1 \chapter[chap:normalization]{Normalization}
2 % A helper to print a single example in the half the page width. The example
3 % text should be in a buffer whose name is given in an argument.
5 % The align=right option really does left-alignment, but without the program
6 % will end up on a single line. The strut=no option prevents a bunch of empty
7 % space at the start of the frame.
9 \framed[offset=1mm,align=right,strut=no,background=box,frame=off]{
10 \setuptyping[option=LAM,style=sans,before=,after=,strip=auto]
12 \setuptyping[option=none,style=\tttf,strip=auto]
16 \define[4]\transexample{
17 \placeexample[here][ex:trans:#1]{#2}
18 \startcombination[2*1]
19 {\example{#3}}{Original program}
20 {\example{#4}}{Transformed program}
24 The first step in the core to \small{VHDL} translation process, is normalization. We
25 aim to bring the core description into a simpler form, which we can
26 subsequently translate into \small{VHDL} easily. This normal form is needed because
27 the full core language is more expressive than \small{VHDL} in some areas and because
28 core can describe expressions that do not have a direct hardware
32 The transformations described here have a well-defined goal: To bring the
33 program in a well-defined form that is directly translatable to hardware,
34 while fully preserving the semantics of the program. We refer to this form as
35 the \emph{normal form} of the program. The formal definition of this normal
38 \placedefinition{}{A program is in \emph{normal form} if none of the
39 transformations from this chapter apply.}
41 Of course, this is an \quote{easy} definition of the normal form, since our
42 program will end up in normal form automatically. The more interesting part is
43 to see if this normal form actually has the properties we would like it to
46 But, before getting into more definitions and details about this normal form,
47 let's try to get a feeling for it first. The easiest way to do this is by
48 describing the things we want to not have in a normal form.
51 \item Any \emph{polymorphism} must be removed. When laying down hardware, we
52 can't generate any signals that can have multiple types. All types must be
53 completely known to generate hardware.
55 \item Any \emph{higher order} constructions must be removed. We can't
56 generate a hardware signal that contains a function, so all values,
57 arguments and returns values used must be first order.
59 \item Any complex \emph{nested scopes} must be removed. In the \small{VHDL}
60 description, every signal is in a single scope. Also, full expressions are
61 not supported everywhere (in particular port maps can only map signal
62 names and constants, not complete expressions). To make the \small{VHDL}
63 generation easy, a separate binder must be bound to ever application or
67 \todo{Intermezzo: functions vs plain values}
69 A very simple example of a program in normal form is given in
70 \in{example}[ex:MulSum]. As you can see, all arguments to the function (which
71 will become input ports in the final hardware) are at the outer level.
72 This means that the body of the inner lambda abstraction is never a
73 function, but always a plain value.
75 As the body of the inner lambda abstraction, we see a single (recursive)
76 let expression, that binds two variables (\lam{mul} and \lam{sum}). These
77 variables will be signals in the final hardware, bound to the output port
78 of the \lam{*} and \lam{+} components.
80 The final line (the \quote{return value} of the function) selects the
81 \lam{sum} signal to be the output port of the function. This \quote{return
82 value} can always only be a variable reference, never a more complex
85 \todo{Add generated VHDL}
88 alu :: Bit -> Word -> Word -> Word
97 \startuseMPgraphic{MulSum}
98 save a, b, c, mul, add, sum;
101 newCircle.a(btex $a$ etex) "framed(false)";
102 newCircle.b(btex $b$ etex) "framed(false)";
103 newCircle.c(btex $c$ etex) "framed(false)";
104 newCircle.sum(btex $res$ etex) "framed(false)";
107 newCircle.mul(btex * etex);
108 newCircle.add(btex + etex);
110 a.c - b.c = (0cm, 2cm);
111 b.c - c.c = (0cm, 2cm);
112 add.c = c.c + (2cm, 0cm);
113 mul.c = midpoint(a.c, b.c) + (2cm, 0cm);
114 sum.c = add.c + (2cm, 0cm);
117 % Draw objects and lines
118 drawObj(a, b, c, mul, add, sum);
120 ncarc(a)(mul) "arcangle(15)";
121 ncarc(b)(mul) "arcangle(-15)";
127 \placeexample[here][ex:MulSum]{Simple architecture consisting of a
128 multiplier and a subtractor.}
129 \startcombination[2*1]
130 {\typebufferlam{MulSum}}{Core description in normal form.}
131 {\boxedgraphic{MulSum}}{The architecture described by the normal form.}
134 The previous example described composing an architecture by calling other
135 functions (operators), resulting in a simple architecture with components and
136 connections. There is of course also some mechanism for choice in the normal
137 form. In a normal Core program, the \emph{case} expression can be used in a
138 few different ways to describe choice. In normal form, this is limited to a
141 \in{Example}[ex:AddSubAlu] shows an example describing a
142 simple \small{ALU}, which chooses between two operations based on an opcode
143 bit. The main structure is similar to \in{example}[ex:MulSum], but this
144 time the \lam{res} variable is bound to a case expression. This case
145 expression scrutinizes the variable \lam{opcode} (and scrutinizing more
146 complex expressions is not supported). The case expression can select a
147 different variable based on the constructor of \lam{opcode}.
149 \startbuffer[AddSubAlu]
150 alu :: Bit -> Word -> Word -> Word
162 \startuseMPgraphic{AddSubAlu}
163 save opcode, a, b, add, sub, mux, res;
166 newCircle.opcode(btex $opcode$ etex) "framed(false)";
167 newCircle.a(btex $a$ etex) "framed(false)";
168 newCircle.b(btex $b$ etex) "framed(false)";
169 newCircle.res(btex $res$ etex) "framed(false)";
171 newCircle.add(btex + etex);
172 newCircle.sub(btex - etex);
175 opcode.c - a.c = (0cm, 2cm);
176 add.c - a.c = (4cm, 0cm);
177 sub.c - b.c = (4cm, 0cm);
178 a.c - b.c = (0cm, 3cm);
179 mux.c = midpoint(add.c, sub.c) + (1.5cm, 0cm);
180 res.c - mux.c = (1.5cm, 0cm);
183 % Draw objects and lines
184 drawObj(opcode, a, b, res, add, sub, mux);
186 ncline(a)(add) "posA(e)";
187 ncline(b)(sub) "posA(e)";
188 nccurve(a)(sub) "posA(e)", "angleA(0)";
189 nccurve(b)(add) "posA(e)", "angleA(0)";
190 nccurve(add)(mux) "posB(inpa)", "angleB(0)";
191 nccurve(sub)(mux) "posB(inpb)", "angleB(0)";
192 nccurve(opcode)(mux) "posB(n)", "angleA(0)", "angleB(-90)";
193 ncline(mux)(res) "posA(out)";
196 \placeexample[here][ex:AddSubAlu]{Simple \small{ALU} supporting two operations.}
197 \startcombination[2*1]
198 {\typebufferlam{AddSubAlu}}{Core description in normal form.}
199 {\boxedgraphic{AddSubAlu}}{The architecture described by the normal form.}
202 As a more complete example, consider \in{example}[ex:NormalComplete]. This
203 example contains everything that is supported in normal form, with the
204 exception of builtin higher order functions. The graphical version of the
205 architecture contains a slightly simplified version, since the state tuple
206 packing and unpacking have been left out. Instead, two seperate registers are
207 drawn. Also note that most synthesis tools will further optimize this
208 architecture by removing the multiplexers at the register input and
209 instead put some gates in front of the register's clock input, but we want
210 to show the architecture as close to the description as possible.
212 As you can see from the previous examples, the generation of the final
213 architecture from the normal form is straightforward. In each of the
214 examples, there is a direct match between the normal form structure,
215 the generated VHDL and the architecture shown in the images.
217 \startbuffer[NormalComplete]
220 -> State (Word, Word)
221 -> (State (Word, Word), Word)
223 -- All arguments are an inital lambda (address, data, packed state)
225 -- There are nested let expressions at top level
227 -- Unpack the state by coercion (\eg, cast from
228 -- State (Word, Word) to (Word, Word))
229 s = sp ▶ (Word, Word)
230 -- Extract both registers from the state
231 r1 = case s of (a, b) -> a
232 r2 = case s of (a, b) -> b
233 -- Calling some other user-defined function.
235 -- Conditional connections
247 -- pack the state by coercion (\eg, cast from
248 -- (Word, Word) to State (Word, Word))
249 sp' = s' ▶ State (Word, Word)
250 -- Pack our return value
257 \startuseMPgraphic{NormalComplete}
258 save a, d, r, foo, muxr, muxout, out;
261 newCircle.a(btex \lam{a} etex) "framed(false)";
262 newCircle.d(btex \lam{d} etex) "framed(false)";
263 newCircle.out(btex \lam{out} etex) "framed(false)";
265 %newCircle.add(btex + etex);
266 newBox.foo(btex \lam{foo} etex);
267 newReg.r1(btex $\lam{r1}$ etex) "dx(4mm)", "dy(6mm)";
268 newReg.r2(btex $\lam{r2}$ etex) "dx(4mm)", "dy(6mm)", "reflect(true)";
270 % Reflect over the vertical axis
271 reflectObj(muxr1)((0,0), (0,1));
274 rotateObj(muxout)(-90);
276 d.c = foo.c + (0cm, 1.5cm);
277 a.c = (xpart r2.c + 2cm, ypart d.c - 0.5cm);
278 foo.c = midpoint(muxr1.c, muxr2.c) + (0cm, 2cm);
279 muxr1.c = r1.c + (0cm, 2cm);
280 muxr2.c = r2.c + (0cm, 2cm);
281 r2.c = r1.c + (4cm, 0cm);
283 muxout.c = midpoint(r1.c, r2.c) - (0cm, 2cm);
284 out.c = muxout.c - (0cm, 1.5cm);
286 % % Draw objects and lines
287 drawObj(a, d, foo, r1, r2, muxr1, muxr2, muxout, out);
290 nccurve(foo)(muxr1) "angleA(-90)", "posB(inpa)", "angleB(180)";
291 nccurve(foo)(muxr2) "angleA(-90)", "posB(inpb)", "angleB(0)";
292 nccurve(muxr1)(r1) "posA(out)", "angleA(180)", "posB(d)", "angleB(0)";
293 nccurve(r1)(muxr1) "posA(out)", "angleA(0)", "posB(inpb)", "angleB(180)";
294 nccurve(muxr2)(r2) "posA(out)", "angleA(0)", "posB(d)", "angleB(180)";
295 nccurve(r2)(muxr2) "posA(out)", "angleA(180)", "posB(inpa)", "angleB(0)";
296 nccurve(r1)(muxout) "posA(out)", "angleA(0)", "posB(inpb)", "angleB(-90)";
297 nccurve(r2)(muxout) "posA(out)", "angleA(180)", "posB(inpa)", "angleB(-90)";
299 nccurve(a)(muxout) "angleA(-90)", "angleB(180)", "posB(sel)";
300 nccurve(a)(muxr1) "angleA(180)", "angleB(-90)", "posB(sel)";
301 nccurve(a)(muxr2) "angleA(180)", "angleB(-90)", "posB(sel)";
302 ncline(muxout)(out) "posA(out)";
305 \todo{Don't split registers in this image?}
306 \placeexample[here][ex:NormalComplete]{Simple architecture consisting of an adder and a
308 \startcombination[2*1]
309 {\typebufferlam{NormalComplete}}{Core description in normal form.}
310 {\boxedgraphic{NormalComplete}}{The architecture described by the normal form.}
315 \subsection[sec:normalization:intendednormalform]{Intended normal form definition}
316 Now we have some intuition for the normal form, we can describe how we want
317 the normal form to look like in a slightly more formal manner. The following
318 EBNF-like description completely captures the intended structure (and
319 generates a subset of GHC's core format).
321 Some clauses have an expression listed in parentheses. These are conditions
322 that need to apply to the clause.
324 \defref{intended normal form definition}
325 \todo{Fix indentation}
327 \italic{normal} := \italic{lambda}
328 \italic{lambda} := λvar.\italic{lambda} (representable(var))
330 \italic{toplet} := letrec [\italic{binding}...] in var (representable(var))
331 \italic{binding} := var = \italic{rhs} (representable(rhs))
332 -- State packing and unpacking by coercion
333 | var0 = var1 ▶ State ty (lvar(var1))
334 | var0 = var1 ▶ ty (var1 :: State ty ∧ lvar(var1))
335 \italic{rhs} := userapp
338 | case var of C a0 ... an -> ai (lvar(var))
340 | case var of (lvar(var))
341 [ DEFAULT -> var ] (lvar(var))
342 C0 w0,0 ... w0,n -> var0
344 Cm wm,0 ... wm,n -> varm (\forall{}i \forall{}j, wi,j \neq vari, lvar(vari))
345 \italic{userapp} := \italic{userfunc}
346 | \italic{userapp} {userarg}
347 \italic{userfunc} := var (gvar(var))
348 \italic{userarg} := var (lvar(var))
349 \italic{builtinapp} := \italic{builtinfunc}
350 | \italic{builtinapp} \italic{builtinarg}
351 \italic{builtinfunc} := var (bvar(var))
352 \italic{builtinarg} := var (representable(var) ∧ lvar(var))
353 | \italic{partapp} (partapp :: a -> b)
354 | \italic{coreexpr} (¬representable(coreexpr) ∧ ¬(coreexpr :: a -> b))
355 \italic{partapp} := \italic{userapp} | \italic{builtinapp}
358 \todo{There can still be other casts around (which the code can handle,
359 e.g., ignore), which still need to be documented here}
361 When looking at such a program from a hardware perspective, the top level
362 lambda's define the input ports. The variable reference in the body of
363 the recursive let expression is the output port. Most function
364 applications bound by the let expression define a component
365 instantiation, where the input and output ports are mapped to local
366 signals or arguments. Some of the others use a builtin construction (\eg
367 the \lam{case} expression) or call a builtin function (\eg \lam{+} or
368 \lam{map}). For these, a hardcoded \small{VHDL} translation is
371 \section[sec:normalization:transformation]{Transformation notation}
372 To be able to concisely present transformations, we use a specific format
373 for them. It is a simple format, similar to one used in logic reasoning.
375 Such a transformation description looks like the following.
380 <original expression>
381 -------------------------- <expression conditions>
382 <transformed expresssion>
387 This format desribes a transformation that applies to \lam{<original
388 expresssion>} and transforms it into \lam{<transformed expression>}, assuming
389 that all conditions apply. In this format, there are a number of placeholders
390 in pointy brackets, most of which should be rather obvious in their meaning.
391 Nevertheless, we will more precisely specify their meaning below:
393 \startdesc{<original expression>} The expression pattern that will be matched
394 against (subexpressions of) the expression to be transformed. We call this a
395 pattern, because it can contain \emph{placeholders} (variables), which match
396 any expression or binder. Any such placeholder is said to be \emph{bound} to
397 the expression it matches. It is convention to use an uppercase letter (\eg
398 \lam{M} or \lam{E}) to refer to any expression (including a simple variable
399 reference) and lowercase letters (\eg \lam{v} or \lam{b}) to refer to
400 (references to) binders.
402 For example, the pattern \lam{a + B} will match the expression
403 \lam{v + (2 * w)} (binding \lam{a} to \lam{v} and \lam{B} to
404 \lam{(2 * w)}), but not \lam{(2 * w) + v}.
407 \startdesc{<expression conditions>}
408 These are extra conditions on the expression that is matched. These
409 conditions can be used to further limit the cases in which the
410 transformation applies, commonly to prevent a transformation from
411 causing a loop with itself or another transformation.
413 Only if these conditions are \emph{all} true, the transformation
417 \startdesc{<context conditions>}
418 These are a number of extra conditions on the context of the function. In
419 particular, these conditions can require some (other) top level function to be
420 present, whose value matches the pattern given here. The format of each of
421 these conditions is: \lam{binder = <pattern>}.
423 Typically, the binder is some placeholder bound in the \lam{<original
424 expression>}, while the pattern contains some placeholders that are used in
425 the \lam{transformed expression}.
427 Only if a top level binder exists that matches each binder and pattern,
428 the transformation applies.
431 \startdesc{<transformed expression>}
432 This is the expression template that is the result of the transformation. If, looking
433 at the above three items, the transformation applies, the \lam{<original
434 expression>} is completely replaced with the \lam{<transformed expression>}.
435 We call this a template, because it can contain placeholders, referring to
436 any placeholder bound by the \lam{<original expression>} or the
437 \lam{<context conditions>}. The resulting expression will have those
438 placeholders replaced by the values bound to them.
440 Any binder (lowercase) placeholder that has no value bound to it yet will be
441 bound to (and replaced with) a fresh binder.
444 \startdesc{<context additions>}
445 These are templates for new functions to add to the context. This is a way
446 to have a transformation create new top level functions.
448 Each addition has the form \lam{binder = template}. As above, any
449 placeholder in the addition is replaced with the value bound to it, and any
450 binder placeholder that has no value bound to it yet will be bound to (and
451 replaced with) a fresh binder.
454 As an example, we'll look at η-abstraction:
458 -------------- \lam{E} does not occur on a function position in an application
459 λx.E x \lam{E} is not a lambda abstraction.
462 η-abstraction is a well known transformation from lambda calculus. What
463 this transformation does, is take any expression that has a function type
464 and turn it into a lambda expression (giving an explicit name to the
465 argument). There are some extra conditions that ensure that this
466 transformation does not apply infinitely (which are not necessarily part
467 of the conventional definition of η-abstraction).
469 Consider the following function, which is a fairly obvious way to specify a
470 simple ALU (Note that \in{example}[ex:AddSubAlu] shows the normal form of this
471 function). The parentheses around the \lam{+} and \lam{-} operators are
472 commonly used in Haskell to show that the operators are used as normal
473 functions, instead of \emph{infix} operators (\eg, the operators appear
474 before their arguments, instead of in between).
477 alu :: Bit -> Word -> Word -> Word
478 alu = λopcode. case opcode of
483 There are a few subexpressions in this function to which we could possibly
484 apply the transformation. Since the pattern of the transformation is only
485 the placeholder \lam{E}, any expression will match that. Whether the
486 transformation applies to an expression is thus solely decided by the
487 conditions to the right of the transformation.
489 We will look at each expression in the function in a top down manner. The
490 first expression is the entire expression the function is bound to.
493 λopcode. case opcode of
498 As said, the expression pattern matches this. The type of this expression is
499 \lam{Bit -> Word -> Word -> Word}, which matches \lam{a -> b} (Note that in
500 this case \lam{a = Bit} and \lam{b = Word -> Word -> Word}).
502 Since this expression is at top level, it does not occur at a function
503 position of an application. However, The expression is a lambda abstraction,
504 so this transformation does not apply.
506 The next expression we could apply this transformation to, is the body of
507 the lambda abstraction:
515 The type of this expression is \lam{Word -> Word -> Word}, which again
516 matches \lam{a -> b}. The expression is the body of a lambda expression, so
517 it does not occur at a function position of an application. Finally, the
518 expression is not a lambda abstraction but a case expression, so all the
519 conditions match. There are no context conditions to match, so the
520 transformation applies.
522 By now, the placeholder \lam{E} is bound to the entire expression. The
523 placeholder \lam{x}, which occurs in the replacement template, is not bound
524 yet, so we need to generate a fresh binder for that. Let's use the binder
525 \lam{a}. This results in the following replacement expression:
533 Continuing with this expression, we see that the transformation does not
534 apply again (it is a lambda expression). Next we look at the body of this
543 Here, the transformation does apply, binding \lam{E} to the entire
544 expression and \lam{x} to the fresh binder \lam{b}, resulting in the
553 Again, the transformation does not apply to this lambda abstraction, so we
554 look at its body. For brevity, we'll put the case statement on one line from
558 (case opcode of Low -> (+); High -> (-)) a b
561 The type of this expression is \lam{Word}, so it does not match \lam{a -> b}
562 and the transformation does not apply. Next, we have two options for the
563 next expression to look at: The function position and argument position of
564 the application. The expression in the argument position is \lam{b}, which
565 has type \lam{Word}, so the transformation does not apply. The expression in
566 the function position is:
569 (case opcode of Low -> (+); High -> (-)) a
572 Obviously, the transformation does not apply here, since it occurs in
573 function position (which makes the second condition false). In the same
574 way the transformation does not apply to both components of this
575 expression (\lam{case opcode of Low -> (+); High -> (-)} and \lam{a}), so
576 we'll skip to the components of the case expression: The scrutinee and
577 both alternatives. Since the opcode is not a function, it does not apply
580 The first alternative is \lam{(+)}. This expression has a function type
581 (the operator still needs two arguments). It does not occur in function
582 position of an application and it is not a lambda expression, so the
583 transformation applies.
585 We look at the \lam{<original expression>} pattern, which is \lam{E}.
586 This means we bind \lam{E} to \lam{(+)}. We then replace the expression
587 with the \lam{<transformed expression>}, replacing all occurences of
588 \lam{E} with \lam{(+)}. In the \lam{<transformed expression>}, the This gives us the replacement expression:
589 \lam{λx.(+) x} (A lambda expression binding \lam{x}, with a body that
590 applies the addition operator to \lam{x}).
592 The complete function then becomes:
594 (case opcode of Low -> λa1.(+) a1; High -> (-)) a
597 Now the transformation no longer applies to the complete first alternative
598 (since it is a lambda expression). It does not apply to the addition
599 operator again, since it is now in function position in an application. It
600 does, however, apply to the application of the addition operator, since
601 that is neither a lambda expression nor does it occur in function
602 position. This means after one more application of the transformation, the
606 (case opcode of Low -> λa1.λb1.(+) a1 b1; High -> (-)) a
609 The other alternative is left as an exercise to the reader. The final
610 function, after applying η-abstraction until it does no longer apply is:
613 alu :: Bit -> Word -> Word -> Word
614 alu = λopcode.λa.b. (case opcode of
615 Low -> λa1.λb1 (+) a1 b1
616 High -> λa2.λb2 (-) a2 b2) a b
619 \subsection{Transformation application}
620 In this chapter we define a number of transformations, but how will we apply
621 these? As stated before, our normal form is reached as soon as no
622 transformation applies anymore. This means our application strategy is to
623 simply apply any transformation that applies, and continuing to do that with
624 the result of each transformation.
626 In particular, we define no particular order of transformations. Since
627 transformation order should not influence the resulting normal form,
628 this leaves the implementation free to choose any application order that
629 results in an efficient implementation. Unfortunately this is not
630 entirely true for the current set of transformations. See
631 \in{section}[sec:normalization:non-determinism] for a discussion of this
634 When applying a single transformation, we try to apply it to every (sub)expression
635 in a function, not just the top level function body. This allows us to
636 keep the transformation descriptions concise and powerful.
638 \subsection{Definitions}
639 In the following sections, we will be using a number of functions and
640 notations, which we will define here.
642 \subsubsection{Concepts}
643 A \emph{global variable} is any variable (binder) that is bound at the
644 top level of a program, or an external module. A \emph{local variable} is any
645 other variable (\eg, variables local to a function, which can be bound by
646 lambda abstractions, let expressions and pattern matches of case
647 alternatives). Note that this is a slightly different notion of global versus
648 local than what \small{GHC} uses internally.
649 \defref{global variable} \defref{local variable}
651 A \emph{hardware representable} (or just \emph{representable}) type or value
652 is (a value of) a type that we can generate a signal for in hardware. For
653 example, a bit, a vector of bits, a 32 bit unsigned word, etc. Values that are
654 not runtime representable notably include (but are not limited to): Types,
655 dictionaries, functions.
656 \defref{representable}
658 A \emph{builtin function} is a function supplied by the Cλash framework, whose
659 implementation is not valid Cλash. The implementation is of course valid
660 Haskell, for simulation, but it is not expressable in Cλash.
661 \defref{builtin function} \defref{user-defined function}
663 For these functions, Cλash has a \emph{builtin hardware translation}, so calls
664 to these functions can still be translated. These are functions like
665 \lam{map}, \lam{hwor} and \lam{length}.
667 A \emph{user-defined} function is a function for which we do have a Cλash
668 implementation available.
670 \subsubsection{Predicates}
671 Here, we define a number of predicates that can be used below to concisely
672 specify conditions.\refdef{global variable}
674 \emph{gvar(expr)} is true when \emph{expr} is a variable that references a
675 global variable. It is false when it references a local variable.
677 \refdef{local variable}\emph{lvar(expr)} is the complement of \emph{gvar}; it is true when \emph{expr}
678 references a local variable, false when it references a global variable.
680 \refdef{representable}\emph{representable(expr)} or \emph{representable(var)} is true when
681 \emph{expr} or \emph{var} is \emph{representable}.
683 \subsection[sec:normalization:uniq]{Binder uniqueness}
684 A common problem in transformation systems, is binder uniqueness. When not
685 considering this problem, it is easy to create transformations that mix up
686 bindings and cause name collisions. Take for example, the following core
690 (λa.λb.λc. a * b * c) x c
693 By applying β-reduction (see \in{section}[sec:normalization:beta]) once,
694 we can simplify this expression to:
700 Now, we have replaced the \lam{a} binder with a reference to the \lam{x}
701 binder. No harm done here. But note that we see multiple occurences of the
702 \lam{c} binder. The first is a binding occurence, to which the second refers.
703 The last, however refers to \emph{another} instance of \lam{c}, which is
704 bound somewhere outside of this expression. Now, if we would apply beta
705 reduction without taking heed of binder uniqueness, we would get:
711 This is obviously not what was supposed to happen! The root of this problem is
712 the reuse of binders: Identical binders can be bound in different scopes, such
713 that only the inner one is \quote{visible} in the inner expression. In the example
714 above, the \lam{c} binder was bound outside of the expression and in the inner
715 lambda expression. Inside that lambda expression, only the inner \lam{c} is
718 There are a number of ways to solve this. \small{GHC} has isolated this
719 problem to their binder substitution code, which performs \emph{deshadowing}
720 during its expression traversal. This means that any binding that shadows
721 another binding on a higher level is replaced by a new binder that does not
722 shadow any other binding. This non-shadowing invariant is enough to prevent
723 binder uniqueness problems in \small{GHC}.
725 In our transformation system, maintaining this non-shadowing invariant is
726 a bit harder to do (mostly due to implementation issues, the prototype doesn't
727 use \small{GHC}'s subsitution code). Also, the following points can be
731 \item Deshadowing does not guarantee overall uniqueness. For example, the
732 following (slightly contrived) expression shows the identifier \lam{x} bound in
733 two seperate places (and to different values), even though no shadowing
737 (let x = 1 in x) + (let x = 2 in x)
740 \item In our normal form (and the resulting \small{VHDL}), all binders
741 (signals) within the same function (entity) will end up in the same
742 scope. To allow this, all binders within the same function should be
745 \item When we know that all binders in an expression are unique, moving around
746 or removing a subexpression will never cause any binder conflicts. If we have
747 some way to generate fresh binders, introducing new subexpressions will not
748 cause any problems either. The only way to cause conflicts is thus to
749 duplicate an existing subexpression.
752 Given the above, our prototype maintains a unique binder invariant. This
753 means that in any given moment during normalization, all binders \emph{within
754 a single function} must be unique. To achieve this, we apply the following
757 \todo{Define fresh binders and unique supplies}
760 \item Before starting normalization, all binders in the function are made
761 unique. This is done by generating a fresh binder for every binder used. This
762 also replaces binders that did not cause any conflict, but it does ensure that
763 all binders within the function are generated by the same unique supply.
764 \refdef{fresh binder}
765 \item Whenever a new binder must be generated, we generate a fresh binder that
766 is guaranteed to be different from \emph{all binders generated so far}. This
767 can thus never introduce duplication and will maintain the invariant.
768 \item Whenever (a part of) an expression is duplicated (for example when
769 inlining), all binders in the expression are replaced with fresh binders
770 (using the same method as at the start of normalization). These fresh binders
771 can never introduce duplication, so this will maintain the invariant.
772 \item Whenever we move part of an expression around within the function, there
773 is no need to do anything special. There is obviously no way to introduce
774 duplication by moving expressions around. Since we know that each of the
775 binders is already unique, there is no way to introduce (incorrect) shadowing
779 \section{Transform passes}
780 In this section we describe the actual transforms.
782 Each transformation will be described informally first, explaining
783 the need for and goal of the transformation. Then, we will formally define
784 the transformation using the syntax introduced in
785 \in{section}[sec:normalization:transformation].
787 \subsection{General cleanup}
788 These transformations are general cleanup transformations, that aim to
789 make expressions simpler. These transformations usually clean up the
790 mess left behind by other transformations or clean up expressions to
791 expose new transformation opportunities for other transformations.
793 Most of these transformations are standard optimizations in other
794 compilers as well. However, in our compiler, most of these are not just
795 optimizations, but they are required to get our program into intended
799 \startframedtext[width=8cm,background=box,frame=no]
800 \startalignment[center]
801 {\tfa Substitution notation}
805 In some of the transformations in this chapter, we need to perform
806 substitution on an expression. Substitution means replacing every
807 occurence of some expression (usually a variable reference) with
810 There have been a lot of different notations used in literature for
811 specifying substitution. The notation that will be used in this report
818 This means expression \lam{E} with all occurences of \lam{A} replaced
823 \defref{beta-reduction}
824 \subsubsection[sec:normalization:beta]{β-reduction}
825 β-reduction is a well known transformation from lambda calculus, where it is
826 the main reduction step. It reduces applications of lambda abstractions,
827 removing both the lambda abstraction and the application.
829 In our transformation system, this step helps to remove unwanted lambda
830 abstractions (basically all but the ones at the top level). Other
831 transformations (application propagation, non-representable inlining) make
832 sure that most lambda abstractions will eventually be reducable by
835 Note that β-reduction also works on type lambda abstractions and type
836 applications as well. This means the substitution below also works on
837 type variables, in the case that the binder is a type variable and teh
838 expression applied to is a type.
855 \transexample{beta}{β-reduction}{from}{to}
865 \transexample{beta-type}{β-reduction for type abstractions}{from}{to}
867 \subsubsection{Empty let removal}
868 This transformation is simple: It removes recursive lets that have no bindings
869 (which usually occurs when unused let binding removal removes the last
872 Note that there is no need to define this transformation for
873 non-recursive lets, since they always contain exactly one binding.
883 \subsubsection[sec:normalization:simplelet]{Simple let binding removal}
884 This transformation inlines simple let bindings, that bind some
885 binder to some other binder instead of a more complex expression (\ie
888 This transformation is not needed to get an expression into intended
889 normal form (since these bindings are part of the intended normal
890 form), but makes the resulting \small{VHDL} a lot shorter.
901 ----------------------------- \lam{b} is a variable reference
902 letrec \lam{ai} ≠ \lam{b}
915 \subsubsection{Unused let binding removal}
916 This transformation removes let bindings that are never used.
917 Occasionally, \GHC's desugarer introduces some unused let bindings.
919 This normalization pass should really be unneeded to get into intended normal form
920 (since unused bindings are not forbidden by the normal form), but in practice
921 the desugarer or simplifier emits some unused bindings that cannot be
922 normalized (e.g., calls to a \type{PatError}\todo{Check this name}). Also,
923 this transformation makes the resulting \small{VHDL} a lot shorter.
925 \todo{Don't use old-style numerals in transformations}
934 M \lam{ai} does not occur free in \lam{M}
935 ---------------------------- \forall j, 0 ≤ j ≤ n, j ≠ i (\lam{ai} does not occur free in \lam{Ej})
949 \subsubsection{Cast propagation / simplification}
950 This transform pushes casts down into the expression as far as possible.
951 Since its exact role and need is not clear yet, this transformation is
954 \todo{Cast propagation}
956 \subsubsection{Top level binding inlining}
957 This transform takes simple top level bindings generated by the
958 \small{GHC} compiler. \small{GHC} sometimes generates very simple
959 \quote{wrapper} bindings, which are bound to just a variable
960 reference, or a partial application to constants or other variable
963 Note that this transformation is completely optional. It is not
964 required to get any function into intended normal form, but it does help making
965 the resulting VHDL output easier to read (since it removes a bunch of
966 components that are really boring).
968 This transform takes any top level binding generated by the compiler,
969 whose normalized form contains only a single let binding.
972 x = λa0 ... λan.let y = E in y
975 -------------------------------------- \lam{x} is generated by the compiler
976 λa0 ... λan.let y = E in y
980 (+) :: Word -> Word -> Word
981 (+) = GHC.Num.(+) @Word \$dNum
986 GHC.Num.(+) @ Alu.Word \$dNum a b
989 \transexample{toplevelinline}{Top level binding inlining}{from}{to}
991 \in{Example}[ex:trans:toplevelinline] shows a typical application of
992 the addition operator generated by \GHC. The type and dictionary
993 arguments used here are described in
994 \in{Section}[section:prototype:polymorphism].
996 Without this transformation, there would be a \lam{(+)} entity
997 in the \VHDL which would just add its inputs. This generates a
998 lot of overhead in the \VHDL, which is particularly annoying
999 when browsing the generated RTL schematic (especially since most
1000 non-alphanumerics, like all characters in \lam{(+)}, are not
1001 allowed in \VHDL architecture names\footnote{Technically, it is
1002 allowed to use non-alphanumerics when using extended
1003 identifiers, but it seems that none of the tooling likes
1004 extended identifiers in filenames, so it effectively doesn't
1005 work.}, so the entity would be called \quote{w7aA7f} or
1006 something similarly unreadable and autogenerated).
1008 \subsection{Program structure}
1009 These transformations are aimed at normalizing the overall structure
1010 into the intended form. This means ensuring there is a lambda abstraction
1011 at the top for every argument (input port or current state), putting all
1012 of the other value definitions in let bindings and making the final
1013 return value a simple variable reference.
1015 \subsubsection[sec:normalization:eta]{η-abstraction}
1016 This transformation makes sure that all arguments of a function-typed
1017 expression are named, by introducing lambda expressions. When combined with
1018 β-reduction and non-representable binding inlining, all function-typed
1019 expressions should be lambda abstractions or global identifiers.
1023 -------------- \lam{E} is not the first argument of an application.
1024 λx.E x \lam{E} is not a lambda abstraction.
1025 \lam{x} is a variable that does not occur free in \lam{E}.
1035 foo = λa.λx.(case a of
1040 \transexample{eta}{η-abstraction}{from}{to}
1042 \subsubsection[sec:normalization:appprop]{Application propagation}
1043 This transformation is meant to propagate application expressions downwards
1044 into expressions as far as possible. This allows partial applications inside
1045 expressions to become fully applied and exposes new transformation
1046 opportunities for other transformations (like β-reduction and
1049 Since all binders in our expression are unique (see
1050 \in{section}[sec:normalization:uniq]), there is no risk that we will
1051 introduce unintended shadowing by moving an expression into a lower
1052 scope. Also, since only move expression into smaller scopes (down into
1053 our expression), there is no risk of moving a variable reference out
1054 of the scope in which it is defined.
1057 (letrec binds in E) M
1058 ------------------------
1078 \transexample{appproplet}{Application propagation for a let expression}{from}{to}
1106 \transexample{apppropcase}{Application propagation for a case expression}{from}{to}
1108 \subsubsection[sec:normalization:letrecurse]{Let recursification}
1109 This transformation makes all non-recursive lets recursive. In the
1110 end, we want a single recursive let in our normalized program, so all
1111 non-recursive lets can be converted. This also makes other
1112 transformations simpler: They can simply assume all lets are
1120 ------------------------------------------
1127 \subsubsection{Let flattening}
1128 This transformation puts nested lets in the same scope, by lifting the
1129 binding(s) of the inner let into the outer let. Eventually, this will
1130 cause all let bindings to appear in the same scope.
1132 This transformation only applies to recursive lets, since all
1133 non-recursive lets will be made recursive (see
1134 \in{section}[sec:normalization:letrecurse]).
1136 Since we are joining two scopes together, there is no risk of moving a
1137 variable reference out of the scope where it is defined.
1143 ai = (letrec bindings in M)
1148 ------------------------------------------
1183 \transexample{letflat}{Let flattening}{from}{to}
1185 \subsubsection{Return value simplification}
1186 This transformation ensures that the return value of a function is always a
1187 simple local variable reference.
1189 Currently implemented using lambda simplification, let simplification, and
1190 top simplification. Should change into something like the following, which
1191 works only on the result of a function instead of any subexpression. This is
1192 achieved by the contexts, like \lam{x = E}, though this is strictly not
1193 correct (you could read this as "if there is any function \lam{x} that binds
1194 \lam{E}, any \lam{E} can be transformed, while we only mean the \lam{E} that
1195 is bound by \lam{x}. This might need some extra notes or something).
1197 Note that the return value is not simplified if its not representable.
1198 Otherwise, this would cause a direct loop with the inlining of
1199 unrepresentable bindings. If the return value is not
1200 representable because it has a function type, η-abstraction should
1201 make sure that this transformation will eventually apply. If the value
1202 is not representable for other reasons, the function result itself is
1203 not representable, meaning this function is not translatable anyway.
1206 x = E \lam{E} is representable
1207 ~ \lam{E} is not a lambda abstraction
1208 E \lam{E} is not a let expression
1209 --------------------------- \lam{E} is not a local variable reference
1215 ~ \lam{E} is representable
1216 E \lam{E} is not a let expression
1217 --------------------------- \lam{E} is not a local variable reference
1222 x = λv0 ... λvn.let ... in E
1223 ~ \lam{E} is representable
1224 E \lam{E} is not a local variable reference
1225 -----------------------------
1234 x = letrec x = add 1 2 in x
1237 \transexample{retvalsimpl}{Return value simplification}{from}{to}
1239 \todo{More examples}
1241 \subsection[sec:normalization:argsimpl]{Representable arguments simplification}
1242 This section contains just a single transformation that deals with
1243 representable arguments in applications. Non-representable arguments are
1244 handled by the transformations in
1245 \in{section}[sec:normalization:nonrep].
1247 This transformation ensures that all representable arguments will become
1248 references to local variables. This ensures they will become references
1249 to local signals in the resulting \small{VHDL}, which is required due to
1250 limitations in the component instantiation code in \VHDL (one can only
1251 assign a signal or constant to an input port). By ensuring that all
1252 arguments are always simple variable references, we always have a signal
1253 available to map to the input ports.
1255 To reduce a complex expression to a simple variable reference, we create
1256 a new let expression around the application, which binds the complex
1257 expression to a new variable. The original function is then applied to
1260 \refdef{global variable}
1261 Note that references to \emph{global variables} (like a top level
1262 function without arguments, but also an argumentless dataconstructors
1263 like \lam{True}) are also simplified. Only local variables generate
1264 signals in the resulting architecture. Even though argumentless
1265 dataconstructors generate constants in generated \VHDL code and could be
1266 mapped to an input port directly, they are still simplified to make the
1267 normal form more regular.
1269 \refdef{representable}
1272 -------------------- \lam{N} is representable
1273 letrec x = N in M x \lam{N} is not a local variable reference
1275 \refdef{local variable}
1282 letrec x = add a 1 in add x 1
1285 \transexample{argsimpl}{Argument simplification}{from}{to}
1287 \subsection[sec:normalization:builtins]{Builtin functions}
1288 This section deals with (arguments to) builtin functions. In the
1289 intended normal form definition\refdef{intended normal form definition}
1290 we can see that there are three sorts of arguments a builtin function
1294 \item A representable local variable reference. This is the most
1295 common argument to any function. The argument simplification
1296 transformation described in \in{section}[sec:normalization:argsimpl]
1297 makes sure that \emph{any} representable argument to \emph{any}
1298 function (including builtin functions) is turned into a local variable
1300 \item (A partial application of) a top level function (either builtin on
1301 user-defined). The function extraction transformation described in
1302 this section takes care of turning every functiontyped argument into
1303 (a partial application of) a top level function.
1304 \item Any expression that is not representable and does not have a
1305 function type. Since these can be any expression, there is no
1306 transformation needed. Note that this category is exactly all
1307 expressions that are not transformed by the transformations for the
1308 previous two categories. This means that \emph{any} core expression
1309 that is used as an argument to a builtin function will be either
1310 transformed into one of the above categories, or end up in this
1311 categorie. In any case, the result is in normal form.
1314 As noted, the argument simplification will handle any representable
1315 arguments to a builtin function. The following transformation is needed
1316 to handle non-representable arguments with a function type, all other
1317 non-representable arguments don't need any special handling.
1319 \subsubsection[sec:normalization:funextract]{Function extraction}
1320 This transform deals with function-typed arguments to builtin
1322 Since builtin functions cannot be specialized (see
1323 \in{section}[sec:normalization:specialize]) to remove the arguments,
1324 these arguments are extracted into a new global function instead. In
1325 other words, we create a new top level function that has exactly the
1326 extracted argument as its body. This greatly simplifies the
1327 translation rules needed for builtin functions, since they only need
1328 to handle (partial applications of) top level functions.
1330 Any free variables occuring in the extracted arguments will become
1331 parameters to the new global function. The original argument is replaced
1332 with a reference to the new function, applied to any free variables from
1333 the original argument.
1335 This transformation is useful when applying higher order builtin functions
1336 like \hs{map} to a lambda abstraction, for example. In this case, the code
1337 that generates \small{VHDL} for \hs{map} only needs to handle top level functions and
1338 partial applications, not any other expression (such as lambda abstractions or
1339 even more complicated expressions).
1342 M N \lam{M} is (a partial aplication of) a builtin function.
1343 --------------------- \lam{f0 ... fn} are all free local variables of \lam{N}
1344 M (x f0 ... fn) \lam{N :: a -> b}
1345 ~ \lam{N} is not a (partial application of) a top level function
1350 addList = λb.λxs.map (λa . add a b) xs
1354 addList = λb.λxs.map (f b) xs
1359 \transexample{funextract}{Function extraction}{from}{to}
1361 Note that the function \lam{f} will still need normalization after
1364 \subsection{Case normalisation}
1365 \subsubsection{Scrutinee simplification}
1366 This transform ensures that the scrutinee of a case expression is always
1367 a simple variable reference.
1372 ----------------- \lam{E} is not a local variable reference
1391 \transexample{letflat}{Case normalisation}{from}{to}
1394 \subsubsection{Case simplification}
1395 This transformation ensures that all case expressions become normal form. This
1396 means they will become one of:
1398 \item An extractor case with a single alternative that picks a single field
1399 from a datatype, \eg \lam{case x of (a, b) -> a}.
1400 \item A selector case with multiple alternatives and only wild binders, that
1401 makes a choice between expressions based on the constructor of another
1402 expression, \eg \lam{case x of Low -> a; High -> b}.
1405 \defref{wild binder}
1408 C0 v0,0 ... v0,m -> E0
1410 Cn vn,0 ... vn,m -> En
1411 --------------------------------------------------- \forall i \forall j, 0 ≤ i ≤ n, 0 ≤ i < m (\lam{wi,j} is a wild (unused) binder)
1413 v0,0 = case E of C0 v0,0 .. v0,m -> v0,0
1415 v0,m = case E of C0 v0,0 .. v0,m -> v0,m
1417 vn,m = case E of Cn vn,0 .. vn,m -> vn,m
1423 C0 w0,0 ... w0,m -> x0
1425 Cn wn,0 ... wn,m -> xn
1427 \todo{Check the subscripts of this transformation}
1429 Note that this transformation applies to case statements with any
1430 scrutinee. If the scrutinee is a complex expression, this might result
1431 in duplicate hardware. An extra condition to only apply this
1432 transformation when the scrutinee is already simple (effectively
1433 causing this transformation to be only applied after the scrutinee
1434 simplification transformation) might be in order.
1436 \fxnote{This transformation specified like this is complicated and misses
1437 conditions to prevent looping with itself. Perhaps it should be split here for
1456 \transexample{selcasesimpl}{Selector case simplification}{from}{to}
1464 b = case a of (,) b c -> b
1465 c = case a of (,) b c -> c
1472 \transexample{excasesimpl}{Extractor case simplification}{from}{to}
1474 \refdef{selector case}
1475 In \in{example}[ex:trans:excasesimpl] the case expression is expanded
1476 into multiple case expressions, including a pretty useless expression
1477 (that is neither a selector or extractor case). This case can be
1478 removed by the Case removal transformation in
1479 \in{section}[sec:transformation:caseremoval].
1481 \subsubsection[sec:transformation:caseremoval]{Case removal}
1482 This transform removes any case statements with a single alternative and
1485 These "useless" case statements are usually leftovers from case simplification
1486 on extractor case (see the previous example).
1491 ---------------------- \lam{\forall i, 0 ≤ i ≤ m} (\lam{vi} does not occur free in E)
1504 \transexample{caserem}{Case removal}{from}{to}
1506 \subsection[sec:normalization:nonrep]{Removing unrepresentable values}
1507 The transformations in this section are aimed at making all the
1508 values used in our expression representable. There are two main
1509 transformations that are applied to \emph{all} unrepresentable let
1510 bindings and function arguments. These are meant to address three
1511 different kinds of unrepresentable values: Polymorphic values, higher
1512 order values and literals. The transformation are described generically:
1513 They apply to all non-representable values. However, non-representable
1514 values that don't fall into one of these three categories will be moved
1515 around by these transformations but are unlikely to completely
1516 disappear. They usually mean the program was not valid in the first
1517 place, because unsupported types were used (for example, a program using
1520 Each of these three categories will be detailed below, followed by the
1521 actual transformations.
1523 \subsubsection{Removing Polymorphism}
1524 As noted in \in{section}[sec:prototype:polymporphism],
1525 polymorphism is made explicit in Core through type and
1526 dictionary arguments. To remove the polymorphism from a
1527 function, we can simply specialize the polymorphic function for
1528 the particular type applied to it. The same goes for dictionary
1529 arguments. To remove polymorphism from let bound values, we
1530 simply inline the let bindings that have a polymorphic type,
1531 which should (eventually) make sure that the polymorphic
1532 expression is applied to a type and/or dictionary, which can
1533 then be removed by β-reduction (\in{section}[sec:normalization:beta]).
1535 Since both type and dictionary arguments are not representable,
1536 \refdef{representable}
1537 the non-representable argument specialization and
1538 non-representable let binding inlining transformations below
1539 take care of exactly this.
1541 There is one case where polymorphism cannot be completely
1542 removed: Builtin functions are still allowed to be polymorphic
1543 (Since we have no function body that we could properly
1544 specialize). However, the code that generates \VHDL for builtin
1545 functions knows how to handle this, so this is not a problem.
1547 \subsubsection{Defunctionalization}
1548 These transformations remove higher order expressions from our
1549 program, making all values first-order.
1551 Higher order values are always introduced by lambda abstractions, none
1552 of the other Core expression elements can introduce a function type.
1553 However, other expressions can \emph{have} a function type, when they
1554 have a lambda expression in their body.
1556 For example, the following expression is a higher order expression
1557 that is not a lambda expression itself:
1559 \refdef{id function}
1566 The reference to the \lam{id} function shows that we can introduce a
1567 higher order expression in our program without using a lambda
1568 expression directly. However, inside the definition of the \lam{id}
1569 function, we can be sure that a lambda expression is present.
1571 Looking closely at the definition of our normal form in
1572 \in{section}[sec:normalization:intendednormalform], we can see that
1573 there are three possibilities for higher order values to appear in our
1574 intended normal form:
1577 \item[item:toplambda] Lambda abstractions can appear at the highest level of a
1578 top level function. These lambda abstractions introduce the
1579 arguments (input ports / current state) of the function.
1580 \item[item:builtinarg] (Partial applications of) top level functions can appear as an
1581 argument to a builtin function.
1582 \item[item:completeapp] (Partial applications of) top level functions can appear in
1583 function position of an application. Since a partial application
1584 cannot appear anywhere else (except as builtin function arguments),
1585 all partial applications are applied, meaning that all applications
1586 will become complete applications. However, since application of
1587 arguments happens one by one, in the expression:
1591 the subexpression \lam{f 1} has a function type. But this is
1592 allowed, since it is inside a complete application.
1595 We will take a typical function with some higher order values as an
1596 example. The following function takes two arguments: a \lam{Bit} and a
1597 list of numbers. Depending on the first argument, each number in the
1598 list is doubled, or the list is returned unmodified. For the sake of
1599 the example, no polymorphism is shown. In reality, at least map would
1603 λy.let double = λx. x + x in
1609 This example shows a number of higher order values that we cannot
1610 translate to \VHDL directly. The \lam{double} binder bound in the let
1611 expression has a function type, as well as both of the alternatives of
1612 the case expression. The first alternative is a partial application of
1613 the \lam{map} builtin function, whereas the second alternative is a
1616 To reduce all higher order values to one of the above items, a number
1617 of transformations we've already seen are used. The η-abstraction
1618 transformation from \in{section}[sec:normalization:eta] ensures all
1619 function arguments are introduced by lambda abstraction on the highest
1620 level of a function. These lambda arguments are allowed because of
1621 \in{item}[item:toplambda] above. After η-abstraction, our example
1622 becomes a bit bigger:
1625 λy.λq.(let double = λx. x + x in
1632 η-abstraction also introduces extra applications (the application of
1633 the let expression to \lam{q} in the above example). These
1634 applications can then propagated down by the application propagation
1635 transformation (\in{section}[sec:normalization:appprop]). In our
1636 example, the \lam{q} and \lam{r} variable will be propagated into the
1637 let expression and then into the case expression:
1640 λy.λq.let double = λx. x + x in
1646 This propagation makes higher order values become applied (in
1647 particular both of the alternatives of the case now have a
1648 representable type. Completely applied top level functions (like the
1649 first alternative) are now no longer invalid (they fall under
1650 \in{item}[item:completeapp] above). (Completely) applied lambda
1651 abstractions can be removed by β-abstraction. For our example,
1652 applying β-abstraction results in the following:
1655 λy.λq.let double = λx. x + x in
1661 As you can see in our example, all of this moves applications towards
1662 the higher order values, but misses higher order functions bound by
1663 let expressions. The applications cannot be moved towards these values
1664 (since they can be used in multiple places), so the values will have
1665 to be moved towards the applications. This is achieved by inlining all
1666 higher order values bound by let applications, by the
1667 non-representable binding inlining transformation below. When applying
1668 it to our example, we get the following:
1672 Low -> map (λx. x + x) q
1676 We've nearly eliminated all unsupported higher order values from this
1677 expressions. The one that's remaining is the first argument to the
1678 \lam{map} function. Having higher order arguments to a builtin
1679 function like \lam{map} is allowed in the intended normal form, but
1680 only if the argument is a (partial application) of a top level
1681 function. This is easily done by introducing a new top level function
1682 and put the lambda abstraction inside. This is done by the function
1683 extraction transformation from
1684 \in{section}[sec:normalization:funextract].
1692 This also introduces a new function, that we have called \lam{func}:
1698 Note that this does not actually remove the lambda, but now it is a
1699 lambda at the highest level of a function, which is allowed in the
1700 intended normal form.
1702 There is one case that has not been discussed yet. What if the
1703 \lam{map} function in the example above was not a builtin function
1704 but a user-defined function? Then extracting the lambda expression
1705 into a new function would not be enough, since user-defined functions
1706 can never have higher order arguments. For example, the following
1707 expression shows an example:
1710 twice :: (Word -> Word) -> Word -> Word
1711 twice = λf.λa.f (f a)
1713 main = λa.app (λx. x + x) a
1716 This example shows a function \lam{twice} that takes a function as a
1717 first argument and applies that function twice to the second argument.
1718 Again, we've made the function monomorphic for clarity, even though
1719 this function would be a lot more useful if it was polymorphic. The
1720 function \lam{main} uses \lam{twice} to apply a lambda epression twice.
1722 When faced with a user defined function, a body is available for that
1723 function. This means we could create a specialized version of the
1724 function that only works for this particular higher order argument
1725 (\ie, we can just remove the argument and call the specialized
1726 function without the argument). This transformation is detailed below.
1727 Applying this transformation to the example gives:
1730 twice' :: Word -> Word
1731 twice' = λb.(λf.λa.f (f a)) (λx. x + x) b
1736 The \lam{main} function is now in normal form, since the only higher
1737 order value there is the top level lambda expression. The new
1738 \lam{twice'} function is a bit complex, but the entire original body of
1739 the original \lam{twice} function is wrapped in a lambda abstraction
1740 and applied to the argument we've specialized for (\lam{λx. x + x})
1741 and the other arguments. This complex expression can fortunately be
1742 effectively reduced by repeatedly applying β-reduction:
1745 twice' :: Word -> Word
1746 twice' = λb.(b + b) + (b + b)
1749 This example also shows that the resulting normal form might not be as
1750 efficient as we might hope it to be (it is calculating \lam{(b + b)}
1751 twice). This is discussed in more detail in
1752 \in{section}[sec:normalization:duplicatework].
1754 \subsubsection{Literals}
1755 There are a limited number of literals available in Haskell and Core.
1756 \refdef{enumerated types} When using (enumerating) algebraic
1757 datatypes, a literal is just a reference to the corresponding data
1758 constructor, which has a representable type (the algebraic datatype)
1759 and can be translated directly. This also holds for literals of the
1760 \hs{Bool} Haskell type, which is just an enumerated type.
1762 There is, however, a second type of literal that does not have a
1763 representable type: Integer literals. Cλash supports using integer
1764 literals for all three integer types supported (\hs{SizedWord},
1765 \hs{SizedInt} and \hs{RangedWord}). This is implemented using
1766 Haskell's \hs{Num} typeclass, which offers a \hs{fromInteger} method
1767 that converts any \hs{Integer} to the Cλash datatypes.
1769 When \GHC sees integer literals, it will automatically insert calls to
1770 the \hs{fromInteger} method in the resulting Core expression. For
1771 example, the following expression in Haskell creates a 32 bit unsigned
1772 word with the value 1. The explicit type signature is needed, since
1773 there is no context for \GHC to determine the type from otherwise.
1779 This Haskell code results in the following Core expression:
1782 fromInteger @(SizedWord D32) \$dNum (smallInteger 10)
1785 The literal 10 will have the type \lam{GHC.Prim.Int\#}, which is
1786 converted into an \lam{Integer} by \lam{smallInteger}. Finally, the
1787 \lam{fromInteger} function will finally convert this into a
1788 \lam{SizedWord D32}.
1790 Both the \lam{GHC.Prim.Int\#} and \lam{Integer} types are not
1791 representable, and cannot be translated directly. Fortunately, there
1792 is no need to translate them, since \lam{fromInteger} is a builtin
1793 function that knows how to handle these values. However, this does
1794 require that the \lam{fromInteger} function is directly applied to
1795 these non-representable literal values, otherwise errors will occur.
1796 For example, the following expression is not in the intended normal
1797 form, since one of the let bindings has an unrepresentable type
1801 let l = smallInteger 10 in fromInteger @(SizedWord D32) \$dNum l
1804 By inlining these let-bindings, we can ensure that unrepresentable
1805 literals bound by a let binding end up in an application of the
1806 appropriate builtin function, where they are allowed. Since it is
1807 possible that the application of that function is in a different
1808 function than the definition of the literal value, we will always need
1809 to specialize away any unrepresentable literals that are used as
1810 function arguments. The following two transformations do exactly this.
1812 \subsubsection{Non-representable binding inlining}
1813 This transform inlines let bindings that are bound to a
1814 non-representable value. Since we can never generate a signal
1815 assignment for these bindings (we cannot declare a signal assignment
1816 with a non-representable type, for obvious reasons), we have no choice
1817 but to inline the binding to remove it.
1819 As we have seen in the previous sections, inlining these bindings
1820 solves (part of) the polymorphism, higher order values and
1821 unrepresentable literals in an expression.
1832 -------------------------- \lam{Ei} has a non-representable type.
1834 a0 = E0 [ai=>Ei] \vdots
1835 ai-1 = Ei-1 [ai=>Ei]
1836 ai+1 = Ei+1 [ai=>Ei]
1855 x = fromInteger (smallInteger 10)
1857 (λb -> add b 1) (add 1 x)
1860 \transexample{nonrepinline}{Nonrepresentable binding inlining}{from}{to}
1862 \subsubsection[sec:normalization:specialize]{Function specialization}
1863 This transform removes arguments to user-defined functions that are
1864 not representable at runtime. This is done by creating a
1865 \emph{specialized} version of the function that only works for one
1866 particular value of that argument (in other words, the argument can be
1869 Specialization means to create a specialized version of the called
1870 function, with one argument already filled in. As a simple example, in
1871 the following program (this is not actual Core, since it directly uses
1872 a literal with the unrepresentable type \lam{GHC.Prim.Int\#}).
1879 We could specialize the function \lam{f} against the literal argument
1880 1, with the following result:
1887 In some way, this transformation is similar to β-reduction, but it
1888 operates across function boundaries. It is also similar to
1889 non-representable let binding inlining above, since it sort of
1890 \quote{inlines} an expression into a called function.
1892 Special care must be taken when the argument has any free variables.
1893 If this is the case, the original argument should not be removed
1894 completely, but replaced by all the free variables of the expression.
1895 In this way, the original expression can still be evaluated inside the
1898 To prevent us from propagating the same argument over and over, a
1899 simple local variable reference is not propagated (since is has
1900 exactly one free variable, itself, we would only replace that argument
1903 This shows that any free local variables that are not runtime
1904 representable cannot be brought into normal form by this transform. We
1905 rely on an inlining or β-reduction transformation to replace such a
1906 variable with an expression we can propagate again.
1911 x Y0 ... Yi ... Yn \lam{Yi} is not representable
1912 --------------------------------------------- \lam{Yi} is not a local variable reference
1913 x' y0 ... yi-1 f0 ... fm Yi+1 ... Yn \lam{f0 ... fm} are all free local vars of \lam{Yi}
1914 ~ \lam{T0 ... Tn} are the types of \lam{Y0 ... Yn}
1915 x' = λ(y0 :: T0) ... λ(yi-1 :: Ty-1). λf0 ... λfm. λ(yi+1 :: Ty+1) ... λ(yn :: Tn).
1916 E y0 ... yi-1 Yi yi+1 ... yn
1919 This is a bit of a complex transformation. It transforms an
1920 application of the function \lam{x}, where one of the arguments
1921 (\lam{Y_i}) is not representable. A new
1922 function \lam{x'} is created that wraps the body of the old function.
1923 The body of the new function becomes a number of nested lambda
1924 abstractions, one for each of the original arguments that are left
1927 The ith argument is replaced with the free variables of
1928 \lam{Y_i}. Note that we reuse the same binders as those used in
1929 \lam{Y_i}, since we can then just use \lam{Y_i} inside the new
1930 function body and have all of the variables it uses be in scope.
1932 The argument that we are specializing for, \lam{Y_i}, is put inside
1933 the new function body. The old function body is applied to it. Since
1934 we use this new function only in place of an application with that
1935 particular argument \lam{Y_i}, behaviour should not change.
1937 Note that the types of the arguments of our new function are taken
1938 from the types of the \emph{actual} arguments (\lam{T0 ... Tn}). This
1939 means that any polymorphism in the arguments is removed, even when the
1940 corresponding explicit type lambda is not removed
1941 yet.\refdef{type lambda}
1943 \todo{Examples. Perhaps reference the previous sections}
1946 \section{Unsolved problems}
1947 The above system of transformations has been implemented in the prototype
1948 and seems to work well to compile simple and more complex examples of
1949 hardware descriptions. \todo{Ref christiaan?} However, this normalization
1950 system has not seen enough review and work to be complete and work for
1951 every Core expression that is supplied to it. A number of problems
1952 have already been identified and are discussed in this section.
1954 \subsection[sec:normalization:duplicatework]{Work duplication}
1955 A possible problem of β-reduction is that it could duplicate work.
1956 When the expression applied is not a simple variable reference, but
1957 requires calculation and the binder the lambda abstraction binds to
1958 is used more than once, more hardware might be generated than strictly
1961 As an example, consider the expression:
1967 When applying β-reduction to this expression, we get:
1973 which of course calculates \lam{(a * b)} twice.
1975 A possible solution to this would be to use the following alternative
1976 transformation, which is of course no longer normal β-reduction. The
1977 followin transformation has not been tested in the prototype, but is
1978 given here for future reference:
1986 This doesn't seem like much of an improvement, but it does get rid of
1987 the lambda expression (and the associated higher order value), while
1988 at the same time introducing a new let binding. Since the result of
1989 every application or case expression must be bound by a let expression
1990 in the intended normal form anyway, this is probably not a problem. If
1991 the argument happens to be a variable reference, then simple let
1992 binding removal (\in{section}[sec:normalization:simplelet]) will
1993 remove it, making the result identical to that of the original
1994 β-reduction transformation.
1996 When also applying argument simplification to the above example, we
1997 get the following expression:
2005 Looking at this, we could imagine an alternative approach: Create a
2006 transformation that removes let bindings that bind identical values.
2007 In the above expression, the \lam{y} and \lam{z} variables could be
2008 merged together, resulting in the more efficient expression:
2011 let y = (a * b) in y + y
2014 \subsection[sec:normalization:non-determinism]{Non-determinism}
2015 As an example, again consider the following expression:
2021 We can apply both β-reduction (\in{section}[sec:normalization:beta])
2022 as well as argument simplification
2023 (\in{section}[sec:normalization:argsimpl]) to this expression.
2025 When applying argument simplification first and then β-reduction, we
2026 get the following expression:
2029 let y = (a * b) in y + y
2032 When applying β-reduction first and then argument simplification, we
2033 get the following expression:
2041 As you can see, this is a different expression. This means that the
2042 order of expressions, does in fact change the resulting normal form,
2043 which is something that we would like to avoid. In this particular
2044 case one of the alternatives is even clearly more efficient, so we
2045 would of course like the more efficient form to be the normal form.
2047 For this particular problem, the solutions for duplication of work
2048 seem from the previous section seem to fix the determinism of our
2049 transformation system as well. However, it is likely that there are
2050 other occurences of this problem.
2053 We do not fully understand the use of cast expressions in Core, so
2054 there are probably expressions involving cast expressions that cannot
2055 be brought into intended normal form by this transformation system.
2057 The uses of casts in the core system should be investigated more and
2058 transformations will probably need updating to handle them in all
2061 \section[sec:normalization:properties]{Provable properties}
2062 When looking at the system of transformations outlined above, there are a
2063 number of questions that we can ask ourselves. The main question is of course:
2064 \quote{Does our system work as intended?}. We can split this question into a
2065 number of subquestions:
2068 \item[q:termination] Does our system \emph{terminate}? Since our system will
2069 keep running as long as transformations apply, there is an obvious risk that
2070 it will keep running indefinitely. This typically happens when one
2071 transformation produces a result that is transformed back to the original
2072 by another transformation, or when one or more transformations keep
2073 expanding some expression.
2074 \item[q:soundness] Is our system \emph{sound}? Since our transformations
2075 continuously modify the expression, there is an obvious risk that the final
2076 normal form will not be equivalent to the original program: Its meaning could
2078 \item[q:completeness] Is our system \emph{complete}? Since we have a complex
2079 system of transformations, there is an obvious risk that some expressions will
2080 not end up in our intended normal form, because we forgot some transformation.
2081 In other words: Does our transformation system result in our intended normal
2082 form for all possible inputs?
2083 \item[q:determinism] Is our system \emph{deterministic}? Since we have defined
2084 no particular order in which the transformation should be applied, there is an
2085 obvious risk that different transformation orderings will result in
2086 \emph{different} normal forms. They might still both be intended normal forms
2087 (if our system is \emph{complete}) and describe correct hardware (if our
2088 system is \emph{sound}), so this property is less important than the previous
2089 three: The translator would still function properly without it.
2092 Unfortunately, the final transformation system has only been
2093 developed in the final part of the research, leaving no more time
2094 for verifying these properties. In fact, it is likely that the
2095 current transformation system still violates some of these
2096 properties in some cases and should be improved (or extra conditions
2097 on the input hardware descriptions should be formulated).
2099 This is most likely the case with the completeness and determinism
2100 properties, perhaps als the termination property. The soundness
2101 property probably holds, since it is easier to manually verify (each
2102 transformation can be reviewed separately).
2104 Even though no complete proofs have been made, some ideas for
2105 possible proof strategies are shown below.
2107 \subsection{Graph representation}
2108 Before looking into how to prove these properties, we'll look at our
2109 transformation system from a graph perspective. The nodes of the graph are
2110 all possible Core expressions. The (directed) edges of the graph are
2111 transformations. When a transformation α applies to an expression \lam{A} to
2112 produce an expression \lam{B}, we add an edge from the node for \lam{A} to the
2113 node for \lam{B}, labeled α.
2115 \startuseMPgraphic{TransformGraph}
2119 newCircle.a(btex \lam{(λx.λy. (+) x y) 1} etex);
2120 newCircle.b(btex \lam{λy. (+) 1 y} etex);
2121 newCircle.c(btex \lam{(λx.(+) x) 1} etex);
2122 newCircle.d(btex \lam{(+) 1} etex);
2125 c.c = b.c + (4cm, 0cm);
2126 a.c = midpoint(b.c, c.c) + (0cm, 4cm);
2127 d.c = midpoint(b.c, c.c) - (0cm, 3cm);
2129 % β-conversion between a and b
2130 ncarc.a(a)(b) "name(bred)";
2131 ObjLabel.a(btex $\xrightarrow[normal]{}{β}$ etex) "labpathname(bred)", "labdir(rt)";
2132 ncarc.b(b)(a) "name(bexp)", "linestyle(dashed withdots)";
2133 ObjLabel.b(btex $\xleftarrow[normal]{}{β}$ etex) "labpathname(bexp)", "labdir(lft)";
2135 % η-conversion between a and c
2136 ncarc.a(a)(c) "name(ered)";
2137 ObjLabel.a(btex $\xrightarrow[normal]{}{η}$ etex) "labpathname(ered)", "labdir(rt)";
2138 ncarc.c(c)(a) "name(eexp)", "linestyle(dashed withdots)";
2139 ObjLabel.c(btex $\xleftarrow[normal]{}{η}$ etex) "labpathname(eexp)", "labdir(lft)";
2141 % η-conversion between b and d
2142 ncarc.b(b)(d) "name(ered)";
2143 ObjLabel.b(btex $\xrightarrow[normal]{}{η}$ etex) "labpathname(ered)", "labdir(rt)";
2144 ncarc.d(d)(b) "name(eexp)", "linestyle(dashed withdots)";
2145 ObjLabel.d(btex $\xleftarrow[normal]{}{η}$ etex) "labpathname(eexp)", "labdir(lft)";
2147 % β-conversion between c and d
2148 ncarc.c(c)(d) "name(bred)";
2149 ObjLabel.c(btex $\xrightarrow[normal]{}{β}$ etex) "labpathname(bred)", "labdir(rt)";
2150 ncarc.d(d)(c) "name(bexp)", "linestyle(dashed withdots)";
2151 ObjLabel.d(btex $\xleftarrow[normal]{}{β}$ etex) "labpathname(bexp)", "labdir(lft)";
2153 % Draw objects and lines
2154 drawObj(a, b, c, d);
2157 \placeexample[right][ex:TransformGraph]{Partial graph of a lambda calculus
2158 system with β and η reduction (solid lines) and expansion (dotted lines).}
2159 \boxedgraphic{TransformGraph}
2161 Of course our graph is unbounded, since we can construct an infinite amount of
2162 Core expressions. Also, there might potentially be multiple edges between two
2163 given nodes (with different labels), though seems unlikely to actually happen
2166 See \in{example}[ex:TransformGraph] for the graph representation of a very
2167 simple lambda calculus that contains just the expressions \lam{(λx.λy. (+) x
2168 y) 1}, \lam{λy. (+) 1 y}, \lam{(λx.(+) x) 1} and \lam{(+) 1}. The
2169 transformation system consists of β-reduction and η-reduction (solid edges) or
2170 β-expansion and η-expansion (dotted edges).
2172 \todo{Define β-reduction and η-reduction?}
2174 Note that the normal form of such a system consists of the set of nodes
2175 (expressions) without outgoing edges, since those are the expression to which
2176 no transformation applies anymore. We call this set of nodes the \emph{normal
2177 set}. The set of nodes containing expressions in intended normal
2178 form \refdef{intended normal form} is called the \emph{intended
2181 From such a graph, we can derive some properties easily:
2183 \item A system will \emph{terminate} if there is no path of infinite length
2184 in the graph (this includes cycles, but can also happen without cycles).
2185 \item Soundness is not easily represented in the graph.
2186 \item A system is \emph{complete} if all of the nodes in the normal set have
2187 the intended normal form. The inverse (that all of the nodes outside of
2188 the normal set are \emph{not} in the intended normal form) is not
2189 strictly required. In other words, our normal set must be a
2190 subset of the intended normal form, but they do not need to be
2193 \item A system is deterministic if all paths starting at a particular
2194 node, which end in a node in the normal set, end at the same node.
2197 When looking at the \in{example}[ex:TransformGraph], we see that the system
2198 terminates for both the reduction and expansion systems (but note that, for
2199 expansion, this is only true because we've limited the possible
2200 expressions. In comlete lambda calculus, there would be a path from
2201 \lam{(λx.λy. (+) x y) 1} to \lam{(λx.λy.(λz.(+) z) x y) 1} to
2202 \lam{(λx.λy.(λz.(λq.(+) q) z) x y) 1} etc.)
2204 If we would consider the system with both expansion and reduction, there
2205 would no longer be termination either, since there would be cycles all
2208 The reduction and expansion systems have a normal set of containing just
2209 \lam{(+) 1} or \lam{(λx.λy. (+) x y) 1} respectively. Since all paths in
2210 either system end up in these normal forms, both systems are \emph{complete}.
2211 Also, since there is only one node in the normal set, it must obviously be
2212 \emph{deterministic} as well.
2214 \todo{Add content to these sections}
2215 \subsection{Termination}
2216 In general, proving termination of an arbitrary program is a very
2217 hard problem. \todo{Ref about arbitrary termination} Fortunately,
2218 we only have to prove termination for our specific transformation
2221 A common approach for these kinds of proofs is to associate a
2222 measure with each possible expression in our system. If we can
2223 show that each transformation strictly decreases this measure
2224 (\ie, the expression transformed to has a lower measure than the
2225 expression transformed from). \todo{ref about measure-based
2226 termination proofs / analysis}
2228 A good measure for a system consisting of just β-reduction would
2229 be the number of lambda expressions in the expression. Since every
2230 application of β-reduction removes a lambda abstraction (and there
2231 is always a bounded number of lambda abstractions in every
2232 expression) we can easily see that a transformation system with
2233 just β-reduction will always terminate.
2235 For our complete system, this measure would be fairly complex
2236 (probably the sum of a lot of things). Since the (conditions on)
2237 our transformations are pretty complex, we would need to include
2238 both simple things like the number of let expressions as well as
2239 more complex things like the number of case expressions that are
2240 not yet in normal form.
2242 No real attempt has been made at finding a suitable measure for
2245 \subsection{Soundness}
2246 Soundness is a property that can be proven for each transformation
2247 separately. Since our system only runs separate transformations
2248 sequentially, if each of our transformations leaves the
2249 \emph{meaning} of the expression unchanged, then the entire system
2250 will of course leave the meaning unchanged and is thus
2253 The current prototype has only been verified in an ad-hoc fashion
2254 by inspecting (the code for) each transformation. A more formal
2255 verification would be more appropriate.
2257 To be able to formally show that each transformation properly
2258 preserves the meaning of every expression, we require an exact
2259 definition of the \emph{meaning} of every expression, so we can
2260 compare them. Currently there seems to be no formal definition of
2261 the meaning or semantics of \GHC's core language, only informal
2262 descriptions are available.
2264 It should be possible to have a single formal definition of
2265 meaning for Core for both normal Core compilation by \GHC and for
2266 our compilation to \VHDL. The main difference seems to be that in
2267 hardware every expression is always evaluated, while in software
2268 it is only evaluated if needed, but it should be possible to
2269 assign a meaning to core expressions that assumes neither.
2271 Since each of the transformations can be applied to any
2272 subexpression as well, there is a constraint on our meaning
2273 definition: The meaning of an expression should depend only on the
2274 meaning of subexpressions, not on the expressions themselves. For
2275 example, the meaning of the application in \lam{f (let x = 4 in
2276 x)} should be the same as the meaning of the application in \lam{f
2277 4}, since the argument subexpression has the same meaning (though
2278 the actual expression is different).
2280 \subsection{Completeness}
2281 Proving completeness is probably not hard, but it could be a lot
2282 of work. We have seen above that to prove completeness, we must
2283 show that the normal set of our graph representation is a subset
2284 of the intended normal set.
2286 However, it is hard to systematically generate or reason about the
2287 normal set, since it is defined as any nodes to which no
2288 transformation applies. To determine this set, each transformation
2289 must be considered and when a transformation is added, the entire
2290 set should be re-evaluated. This means it is hard to show that
2291 each node in the normal set is also in the intended normal set.
2292 Reasoning about our intended normal set is easier, since we know
2293 how to generate it from its definition. \refdef{intended normal
2296 Fortunately, we can also prove the complement (which is
2297 equivalent, since $A \subseteq B \Leftrightarrow \overline{B}
2298 \subseteq \overline{A}$): Show that the set of nodes not in
2299 intended normal form is a subset of the set of nodes not in normal
2300 form. In other words, show that for every expression that is not
2301 in intended normal form, that there is at least one transformation
2302 that applies to it (since that means it is not in normal form
2303 either and since $A \subseteq C \Leftrightarrow \forall x (x \in A
2304 \rightarrow x \in C)$).
2306 By systematically reviewing the entire Core language definition
2307 along with the intended normal form definition (both of which have
2308 a similar structure), it should be possible to identify all
2309 possible (sets of) core expressions that are not in intended
2310 normal form and identify a transformation that applies to it.
2312 This approach is especially useful for proving completeness of our
2313 system, since if expressions exist to which none of the
2314 transformations apply (\ie if the system is not yet complete), it
2315 is immediately clear which expressions these are and adding
2316 (or modifying) transformations to fix this should be relatively
2319 As observed above, applying this approach is a lot of work, since
2320 we need to check every (set of) transformation(s) separately.
2322 \todo{Perhaps do a few steps of the proofs as proof-of-concept}
2324 % vim: set sw=2 sts=2 expandtab: