $D' \subset D$. This subset can be chosen in all
597 sorts of ways. Any subset is valid for the general definition of
598 specialization, but in practice only some of them provide useful
599 optimization opportunities.
601 Common subsets include limiting a polymorphic argument to a single type
602 (\ie, removing polymorphism) or limiting an argument to just a single
603 value (\ie, cross-function constant propagation, effectively removing
606 Since we limit the argument domain of the specialized function, its
607 definition can often be optimized further (since now more types or even
608 values of arguments are already known). By replacing any application of
609 the function that falls within the reduced domain by an application of
610 the specialized version, the code gets faster (but the code also gets
611 bigger, since we now have two versions instead of one). If we apply
612 this technique often enough, we can often replace all applications of a
613 function by specialized versions, allowing the original function to be
614 removed (in some cases, this can even give a net reduction of the code
615 compared to the non-specialized version).
617 Specialization is useful for our hardware descriptions for functions
618 that contain arguments that cannot be translated to hardware directly
619 (polymorphic or higher-order arguments, for example). If we can create
620 specialized functions that remove the argument, or make it translatable,
621 we can use specialization to make the original, untranslatable, function
624 \section{Higher order values}
625 What holds for partial application, can be easily generalized to any
626 higher-order expression. This includes partial applications, plain
627 variables (e.g., a binder referring to a top level function), lambda
628 expressions and more complex expressions with a function type (a \hs{case}
629 expression returning lambda's, for example).
631 Each of these values cannot be directly represented in hardware (just like
632 partial applications). Also, to make them representable, they need to be
633 applied: function variables and partial applications will then eventually
634 become complete applications, applied lambda expressions disappear by
635 applying β-reduction, etc.
637 So any higher-order value will be \quote{pushed down} towards its
638 application just like partial applications. Whenever a function boundary
639 needs to be crossed, the called function can be specialized.
641 \fxnote{This section needs improvement and an example}
643 \section{Polymorphism}
644 In Haskell, values can be \emph{polymorphic}: They can have multiple types. For
645 example, the function \hs{fst :: (a, b) -> a} is an example of a
646 polymorphic function: It works for tuples with any two element types. Haskell
647 type classes allow a function to work on a specific set of types, but the
648 general idea is the same. The opposite of this is a \emph{monomorphic}
649 value, which has a single, fixed, type.
651 % A type class is a collection of types for which some operations are
652 % defined. It is thus possible for a value to be polymorphic while having
653 % any number of \emph{class constraints}: The value is not defined for
654 % every type, but only for types in the type class. An example of this is
655 % the \hs{even :: (Integral a) => a -> Bool} function, which can map any
656 % value of a type that is member of the \hs{Integral} type class
658 When generating hardware, polymorphism cannot be easily translated. How
659 many wires will you lay down for a value that could have any type? When
660 type classes are involved, what hardware components will you lay down for
661 a class method (whose behaviour depends on the type of its arguments)?
662 Note that Cλash currently does not allow user-defined type classes,
663 but does partly support some of the built-in type classes (like \hs{Num}).
665 Fortunately, we can again use the principle of specialization: Since every
666 function application generates a separate piece of hardware, we can know
667 the types of all arguments exactly. Provided that existential typing
668 (which is a \GHC\ extension) is not used typing, all of the
669 polymorphic types in a function must depend on the types of the
670 arguments (In other words, the only way to introduce a type variable
671 is in a lambda abstraction).
673 If a function is monomorphic, all values inside it are monomorphic as
674 well, so any function that is applied within the function can only be
675 applied to monomorphic values. The applied functions can then be
676 specialized to work just for these specific types, removing the
677 polymorphism from the applied functions as well.
679 \defref{entry function}The entry function must not have a
680 polymorphic type (otherwise no hardware interface could be generated
681 for the entry function).
683 By induction, this means that all functions that are (indirectly) called
684 by our top level function (meaning all functions that are translated in
685 the final hardware) become monomorphic.
688 A very important concept in hardware designs is \emph{state}. In a
689 stateless (or, \emph{combinational}) design, every output is directly and solely dependent on the
690 inputs. In a stateful design, the outputs can depend on the history of
691 inputs, or the \emph{state}. State is usually stored in \emph{registers},
692 which retain their value during a clockcycle, and are typically updated at
693 the start of every clockcycle. Since the updating of the state is tightly
694 coupled (synchronized) to the clock signal, these state updates are often
695 called \emph{synchronous} behaviour.
697 \todo{Sidenote? Registers can contain any (complex) type}
699 To make Cλash useful to describe more than simple combinational
700 designs, it needs to be able to describe state in some way.
702 \subsection{Approaches to state}
703 In Haskell, functions are always pure (except when using unsafe
704 functions with the \hs{IO} monad, which is not supported by Cλash). This
705 means that the output of a function solely depends on its inputs. If you
706 evaluate a given function with given inputs, it will always provide the
711 \startframedtext[width=8cm,background=box,frame=no]
712 \startalignment[center]
717 A function is said to be pure if it satisfies two conditions:
720 \item When a pure function is called with the same arguments twice, it should
721 return the same value in both cases.
722 \item When a pure function is called, it should have not
723 observable side-effects.
726 Purity is an important property in functional languages, since
727 it enables all kinds of mathematical reasoning and
728 optimizattions with pure functions, that are not guaranteed to
729 be correct for impure functions.
731 An example of a pure function is the square root function
732 \hs{sqrt}. An example of an impure function is the \hs{today}
733 function that returns the current date (which of course cannot
734 exist like this in Haskell).
738 This is a perfect match for a combinational circuit, where the output
739 also solely depends on the inputs. However, when state is involved, this
740 no longer holds. Of course this purity constraint cannot just be
741 removed from Haskell. But even when designing a completely new (hardware
742 description) language, it does not seem to be a good idea to
743 remove this purity. This would that all kinds of interesting properties of
744 the functional language get lost, and all kinds of transformations
745 and optimizations are no longer be meaning preserving.
747 So our functions must remain pure, meaning the current state has
748 to be present in the function's arguments in some way. There seem
749 to be two obvious ways to do this: Adding the current state as an
750 argument, or including the full history of each argument.
752 \subsubsection{Stream arguments and results}
753 Including the entire history of each input (\eg, the value of that
754 input for each previous clockcycle) is an obvious way to make outputs
755 depend on all previous input. This is easily done by making every
756 input a list instead of a single value, containing all previous values
757 as well as the current value.
759 An obvious downside of this solution is that on each cycle, all the
760 previous cycles must be resimulated to obtain the current state. To do
761 this, it might be needed to have a recursive helper function as well,
762 which might be hard to be properly analyzed by the compiler.
764 A slight variation on this approach is one taken by some of the other
765 functional \small{HDL}s in the field: \todo{References to Lava,
766 ForSyDe, ...} Make functions operate on complete streams. This means
767 that a function is no longer called on every cycle, but just once. It
768 takes stream as inputs instead of values, where each stream contains
769 all the values for every clockcycle since system start. This is easily
770 modeled using an (infinite) list, with one element for each clock
771 cycle. Since the function is only evaluated once, its output must also
772 be a stream. Note that, since we are working with infinite lists and
773 still want to be able to simulate the system cycle-by-cycle, this
774 relies heavily on the lazy semantics of Haskell.
776 Since our inputs and outputs are streams, all other (intermediate)
777 values must be streams. All of our primitive operators (\eg, addition,
778 substraction, bitwise operations, etc.) must operate on streams as
779 well (note that changing a single-element operation to a stream
780 operation can done with \hs{map}, \hs{zipwith}, etc.).
782 This also means that primitive operations from an existing
783 language such as Haskell cannot be used directly (including some
784 syntax constructs, like \hs{case} expressions and \hs{if}
785 expressions). This mkes this approach well suited for use in
786 \small{EDSL}s, since those already impose these same
787 limitations. \refdef{EDSL}
789 Note that the concept of \emph{state} is no more than having some way
790 to communicate a value from one cycle to the next. By introducing a
791 \hs{delay} function, we can do exactly that: Delay (each value in) a
792 stream so that we can "look into" the past. This \hs{delay} function
793 simply outputs a stream where each value is the same as the input
794 value, but shifted one cycle. This causes a \quote{gap} at the
795 beginning of the stream: What is the value of the delay output in the
796 first cycle? For this, the \hs{delay} function has a second input
797 (which is a value, not a stream!).
799 \in{Example}[ex:DelayAcc] shows a simple accumulator expressed in this
802 \startbuffer[DelayAcc]
803 acc :: Stream Word -> Stream Word
806 out = (delay out 0) + in
809 \startuseMPgraphic{DelayAcc}
810 save in, out, add, reg;
813 newCircle.in(btex $in$ etex) "framed(false)";
814 newCircle.out(btex $out$ etex) "framed(false)";
817 newReg.reg("") "dx(4mm)", "dy(6mm)", "reflect(true)";
818 newCircle.add(btex + etex);
821 add.c = in.c + (2cm, 0cm);
822 out.c = add.c + (2cm, 0cm);
823 reg.c = add.c + (0cm, 2cm);
825 % Draw objects and lines
826 drawObj(in, out, add, reg);
828 nccurve(add)(reg) "angleA(0)", "angleB(180)", "posB(d)";
829 nccurve(reg)(add) "angleA(180)", "angleB(-45)", "posA(out)";
835 \placeexample[here][ex:DelayAcc]{Simple accumulator architecture.}
836 \startcombination[2*1]
837 {\typebufferhs{DelayAcc}}{Haskell description using streams.}
838 {\boxedgraphic{DelayAcc}}{The architecture described by the Haskell description.}
842 This notation can be confusing (especially due to the loop in the
843 definition of out), but is essentially easy to interpret. There is a
844 single call to delay, resulting in a circuit with a single register,
845 whose input is connected to \hs{out} (which is the output of the
846 adder), and its output is the expression \hs{delay out 0} (which is
847 connected to one of the adder inputs).
849 \subsubsection{Explicit state arguments and results}
850 A more explicit way to model state, is to simply add an extra argument
851 containing the current state value. This allows an output to depend on
852 both the inputs as well as the current state while keeping the
853 function pure (letting the result depend only on the arguments), since
854 the current state is now an argument.
856 In Haskell, this would look like
857 \in{example}[ex:ExplicitAcc]\footnote[notfinalsyntax]{This
858 example is not in the final Cλash syntax}. \todo{Referencing
859 notfinalsyntax from Introduction.tex doesn't work}
861 \startbuffer[ExplicitAcc]
862 -- input -> current state -> (new state, output)
863 acc :: Word -> Word -> (Word, Word)
870 \placeexample[here][ex:ExplicitAcc]{Simple accumulator architecture.}
871 \startcombination[2*1]
872 {\typebufferhs{ExplicitAcc}}{Haskell description using explicit state arguments.}
873 % Picture is identical to the one we had just now.
874 {\boxedgraphic{DelayAcc}}{The architecture described by the Haskell description.}
877 This approach makes a function's state very explicit, which state
878 variables are used by a function can be completely determined from its
879 type signature (as opposed to the stream approach, where a function
880 looks the same from the outside, regardless of what state variables it
881 uses or whether it is stateful at all).
883 This approach to state has been one of the initial drives behind
884 this research. Unlike a stream based approach it is well suited
885 to completely use existing code and language features (like
886 \hs{if} and \hs{case} expressions) because it operates on normal
887 values. Because of these reasons, this is the approach chosen
888 for Cλash. It will be examined more closely below.
890 \subsection{Explicit state specification}
891 The concept of explicit state has been introduced with some
892 examples above, but what are the implications of this approach?
894 \subsubsection{Substates}
895 Since a function's state is reflected directly in its type signature,
896 if a function calls other stateful functions (\eg, has subcircuits), it
897 has to somehow know the current state for these called functions. The
898 only way to do this, is to put these \emph{substates} inside the
899 caller's state. This means that a function's state is the sum of the
900 states of all functions it calls, and its own state. This sum
901 can be obtained using something simple like a tuple, or possibly
902 custom algebraic types for clarity.
904 This also means that the type of a function (at least the "state"
905 part) is dependent on its own implementation and of the functions it
908 This is the major downside of this approach: The separation between
909 interface and implementation is limited. However, since Cλash is not
910 very suitable for separate compilation (see
911 \in{section}[sec:prototype:separate]) this is not a big problem in
914 Additionally, when using a type synonym for the state type
915 of each function, we can still provide explicit type signatures
916 while keeping the state specification for a function near its
920 \subsubsection{Which arguments and results are stateful?}
921 \fxnote{This section should get some examples}
922 We need some way to know which arguments should become input ports and
923 which argument(s?) should become the current state (\eg, be bound to
924 the register outputs). This does not hold just for the top
925 level function, but also for any subfunction. Or could we perhaps
926 deduce the statefulness of subfunctions by analyzing the flow of data
927 in the calling functions?
929 To explore this matter, the following observeration is interesting: We
930 get completely correct behaviour when we put all state registers in
931 the top level entity (or even outside of it). All of the state
932 arguments and results on subfunctions are treated as normal input and
933 output ports. Effectively, a stateful function results in a stateless
934 hardware component that has one of its input ports connected to the
935 output of a register and one of its output ports connected to the
936 input of the same register.
940 Of course, even though the hardware described like this has the
941 correct behaviour, unless the layout tool does smart optimizations,
942 there will be a lot of extra wire in the design (since registers will
943 not be close to the component that uses them). Also, when working with
944 the generated \small{VHDL} code, there will be a lot of extra ports
945 just to pass on state values, which can get quite confusing.
947 To fix this, we can simply \quote{push} the registers down into the
948 subcircuits. When we see a register that is connected directly to a
949 subcircuit, we remove the corresponding input and output port and put
950 the register inside the subcircuit instead. This is slightly less
951 trivial when looking at the Haskell code instead of the resulting
952 circuit, but the idea is still the same.
956 However, when applying this technique, we might push registers down
957 too far. When you intend to store a result of a stateless subfunction
958 in the caller's state and pass the current value of that state
959 variable to that same function, the register might get pushed down too
960 far. It is impossible to distinguish this case from similar code where
961 the called function is in fact stateful. From this we can conclude
962 that we have to either:
964 \todo{Example of wrong downpushing}
967 \item accept that the generated hardware might not be exactly what we
968 intended, in some specific cases. In most cases, the hardware will be
970 \item explicitly annotate state arguments and results in the input
974 The first option causes (non-obvious) exceptions in the language
975 intepretation. Also, automatically determining where registers should
976 end up is easier to implement correctly with explicit annotations, so
977 for these reasons we will look at how this annotations could work.
979 \todo{Sidenote: One or more state arguments?}
981 \subsection[sec:description:stateann]{Explicit state annotation}
982 To make our stateful descriptions unambigious and easier to translate,
983 we need some way for the developer to describe which arguments and
984 results are intended to become stateful.
986 Roughly, we have two ways to achieve this:
988 \item Use some kind of annotation method or syntactic construction in
989 the language to indicate exactly which argument and (part of the)
990 result is stateful. This means that the annotation lives
991 \quote{outside} of the function, it is completely invisible when
992 looking at the function body.
993 \item Use some kind of annotation on the type level, \ie\ give stateful
994 arguments and stateful (parts of) results a different type. This has the
995 potential to make this annotation visible inside the function as well,
996 such that when looking at a value inside the function body you can
997 tell if it is stateful by looking at its type. This could possibly make
998 the translation process a lot easier, since less analysis of the
999 program flow might be required.
1002 From these approaches, the type level \quote{annotations} have been
1003 implemented in Cλash. \in{Section}[sec:prototype:statetype] expands on
1004 the possible ways this could have been implemented.
1006 \todo{Note about conditions on state variables and checking them}
1008 \section[sec:recursion]{Recursion}
1009 An important concept in functional languages is recursion. In its most basic
1010 form, recursion is a definition that is described in terms of itself. A
1011 recursive function is thus a function that uses itself in its body. This
1012 usually requires multiple evaluations of this function, with changing
1013 arguments, until eventually an evaluation of the function no longer requires
1016 Given the notion that each function application will translate to a
1017 component instantiation, we are presented with a problem. A recursive
1018 function would translate to a component that contains itself. Or, more
1019 precisely, that contains an instance of itself. This instance would again
1020 contain an instance of itself, and again, into infinity. This is obviously a
1021 problem for generating hardware.
1023 This is expected for functions that describe infinite recursion. In that
1024 case, we cannot generate hardware that shows correct behaviour in a single
1025 cycle (at best, we could generate hardware that needs an infinite number of
1026 cycles to complete).
1028 However, most recursive definitions will describe finite
1029 recursion. This is because the recursive call is done conditionally. There
1030 is usually a \hs{case} expression where at least one alternative does not contain
1031 the recursive call, which we call the "base case". If, for each call to the
1032 recursive function, we would be able to detect at compile time which
1033 alternative applies, we would be able to remove the \hs{case} expression and
1034 leave only the base case when it applies. This will ensure that expanding
1035 the recursive functions will terminate after a bounded number of expansions.
1037 This does imply the extra requirement that the base case is detectable at
1038 compile time. In particular, this means that the decision between the base
1039 case and the recursive case must not depend on runtime data.
1041 \subsection{List recursion}
1042 The most common deciding factor in recursion is the length of a list that is
1043 passed in as an argument. Since we represent lists as vectors that encode
1044 the length in the vector type, it seems easy to determine the base case. We
1045 can simply look at the argument type for this. However, it turns out that
1046 this is rather non-trivial to write down in Haskell already, not even
1047 looking at translation. As an example, we would like to write down something
1051 sum :: Vector n Word -> Word
1052 sum xs = case null xs of
1054 False -> head xs + sum (tail xs)
1057 However, the Haskell typechecker will now use the following reasoning.
1058 For simplicity, the element type of a vector is left out, all vectors
1059 are assumed to have the same element type. Below, we write conditions
1060 on type variables before the \hs{=>} operator. This is not completely
1061 valid Haskell syntax, but serves to illustrate the typechecker
1062 reasoning. Also note that a vector can never have a negative length,
1063 so \hs{Vector n} implicitly means \hs{(n >= 0) => Vector n}.
1065 \todo{This typechecker disregards the type signature}
1067 \item tail has the type \hs{(n > 0) => Vector n -> Vector (n - 1)}
1068 \item This means that xs must have the type \hs{(n > 0) => Vector n}
1069 \item This means that sum must have the type \hs{(n > 0) => Vector n -> a}
1070 \item sum is called with the result of tail as an argument, which has the
1071 type \hs{Vector n} (since \hs{(n > 0) => Vector (n - 1)} is the same as \hs{(n >= 0)
1072 => Vector n}, which is the same as just \hs{Vector n}).
1073 \item This means that sum must have the type \hs{Vector n -> a}
1074 \item This is a contradiction between the type deduced from the body of sum
1075 (the input vector must be non-empty) and the use of sum (the input vector
1076 could have any length).
1079 As you can see, using a simple \hs{case} expression at value level causes
1080 the type checker to always typecheck both alternatives, which cannot be
1081 done. The typechecker is unable to distinguish the two case
1082 alternatives (this is partly possible using \small{GADT}s, but that
1083 approach faced other problems \todo{ref christiaan?}).
1085 This is a fundamental problem, that would seem perfectly suited for a
1086 type class. Considering that we need to switch between to
1087 implementations of the sum function, based on the type of the
1088 argument, this sounds like the perfect problem to solve with a type
1089 class. However, this approach has its own problems (not the least of
1090 them that you need to define a new type class for every recursive
1091 function you want to define).
1093 \todo{This should reference Christiaan}
1095 \subsection{General recursion}
1096 Of course there are other forms of recursion, that do not depend on the
1097 length (and thus type) of a list. For example, simple recursion using a
1098 counter could be expressed, but only translated to hardware for a fixed
1099 number of iterations. Also, this would require extensive support for compile
1100 time simplification (constant propagation) and compile time evaluation
1101 (evaluation of constant comparisons), to ensure non-termination.
1102 Supporting general recursion will probably require strict conditions
1103 on the input descriptions. Even then, it will be hard (if not
1104 impossible) to really guarantee termination, since the user (or \GHC\
1105 desugarer) might use some obscure notation that results in a corner
1106 case of the simplifier that is not caught and thus non-termination.
1108 Evaluating all possible (and non-possible) ways to add recursion to
1109 our descriptions, it seems better to limit the scope of this research
1110 to exclude recursion. This allows for focusing on other interesting
1111 areas instead. By including (built-in) support for a number of
1112 higher-order functions like \hs{map} and \hs{fold}, we can still
1113 express most of the things we would use (list) recursion for.
1116 % vim: set sw=2 sts=2 expandtab: