1 \chapter[chap:prototype]{Prototype}
2 An important step in this research is the creation of a prototype compiler.
3 Having this prototype allows us to apply the ideas from the previous chapter
4 to actual hardware descriptions and evaluate their usefulness. Having a
5 prototype also helps to find new techniques and test possible
8 Obviously the prototype was not created after all research
9 ideas were formed, but its implementation has been interleaved with the
10 research itself. Also, the prototype described here is the final version, it
11 has gone through a number of design iterations which we will not completely
14 \section[sec:prototype:input]{Input language}
15 When implementing this prototype, the first question to ask is:
16 Which (functional) language will be used to describe our hardware?
17 (Note that this does not concern the \emph{implementation language}
18 of the compiler, just the language \emph{translated by} the
21 Initially, we have two choices:
24 \item Create a new functional language from scratch. This has the
25 advantage of having a language that contains exactly those elements that
26 are convenient for describing hardware and can contain special
27 constructs that allows our hardware descriptions to be more powerful or
29 \item Use an existing language and create a new backend for it. This has
30 the advantage that existing tools can be reused, which will speed up
36 \startframedtext[width=8cm,background=box,frame=no]
37 \startalignment[center]
38 {\tfa No \small{EDSL} or Template Haskell}
42 Note that in this consideration, embedded domain-specific
43 languages (\small{EDSL}) and Template Haskell (\small{TH})
44 approaches have not been included. As we have seen in
45 \in{section}[sec:context:fhdls], these approaches have all kinds
46 of limitations on the description language that we would like to
50 Considering that we required a prototype which should be working quickly,
51 and that implementing parsers, semantic checkers and especially
52 typcheckers is not exactly the core of this research (but it is lots and
53 lots of work!), using an existing language is the obvious choice. This
54 also has the advantage that a large set of language features is available
55 to experiment with and it is easy to find which features apply well and
56 which do not. Another import advantage of using an existing language, is
57 that simulation of the code becomes trivial. Since there are existing
58 compilers and interpreters that can run the hardware description directly,
59 it can be simulated without also having to write an interpreter for the
62 A possible second prototype could use a custom language with just the useful
63 features (and possibly extra features that are specific to
64 the domain of hardware description as well).
66 The second choice to be made is which of the many existing languages to use. As
67 mentioned before, the chosen language is Haskell. This choice has not been the
68 result of a thorough comparison of languages, for the simple reason that
69 the requirements on the language were completely unclear at the start of
70 this research. The fact that Haskell is a language with a broad spectrum
71 of features, that it is commonly used in research projects and that the
72 primary compiler, \GHC, provides a high level API to its internals, made
73 Haskell an obvious choice.
75 \section[sec:prototype:output]{Output format}
76 The second important question is: What will be our output format?
77 This output format should at least allow for programming the
78 hardware design into a field-programmable gate array (\small{FPGA}).
79 The choice of output format is thus limited by what hardware
80 synthesis and programming tools can process.
82 Looking at other tools in the industry, the Electronic Design Interchange
83 Format (\small{EDIF}) is commonly used for storing intermediate
84 \emph{netlists} (lists of components and connections between these
85 components) and is commonly the target for \small{VHDL} and Verilog
88 However, \small{EDIF} is not completely tool-independent. It specifies a
89 meta-format, but the hardware components that can be used vary between
90 various tool and hardware vendors, as well as the interpretation of the
91 \small{EDIF} standard. \cite[li89]
93 This means that when working with \small{EDIF}, our prototype would become
94 technology dependent (\eg only work with \small{FPGA}s of a specific
95 vendor, or even only with specific chips). This limits the applicability
96 of our prototype. Also, the tools we would like to use for verifying,
97 simulating and draw pretty pictures of our output (like Precision, or
98 QuestaSim) are designed for \small{VHDL} or Verilog input.
100 For these reasons, we will not use \small{EDIF}, but \small{VHDL} as our
101 output language. We choose \VHDL over Verilog simply because we are
102 familiar with \small{VHDL} already. The differences between \small{VHDL}
103 and Verilog are on the higher level, while we will be using \small{VHDL}
104 mainly to write low level, netlist-like descriptions anyway.
106 An added advantage of using VHDL is that we can profit from existing
107 optimizations in VHDL synthesizers. A lot of optimizations are done on the
108 VHDL level by existing tools. These tools have been under
109 development for years, so it would not be reasonable to assume we
110 could achieve a similar amount of optimization in our prototype (nor
111 should it be a goal, considering this is just a prototype).
114 \startframedtext[width=8cm,background=box,frame=no]
115 \startalignment[center]
116 {\tfa Translation vs. compilation vs. synthesis}
119 In this thesis the words \emph{translation}, \emph{compilation} and
120 sometimes \emph{synthesis} will be used interchangedly to refer to the
121 process of translating the hardware description from the Haskell
122 language to the \VHDL language.
124 Similarly, the prototype created is referred to as both the
125 \emph{translator} as well as the \emph{compiler}.
127 The final part of this process is usually referred to as \emph{\VHDL
132 Note that we will be using \small{VHDL} as our output language, but will
133 not use its full expressive power. Our output will be limited to using
134 simple, structural descriptions, without any complex behavioural
135 descriptions like arbitrary sequential statements (which might not
136 be supported by all tools). This ensures that any tool that works
137 with \VHDL will understand our output (most tools do not support
138 synthesis of more complex \VHDL). This also leaves open the option
139 to switch to \small{EDIF} in the future, with minimal changes to the
142 \section{Simulation and synthesis}
143 As mentioned above, by using the Haskell language, we get simulation of
144 our hardware descriptions almost for free. The only thing that is needed
145 is to provide a Haskell implementation of all built-in functions that can
146 be used by the Haskell interpreter to simulate them.
148 The main topic of this thesis is therefore the path from the Haskell
149 hardware descriptions to \small{FPGA} synthesis, focusing of course on the
150 \VHDL generation. Since the \VHDL generation process preserves the meaning
151 of the Haskell description exactly, any simulation done in Haskell
152 \emph{should} produce identical results as the synthesized hardware.
154 \section[sec:prototype:design]{Prototype design}
155 As suggested above, we will use the Glasgow Haskell Compiler (\small{GHC}) to
156 implement our prototype compiler. To understand the design of the
157 compiler, we will first dive into the \small{GHC} compiler a bit. Its
158 compilation consists of the following steps (slightly simplified):
160 \startuseMPgraphic{ghc-pipeline}
162 save inp, front, desugar, simpl, back, out;
163 newEmptyBox.inp(0,0);
164 newBox.front(btex Frontend etex);
165 newBox.desugar(btex Desugarer etex);
166 newBox.simpl(btex Simplifier etex);
167 newBox.back(btex Backend etex);
168 newEmptyBox.out(0,0);
170 % Space the boxes evenly
171 inp.c - front.c = front.c - desugar.c = desugar.c - simpl.c
172 = simpl.c - back.c = back.c - out.c = (0, 1.5cm);
175 % Draw lines between the boxes. We make these lines "deferred" and give
176 % them a name, so we can use ObjLabel to draw a label beside them.
177 ncline.inp(inp)(front) "name(haskell)";
178 ncline.front(front)(desugar) "name(ast)";
179 ncline.desugar(desugar)(simpl) "name(core)";
180 ncline.simpl(simpl)(back) "name(simplcore)";
181 ncline.back(back)(out) "name(native)";
182 ObjLabel.inp(btex Haskell source etex) "labpathname(haskell)", "labdir(rt)";
183 ObjLabel.front(btex Haskell AST etex) "labpathname(ast)", "labdir(rt)";
184 ObjLabel.desugar(btex Core etex) "labpathname(core)", "labdir(rt)";
185 ObjLabel.simpl(btex Simplified core etex) "labpathname(simplcore)", "labdir(rt)";
186 ObjLabel.back(btex Native code etex) "labpathname(native)", "labdir(rt)";
188 % Draw the objects (and deferred labels)
189 drawObj (inp, front, desugar, simpl, back, out);
191 \placefigure[right]{GHC compiler pipeline}{\useMPgraphic{ghc-pipeline}}
194 This step takes the Haskell source files and parses them into an
195 abstract syntax tree (\small{AST}). This \small{AST} can express the
196 complete Haskell language and is thus a very complex one (in contrast
197 with the Core \small{AST}, later on). All identifiers in this
198 \small{AST} are resolved by the renamer and all types are checked by the
201 \startdesc{Desugaring}
202 This steps takes the full \small{AST} and translates it to the
203 \emph{Core} language. Core is a very small functional language with lazy
204 semantics, that can still express everything Haskell can express. Its
205 simpleness makes Core very suitable for further simplification and
206 translation. Core is the language we will be working with as well.
208 \startdesc{Simplification}
209 Through a number of simplification steps (such as inlining, common
210 subexpression elimination, etc.) the Core program is simplified to make
211 it faster or easier to process further.
214 This step takes the simplified Core program and generates an actual
215 runnable program for it. This is a big and complicated step we will not
216 discuss it any further, since it is not required for our prototype.
219 In this process, there are a number of places where we can start our work.
220 Assuming that we do not want to deal with (or modify) parsing, typechecking
221 and other frontend business and that native code is not really a useful
222 format anymore, we are left with the choice between the full Haskell
223 \small{AST}, or the smaller (simplified) core representation.
225 The advantage of taking the full \small{AST} is that the exact structure
226 of the source program is preserved. We can see exactly what the hardware
227 description looks like and which syntax constructs were used. However,
228 the full \small{AST} is a very complicated datastructure. If we are to
229 handle everything it offers, we will quickly get a big compiler.
231 Using the core representation gives us a much more compact datastructure
232 (a core expression only uses 9 constructors). Note that this does not mean
233 that the core representation itself is smaller, on the contrary.
234 Since the core language has less constructs, most Core expressions
235 are larger than the equivalent versions in Haskell.
237 However, the fact that the core language is so much smaller, means it is a
238 lot easier to analyze and translate it into something else. For the same
239 reason, \small{GHC} runs its simplifications and optimizations on the core
240 representation as well \cite[jones96].
242 However, we will use the normal core representation, not the simplified
243 core. Reasons for this are detailed below. \todo{Ref}
245 The final prototype roughly consists of three steps:
247 \startuseMPgraphic{clash-pipeline}
249 save inp, front, norm, vhdl, out;
250 newEmptyBox.inp(0,0);
251 newBox.front(btex \small{GHC} frontend etex);
252 newBox.norm(btex Normalization etex);
253 newBox.vhdl(btex \small{VHDL} generation etex);
254 newEmptyBox.out(0,0);
256 % Space the boxes evenly
257 inp.c - front.c = front.c - norm.c = norm.c - vhdl.c
258 = vhdl.c - out.c = (0, 1.5cm);
261 % Draw lines between the boxes. We make these lines "deferred" and give
262 % them a name, so we can use ObjLabel to draw a label beside them.
263 ncline.inp(inp)(front) "name(haskell)";
264 ncline.front(front)(norm) "name(core)";
265 ncline.norm(norm)(vhdl) "name(normal)";
266 ncline.vhdl(vhdl)(out) "name(vhdl)";
267 ObjLabel.inp(btex Haskell source etex) "labpathname(haskell)", "labdir(rt)";
268 ObjLabel.front(btex Core etex) "labpathname(core)", "labdir(rt)";
269 ObjLabel.norm(btex Normalized core etex) "labpathname(normal)", "labdir(rt)";
270 ObjLabel.vhdl(btex \small{VHDL} description etex) "labpathname(vhdl)", "labdir(rt)";
272 % Draw the objects (and deferred labels)
273 drawObj (inp, front, norm, vhdl, out);
275 \placefigure[right]{Cλash compiler pipeline}{\useMPgraphic{clash-pipeline}}
278 This is exactly the frontend from the \small{GHC} pipeline, that
279 translates Haskell sources to a typed Core representation.
281 \startdesc{Normalization}
282 This is a step that transforms the core representation into a normal
283 form. This normal form is still expressed in the core language, but has
284 to adhere to an additional set of constraints. This normal form is less
285 expressive than the full core language (e.g., it can have limited
286 higher-order expressions, has a specific structure, etc.), but is
287 also very close to directly describing hardware.
289 \startdesc{\small{VHDL} generation}
290 The last step takes the normal formed core representation and generates
291 \small{VHDL} for it. Since the normal form has a specific, hardware-like
292 structure, this final step is very straightforward.
295 The most interesting step in this process is the normalization step. That
296 is where more complicated functional constructs, which have no direct
297 hardware interpretation, are removed and translated into hardware
298 constructs. This step is described in a lot of detail at
299 \in{chapter}[chap:normalization].
302 \defref{entry function}Translation of a hardware description always
303 starts at a single function, which is referred to as the \emph{entry
304 function}. \VHDL is generated for this function first, followed by
305 any functions used by the entry functions (recursively).
307 \section[sec:prototype:core]{The Core language}
308 \defreftxt{core}{the Core language}
309 Most of the prototype deals with handling the program in the Core
310 language. In this section we will show what this language looks like and
313 The Core language is a functional language that describes
314 \emph{expressions}. Every identifier used in Core is called a
315 \emph{binder}, since it is bound to a value somewhere. On the highest
316 level, a Core program is a collection of functions, each of which bind a
317 binder (the function name) to an expression (the function value, which has
320 The Core language itself does not prescribe any program structure
321 (like modules, declarations, imports, etc.), only expression
322 structure. In the \small{GHC} compiler, the Haskell module structure
323 is used for the resulting Core code as well. Since this is not so
324 relevant for understanding the Core language or the Normalization
325 process, we will only look at the Core expression language here.
327 Each Core expression consists of one of these possible expressions.
329 \startdesc{Variable reference}
330 \defref{variable reference}
334 This is a reference to a binder. It is written down as the
335 name of the binder that is being referred to along with its type. The
336 binder name should of course be bound in a containing scope
337 (including top level scope, so a reference to a top level function
338 is also a variable reference). Additionally, constructors from
339 algebraic datatypes also become variable references.
341 In our examples, binders will commonly consist of a single
342 characters, but they can have any length.
344 The value of this expression is the value bound to the given
347 Each binder also carries around its type (explicitly shown above), but
348 this is usually not shown in the Core expressions. Only when the type is
349 relevant (when a new binder is introduced, for example) will it be
350 shown. In other cases, the binder is either not relevant, or easily
351 derived from the context of the expression. \todo{Ref sidenote on type
360 This is a literal. Only primitive types are supported, like
361 chars, strings, ints and doubles. The types of these literals are the
362 \quote{primitive}, unboxed versions, like \lam{Char\#} and \lam{Word\#}, not the
363 normal Haskell versions (but there are built-in conversion
364 functions). Without going into detail about these types, note that
365 a few conversion functions exist to convert these to the normal
366 (boxed) Haskell equivalents.
369 \startdesc{Application}
374 This is function application. Each application consists of two
375 parts: The function part and the argument part. Applications are used
376 for normal function \quote{calls}, but also for applying type
377 abstractions and data constructors.
379 In core, there is no distinction between an operator and a
380 function. This means that, for example the addition of two numbers
381 looks like the following in Core:
387 Where the function \quote{\lam{(+)}} is applied to the numbers 1
388 and 2. However, to increase readability, an application of an
389 operator like \lam{(+)} is sometimes written infix. In this case,
390 the parenthesis are also left out, just like in Haskell. In other
391 words, the following means exactly the same as the addition above:
397 The value of an application is the value of the function part, with the
398 first argument binder bound to the argument part.
401 \startdesc{Lambda abstraction}
402 \defref{lambda abstraction}
406 This is the basic lambda abstraction, as it occurs in lambda calculus.
407 It consists of a binder part and a body part. A lambda abstraction
408 creates a function, that can be applied to an argument. The binder is
409 usually a value binder, but it can also be a \emph{type binder} (or
410 \emph{type variable}). The latter case introduces a new polymorphic
411 variable, which can be used in types later on. See
412 \in{section}[sec:prototype:coretypes] for details.
414 The body of a lambda abstraction extends all the way to the end of
415 the expression, or the closing bracket surrounding the lambda. In
416 other words, the lambda abstraction \quote{operator} has the
417 lowest priority of all.
419 The value of an application is the value of the body part, with the
420 binder bound to the value the entire lambda abstraction is applied to.
423 \startdesc{Non-recursive let expression}
424 \defref{let expression}
426 let bndr = value in body
428 A let expression allows you to bind a binder to some value, while
429 evaluating to some other value (for which that binder is in scope). This
430 allows for sharing of subexpressions (you can use a binder twice) and
431 explicit \quote{naming} of arbitrary expressions. A binder is not
432 in scope in the value bound it is bound to, so it is not possible
433 to make recursive definitions with a non-recursive let expression
434 (see the recursive form below).
436 Even though this let expression is an extension on the basic lambda
437 calculus, it is easily translated to a lambda abstraction. The let
438 expression above would then become:
444 This notion might be useful for verifying certain properties on
445 transformations, since a lot of verification work has been done on
446 lambda calculus already.
448 The value of a let expression is the value of the body part, with the
449 binder bound to the value.
452 \startdesc{Recursive let expression}
461 This is the recursive version of the let expression. In \small{GHC}'s
462 Core implementation, non-recursive and recursive lets are not so
463 distinct as we present them here, but this provides a clearer overview.
465 The main difference with the normal let expression is that it can
466 contain multiple bindings (or even none) and each of the binders
467 is in scope in each of the values, in addition to the body. This
468 allows for self-recursive or mutually recursive definitions.
470 It is also possible to express a recursive let expression using
471 normal lambda calculus, if we use the \emph{least fixed-point
472 operator}, \lam{Y} (but the details are too complicated to help
473 clarify the let expression, so this will not be explored further).
477 \startframedtext[width=8cm,background=box,frame=no]
478 \startalignment[center]
479 {\tfa Weak head normal form (\small{WHNF})}
482 An expression is in weak head normal form if it is either an
483 constructor application or lambda abstraction. \todo{How about
486 Without going into detail about the differences with head
487 normal form and normal form, note that evaluating the scrutinee
488 of a case expression to normal form (evaluating any function
489 applications, variable references and case expressions) is
490 sufficient to decide which case alternatives should be chosen.
496 \startdesc{Case expression}
497 \defref{case expression}
499 case scrutinee of bndr
500 DEFAULT -> defaultbody
501 C0 bndr0,0 ... bndr0,m -> body0
503 Cn bndrn,0 ... bndrn,m -> bodyn
506 A case expression is the only way in Core to choose between values. All
507 \hs{if} expressions and pattern matchings from the original Haskell
508 PRogram have been translated to case expressions by the desugarer.
510 A case expression evaluates its scrutinee, which should have an
511 algebraic datatype, into weak head normal form (\small{WHNF}) and
512 (optionally) binds it to \lam{bndr}. Every alternative lists a
513 single constructor (\lam{C0 ... Cn}). Based on the actual
514 constructor of the scrutinee, the corresponding alternative is
515 chosen. The binders in the chosen alternative (\lam{bndr0,0 ....
516 bndr0,m} are bound to the actual arguments to the constructor in
519 This is best illustrated with an example. Assume
520 there is an algebraic datatype declared as follows\footnote{This
521 datatype is not suported by the current Cλash implementation, but
522 serves well to illustrate the case expression}:
525 data D = A Word | B Bit
528 This is an algebraic datatype with two constructors, each getting
529 a single argument. A case expression scrutinizing this datatype
530 could look like the following:
538 What this expression does is check the constructor of the
539 scrutinee \lam{s}. If it is \lam{A}, it always evaluates to
540 \lam{High}. If the constructor is \lam{B}, the binder \lam{bit} is
541 bound to the argument passed to \lam{B} and the case expression
542 evaluates to this bit.
544 If none of the alternatives match, the \lam{DEFAULT} alternative
545 is chosen. A case expression must always be exhaustive, \ie it
546 must cover all possible constructors that the scrutinee can have
547 (if all of them are covered explicitly, the \lam{DEFAULT}
548 alternative can be left out).
550 Since we can only match the top level constructor, there can be no overlap
551 in the alternatives and thus order of alternatives is not relevant (though
552 the \lam{DEFAULT} alternative must appear first for implementation
555 To support strictness, the scrutinee is always evaluated into
556 \small{WHNF}, even when there is only a \lam{DEFAULT} alternative. This
557 allows aplication of the strict function \lam{f} to the argument \lam{a}
561 f (case a of arg DEFAULT -> arg)
564 According to the \GHC documentation, this is the only use for the extra
565 binder to which the scrutinee is bound. When not using strictness
566 annotations (which is rather pointless in hardware descriptions),
567 \small{GHC} seems to never generate any code making use of this binder.
568 In fact, \GHC has never been observed to generate code using this
569 binder, even when strictness was involved. Nonetheless, the prototype
570 handles this binder as expected.
572 Note that these case expressions are less powerful than the full Haskell
573 case expressions. In particular, they do not support complex patterns like
574 in Haskell. Only the constructor of an expression can be matched,
575 complex patterns are implemented using multiple nested case expressions.
577 Case expressions are also used for unpacking of algebraic datatypes, even
578 when there is only a single constructor. For examples, to add the elements
579 of a tuple, the following Core is generated:
582 sum = λtuple.case tuple of
586 Here, there is only a single alternative (but no \lam{DEFAULT}
587 alternative, since the single alternative is already exhaustive). When
588 its body is evaluated, the arguments to the tuple constructor \lam{(,)}
589 (\eg, the elements of the tuple) are bound to \lam{a} and \lam{b}.
592 \startdesc{Cast expression}
593 \defref{cast expression}
597 A cast expression allows you to change the type of an expression to an
598 equivalent type. Note that this is not meant to do any actual work, like
599 conversion of data from one format to another, or force a complete type
600 change. Instead, it is meant to change between different representations
601 of the same type, \eg switch between types that are provably equal (but
604 In our hardware descriptions, we typically see casts to change between a
605 Haskell newtype and its contained type, since those are effectively
606 different types (so a cast is needed) with the same representation (but
607 no work is done by the cast).
609 More complex are types that are proven to be equal by the typechecker,
610 but look different at first glance. To ensure that, once the typechecker
611 has proven equality, this information sticks around, explicit casts are
612 added. In our notation we only write the target type, but in reality a
613 cast expressions carries around a \emph{coercion}, which can be seen as a
614 proof of equality. \todo{Example}
616 The value of a cast is the value of its body, unchanged. The type of this
617 value is equal to the target type, not the type of its body.
619 \todo{Move and update this paragraph}
620 Note that this syntax is also used sometimes to indicate that a particular
621 expression has a particular type, even when no cast expression is
622 involved. This is then purely informational, since the only elements that
623 are explicitly typed in the Core language are the binder references and
624 cast expressions, the types of all other elements are determined at
629 The Core language in \small{GHC} allows adding \emph{notes}, which serve
630 as hints to the inliner or add custom (string) annotations to a core
631 expression. These should not be generated normally, so these are not
632 handled in any way in the prototype.
636 \defref{type expression}
640 It is possibly to use a Core type as a Core expression. To prevent
641 confusion between types and values, the \lam{@} sign is used to
642 explicitly mark a type that is used in a Core expression.
644 For the actual types supported by Core, see
645 \in{section}[sec:prototype:coretypes]. This \quote{lifting} of a
646 type into the value domain is done to allow for type abstractions
647 and applications to be handled as normal lambda abstractions and
648 applications above. This means that a type expression in Core can
649 only ever occur in the argument position of an application, and
650 only if the type of the function that is applied to expects a type
651 as the first argument. This happens in applications of all
652 polymorphic functions. Consider the \lam{fst} function:
655 fst :: \forall t1. \forall t2. (t1, t2) ->t1
656 fst = λt1.λt2.λ(tup :: (t1, t2)). case tup of (,) a b -> a
658 fstint :: (Int, Int) -> Int
659 fstint = λa.λb.fst @Int @Int a b
662 The type of \lam{fst} has two universally quantified type variables. When
663 \lam{fst} is applied in \lam{fstint}, it is first applied to two types.
664 (which are substitued for \lam{t1} and \lam{t2} in the type of \lam{fst}, so
665 the actual type of arguments and result of \lam{fst} can be found:
666 \lam{fst @Int @Int :: (Int, Int) -> Int}).
669 \subsection[sec:prototype:coretypes]{Core type system}
670 Whereas the expression syntax of Core is very simple, its type system is
671 a bit more complicated. It turns out it is harder to \quote{desugar}
672 Haskell's complex type system into something more simple. Most of the
673 type system is thus very similar to that of Haskell.
675 We will slightly limit our view on Core's type system, since the more
676 complicated parts of it are only meant to support Haskell's (or rather,
677 \GHC's) type extensions, such as existential types, \small{GADT}s, type
678 families and other non-standard Haskell stuff which we do not (plan to)
681 In Core, every expression is typed. The translation to Core happens
682 after the typechecker, so types in Core are always correct as well
683 (though you could of course construct invalidly typed expressions
684 through the \GHC API).
686 Any type in core is one of the following:
688 \startdesc{A type variable}
693 This is a reference to a type defined elsewhere. This can either be a
694 polymorphic type (like the latter two \lam{t}'s in \lam{id :: \forall t.
695 t -> t}), or a type constructor (like \lam{Bool} in \lam{not :: Bool ->
696 Bool}). Like in Haskell, polymorphic type variables always
697 start with a lowercase letter, while type constructors always start
698 with an uppercase letter.
700 \todo{How to define (new) type constructors?}
702 A special case of a type constructor is the \emph{function type
703 constructor}, \lam{->}. This is a type constructor taking two arguments
704 (using application below). The function type constructor is commonly
705 written inline, so we write \lam{a -> b} when we really mean \lam{-> a
706 b}, the function type constructor applied to \lam{a} and \lam{b}.
708 Polymorphic type variables can only be defined by a lambda
709 abstraction, see the forall type below.
712 \startdesc{A type application}
717 This applies some type to another type. This is particularly used to
718 apply type variables (type constructors) to their arguments.
720 As mentioned above, applications of some type constructors have
721 special notation. In particular, these are applications of the
722 \emph{function type constructor} and \emph{tuple type constructors}:
727 bar' :: (,,) t1 t2 t3
731 \startdesc{The forall type}
733 id :: \forall t. t -> t
735 The forall type introduces polymorphism. It is the only way to
736 introduce new type variables, which are completely unconstrained (Any
737 possible type can be assigned to it). Constraints can be added later
738 using predicate types, see below.
740 A forall type is always (and only) introduced by a type lambda
741 expression. For example, the Core translation of the
747 Here, the type of the binder \lam{x} is \lam{t}, referring to the
748 binder in the topmost lambda.
750 When using a value with a forall type, the actual type
751 used must be applied first. For example Haskell expression \hs{id
752 True} (the function \hs{id} appleid to the dataconstructor \hs{True})
753 translates to the following Core:
759 Here, id is first applied to the type to work with. Note that the type
760 then changes from \lam{id :: \forall t. t -> t} to \lam{id @Bool ::
761 Bool -> Bool}. Note that the type variable \lam{a} has been
762 substituted with the actual type.
764 In Haskell, forall types are usually not explicitly specified (The use
765 of a lowercase type variable implicitly introduces a forall type for
766 that variable). In fact, in standard Haskell there is no way to
767 explicitly specify forall types. Through a language extension, the
768 \hs{forall} keyword is available, but still optional for normal forall
769 types (it is needed for \emph{existentially quantified types}, which
770 Cλash does not support).
773 \startdesc{Predicate type}
775 show :: \forall t. Show t ⇒ t → String
778 \todo{Sidenote: type classes?}
780 A predicate type introduces a constraint on a type variable introduced
781 by a forall type (or type lambda). In the example above, the type
782 variable \lam{t} can only contain types that are an \emph{instance} of
783 the \emph{type class} \lam{Show}. \refdef{type class}
785 There are other sorts of predicate types, used for the type families
786 extension, which we will not discuss here.
788 A predicate type is introduced by a lambda abstraction. Unlike with
789 the forall type, this is a value lambda abstraction, that must be
790 applied to a value. We call this value a \emph{dictionary}.
792 Without going into the implementation details, a dictionary can be
793 seen as a lookup table all the methods for a given (single) type class
794 instance. This means that all the dictionaries for the same type class
795 look the same (\eg contain methods with the same names). However,
796 dictionaries for different instances of the same class contain
797 different methods, of course.
799 A dictionary is introduced by \small{GHC} whenever it encounters an
800 instance declaration. This dictionary, as well as the binder
801 introduced by a lambda that introduces a dictionary, have the
802 predicate type as their type. These binders are usually named starting
803 with a \lam{\$}. Usually the name of the type concerned is not
804 reflected in the name of the dictionary, but the name of the type
805 class is. The Haskell expression \hs{show True} thus becomes:
808 show @Bool \$dShow True
812 Using this set of types, all types in basic Haskell can be represented.
814 \todo{Overview of polymorphism with more examples (or move examples
817 \section[sec:prototype:statetype]{State annotations in Haskell}
818 As noted in \in{section}[sec:description:stateann], Cλash needs some
819 way to let the programmer explicitly specify which of a function's
820 arguments and which part of a function's result represent the
823 Using the Haskell type systems, there are a few ways we can tackle this.
825 \subsection{Type synonyms}
826 Haskell provides type synonyms as a way to declare a new type that is
827 equal to an existing type (or rather, a new name for an existing type).
828 This allows both the original type and the synonym to be used
829 interchangedly in a Haskell program. This means no explicit conversion
830 is needed either. For example, a simple accumulator would become:
834 acc :: Word -> State Word -> (State Word, Word)
835 acc i s = let sum = s + i in (sum, sum)
838 This looks nice in Haskell, but turns out to be hard to implement. There
839 are no explicit conversion in Haskell, but not in Core either. This
840 means the type of a value might be show as \hs{AccState} in some places,
841 but \hs{Word} in others (and this can even change due to
842 transformations). Since every binder has an explicit type associated
843 with it, the type of every function type will be properly preserved and
844 could be used to track down the statefulness of each value by the
845 compiler. However, this makes the implementation a lot more complicated
846 than it currently is using \hs{newtypes}.
848 % Use \type instead of \hs here, since the latter breaks inside
850 \subsection{Type renaming (\type{newtype})}
851 Haskell also supports type renamings as a way to declare a new type that
852 has the same (runtime) representation as an existing type (but is in
853 fact a different type to the typechecker). With type renaming, an
854 explicit conversion between values of the two types is needed. The
855 accumulator would then become:
858 newtype State s = State s
859 acc :: Word -> State Word -> (State Word, Word)
860 acc i (State s) = let sum = s + i in (State sum, sum)
863 The \hs{newtype} line declares a new type \hs{State} that has one type
864 argument, \hs{s}. This type contains one \quote{constructor} \hs{State}
865 with a single argument of type \hs{s}. It is customary to name the
866 constructor the same as the type, which is allowed (since types can
867 never cause name collisions with values). The difference with the type
868 synonym example is in the explicit conversion between the \hs{State
869 Word} and \hs{Word} types by pattern matching and by using the explicit
870 the \hs{State constructor}.
872 This explicit conversion makes the \VHDL generation easier: Whenever we
873 remove (unpack) the \hs{State} type, this means we are accessing the
874 current state (\eg, accessing the register output). Whenever we are a
875 adding (packing) the \hs{State} type, we are producing a new value for
876 the state (\eg, providing the register input).
878 When dealing with nested states (a stateful function that calls stateful
879 functions, which might call stateful functions, etc.) the state type
880 could quickly grow complex because of all the \hs{State} type constructors
881 needed. For example, consider the following state type (this is just the
882 state type, not the entire function type):
885 State (State Bit, State (State Word, Bit), Word)
888 We cannot leave all these \hs{State} type constructors out, since that
889 would change the type (unlike when using type synonyms). However, when
890 using type synonyms to hide away substates (see
891 \in{section}[sec:prototype:substatesynonyms] below), this
892 disadvantage should be limited.
894 \subsubsection{Different input and output types}
895 An alternative could be to use different types for input and output
896 state (\ie current and updated state). The accumulator example would
897 then become something like:
900 newtype StateIn s = StateIn s
901 newtype StateOut s = StateOut s
902 acc :: Word -> StateIn Word -> (StateIn Word, Word)
903 acc i (StateIn s) = let sum = s + i in (StateIn sum, sum)
906 This could make the implementation easier and the hardware
907 descriptions less errorprone (you can no longer \quote{forget} to
908 unpack and repack a state variable and just return it directly, which
909 can be a problem in the current prototype). However, it also means we
910 need twice as many type synonyms to hide away substates, making this
911 approach a bit cumbersome. It also makes it harder to copmare input
912 and output state types, possible reducing the type safety of the
915 \subsection[sec:prototype:substatesynonyms]{Type synonyms for substates}
916 As noted above, when using nested (hierarchical) states, the state types
917 of the \quote{upper} functions (those that call other functions, which
918 call other functions, etc.) quickly becomes complicated. Also, when the
919 state type of one of the \quote{lower} functions changes, the state
920 types of all the upper functions changes as well. If the state type for
921 each function is explicitly and completely specified, this means that a
922 lot of code needs updating whenever a state type changes.
924 To prevent this, it is recommended (but not enforced) to use a type
925 synonym for the state type of every function. Every function calling
926 other functions will then use the state type synonym of the called
927 functions in its own type, requiring no code changes when the state type
928 of a called function changes. This approach is used in
929 \in{example}[ex:AvgState] below. The \hs{AccState} and \hs{AvgState}
930 are examples of such state type synonyms.
932 \subsection{Chosen approach}
933 To keep implementation simple, the current prototype uses the type
934 renaming approach, with a single type for both input and output
935 states. In the future, it might be worthwhile to revisit this
936 approach if more complicated flow analysis is implemented for
937 state variables. This analysis is needed to add proper error
938 checking anyway and might allow the use of type synonyms without
939 losing any expressivity.
941 \subsubsection{Example}
942 As an example of the used approach, there is a simple averaging circuit in
943 \in{example}[ex:AvgState]. This circuit lets the accumulation of the
944 inputs be done by a subcomponent, \hs{acc}, but keeps a count of value
945 accumulated in its own state.\footnote{Currently, the prototype
946 is not able to compile this example, since the built-in function
947 for division has not been added.}
949 \startbuffer[AvgState]
950 -- The state type annotation
951 newtype State s = State s
953 -- The accumulator state type
954 type AccState = State Word
956 acc :: Word -> AccState -> (AccState, Word)
957 acc i (State s) = let sum = s + i in (State sum, sum)
959 -- The averaging circuit state type
960 type AvgState = State (AccState, Word)
961 -- The averaging circuit
962 avg :: Word -> AvgState -> (AvgState, Word)
963 avg i (State s) = (State s', o)
966 -- Pass our input through the accumulator, which outputs a sum
967 (accs', sum) = acc i accs
968 -- Increment the count (which will be our new state)
970 -- Compute the average
975 \placeexample[here][ex:AvgState]{Simple stateful averaging circuit.}
976 %\startcombination[2*1]
977 {\typebufferhs{AvgState}}%{Haskell description using function applications.}
978 % {\boxedgraphic{AvgState}}{The architecture described by the Haskell description.}
982 \section{Implementing state}
983 Now its clear how to put state annotations in the Haskell source,
984 there is the question of how to implement this state translation. As
985 we have seen in \in{section}[sec:prototype:design], the translation to
986 \VHDL happens as a simple, final step in the compilation process.
987 This step works on a core expression in normal form. The specifics
988 of normal form will be explained in
989 \in{chapter}[chap:normalization], but the examples given should be
990 easy to understand using the definitin of Core given above.
992 \startbuffer[AvgStateNormal]
995 -- Remove the State newtype
999 -- Add the State newtype again
1000 spacked' = s' ▶ State Word
1007 s = spacked ▶ (AccState, Word)
1008 accs = case s of (accs, _) -> accs
1009 count = case s of (_, count) -> count
1011 accs' = case accres of (accs', _) -> accs'
1012 sum = case accres of (_, sum) -> sum
1015 s' = (accs', count')
1016 spacked' = s' ▶ State (AccState, Word)
1022 \placeexample[here][ex:AvgStateNormal]{Normalized version of \in{example}[ex:AvgState]}
1023 {\typebufferlam{AvgStateNormal}}
1025 \subsection[sec:prototype:statelimits]{State in normal form}
1026 Before describing how to translate state from normal form to
1027 \VHDL, we will first see how state handling looks in normal form.
1028 What limitations are there on their use to guarantee that proper
1029 \VHDL can be generated?
1031 We will try to formulate a number of rules about what operations are
1032 allowed with state variables. These rules apply to the normalized Core
1033 representation, but will in practice apply to the original Haskell
1034 hardware description as well. Ideally, these rules would become part
1035 of the intended normal form definition \refdef{intended normal form
1036 definition}, but this is not the case right now. This can cause some
1037 problems, which are detailed in
1038 \in{section}[sec:normalization:stateproblems].
1040 In these rules we use the terms \emph{state variable} to refer to any
1041 variable that has a \lam{State} type. A \emph{state-containing
1042 variable} is any variable whose type contains a \lam{State} type,
1043 but is not one itself (like \lam{(AccState, Word)} in the example,
1044 which is a tuple type, but contains \lam{AccState}, which is again
1045 equal to \lam{State Word}).
1047 We also use a distinction between \emph{input} and \emph{output
1048 (state) variables} and \emph{substate variables}, which will be
1049 defined in the rules themselves.
1051 \startdesc{State variables can appear as an argument.}
1053 avg = λi.λspacked. ...
1056 Any lambda that binds a variable with a state type, creates a new
1057 input state variable.
1060 \startdesc{Input state variables can be unpacked.}
1062 s = spacked ▶ (AccState, Word)
1065 An input state variable may be unpacked using a cast operation. This
1066 removes the \lam{State} type renaming and the result has no longer a
1069 If the result of this unpacking does not have a state type and does
1070 not contain state variables, there are no limitations on its use.
1071 Otherwise if it does not have a state type but does contain
1072 substates, we refer to it as a \emph{state-containing input
1073 variable} and the limitations below apply. If it has a state type
1074 itself, we refer to it as an \emph{input substate variable} and the
1075 below limitations apply as well.
1077 It may seem strange to consider a variable that still has a state
1078 type directly after unpacking, but consider the case where a
1079 function does not have any state of its own, but does call a single
1080 stateful function. This means it must have a state argument that
1081 contains just a substate. The function signature of such a function
1085 type FooState = State AccState
1088 Which is of course equivalent to \lam{State (State Word)}.
1091 \startdesc{Variables can be extracted from state-containing input variables.}
1093 accs = case s of (accs, _) -> accs
1096 A state-containing input variable is typically a tuple containing
1097 multiple elements (like the current function's state, substates or
1098 more tuples containing substates). All of these can be extracted
1099 from an input variable using an extractor case (or possibly
1100 multiple, when the input variable is nested).
1102 If the result has no state type and does not contain any state
1103 variables either, there are no further limitations on its use. If
1104 the result has no state type but does contain state variables we
1105 refer to it as a \emph{state-containing input variable} and this
1106 limitation keeps applying. If the variable has a state type itself,
1107 we refer to it as an \emph{input substate variable} and below
1110 \startdesc{Input substate variables can be passed to functions.}
1113 accs' = case accres of (accs', _) -> accs'
1116 An input substate variable can (only) be passed to a function.
1117 Additionally, every input substate variable must be used in exactly
1118 \emph{one} application, no more and no less.
1120 The function result should contain exactly one state variable, which
1121 can be extracted using (multiple) case expressions. The extracted
1122 state variable is referred to the \emph{output substate}
1124 The type of this output substate must be identical to the type of
1125 the input substate passed to the function.
1128 \startdesc{Variables can be inserted into a state-containing output variable.}
1130 s' = (accs', count')
1133 A function's output state is usually a tuple containing its own
1134 updated state variables and all output substates. This result is
1135 built up using any single-constructor algebraic datatype.
1137 The result of these expressions is referred to as a
1138 \emph{state-containing output variable}, which are subject to these
1142 \startdesc{State containing output variables can be packed.}
1144 spacked' = s' ▶ State (AccState, Word)
1147 As soon as all a functions own update state and output substate
1148 variables have been joined together, the resulting
1149 state-containing output variable can be packed into an output
1150 state variable. Packing is done by casting into a state type.
1153 \startdesc{Output state variables can appear as (part of) a function result.}
1162 When the output state is packed, it can be returned as a part
1163 of the function result. Nothing else can be done with this
1164 value (or any value that contains it).
1167 There is one final limitation that is hard to express in the above
1168 itemization. Whenever substates are extracted from the input state
1169 to be passed to functions, the corresponding output substates
1170 should be inserted into the output state in the same way. In other
1171 words, each pair of corresponding substates in the input and
1172 output states should be passed / returned from the same called
1175 The prototype currently does not check much of the above
1176 conditions. This means that if the conditions are violated,
1177 sometimes a compile error is generated, but in other cases output
1178 can be generated that is not valid \VHDL or at the very least does
1179 not correspond to the input.
1181 \subsection{Translating to \VHDL}
1182 As noted above, the basic approach when generating \VHDL for stateful
1183 functions is to generate a single register for every stateful function.
1184 We look around the normal form to find the let binding that removes the
1185 \lam{State} newtype (using a cast). We also find the let binding that
1186 adds a \lam{State} type. These are connected to the output and the input
1187 of the generated let binding respectively. This means that there can
1188 only be one let binding that adds and one that removes the \lam{State}
1189 type. It is easy to violate this constraint. This problem is detailed in
1190 \in{section}[sec:normalization:stateproblems].
1192 This approach seems simple enough, but will this also work for more
1193 complex stateful functions involving substates? Observe that any
1194 component of a function's state that is a substate, \ie passed on as
1195 the state of another function, should have no influence on the
1196 hardware generated for the calling function. Any state-specific
1197 \small{VHDL} for this component can be generated entirely within the
1198 called function. So, we can completely ignore substates when
1199 generating \VHDL for a function.
1201 From this observation it might seem logical to remove the
1202 substates from a function's states altogether and leave only the
1203 state components which are actual states of the current function.
1204 While doing this would not remove any information needed to
1205 generate \small{VHDL} from the function, it would cause the
1206 function definition to become invalid (since we will not have any
1207 substate to pass to the functions anymore). We could solve the
1208 syntactic problems by passing \type{undefined} for state
1209 variables, but that would still break the code on the semantic
1210 level (\ie, the function would no longer be semantically
1211 equivalent to the original input).
1213 To keep the function definition correct until the very end of the
1214 process, we will not deal with (sub)states until we get to the
1215 \small{VHDL} generation. Then, we are translating from Core to
1216 \small{VHDL}, and we can simply ignore substates, effectively removing
1217 the substate components altogether.
1219 But, how will we know what exactly is a substate? Since any state
1220 argument or return value that represents state must be of the
1221 \type{State} type, we can look at the type of a value. However, we
1222 must be careful to ignore only \emph{substates}, and not a
1223 function's own state.
1225 In \in{example}[ex:AvgStateNorm] above, we should generate a register
1226 connected with its output connected to \lam{s} and its input connected
1227 to \lam{s'}. However, \lam{s'} is build up from both \lam{accs'} and
1228 \lam{count'}, while only \lam{count'} should end up in the register.
1229 \lam{accs'} is a substate for the \lam{acc} function, for which a
1230 register will be created when generating \VHDL for the \lam{acc}
1233 Fortunately, the \lam{accs'} variable (and any other substate) has a
1234 property that we can easily check: It has a \lam{State} type
1235 annotation. This means that whenever \VHDL is generated for a tuple
1236 (or other algebraic type), we can simply leave out all elements that
1237 have a \lam{State} type. This will leave just the parts of the state
1238 that do not have a \lam{State} type themselves, like \lam{count'},
1239 which is exactly a function's own state. This approach also means that
1240 the state part of the result is automatically excluded when generating
1241 the output port, which is also required.
1243 We can formalize this translation a bit, using the following
1247 \item A state unpack operation should not generate any \small{VHDL}.
1248 The binder to which the unpacked state is bound should still be
1249 declared, this signal will become the register and will hold the
1251 \item A state pack operation should not generate any \small{VHDL}.
1252 The binder to which the packed state is bound should not be
1253 declared. The binder that is packed is the signal that will hold the
1255 \item Any values of a State type should not be translated to
1256 \small{VHDL}. In particular, State elements should be removed from
1257 tuples (and other datatypes) and arguments with a state type should
1259 \item To make the state actually work, a simple \small{VHDL}
1260 (sequential) process should be generated. This process updates
1261 the state at every clockcycle, by assigning the new state to the
1262 current state. This will be recognized by synthesis tools as a
1263 register specification.
1266 When applying these rules to the description in
1267 \in{example}[ex:AvgStateNormal], we be left with the description
1268 in \in{example}[ex:AvgStateRemoved]. All the parts that do not
1269 generate any \VHDL directly are crossed out, leaving just the
1270 actual flow of values in the final hardware.
1273 avg = iλ.λ--spacked.--
1275 s = --spacked ▶ (AccState, Word)--
1276 --accs = case s of (accs, _) -> accs--
1277 count = case s of (--_,-- count) -> count
1278 accres = acc i --accs--
1279 --accs' = case accres of (accs', _) -> accs'--
1280 sum = case accres of (--_,-- sum) -> sum
1283 s' = (--accs',-- count')
1284 --spacked' = s' ▶ State (AccState, Word)--
1285 res = (--spacked',-- o)
1290 When we would really leave out the crossed out parts, we get a slightly
1291 weird program: There is a variable \lam{s} which has no value, and there
1292 is a variable \lam{s'} that is never used. Together, these two will form
1293 the state process of the function. \lam{s} contains the "current" state,
1294 \lam{s'} is assigned the "next" state. So, at the end of each clock
1295 cycle, \lam{s'} should be assigned to \lam{s}.
1297 In the example the definition of \lam{s'} is still present, since
1298 it does not have a state type. The \lam{accums'} substate has been
1299 removed, leaving us just with the state of \lam{avg} itself.
1301 As an illustration of the result of this function,
1302 \in{example}[ex:AccStateVHDL] and \in{example}[ex:AvgStateVHDL] show the the \VHDL that is
1303 generated from the examples is this section.
1305 \startbuffer[AvgStateVHDL]
1306 entity avgComponent_0 is
1307 port (\izAlE2\ : in \unsigned_31\;
1308 \foozAo1zAo12\ : out \(,)unsigned_31\;
1309 clock : in std_logic;
1310 resetn : in std_logic);
1311 end entity avgComponent_0;
1314 architecture structural of avgComponent_0 is
1315 signal \szAlG2\ : \(,)unsigned_31\;
1316 signal \countzAlW2\ : \unsigned_31\;
1317 signal \dszAm62\ : \(,)unsigned_31\;
1318 signal \sumzAmk3\ : \unsigned_31\;
1319 signal \reszAnCzAnM2\ : \unsigned_31\;
1320 signal \foozAnZzAnZ2\ : \unsigned_31\;
1321 signal \reszAnfzAnj3\ : \unsigned_31\;
1322 signal \s'zAmC2\ : \(,)unsigned_31\;
1324 \countzAlW2\ <= \szAlG2\.A;
1326 \comp_ins_dszAm62\ : entity accComponent_1
1327 port map (\izAob3\ => \izAlE2\,
1328 \foozAoBzAoB2\ => \dszAm62\,
1332 \sumzAmk3\ <= \dszAm62\.A;
1334 \reszAnCzAnM2\ <= to_unsigned(1, 32);
1336 \foozAnZzAnZ2\ <= \countzAlW2\ + \reszAnCzAnM2\;
1338 \reszAnfzAnj3\ <= \sumzAmk3\ * \foozAnZzAnZ2\;
1340 \s'zAmC2\.A <= \foozAnZzAnZ2\;
1342 \foozAo1zAo12\.A <= \reszAnfzAnj3\;
1344 state : process (clock, resetn)
1346 if resetn = '0' then
1347 elseif rising_edge(clock) then
1348 \szAlG2\ <= \s'zAmC2\;
1351 end architecture structural;
1353 \startbuffer[AccStateVHDL]
1354 entity accComponent_1 is
1355 port (\izAob3\ : in \unsigned_31\;
1356 \foozAoBzAoB2\ : out \(,)unsigned_31\;
1357 clock : in std_logic;
1358 resetn : in std_logic);
1359 end entity accComponent_1;
1362 architecture structural of accComponent_1 is
1363 signal \szAod3\ : \unsigned_31\;
1364 signal \reszAonzAor3\ : \unsigned_31\;
1366 \reszAonzAor3\ <= \szAod3\ + \izAob3\;
1368 \foozAoBzAoB2\.A <= \reszAonzAor3\;
1370 state : process (clock, resetn)
1372 if resetn = '0' then
1373 elseif rising_edge(clock) then
1374 \szAod3\ <= \reszAonzAor3\;
1377 end architecture structural;
1380 \placeexample[][ex:AccStateVHDL]{\VHDL generated for acc from \in{example}[ex:AvgState]}
1381 {\typebuffer[AccStateVHDL]}
1382 \placeexample[][ex:AvgStateVHDL]{\VHDL generated for avg from \in{example}[ex:AvgState]}
1383 {\typebuffer[AvgStateVHDL]}
1384 % \subsection{Initial state}
1385 % How to specify the initial state? Cannot be done inside a hardware
1386 % function, since the initial state is its own state argument for the first
1387 % call (unless you add an explicit, synchronous reset port).
1389 % External init state is natural for simulation.
1391 % External init state works for hardware generation as well.
1393 % Implementation issues: state splitting, linking input to output state,
1394 % checking usage constraints on state variables.
1396 % \todo{Implementation issues: Separate compilation, simplified core.}
1398 % vim: set sw=2 sts=2 expandtab: