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 We will use the normal Core representation, not the simplified Core. Even
243 though the simplified Core version is an equivalent, but simpler
244 definition, some problems were encountered with it in practice. The
245 simplifier restructures some (stateful) functions in a way the normalizer
246 and the \VHDL\ generation cannot handle, leading to uncompilable programs
247 (whereas the non-simplified version more closely resembles the original
248 program, allowing the original to be written in a way that can be
249 handled). This problem is further discussed in
250 \in{section}[sec:normalization:stateproblems].
252 The final prototype roughly consists of three steps:
254 \startuseMPgraphic{clash-pipeline}
256 save inp, front, norm, vhdl, out;
257 newEmptyBox.inp(0,0);
258 newBox.front(btex \small{GHC} frontend etex);
259 newBox.norm(btex Normalization etex);
260 newBox.vhdl(btex \small{VHDL} generation etex);
261 newEmptyBox.out(0,0);
263 % Space the boxes evenly
264 inp.c - front.c = front.c - norm.c = norm.c - vhdl.c
265 = vhdl.c - out.c = (0, 1.5cm);
268 % Draw lines between the boxes. We make these lines "deferred" and give
269 % them a name, so we can use ObjLabel to draw a label beside them.
270 ncline.inp(inp)(front) "name(haskell)";
271 ncline.front(front)(norm) "name(core)";
272 ncline.norm(norm)(vhdl) "name(normal)";
273 ncline.vhdl(vhdl)(out) "name(vhdl)";
274 ObjLabel.inp(btex Haskell source etex) "labpathname(haskell)", "labdir(rt)";
275 ObjLabel.front(btex Core etex) "labpathname(core)", "labdir(rt)";
276 ObjLabel.norm(btex Normalized core etex) "labpathname(normal)", "labdir(rt)";
277 ObjLabel.vhdl(btex \small{VHDL} description etex) "labpathname(vhdl)", "labdir(rt)";
279 % Draw the objects (and deferred labels)
280 drawObj (inp, front, norm, vhdl, out);
282 \placefigure[right]{Cλash compiler pipeline}{\useMPgraphic{clash-pipeline}}
285 This is exactly the frontend from the \small{GHC} pipeline, that
286 translates Haskell sources to a typed Core representation.
288 \startdesc{Normalization}
289 This is a step that transforms the core representation into a normal
290 form. This normal form is still expressed in the core language, but has
291 to adhere to an additional set of constraints. This normal form is less
292 expressive than the full core language (e.g., it can have limited
293 higher-order expressions, has a specific structure, etc.), but is
294 also very close to directly describing hardware.
296 \startdesc{\small{VHDL} generation}
297 The last step takes the normal formed core representation and generates
298 \small{VHDL} for it. Since the normal form has a specific, hardware-like
299 structure, this final step is very straightforward.
302 The most interesting step in this process is the normalization step. That
303 is where more complicated functional constructs, which have no direct
304 hardware interpretation, are removed and translated into hardware
305 constructs. This step is described in a lot of detail at
306 \in{chapter}[chap:normalization].
309 \defref{entry function}Translation of a hardware description always
310 starts at a single function, which is referred to as the \emph{entry
311 function}. \VHDL\ is generated for this function first, followed by
312 any functions used by the entry functions (recursively).
314 \section[sec:prototype:core]{The Core language}
315 \defreftxt{core}{the Core language}
316 Most of the prototype deals with handling the program in the Core
317 language. In this section we will show what this language looks like and
320 The Core language is a functional language that describes
321 \emph{expressions}. Every identifier used in Core is called a
322 \emph{binder}, since it is bound to a value somewhere. On the highest
323 level, a Core program is a collection of functions, each of which bind a
324 binder (the function name) to an expression (the function value, which has
327 The Core language itself does not prescribe any program structure
328 (like modules, declarations, imports, etc.), only expression
329 structure. In the \small{GHC} compiler, the Haskell module structure
330 is used for the resulting Core code as well. Since this is not so
331 relevant for understanding the Core language or the Normalization
332 process, we will only look at the Core expression language here.
334 Each Core expression consists of one of these possible expressions.
336 \startdesc{Variable reference}
337 \defref{variable reference}
341 This is a reference to a binder. It is written down as the
342 name of the binder that is being referred to along with its type. The
343 binder name should of course be bound in a containing scope
344 (including top level scope, so a reference to a top level function
345 is also a variable reference). Additionally, constructors from
346 algebraic datatypes also become variable references.
348 In our examples, binders will commonly consist of a single
349 characters, but they can have any length.
351 The value of this expression is the value bound to the given
354 Each binder also carries around its type (explicitly shown above), but
355 this is usually not shown in the Core expressions. Only when the type is
356 relevant (when a new binder is introduced, for example) will it be
357 shown. In other cases, the binder is either not relevant, or easily
358 derived from the context of the expression. \todo{Ref sidenote on type
367 This is a literal. Only primitive types are supported, like
368 chars, strings, ints and doubles. The types of these literals are the
369 \quote{primitive}, unboxed versions, like \lam{Char\#} and \lam{Word\#}, not the
370 normal Haskell versions (but there are built-in conversion
371 functions). Without going into detail about these types, note that
372 a few conversion functions exist to convert these to the normal
373 (boxed) Haskell equivalents.
376 \startdesc{Application}
381 This is function application. Each application consists of two
382 parts: The function part and the argument part. Applications are used
383 for normal function \quote{calls}, but also for applying type
384 abstractions and data constructors.
386 In core, there is no distinction between an operator and a
387 function. This means that, for example the addition of two numbers
388 looks like the following in Core:
394 Where the function \quote{\lam{(+)}} is applied to the numbers 1
395 and 2. However, to increase readability, an application of an
396 operator like \lam{(+)} is sometimes written infix. In this case,
397 the parenthesis are also left out, just like in Haskell. In other
398 words, the following means exactly the same as the addition above:
404 The value of an application is the value of the function part, with the
405 first argument binder bound to the argument part.
408 \startdesc{Lambda abstraction}
409 \defref{lambda abstraction}
413 This is the basic lambda abstraction, as it occurs in lambda calculus.
414 It consists of a binder part and a body part. A lambda abstraction
415 creates a function, that can be applied to an argument. The binder is
416 usually a value binder, but it can also be a \emph{type binder} (or
417 \emph{type variable}). The latter case introduces a new polymorphic
418 variable, which can be used in types later on. See
419 \in{section}[sec:prototype:coretypes] for details.
421 The body of a lambda abstraction extends all the way to the end of
422 the expression, or the closing bracket surrounding the lambda. In
423 other words, the lambda abstraction \quote{operator} has the
424 lowest priority of all.
426 The value of an application is the value of the body part, with the
427 binder bound to the value the entire lambda abstraction is applied to.
430 \startdesc{Non-recursive let expression}
431 \defref{let expression}
433 let bndr = value in body
435 A let expression allows you to bind a binder to some value, while
436 evaluating to some other value (for which that binder is in scope). This
437 allows for sharing of subexpressions (you can use a binder twice) and
438 explicit \quote{naming} of arbitrary expressions. A binder is not
439 in scope in the value bound it is bound to, so it is not possible
440 to make recursive definitions with a non-recursive let expression
441 (see the recursive form below).
443 Even though this let expression is an extension on the basic lambda
444 calculus, it is easily translated to a lambda abstraction. The let
445 expression above would then become:
451 This notion might be useful for verifying certain properties on
452 transformations, since a lot of verification work has been done on
453 lambda calculus already.
455 The value of a let expression is the value of the body part, with the
456 binder bound to the value.
459 \startdesc{Recursive let expression}
468 This is the recursive version of the let expression. In \small{GHC}'s
469 Core implementation, non-recursive and recursive lets are not so
470 distinct as we present them here, but this provides a clearer overview.
472 The main difference with the normal let expression is that it can
473 contain multiple bindings (or even none) and each of the binders
474 is in scope in each of the values, in addition to the body. This
475 allows for self-recursive or mutually recursive definitions.
477 It is also possible to express a recursive let expression using
478 normal lambda calculus, if we use the \emph{least fixed-point
479 operator}, \lam{Y} (but the details are too complicated to help
480 clarify the let expression, so this will not be explored further).
484 \startframedtext[width=8cm,background=box,frame=no]
485 \startalignment[center]
486 {\tfa Weak head normal form (\small{WHNF})}
489 An expression is in weak head normal form if it is either an
490 constructor application or lambda abstraction. \todo{How about
493 Without going into detail about the differences with head
494 normal form and normal form, note that evaluating the scrutinee
495 of a case expression to normal form (evaluating any function
496 applications, variable references and case expressions) is
497 sufficient to decide which case alternatives should be chosen.
503 \startdesc{Case expression}
504 \defref{case expression}
506 case scrutinee of bndr
507 DEFAULT -> defaultbody
508 C0 bndr0,0 ... bndr0,m -> body0
510 Cn bndrn,0 ... bndrn,m -> bodyn
513 A case expression is the only way in Core to choose between values. All
514 \hs{if} expressions and pattern matchings from the original Haskell
515 PRogram have been translated to case expressions by the desugarer.
517 A case expression evaluates its scrutinee, which should have an
518 algebraic datatype, into weak head normal form (\small{WHNF}) and
519 (optionally) binds it to \lam{bndr}. Every alternative lists a
520 single constructor (\lam{C0 ... Cn}). Based on the actual
521 constructor of the scrutinee, the corresponding alternative is
522 chosen. The binders in the chosen alternative (\lam{bndr0,0 ....
523 bndr0,m} are bound to the actual arguments to the constructor in
526 This is best illustrated with an example. Assume
527 there is an algebraic datatype declared as follows\footnote{This
528 datatype is not suported by the current Cλash implementation, but
529 serves well to illustrate the case expression}:
532 data D = A Word | B Bit
535 This is an algebraic datatype with two constructors, each getting
536 a single argument. A case expression scrutinizing this datatype
537 could look like the following:
545 What this expression does is check the constructor of the
546 scrutinee \lam{s}. If it is \lam{A}, it always evaluates to
547 \lam{High}. If the constructor is \lam{B}, the binder \lam{bit} is
548 bound to the argument passed to \lam{B} and the case expression
549 evaluates to this bit.
551 If none of the alternatives match, the \lam{DEFAULT} alternative
552 is chosen. A case expression must always be exhaustive, \ie\ it
553 must cover all possible constructors that the scrutinee can have
554 (if all of them are covered explicitly, the \lam{DEFAULT}
555 alternative can be left out).
557 Since we can only match the top level constructor, there can be no overlap
558 in the alternatives and thus order of alternatives is not relevant (though
559 the \lam{DEFAULT} alternative must appear first for implementation
562 To support strictness, the scrutinee is always evaluated into
563 \small{WHNF}, even when there is only a \lam{DEFAULT} alternative. This
564 allows aplication of the strict function \lam{f} to the argument \lam{a}
568 f (case a of arg DEFAULT -> arg)
571 According to the \GHC\ documentation, this is the only use for the extra
572 binder to which the scrutinee is bound. When not using strictness
573 annotations (which is rather pointless in hardware descriptions),
574 \small{GHC} seems to never generate any code making use of this binder.
575 In fact, \GHC\ has never been observed to generate code using this
576 binder, even when strictness was involved. Nonetheless, the prototype
577 handles this binder as expected.
579 Note that these case expressions are less powerful than the full Haskell
580 case expressions. In particular, they do not support complex patterns like
581 in Haskell. Only the constructor of an expression can be matched,
582 complex patterns are implemented using multiple nested case expressions.
584 Case expressions are also used for unpacking of algebraic datatypes, even
585 when there is only a single constructor. For examples, to add the elements
586 of a tuple, the following Core is generated:
589 sum = λtuple.case tuple of
593 Here, there is only a single alternative (but no \lam{DEFAULT}
594 alternative, since the single alternative is already exhaustive). When
595 its body is evaluated, the arguments to the tuple constructor \lam{(,)}
596 (\eg, the elements of the tuple) are bound to \lam{a} and \lam{b}.
599 \startdesc{Cast expression}
600 \defref{cast expression}
604 A cast expression allows you to change the type of an expression to an
605 equivalent type. Note that this is not meant to do any actual work, like
606 conversion of data from one format to another, or force a complete type
607 change. Instead, it is meant to change between different representations
608 of the same type, \eg\ switch between types that are provably equal (but
611 In our hardware descriptions, we typically see casts to change between a
612 Haskell newtype and its contained type, since those are effectively
613 different types (so a cast is needed) with the same representation (but
614 no work is done by the cast).
616 More complex are types that are proven to be equal by the typechecker,
617 but look different at first glance. To ensure that, once the typechecker
618 has proven equality, this information sticks around, explicit casts are
619 added. In our notation we only write the target type, but in reality a
620 cast expressions carries around a \emph{coercion}, which can be seen as a
621 proof of equality. \todo{Example}
623 The value of a cast is the value of its body, unchanged. The type of this
624 value is equal to the target type, not the type of its body.
628 The Core language in \small{GHC} allows adding \emph{notes}, which serve
629 as hints to the inliner or add custom (string) annotations to a core
630 expression. These should not be generated normally, so these are not
631 handled in any way in the prototype.
635 \defref{type expression}
639 It is possibly to use a Core type as a Core expression. To prevent
640 confusion between types and values, the \lam{@} sign is used to
641 explicitly mark a type that is used in a Core expression.
643 For the actual types supported by Core, see
644 \in{section}[sec:prototype:coretypes]. This \quote{lifting} of a
645 type into the value domain is done to allow for type abstractions
646 and applications to be handled as normal lambda abstractions and
647 applications above. This means that a type expression in Core can
648 only ever occur in the argument position of an application, and
649 only if the type of the function that is applied to expects a type
650 as the first argument. This happens in applications of all
651 polymorphic functions. Consider the \lam{fst} function:
654 fst :: \forall t1. \forall t2. (t1, t2) ->t1
655 fst = λt1.λt2.λ(tup :: (t1, t2)). case tup of (,) a b -> a
657 fstint :: (Int, Int) -> Int
658 fstint = λa.λb.fst @Int @Int a b
661 The type of \lam{fst} has two universally quantified type variables. When
662 \lam{fst} is applied in \lam{fstint}, it is first applied to two types.
663 (which are substitued for \lam{t1} and \lam{t2} in the type of \lam{fst}, so
664 the actual type of arguments and result of \lam{fst} can be found:
665 \lam{fst @Int @Int :: (Int, Int) -> Int}).
668 \subsection[sec:prototype:coretypes]{Core type system}
669 Whereas the expression syntax of Core is very simple, its type system is
670 a bit more complicated. It turns out it is harder to \quote{desugar}
671 Haskell's complex type system into something more simple. Most of the
672 type system is thus very similar to that of Haskell.
674 We will slightly limit our view on Core's type system, since the more
675 complicated parts of it are only meant to support Haskell's (or rather,
676 \GHC's) type extensions, such as existential types, \small{GADT}s, type
677 families and other non-standard Haskell stuff which we do not (plan to)
680 In Core, every expression is typed. The translation to Core happens
681 after the typechecker, so types in Core are always correct as well
682 (though you could of course construct invalidly typed expressions
683 through the \GHC\ API).
685 Any type in core is one of the following:
687 \startdesc{A type variable}
692 This is a reference to a type defined elsewhere. This can either be a
693 polymorphic type (like the latter two \lam{t}'s in \lam{id :: \forall t.
694 t -> t}), or a type constructor (like \lam{Bool} in \lam{not :: Bool ->
695 Bool}). Like in Haskell, polymorphic type variables always
696 start with a lowercase letter, while type constructors always start
697 with an uppercase letter.
699 \todo{How to define (new) type constructors?}
701 A special case of a type constructor is the \emph{function type
702 constructor}, \lam{->}. This is a type constructor taking two arguments
703 (using application below). The function type constructor is commonly
704 written inline, so we write \lam{a -> b} when we really mean \lam{-> a
705 b}, the function type constructor applied to \lam{a} and \lam{b}.
707 Polymorphic type variables can only be defined by a lambda
708 abstraction, see the forall type below.
711 \startdesc{A type application}
716 This applies some type to another type. This is particularly used to
717 apply type variables (type constructors) to their arguments.
719 As mentioned above, applications of some type constructors have
720 special notation. In particular, these are applications of the
721 \emph{function type constructor} and \emph{tuple type constructors}:
726 bar' :: (,,) t1 t2 t3
730 \startdesc{The forall type}
732 id :: \forall t. t -> t
734 The forall type introduces polymorphism. It is the only way to
735 introduce new type variables, which are completely unconstrained (Any
736 possible type can be assigned to it). Constraints can be added later
737 using predicate types, see below.
739 A forall type is always (and only) introduced by a type lambda
740 expression. For example, the Core translation of the
746 Here, the type of the binder \lam{x} is \lam{t}, referring to the
747 binder in the topmost lambda.
749 When using a value with a forall type, the actual type
750 used must be applied first. For example Haskell expression \hs{id
751 True} (the function \hs{id} appleid to the dataconstructor \hs{True})
752 translates to the following Core:
758 Here, id is first applied to the type to work with. Note that the type
759 then changes from \lam{id :: \forall t. t -> t} to \lam{id @Bool ::
760 Bool -> Bool}. Note that the type variable \lam{a} has been
761 substituted with the actual type.
763 In Haskell, forall types are usually not explicitly specified (The use
764 of a lowercase type variable implicitly introduces a forall type for
765 that variable). In fact, in standard Haskell there is no way to
766 explicitly specify forall types. Through a language extension, the
767 \hs{forall} keyword is available, but still optional for normal forall
768 types (it is needed for \emph{existentially quantified types}, which
769 Cλash does not support).
772 \startdesc{Predicate type}
774 show :: \forall t. Show t ⇒ t → String
777 \todo{Sidenote: type classes?}
779 A predicate type introduces a constraint on a type variable introduced
780 by a forall type (or type lambda). In the example above, the type
781 variable \lam{t} can only contain types that are an \emph{instance} of
782 the \emph{type class} \lam{Show}. \refdef{type class}
784 There are other sorts of predicate types, used for the type families
785 extension, which we will not discuss here.
787 A predicate type is introduced by a lambda abstraction. Unlike with
788 the forall type, this is a value lambda abstraction, that must be
789 applied to a value. We call this value a \emph{dictionary}.
791 Without going into the implementation details, a dictionary can be
792 seen as a lookup table all the methods for a given (single) type class
793 instance. This means that all the dictionaries for the same type class
794 look the same (\eg\ contain methods with the same names). However,
795 dictionaries for different instances of the same class contain
796 different methods, of course.
798 A dictionary is introduced by \small{GHC} whenever it encounters an
799 instance declaration. This dictionary, as well as the binder
800 introduced by a lambda that introduces a dictionary, have the
801 predicate type as their type. These binders are usually named starting
802 with a \lam{\$}. Usually the name of the type concerned is not
803 reflected in the name of the dictionary, but the name of the type
804 class is. The Haskell expression \hs{show True} thus becomes:
807 show @Bool \$dShow True
811 Using this set of types, all types in basic Haskell can be represented.
813 \todo{Overview of polymorphism with more examples (or move examples
816 \section[sec:prototype:statetype]{State annotations in Haskell}
817 As noted in \in{section}[sec:description:stateann], Cλash needs some
818 way to let the programmer explicitly specify which of a function's
819 arguments and which part of a function's result represent the
822 Using the Haskell type systems, there are a few ways we can tackle this.
824 \subsection{Type synonyms}
825 Haskell provides type synonyms as a way to declare a new type that is
826 equal to an existing type (or rather, a new name for an existing type).
827 This allows both the original type and the synonym to be used
828 interchangedly in a Haskell program. This means no explicit conversion
829 is needed either. For example, a simple accumulator would become:
833 acc :: Word -> State Word -> (State Word, Word)
834 acc i s = let sum = s + i in (sum, sum)
837 This looks nice in Haskell, but turns out to be hard to implement. There
838 are no explicit conversion in Haskell, but not in Core either. This
839 means the type of a value might be show as \hs{AccState} in some places,
840 but \hs{Word} in others (and this can even change due to
841 transformations). Since every binder has an explicit type associated
842 with it, the type of every function type will be properly preserved and
843 could be used to track down the statefulness of each value by the
844 compiler. However, this makes the implementation a lot more complicated
845 than it currently is using \hs{newtypes}.
847 % Use \type instead of \hs here, since the latter breaks inside
849 \subsection{Type renaming (\type{newtype})}
850 Haskell also supports type renamings as a way to declare a new type that
851 has the same (runtime) representation as an existing type (but is in
852 fact a different type to the typechecker). With type renaming, an
853 explicit conversion between values of the two types is needed. The
854 accumulator would then become:
857 newtype State s = State s
858 acc :: Word -> State Word -> (State Word, Word)
859 acc i (State s) = let sum = s + i in (State sum, sum)
862 The \hs{newtype} line declares a new type \hs{State} that has one type
863 argument, \hs{s}. This type contains one \quote{constructor} \hs{State}
864 with a single argument of type \hs{s}. It is customary to name the
865 constructor the same as the type, which is allowed (since types can
866 never cause name collisions with values). The difference with the type
867 synonym example is in the explicit conversion between the \hs{State
868 Word} and \hs{Word} types by pattern matching and by using the explicit
869 the \hs{State constructor}.
871 This explicit conversion makes the \VHDL\ generation easier: Whenever we
872 remove (unpack) the \hs{State} type, this means we are accessing the
873 current state (\eg, accessing the register output). Whenever we are a
874 adding (packing) the \hs{State} type, we are producing a new value for
875 the state (\eg, providing the register input).
877 When dealing with nested states (a stateful function that calls stateful
878 functions, which might call stateful functions, etc.) the state type
879 could quickly grow complex because of all the \hs{State} type constructors
880 needed. For example, consider the following state type (this is just the
881 state type, not the entire function type):
884 State (State Bit, State (State Word, Bit), Word)
887 We cannot leave all these \hs{State} type constructors out, since that
888 would change the type (unlike when using type synonyms). However, when
889 using type synonyms to hide away substates (see
890 \in{section}[sec:prototype:substatesynonyms] below), this
891 disadvantage should be limited.
893 \subsubsection{Different input and output types}
894 An alternative could be to use different types for input and output
895 state (\ie\ current and updated state). The accumulator example would
896 then become something like:
899 newtype StateIn s = StateIn s
900 newtype StateOut s = StateOut s
901 acc :: Word -> StateIn Word -> (StateIn Word, Word)
902 acc i (StateIn s) = let sum = s + i in (StateIn sum, sum)
905 This could make the implementation easier and the hardware
906 descriptions less errorprone (you can no longer \quote{forget} to
907 unpack and repack a state variable and just return it directly, which
908 can be a problem in the current prototype). However, it also means we
909 need twice as many type synonyms to hide away substates, making this
910 approach a bit cumbersome. It also makes it harder to copmare input
911 and output state types, possible reducing the type safety of the
914 \subsection[sec:prototype:substatesynonyms]{Type synonyms for substates}
915 As noted above, when using nested (hierarchical) states, the state types
916 of the \quote{upper} functions (those that call other functions, which
917 call other functions, etc.) quickly becomes complicated. Also, when the
918 state type of one of the \quote{lower} functions changes, the state
919 types of all the upper functions changes as well. If the state type for
920 each function is explicitly and completely specified, this means that a
921 lot of code needs updating whenever a state type changes.
923 To prevent this, it is recommended (but not enforced) to use a type
924 synonym for the state type of every function. Every function calling
925 other functions will then use the state type synonym of the called
926 functions in its own type, requiring no code changes when the state type
927 of a called function changes. This approach is used in
928 \in{example}[ex:AvgState] below. The \hs{AccState} and \hs{AvgState}
929 are examples of such state type synonyms.
931 \subsection{Chosen approach}
932 To keep implementation simple, the current prototype uses the type
933 renaming approach, with a single type for both input and output
934 states. In the future, it might be worthwhile to revisit this
935 approach if more complicated flow analysis is implemented for
936 state variables. This analysis is needed to add proper error
937 checking anyway and might allow the use of type synonyms without
938 losing any expressivity.
940 \subsubsection{Example}
941 As an example of the used approach, there is a simple averaging circuit in
942 \in{example}[ex:AvgState]. This circuit lets the accumulation of the
943 inputs be done by a subcomponent, \hs{acc}, but keeps a count of value
944 accumulated in its own state.\footnote{Currently, the prototype
945 is not able to compile this example, since the built-in function
946 for division has not been added.}
948 \startbuffer[AvgState]
949 -- The state type annotation
950 newtype State s = State s
952 -- The accumulator state type
953 type AccState = State Word
955 acc :: Word -> AccState -> (AccState, Word)
956 acc i (State s) = let sum = s + i in (State sum, sum)
958 -- The averaging circuit state type
959 type AvgState = State (AccState, Word)
960 -- The averaging circuit
961 avg :: Word -> AvgState -> (AvgState, Word)
962 avg i (State s) = (State s', o)
965 -- Pass our input through the accumulator, which outputs a sum
966 (accs', sum) = acc i accs
967 -- Increment the count (which will be our new state)
969 -- Compute the average
974 \placeexample[here][ex:AvgState]{Simple stateful averaging circuit.}
975 %\startcombination[2*1]
976 {\typebufferhs{AvgState}}%{Haskell description using function applications.}
977 % {\boxedgraphic{AvgState}}{The architecture described by the Haskell description.}
981 \section{Implementing state}
982 Now its clear how to put state annotations in the Haskell source,
983 there is the question of how to implement this state translation. As
984 we have seen in \in{section}[sec:prototype:design], the translation to
985 \VHDL\ happens as a simple, final step in the compilation process.
986 This step works on a core expression in normal form. The specifics
987 of normal form will be explained in
988 \in{chapter}[chap:normalization], but the examples given should be
989 easy to understand using the definitin of Core given above.
991 \startbuffer[AvgStateNormal]
994 -- Remove the State newtype
998 -- Add the State newtype again
999 spacked' = s' ▶ State Word
1006 s = spacked ▶ (AccState, Word)
1007 accs = case s of (accs, _) -> accs
1008 count = case s of (_, count) -> count
1010 accs' = case accres of (accs', _) -> accs'
1011 sum = case accres of (_, sum) -> sum
1014 s' = (accs', count')
1015 spacked' = s' ▶ State (AccState, Word)
1021 \placeexample[here][ex:AvgStateNormal]{Normalized version of \in{example}[ex:AvgState]}
1022 {\typebufferlam{AvgStateNormal}}
1024 \subsection[sec:prototype:statelimits]{State in normal form}
1025 Before describing how to translate state from normal form to
1026 \VHDL, we will first see how state handling looks in normal form.
1027 What limitations are there on their use to guarantee that proper
1028 \VHDL\ can be generated?
1030 We will try to formulate a number of rules about what operations are
1031 allowed with state variables. These rules apply to the normalized Core
1032 representation, but will in practice apply to the original Haskell
1033 hardware description as well. Ideally, these rules would become part
1034 of the intended normal form definition \refdef{intended normal form
1035 definition}, but this is not the case right now. This can cause some
1036 problems, which are detailed in
1037 \in{section}[sec:normalization:stateproblems].
1039 In these rules we use the terms \emph{state variable} to refer to any
1040 variable that has a \lam{State} type. A \emph{state-containing
1041 variable} is any variable whose type contains a \lam{State} type,
1042 but is not one itself (like \lam{(AccState, Word)} in the example,
1043 which is a tuple type, but contains \lam{AccState}, which is again
1044 equal to \lam{State Word}).
1046 We also use a distinction between \emph{input} and \emph{output
1047 (state) variables} and \emph{substate variables}, which will be
1048 defined in the rules themselves.
1050 \startdesc{State variables can appear as an argument.}
1052 avg = λi.λspacked. ...
1055 Any lambda that binds a variable with a state type, creates a new
1056 input state variable.
1059 \startdesc{Input state variables can be unpacked.}
1061 s = spacked ▶ (AccState, Word)
1064 An input state variable may be unpacked using a cast operation. This
1065 removes the \lam{State} type renaming and the result has no longer a
1068 If the result of this unpacking does not have a state type and does
1069 not contain state variables, there are no limitations on its use.
1070 Otherwise if it does not have a state type but does contain
1071 substates, we refer to it as a \emph{state-containing input
1072 variable} and the limitations below apply. If it has a state type
1073 itself, we refer to it as an \emph{input substate variable} and the
1074 below limitations apply as well.
1076 It may seem strange to consider a variable that still has a state
1077 type directly after unpacking, but consider the case where a
1078 function does not have any state of its own, but does call a single
1079 stateful function. This means it must have a state argument that
1080 contains just a substate. The function signature of such a function
1084 type FooState = State AccState
1087 Which is of course equivalent to \lam{State (State Word)}.
1090 \startdesc{Variables can be extracted from state-containing input variables.}
1092 accs = case s of (accs, _) -> accs
1095 A state-containing input variable is typically a tuple containing
1096 multiple elements (like the current function's state, substates or
1097 more tuples containing substates). All of these can be extracted
1098 from an input variable using an extractor case (or possibly
1099 multiple, when the input variable is nested).
1101 If the result has no state type and does not contain any state
1102 variables either, there are no further limitations on its use. If
1103 the result has no state type but does contain state variables we
1104 refer to it as a \emph{state-containing input variable} and this
1105 limitation keeps applying. If the variable has a state type itself,
1106 we refer to it as an \emph{input substate variable} and below
1109 \startdesc{Input substate variables can be passed to functions.}
1112 accs' = case accres of (accs', _) -> accs'
1115 An input substate variable can (only) be passed to a function.
1116 Additionally, every input substate variable must be used in exactly
1117 \emph{one} application, no more and no less.
1119 The function result should contain exactly one state variable, which
1120 can be extracted using (multiple) case expressions. The extracted
1121 state variable is referred to the \emph{output substate}
1123 The type of this output substate must be identical to the type of
1124 the input substate passed to the function.
1127 \startdesc{Variables can be inserted into a state-containing output variable.}
1129 s' = (accs', count')
1132 A function's output state is usually a tuple containing its own
1133 updated state variables and all output substates. This result is
1134 built up using any single-constructor algebraic datatype.
1136 The result of these expressions is referred to as a
1137 \emph{state-containing output variable}, which are subject to these
1141 \startdesc{State containing output variables can be packed.}
1143 spacked' = s' ▶ State (AccState, Word)
1146 As soon as all a functions own update state and output substate
1147 variables have been joined together, the resulting
1148 state-containing output variable can be packed into an output
1149 state variable. Packing is done by casting into a state type.
1152 \startdesc{Output state variables can appear as (part of) a function result.}
1161 When the output state is packed, it can be returned as a part
1162 of the function result. Nothing else can be done with this
1163 value (or any value that contains it).
1166 There is one final limitation that is hard to express in the above
1167 itemization. Whenever substates are extracted from the input state
1168 to be passed to functions, the corresponding output substates
1169 should be inserted into the output state in the same way. In other
1170 words, each pair of corresponding substates in the input and
1171 output states should be passed / returned from the same called
1174 The prototype currently does not check much of the above
1175 conditions. This means that if the conditions are violated,
1176 sometimes a compile error is generated, but in other cases output
1177 can be generated that is not valid \VHDL\ or at the very least does
1178 not correspond to the input.
1180 \subsection{Translating to \VHDL}
1181 As noted above, the basic approach when generating \VHDL\ for stateful
1182 functions is to generate a single register for every stateful function.
1183 We look around the normal form to find the let binding that removes the
1184 \lam{State} newtype (using a cast). We also find the let binding that
1185 adds a \lam{State} type. These are connected to the output and the input
1186 of the generated let binding respectively. This means that there can
1187 only be one let binding that adds and one that removes the \lam{State}
1188 type. It is easy to violate this constraint. This problem is detailed in
1189 \in{section}[sec:normalization:stateproblems].
1191 This approach seems simple enough, but will this also work for more
1192 complex stateful functions involving substates? Observe that any
1193 component of a function's state that is a substate, \ie\ passed on as
1194 the state of another function, should have no influence on the
1195 hardware generated for the calling function. Any state-specific
1196 \small{VHDL} for this component can be generated entirely within the
1197 called function. So, we can completely ignore substates when
1198 generating \VHDL\ for a function.
1200 From this observation it might seem logical to remove the
1201 substates from a function's states altogether and leave only the
1202 state components which are actual states of the current function.
1203 While doing this would not remove any information needed to
1204 generate \small{VHDL} from the function, it would cause the
1205 function definition to become invalid (since we will not have any
1206 substate to pass to the functions anymore). We could solve the
1207 syntactic problems by passing \type{undefined} for state
1208 variables, but that would still break the code on the semantic
1209 level (\ie, the function would no longer be semantically
1210 equivalent to the original input).
1212 To keep the function definition correct until the very end of the
1213 process, we will not deal with (sub)states until we get to the
1214 \small{VHDL} generation. Then, we are translating from Core to
1215 \small{VHDL}, and we can simply ignore substates, effectively removing
1216 the substate components altogether.
1218 But, how will we know what exactly is a substate? Since any state
1219 argument or return value that represents state must be of the
1220 \type{State} type, we can look at the type of a value. However, we
1221 must be careful to ignore only \emph{substates}, and not a
1222 function's own state.
1224 In \in{example}[ex:AvgStateNorm] above, we should generate a register
1225 connected with its output connected to \lam{s} and its input connected
1226 to \lam{s'}. However, \lam{s'} is build up from both \lam{accs'} and
1227 \lam{count'}, while only \lam{count'} should end up in the register.
1228 \lam{accs'} is a substate for the \lam{acc} function, for which a
1229 register will be created when generating \VHDL\ for the \lam{acc}
1232 Fortunately, the \lam{accs'} variable (and any other substate) has a
1233 property that we can easily check: It has a \lam{State} type
1234 annotation. This means that whenever \VHDL\ is generated for a tuple
1235 (or other algebraic type), we can simply leave out all elements that
1236 have a \lam{State} type. This will leave just the parts of the state
1237 that do not have a \lam{State} type themselves, like \lam{count'},
1238 which is exactly a function's own state. This approach also means that
1239 the state part of the result is automatically excluded when generating
1240 the output port, which is also required.
1242 We can formalize this translation a bit, using the following
1246 \item A state unpack operation should not generate any \small{VHDL}.
1247 The binder to which the unpacked state is bound should still be
1248 declared, this signal will become the register and will hold the
1250 \item A state pack operation should not generate any \small{VHDL}.
1251 The binder to which the packed state is bound should not be
1252 declared. The binder that is packed is the signal that will hold the
1254 \item Any values of a State type should not be translated to
1255 \small{VHDL}. In particular, State elements should be removed from
1256 tuples (and other datatypes) and arguments with a state type should
1258 \item To make the state actually work, a simple \small{VHDL}
1259 (sequential) process should be generated. This process updates
1260 the state at every clockcycle, by assigning the new state to the
1261 current state. This will be recognized by synthesis tools as a
1262 register specification.
1265 When applying these rules to the description in
1266 \in{example}[ex:AvgStateNormal], we be left with the description
1267 in \in{example}[ex:AvgStateRemoved]. All the parts that do not
1268 generate any \VHDL\ directly are crossed out, leaving just the
1269 actual flow of values in the final hardware.
1272 avg = iλ.λ--spacked.--
1274 s = --spacked ▶ (AccState, Word)--
1275 --accs = case s of (accs, _) -> accs--
1276 count = case s of (--_,-- count) -> count
1277 accres = acc i --accs--
1278 --accs' = case accres of (accs', _) -> accs'--
1279 sum = case accres of (--_,-- sum) -> sum
1282 s' = (--accs',-- count')
1283 --spacked' = s' ▶ State (AccState, Word)--
1284 res = (--spacked',-- o)
1289 When we would really leave out the crossed out parts, we get a slightly
1290 weird program: There is a variable \lam{s} which has no value, and there
1291 is a variable \lam{s'} that is never used. Together, these two will form
1292 the state process of the function. \lam{s} contains the "current" state,
1293 \lam{s'} is assigned the "next" state. So, at the end of each clock
1294 cycle, \lam{s'} should be assigned to \lam{s}.
1296 In the example the definition of \lam{s'} is still present, since
1297 it does not have a state type. The \lam{accums'} substate has been
1298 removed, leaving us just with the state of \lam{avg} itself.
1300 As an illustration of the result of this function,
1301 \in{example}[ex:AccStateVHDL] and \in{example}[ex:AvgStateVHDL] show the the \VHDL\ that is
1302 generated from the examples is this section.
1304 \startbuffer[AvgStateVHDL]
1305 entity avgComponent_0 is
1306 port (\izAlE2\ : in \unsigned_31\;
1307 \foozAo1zAo12\ : out \(,)unsigned_31\;
1308 clock : in std_logic;
1309 resetn : in std_logic);
1310 end entity avgComponent_0;
1313 architecture structural of avgComponent_0 is
1314 signal \szAlG2\ : \(,)unsigned_31\;
1315 signal \countzAlW2\ : \unsigned_31\;
1316 signal \dszAm62\ : \(,)unsigned_31\;
1317 signal \sumzAmk3\ : \unsigned_31\;
1318 signal \reszAnCzAnM2\ : \unsigned_31\;
1319 signal \foozAnZzAnZ2\ : \unsigned_31\;
1320 signal \reszAnfzAnj3\ : \unsigned_31\;
1321 signal \s'zAmC2\ : \(,)unsigned_31\;
1323 \countzAlW2\ <= \szAlG2\.A;
1325 \comp_ins_dszAm62\ : entity accComponent_1
1326 port map (\izAob3\ => \izAlE2\,
1327 \foozAoBzAoB2\ => \dszAm62\,
1331 \sumzAmk3\ <= \dszAm62\.A;
1333 \reszAnCzAnM2\ <= to_unsigned(1, 32);
1335 \foozAnZzAnZ2\ <= \countzAlW2\ + \reszAnCzAnM2\;
1337 \reszAnfzAnj3\ <= \sumzAmk3\ * \foozAnZzAnZ2\;
1339 \s'zAmC2\.A <= \foozAnZzAnZ2\;
1341 \foozAo1zAo12\.A <= \reszAnfzAnj3\;
1343 state : process (clock, resetn)
1345 if resetn = '0' then
1346 elseif rising_edge(clock) then
1347 \szAlG2\ <= \s'zAmC2\;
1350 end architecture structural;
1352 \startbuffer[AccStateVHDL]
1353 entity accComponent_1 is
1354 port (\izAob3\ : in \unsigned_31\;
1355 \foozAoBzAoB2\ : out \(,)unsigned_31\;
1356 clock : in std_logic;
1357 resetn : in std_logic);
1358 end entity accComponent_1;
1361 architecture structural of accComponent_1 is
1362 signal \szAod3\ : \unsigned_31\;
1363 signal \reszAonzAor3\ : \unsigned_31\;
1365 \reszAonzAor3\ <= \szAod3\ + \izAob3\;
1367 \foozAoBzAoB2\.A <= \reszAonzAor3\;
1369 state : process (clock, resetn)
1371 if resetn = '0' then
1372 elseif rising_edge(clock) then
1373 \szAod3\ <= \reszAonzAor3\;
1376 end architecture structural;
1379 \placeexample[][ex:AccStateVHDL]{\VHDL\ generated for acc from \in{example}[ex:AvgState]}
1380 {\typebuffer[AccStateVHDL]}
1381 \placeexample[][ex:AvgStateVHDL]{\VHDL\ generated for avg from \in{example}[ex:AvgState]}
1382 {\typebuffer[AvgStateVHDL]}
1383 % \subsection{Initial state}
1384 % How to specify the initial state? Cannot be done inside a hardware
1385 % function, since the initial state is its own state argument for the first
1386 % call (unless you add an explicit, synchronous reset port).
1388 % External init state is natural for simulation.
1390 % External init state works for hardware generation as well.
1392 % Implementation issues: state splitting, linking input to output state,
1393 % checking usage constraints on state variables.
1396 % vim: set sw=2 sts=2 expandtab: