1 \chapter[chap:normalization]{Normalization}
3 % A helper to print a single example in the half the page width. The example
4 % text should be in a buffer whose name is given in an argument.
6 % The align=right option really does left-alignment, but without the program
7 % will end up on a single line. The strut=no option prevents a bunch of empty
8 % space at the start of the frame.
10 \framed[offset=1mm,align=right,strut=no,background=box,frame=off]{
11 \setuptyping[option=LAM,style=sans,before=,after=]
13 \setuptyping[option=none,style=\tttf]
18 % A transformation example
19 \definefloat[example][examples]
20 \setupcaption[example][location=top] % Put captions on top
22 \define[3]\transexample{
23 \placeexample[here]{#1}
24 \startcombination[2*1]
25 {\example{#2}}{Original program}
26 {\example{#3}}{Transformed program}
30 %\define[3]\transexampleh{
31 %% \placeexample[here]{#1}
32 %% \startcombination[1*2]
33 %% {\example{#2}}{Original program}
34 %% {\example{#3}}{Transformed program}
38 The first step in the core to \small{VHDL} translation process, is normalization. We
39 aim to bring the core description into a simpler form, which we can
40 subsequently translate into \small{VHDL} easily. This normal form is needed because
41 the full core language is more expressive than \small{VHDL} in some areas and because
42 core can describe expressions that do not have a direct hardware
45 TODO: Describe core properties not supported in \small{VHDL}, and describe how the
46 \small{VHDL} we want to generate should look like.
49 The transformations described here have a well-defined goal: To bring the
50 program in a well-defined form that is directly translatable to hardware,
51 while fully preserving the semantics of the program. We refer to this form as
52 the \emph{normal form} of the program. The formal definition of this normal
55 \placedefinition{}{A program is in \emph{normal form} if none of the
56 transformations from this chapter apply.}
58 Of course, this is an \quote{easy} definition of the normal form, since our
59 program will end up in normal form automatically. The more interesting part is
60 to see if this normal form actually has the properties we would like it to
63 But, before getting into more definitions and details about this normal form,
64 let's try to get a feeling for it first. The easiest way to do this is by
65 describing the things we want to not have in a normal form.
68 \item Any \emph{polymorphism} must be removed. When laying down hardware, we
69 can't generate any signals that can have multiple types. All types must be
70 completely known to generate hardware.
72 \item Any \emph{higher order} constructions must be removed. We can't
73 generate a hardware signal that contains a function, so all values,
74 arguments and returns values used must be first order.
76 \item Any complex \emph{nested scopes} must be removed. In the \small{VHDL}
77 description, every signal is in a single scope. Also, full expressions are
78 not supported everywhere (in particular port maps can only map signal names,
79 not expressions). To make the \small{VHDL} generation easy, all values must be bound
80 on the \quote{top level}.
83 TODO: Intermezzo: functions vs plain values
85 A very simple example of a program in normal form is given in
86 \in{example}[ex:MulSum]. As you can see, all arguments to the function (which
87 will become input ports in the final hardware) are at the top. This means that
88 the body of the final lambda abstraction is never a function, but always a
91 After the lambda abstractions, we see a single let expression, that binds two
92 variables (\lam{mul} and \lam{sum}). These variables will be signals in the
93 final hardware, bound to the output port of the \lam{*} and \lam{+}
96 The final line (the \quote{return value} of the function) selects the
97 \lam{sum} signal to be the output port of the function. This \quote{return
98 value} can always only be a variable reference, never a more complex
102 alu :: Bit -> Word -> Word -> Word
111 \startuseMPgraphic{MulSum}
112 save a, b, c, mul, add, sum;
115 newCircle.a(btex $a$ etex) "framed(false)";
116 newCircle.b(btex $b$ etex) "framed(false)";
117 newCircle.c(btex $c$ etex) "framed(false)";
118 newCircle.sum(btex $res$ etex) "framed(false)";
121 newCircle.mul(btex - etex);
122 newCircle.add(btex + etex);
124 a.c - b.c = (0cm, 2cm);
125 b.c - c.c = (0cm, 2cm);
126 add.c = c.c + (2cm, 0cm);
127 mul.c = midpoint(a.c, b.c) + (2cm, 0cm);
128 sum.c = add.c + (2cm, 0cm);
131 % Draw objects and lines
132 drawObj(a, b, c, mul, add, sum);
134 ncarc(a)(mul) "arcangle(15)";
135 ncarc(b)(mul) "arcangle(-15)";
141 \placeexample[here][ex:MulSum]{Simple architecture consisting of an adder and a
143 \startcombination[2*1]
144 {\typebufferlam{MulSum}}{Core description in normal form.}
145 {\boxedgraphic{MulSum}}{The architecture described by the normal form.}
148 The previous example described composing an architecture by calling other
149 functions (operators), resulting in a simple architecture with component and
150 connection. There is of course also some mechanism for choice in the normal
151 form. In a normal Core program, the \emph{case} expression can be used in a
152 few different ways to describe choice. In normal form, this is limited to a
155 \in{Example}[ex:AddSubAlu] shows an example describing a
156 simple \small{ALU}, which chooses between two operations based on an opcode
157 bit. The main structure is the same as in \in{example}[ex:MulSum], but this
158 time the \lam{res} variable is bound to a case expression. This case
159 expression scrutinizes the variable \lam{opcode} (and scrutinizing more
160 complex expressions is not supported). The case expression can select a
161 different variable based on the constructor of \lam{opcode}.
163 \startbuffer[AddSubAlu]
164 alu :: Bit -> Word -> Word -> Word
176 \startuseMPgraphic{AddSubAlu}
177 save opcode, a, b, add, sub, mux, res;
180 newCircle.opcode(btex $opcode$ etex) "framed(false)";
181 newCircle.a(btex $a$ etex) "framed(false)";
182 newCircle.b(btex $b$ etex) "framed(false)";
183 newCircle.res(btex $res$ etex) "framed(false)";
185 newCircle.add(btex + etex);
186 newCircle.sub(btex - etex);
189 opcode.c - a.c = (0cm, 2cm);
190 add.c - a.c = (4cm, 0cm);
191 sub.c - b.c = (4cm, 0cm);
192 a.c - b.c = (0cm, 3cm);
193 mux.c = midpoint(add.c, sub.c) + (1.5cm, 0cm);
194 res.c - mux.c = (1.5cm, 0cm);
197 % Draw objects and lines
198 drawObj(opcode, a, b, res, add, sub, mux);
200 ncline(a)(add) "posA(e)";
201 ncline(b)(sub) "posA(e)";
202 nccurve(a)(sub) "posA(e)", "angleA(0)";
203 nccurve(b)(add) "posA(e)", "angleA(0)";
204 nccurve(add)(mux) "posB(inpa)", "angleB(0)";
205 nccurve(sub)(mux) "posB(inpb)", "angleB(0)";
206 nccurve(opcode)(mux) "posB(n)", "angleA(0)", "angleB(-90)";
207 ncline(mux)(res) "posA(out)";
210 \placeexample[here][ex:AddSubAlu]{Simple \small{ALU} supporting two operations.}
211 \startcombination[2*1]
212 {\typebufferlam{AddSubAlu}}{Core description in normal form.}
213 {\boxedgraphic{AddSubAlu}}{The architecture described by the normal form.}
216 As a more complete example, consider \in{example}[ex:NormalComplete]. This
217 example contains everything that is supported in normal form, with the
218 exception of builtin higher order functions. The graphical version of the
219 architecture contains a slightly simplified version, since the state tuple
220 packing and unpacking have been left out. Instead, two seperate registers are
221 drawn. Also note that most synthesis tools will further optimize this
222 architecture by removing the multiplexers at the register input and replace
223 them with some logic in the clock inputs, but we want to show the architecture
224 as close to the description as possible.
226 \startbuffer[NormalComplete]
229 -> State (Word, Word)
230 -> (State (Word, Word), Word)
232 -- All arguments are an inital lambda
234 -- There are nested let expressions at top level
236 -- Unpack the state by coercion (\eg, cast from
237 -- State (Word, Word) to (Word, Word))
238 s = sp :: (Word, Word)
239 -- Extract both registers from the state
240 r1 = case s of (fst, snd) -> fst
241 r2 = case s of (fst, snd) -> snd
242 -- Calling some other user-defined function.
244 -- Conditional connections
256 -- pack the state by coercion (\eg, cast from
257 -- (Word, Word) to State (Word, Word))
258 sp' = s' :: State (Word, Word)
259 -- Pack our return value
266 \startuseMPgraphic{NormalComplete}
267 save a, d, r, foo, muxr, muxout, out;
270 newCircle.a(btex \lam{a} etex) "framed(false)";
271 newCircle.d(btex \lam{d} etex) "framed(false)";
272 newCircle.out(btex \lam{out} etex) "framed(false)";
274 %newCircle.add(btex + etex);
275 newBox.foo(btex \lam{foo} etex);
276 newReg.r1(btex $\lam{r1}$ etex) "dx(4mm)", "dy(6mm)";
277 newReg.r2(btex $\lam{r2}$ etex) "dx(4mm)", "dy(6mm)", "reflect(true)";
279 % Reflect over the vertical axis
280 reflectObj(muxr1)((0,0), (0,1));
283 rotateObj(muxout)(-90);
285 d.c = foo.c + (0cm, 1.5cm);
286 a.c = (xpart r2.c + 2cm, ypart d.c - 0.5cm);
287 foo.c = midpoint(muxr1.c, muxr2.c) + (0cm, 2cm);
288 muxr1.c = r1.c + (0cm, 2cm);
289 muxr2.c = r2.c + (0cm, 2cm);
290 r2.c = r1.c + (4cm, 0cm);
292 muxout.c = midpoint(r1.c, r2.c) - (0cm, 2cm);
293 out.c = muxout.c - (0cm, 1.5cm);
295 % % Draw objects and lines
296 drawObj(a, d, foo, r1, r2, muxr1, muxr2, muxout, out);
299 nccurve(foo)(muxr1) "angleA(-90)", "posB(inpa)", "angleB(180)";
300 nccurve(foo)(muxr2) "angleA(-90)", "posB(inpb)", "angleB(0)";
301 nccurve(muxr1)(r1) "posA(out)", "angleA(180)", "posB(d)", "angleB(0)";
302 nccurve(r1)(muxr1) "posA(out)", "angleA(0)", "posB(inpb)", "angleB(180)";
303 nccurve(muxr2)(r2) "posA(out)", "angleA(0)", "posB(d)", "angleB(180)";
304 nccurve(r2)(muxr2) "posA(out)", "angleA(180)", "posB(inpa)", "angleB(0)";
305 nccurve(r1)(muxout) "posA(out)", "angleA(0)", "posB(inpb)", "angleB(-90)";
306 nccurve(r2)(muxout) "posA(out)", "angleA(180)", "posB(inpa)", "angleB(-90)";
308 nccurve(a)(muxout) "angleA(-90)", "angleB(180)", "posB(sel)";
309 nccurve(a)(muxr1) "angleA(180)", "angleB(-90)", "posB(sel)";
310 nccurve(a)(muxr2) "angleA(180)", "angleB(-90)", "posB(sel)";
311 ncline(muxout)(out) "posA(out)";
314 \placeexample[here][ex:NormalComplete]{Simple architecture consisting of an adder and a
316 \startcombination[2*1]
317 {\typebufferlam{NormalComplete}}{Core description in normal form.}
318 {\boxedgraphic{NormalComplete}}{The architecture described by the normal form.}
321 \subsection{Normal form definition}
322 Now we have some intuition for the normal form, we can describe how we want
323 the normal form to look like in a slightly more formal manner. The following
324 EBNF-like description completely captures the intended structure (and
325 generates a subset of GHC's core format).
327 Some clauses have an expression listed in parentheses. These are conditions
328 that need to apply to the clause.
331 \italic{normal} = \italic{lambda}
332 \italic{lambda} = λvar.\italic{lambda} (representable(var))
334 \italic{toplet} = let \italic{binding} in \italic{toplet}
335 | letrec [\italic{binding}] in \italic{toplet}
336 | var (representable(varvar))
337 \italic{binding} = var = \italic{rhs} (representable(rhs))
338 -- State packing and unpacking by coercion
339 | var0 = var1 :: State ty (lvar(var1))
340 | var0 = var1 :: ty (var0 :: State ty) (lvar(var1))
341 \italic{rhs} = userapp
344 | case var of C a0 ... an -> ai (lvar(var))
346 | case var of (lvar(var))
347 DEFAULT -> var0 (lvar(var0))
348 C w0 ... wn -> resvar (\forall{}i, wi \neq resvar, lvar(resvar))
349 \italic{userapp} = \italic{userfunc}
350 | \italic{userapp} {userarg}
351 \italic{userfunc} = var (gvar(var))
352 \italic{userarg} = var (lvar(var))
353 \italic{builtinapp} = \italic{builtinfunc}
354 | \italic{builtinapp} \italic{builtinarg}
355 \italic{builtinfunc} = var (bvar(var))
356 \italic{builtinarg} = \italic{coreexpr}
359 -- TODO: Limit builtinarg further
361 -- TODO: There can still be other casts around (which the code can handle,
362 e.g., ignore), which still need to be documented here.
364 -- TODO: Note about the selector case. It just supports Bit and Bool
365 currently, perhaps it should be generalized in the normal form?
367 When looking at such a program from a hardware perspective, the top level
368 lambda's define the input ports. The value produced by the let expression is
369 the output port. Most function applications bound by the let expression
370 define a component instantiation, where the input and output ports are mapped
371 to local signals or arguments. Some of the others use a builtin
372 construction (\eg the \lam{case} statement) or call a builtin function
373 (\eg \lam{add} or \lam{sub}). For these, a hardcoded \small{VHDL} translation is
376 \subsection{Definitions}
377 In the following sections, we will be using a number of functions and
378 notations, which we will define here.
380 \subsubsection{Transformation notation}
381 To be able to concisely present transformations, we use a specific format to
382 them. It is a simple format, similar to one used in logic reasoning.
384 Such a transformation description looks like the following.
389 <original expression>
390 -------------------------- <expression conditions>
391 <transformed expresssion>
396 This format desribes a transformation that applies to \lam{original
397 expresssion} and transforms it into \lam{transformed expression}, assuming
398 that all conditions apply. In this format, there are a number of placeholders
399 in pointy brackets, most of which should be rather obvious in their meaning.
400 Nevertheless, we will more precisely specify their meaning below:
402 \startdesc{<original expression>} The expression pattern that will be matched
403 against (subexpressions of) the expression to be transformed. We call this a
404 pattern, because it can contain \emph{placeholders} (variables), which match
405 any expression or binder. Any such placeholder is said to be \emph{bound} to
406 the expression it matches. It is convention to use an uppercase latter (\eg
407 \lam{M} or \lam{E} to refer to any expression (including a simple variable
408 reference) and lowercase letters (\eg \lam{v} or \lam{b}) to refer to
409 (references to) binders.
411 For example, the pattern \lam{a + B} will match the expression
412 \lam{v + (2 * w)} (and bind \lam{a} to \lam{v} and \lam{B} to
413 \lam{(2 * 2)}), but not \lam{v + (2 * w)}.
416 \startdesc{<expression conditions>}
417 These are extra conditions on the expression that is matched. These
418 conditions can be used to further limit the cases in which the
419 transformation applies, in particular to prevent a transformation from
420 causing a loop with itself or another transformation.
422 Only if these if these conditions are \emph{all} true, this transformation
426 \startdesc{<context conditions>}
427 These are a number of extra conditions on the context of the function. In
428 particular, these conditions can require some other top level function to be
429 present, whose value matches the pattern given here. The format of each of
430 these conditions is: \lam{binder = <pattern>}.
432 Typically, the binder is some placeholder bound in the \lam{<original
433 expression>}, while the pattern contains some placeholders that are used in
434 the \lam{transformed expression}.
436 Only if a top level binder exists that matches each binder and pattern, this
437 transformation applies.
440 \startdesc{<transformed expression>}
441 This is the expression template that is the result of the transformation. If, looking
442 at the above three items, the transformation applies, the \lam{original
443 expression} is completely replaced with the \lam{<transformed expression>}.
444 We call this a template, because it can contain placeholders, referring to
445 any placeholder bound by the \lam{<original expression>} or the
446 \lam{<context conditions>}. The resulting expression will have those
447 placeholders replaced by the values bound to them.
449 Any binder (lowercase) placeholder that has no value bound to it yet will be
450 bound to (and replaced with) a fresh binder.
453 \startdesc{<context additions>}
454 These are templates for new functions to add to the context. This is a way
455 to have a transformation create new top level functiosn.
457 Each addition has the form \lam{binder = template}. As above, any
458 placeholder in the addition is replaced with the value bound to it, and any
459 binder placeholder that has no value bound to it yet will be bound to (and
460 replaced with) a fresh binder.
463 As an example, we'll look at η-abstraction:
467 -------------- \lam{E} does not occur on a function position in an application
468 λx.E x \lam{E} is not a lambda abstraction.
471 Consider the following function, which is a fairly obvious way to specify a
472 simple ALU (Note \at{example}[ex:AddSubAlu] is the normal form of this
476 alu :: Bit -> Word -> Word -> Word
477 alu = λopcode. case opcode of
482 There are a few subexpressions in this function to which we could possibly
483 apply the transformation. Since the pattern of the transformation is only
484 the placeholder \lam{E}, any expression will match that. Whether the
485 transformation applies to an expression is thus solely decided by the
486 conditions to the right of the transformation.
488 We will look at each expression in the function in a top down manner. The
489 first expression is the entire expression the function is bound to.
492 λopcode. case opcode of
497 As said, the expression pattern matches this. The type of this expression is
498 \lam{Bit -> Word -> Word -> Word}, which matches \lam{a -> b} (Note that in
499 this case \lam{a = Bit} and \lam{b = Word -> Word -> Word}).
501 Since this expression is at top level, it does not occur at a function
502 position of an application. However, The expression is a lambda abstraction,
503 so this transformation does not apply.
505 The next expression we could apply this transformation to, is the body of
506 the lambda abstraction:
514 The type of this expression is \lam{Word -> Word -> Word}, which again
515 matches \lam{a -> b}. The expression is the body of a lambda expression, so
516 it does not occur at a function position of an application. Finally, the
517 expression is not a lambda abstraction but a case expression, so all the
518 conditions match. There are no context conditions to match, so the
519 transformation applies.
521 By now, the placeholder \lam{E} is bound to the entire expression. The
522 placeholder \lam{x}, which occurs in the replacement template, is not bound
523 yet, so we need to generate a fresh binder for that. Let's use the binder
524 \lam{a}. This results in the following replacement expression:
532 Continuing with this expression, we see that the transformation does not
533 apply again (it is a lambda expression). Next we look at the body of this
542 Here, the transformation does apply, binding \lam{E} to the entire
543 expression and \lam{x} to the fresh binder \lam{b}, resulting in the
552 Again, the transformation does not apply to this lambda abstraction, so we
553 look at its body. For brevity, we'll put the case statement on one line from
557 (case opcode of Low -> (+); High -> (-)) a b
560 The type of this expression is \lam{Word}, so it does not match \lam{a -> b}
561 and the transformation does not apply. Next, we have two options for the
562 next expression to look at: The function position and argument position of
563 the application. The expression in the argument position is \lam{b}, which
564 has type \lam{Word}, so the transformation does not apply. The expression in
565 the function position is:
568 (case opcode of Low -> (+); High -> (-)) a
571 Obviously, the transformation does not apply here, since it occurs in
572 function position. In the same way the transformation does not apply to both
573 components of this expression (\lam{case opcode of Low -> (+); High -> (-)}
574 and \lam{a}), so we'll skip to the components of the case expression: The
575 scrutinee and both alternatives. Since the opcode is not a function, it does
576 not apply here, and we'll leave both alternatives as an exercise to the
577 reader. The final function, after all these transformations becomes:
580 alu :: Bit -> Word -> Word -> Word
581 alu = λopcode.λa.b. (case opcode of
582 Low -> λa1.λb1 (+) a1 b1
583 High -> λa2.λb2 (-) a2 b2) a b
586 In this case, the transformation does not apply anymore, though this might
587 not always be the case (e.g., the application of a transformation on a
588 subexpression might open up possibilities to apply the transformation
589 further up in the expression).
591 \subsubsection{Transformation application}
592 In this chapter we define a number of transformations, but how will we apply
593 these? As stated before, our normal form is reached as soon as no
594 transformation applies anymore. This means our application strategy is to
595 simply apply any transformation that applies, and continuing to do that with
596 the result of each transformation.
598 In particular, we define no particular order of transformations. Since
599 transformation order should not influence the resulting normal form (see TODO
600 ref), this leaves the implementation free to choose any application order that
601 results in an efficient implementation.
603 When applying a single transformation, we try to apply it to every (sub)expression
604 in a function, not just the top level function. This allows us to keep the
605 transformation descriptions concise and powerful.
607 \subsubsection{Other concepts}
608 A \emph{global variable} is any variable that is bound at the
609 top level of a program, or an external module. A local variable is any other
610 variable (\eg, variables local to a function, which can be bound by lambda
611 abstractions, let expressions and case expressions).
613 A \emph{hardware representable} type is a type that we can generate
614 a signal for in hardware. For example, a bit, a vector of bits, a 32 bit
615 unsigned word, etc. Types that are not runtime representable notably
616 include (but are not limited to): Types, dictionaries, functions.
618 A \emph{builtin function} is a function for which a builtin
619 hardware translation is available, because its actual definition is not
620 translatable. A user-defined function is any other function.
622 \subsubsection{Functions}
623 Here, we define a number of functions that can be used below to concisely
626 \emph{gvar(expr)} is true when \emph{expr} is a variable that references a
627 global variable. It is false when it references a local variable.
629 \emph{lvar(expr)} is the inverse of \emph{gvar}; it is true when \emph{expr}
630 references a local variable, false when it references a global variable.
632 \emph{representable(expr)} or \emph{representable(var)} is true when
633 \emph{expr} or \emph{var} has a type that is representable at runtime.
635 \section{Transform passes}
636 In this section we describe the actual transforms. Here we're using
637 the core language in a notation that resembles lambda calculus.
639 Each of these transforms is meant to be applied to every (sub)expression
640 in a program, for as long as it applies. Only when none of the
641 transformations can be applied anymore, the program is in normal form (by
642 definition). We hope to be able to prove that this form will obey all of the
643 constraints defined above, but this has yet to happen (though it seems likely
646 Each of the transforms will be described informally first, explaining
647 the need for and goal of the transform. Then, a formal definition is
648 given, using a familiar syntax from the world of logic. Each transform
649 is specified as a number of conditions (above the horizontal line) and a
650 number of conclusions (below the horizontal line). The details of using
651 this notation are still a bit fuzzy, so comments are welcom.
653 TODO: Formally describe the "apply to every (sub)expression" in terms of
654 rules with full transformations in the conditions.
656 \subsection{Binder uniqueness}
657 A common problem in transformation systems, is binder uniqueness. When not
658 considering this problem, it is easy to create transformations that mix up
659 bindings and cause name collisions. Take for example, the following core
663 (λa.λb.λc. a * b * c) x c
666 By applying β-reduction (see below) once, we can simplify this expression to:
672 Now, we have replaced the \lam{a} binder with a reference to the \lam{x}
673 binder. No harm done here. But note that we see multiple occurences of the
674 \lam{c} binder. The first is a binding occurence, to which the second refers.
675 The last, however refers to \emph{another} instance of \lam{c}, which is
676 bound somewhere outside of this expression. Now, if we would apply beta
677 reduction without taking heed of binder uniqueness, we would get:
683 This is obviously not what was supposed to happen! The root of this problem is
684 the reuse of binders: Identical binders can be bound in different scopes, such
685 that only the inner one is \quote{visible} in the inner expression. In the example
686 above, the \lam{c} binder was bound outside of the expression and in the inner
687 lambda expression. Inside that lambda expression, only the inner \lam{c} is
690 There are a number of ways to solve this. \small{GHC} has isolated this
691 problem to their binder substitution code, which performs \emph{deshadowing}
692 during its expression traversal. This means that any binding that shadows
693 another binding on a higher level is replaced by a new binder that does not
694 shadow any other binding. This non-shadowing invariant is enough to prevent
695 binder uniqueness problems in \small{GHC}.
697 In our transformation system, maintaining this non-shadowing invariant is
698 a bit harder to do (mostly due to implementation issues, the prototype doesn't
699 use \small{GHC}'s subsitution code). Also, we can observe the following
703 \item Deshadowing does not guarantee overall uniqueness. For example, the
704 following (slightly contrived) expression shows the identifier \lam{x} bound in
705 two seperate places (and to different values), even though no shadowing
709 (let x = 1 in x) + (let x = 2 in x)
712 \item In our normal form (and the resulting \small{VHDL}), all binders
713 (signals) will end up in the same scope. To allow this, all binders within the
714 same function should be unique.
716 \item When we know that all binders in an expression are unique, moving around
717 or removing a subexpression will never cause any binder conflicts. If we have
718 some way to generate fresh binders, introducing new subexpressions will not
719 cause any problems either. The only way to cause conflicts is thus to
720 duplicate an existing subexpression.
723 Given the above, our prototype maintains a unique binder invariant. This
724 meanst that in any given moment during normalization, all binders \emph{within
725 a single function} must be unique. To achieve this, we apply the following
728 TODO: Define fresh binders and unique supplies
731 \item Before starting normalization, all binders in the function are made
732 unique. This is done by generating a fresh binder for every binder used. This
733 also replaces binders that did not pose any conflict, but it does ensure that
734 all binders within the function are generated by the same unique supply. See
735 (TODO: ref fresh binder).
736 \item Whenever a new binder must be generated, we generate a fresh binder that
737 is guaranteed to be different from \emph{all binders generated so far}. This
738 can thus never introduce duplication and will maintain the invariant.
739 \item Whenever (part of) an expression is duplicated (for example when
740 inlining), all binders in the expression are replaced with fresh binders
741 (using the same method as at the start of normalization). These fresh binders
742 can never introduce duplication, so this will maintain the invariant.
743 \item Whenever we move part of an expression around within the function, there
744 is no need to do anything special. There is obviously no way to introduce
745 duplication by moving expressions around. Since we know that each of the
746 binders is already unique, there is no way to introduce (incorrect) shadowing
750 \subsection{η-abstraction}
751 This transformation makes sure that all arguments of a function-typed
752 expression are named, by introducing lambda expressions. When combined with
753 β-reduction and function inlining below, all function-typed expressions should
754 be lambda abstractions or global identifiers.
758 -------------- \lam{E} is not the first argument of an application.
759 λx.E x \lam{E} is not a lambda abstraction.
760 \lam{x} is a variable that does not occur free in \lam{E}.
770 foo = λa.λx.(case a of
775 \transexample{η-abstraction}{from}{to}
777 \subsection{Extended β-reduction}
778 This transformation is meant to propagate application expressions downwards
779 into expressions as far as possible. In lambda calculus, this reduction
780 is known as β-reduction, but it is of course only defined for
781 applications of lambda abstractions. We extend this reduction to also
782 work for the rest of core (case and let expressions).
804 For lambda expressions:
817 b = (let y = 3 in add y) 2
827 b = let y = 3 in add y 2
832 \transexample{Extended β-reduction}{from}{to}
834 \subsection{Let derecursification}
835 This transformation is meant to make lets non-recursive whenever possible.
836 This might allow other optimizations to do their work better. TODO: Why is
839 \subsection{Let flattening}
840 This transformation puts nested lets in the same scope, by lifting the
841 binding(s) of the inner let into a new let around the outer let. Eventually,
842 this will cause all let bindings to appear in the same scope (they will all be
843 in scope for the function return value).
845 Note that this transformation does not try to be smart when faced with
846 recursive lets, it will just leave the lets recursive (possibly joining a
847 recursive and non-recursive let into a single recursive let). The let
848 rederursification transformation will do this instead.
851 letnonrec x = (let bindings in M) in N
852 ------------------------------------------
853 let bindings in (letnonrec x = M) in N
859 x = (let bindings in M)
863 ------------------------------------------
882 b = let c = 3 in a + c
903 \transexample{Let flattening}{from}{to}
905 \subsection{Empty let removal}
906 This transformation is simple: It removes recursive lets that have no bindings
907 (which usually occurs when let derecursification removes the last binding from
916 \subsection{Simple let binding removal}
917 This transformation inlines simple let bindings (\eg a = b).
919 This transformation is not needed to get into normal form, but makes the
920 resulting \small{VHDL} a lot shorter.
946 \subsection{Unused let binding removal}
947 This transformation removes let bindings that are never used. Usually,
948 the desugarer introduces some unused let bindings.
950 This normalization pass should really be unneeded to get into normal form
951 (since ununsed bindings are not forbidden by the normal form), but in practice
952 the desugarer or simplifier emits some unused bindings that cannot be
953 normalized (e.g., calls to a \type{PatError} (TODO: Check this name)). Also,
954 this transformation makes the resulting \small{VHDL} a lot shorter.
958 ---------------------------- \lam{a} does not occur free in \lam{M}
969 ---------------------------- \lam{a} does not occur free in \lam{M}
977 \subsection{Non-representable binding inlining}
978 This transform inlines let bindings that have a non-representable type. Since
979 we can never generate a signal assignment for these bindings (we cannot
980 declare a signal assignment with a non-representable type, for obvious
981 reasons), we have no choice but to inline the binding to remove it.
983 If the binding is non-representable because it is a lambda abstraction, it is
984 likely that it will inlined into an application and β-reduction will remove
985 the lambda abstraction and turn it into a representable expression at the
986 inline site. The same holds for partial applications, which can be turned into
987 full applications by inlining.
989 Other cases of non-representable bindings we see in practice are primitive
990 Haskell types. In most cases, these will not result in a valid normalized
991 output, but then the input would have been invalid to start with. There is one
992 exception to this: When a builtin function is applied to a non-representable
993 expression, things might work out in some cases. For example, when you write a
994 literal \hs{SizedInt} in Haskell, like \hs{1 :: SizedInt D8}, this results in
995 the following core: \lam{fromInteger (smallInteger 10)}, where for example
996 \lam{10 :: GHC.Prim.Int\#} and \lam{smallInteger 10 :: Integer} have
997 non-representable types. TODO: This/these paragraph(s) should probably become a
998 separate discussion somewhere else.
1001 letnonrec a = E in M
1002 -------------------------- \lam{E} has a non-representable type.
1013 -------------------------- \lam{E} has a non-representable type.
1033 x = fromInteger (smallInteger 10)
1035 (λa -> add a 1) (add 1 x)
1038 \transexample{Let flattening}{from}{to}
1040 \subsection{Compiler generated top level binding inlining}
1043 \subsection{Scrutinee simplification}
1044 This transform ensures that the scrutinee of a case expression is always
1045 a simple variable reference.
1050 ----------------- \lam{E} is not a local variable reference
1069 \transexample{Let flattening}{from}{to}
1072 \subsection{Case simplification}
1073 This transformation ensures that all case expressions become normal form. This
1074 means they will become one of:
1076 \item An extractor case with a single alternative that picks a single field
1077 from a datatype, \eg \lam{case x of (a, b) -> a}.
1078 \item A selector case with multiple alternatives and only wild binders, that
1079 makes a choice between expressions based on the constructor of another
1080 expression, \eg \lam{case x of Low -> a; High -> b}.
1085 C0 v0,0 ... v0,m -> E0
1087 Cn vn,0 ... vn,m -> En
1088 --------------------------------------------------- \forall i \forall j, 0 <= i <= n, 0 <= i < m (\lam{wi,j} is a wild (unused) binder)
1090 v0,0 = case x of C0 v0,0 .. v0,m -> v0,0
1092 v0,m = case x of C0 v0,0 .. v0,m -> v0,m
1095 vn,m = case x of Cn vn,0 .. vn,m -> vn,m
1099 C0 w0,0 ... w0,m -> x0
1101 Cn wn,0 ... wn,m -> xn
1104 TODO: This transformation specified like this is complicated and misses
1105 conditions to prevent looping with itself. Perhaps we should split it here for
1124 \transexample{Selector case simplification}{from}{to}
1132 b = case a of (,) b c -> b
1133 c = case a of (,) b c -> c
1140 \transexample{Extractor case simplification}{from}{to}
1142 \subsection{Case removal}
1143 This transform removes any case statements with a single alternative and
1146 These "useless" case statements are usually leftovers from case simplification
1147 on extractor case (see the previous example).
1152 ---------------------- \lam{\forall i, 0 <= i <= m} (\lam{vi} does not occur free in E)
1165 \transexample{Case removal}{from}{to}
1167 \subsection{Argument simplification}
1168 The transforms in this section deal with simplifying application
1169 arguments into normal form. The goal here is to:
1172 \item Make all arguments of user-defined functions (\eg, of which
1173 we have a function body) simple variable references of a runtime
1174 representable type. This is needed, since these applications will be turned
1175 into component instantiations.
1176 \item Make all arguments of builtin functions one of:
1178 \item A type argument.
1179 \item A dictionary argument.
1180 \item A type level expression.
1181 \item A variable reference of a runtime representable type.
1182 \item A variable reference or partial application of a function type.
1186 When looking at the arguments of a user-defined function, we can
1187 divide them into two categories:
1189 \item Arguments of a runtime representable type (\eg bits or vectors).
1191 These arguments can be preserved in the program, since they can
1192 be translated to input ports later on. However, since we can
1193 only connect signals to input ports, these arguments must be
1194 reduced to simple variables (for which signals will be
1195 produced). This is taken care of by the argument extraction
1197 \item Non-runtime representable typed arguments.
1199 These arguments cannot be preserved in the program, since we
1200 cannot represent them as input or output ports in the resulting
1201 \small{VHDL}. To remove them, we create a specialized version of the
1202 called function with these arguments filled in. This is done by
1203 the argument propagation transform.
1205 Typically, these arguments are type and dictionary arguments that are
1206 used to make functions polymorphic. By propagating these arguments, we
1207 are essentially doing the same which GHC does when it specializes
1208 functions: Creating multiple variants of the same function, one for
1209 each type for which it is used. Other common non-representable
1210 arguments are functions, e.g. when calling a higher order function
1211 with another function or a lambda abstraction as an argument.
1213 The reason for doing this is similar to the reasoning provided for
1214 the inlining of non-representable let bindings above. In fact, this
1215 argument propagation could be viewed as a form of cross-function
1219 TODO: Check the following itemization.
1221 When looking at the arguments of a builtin function, we can divide them
1225 \item Arguments of a runtime representable type.
1227 As we have seen with user-defined functions, these arguments can
1228 always be reduced to a simple variable reference, by the
1229 argument extraction transform. Performing this transform for
1230 builtin functions as well, means that the translation of builtin
1231 functions can be limited to signal references, instead of
1232 needing to support all possible expressions.
1234 \item Arguments of a function type.
1236 These arguments are functions passed to higher order builtins,
1237 like \lam{map} and \lam{foldl}. Since implementing these
1238 functions for arbitrary function-typed expressions (\eg, lambda
1239 expressions) is rather comlex, we reduce these arguments to
1240 (partial applications of) global functions.
1242 We can still support arbitrary expressions from the user code,
1243 by creating a new global function containing that expression.
1244 This way, we can simply replace the argument with a reference to
1245 that new function. However, since the expression can contain any
1246 number of free variables we also have to include partial
1247 applications in our normal form.
1249 This category of arguments is handled by the function extraction
1251 \item Other unrepresentable arguments.
1253 These arguments can take a few different forms:
1254 \startdesc{Type arguments}
1255 In the core language, type arguments can only take a single
1256 form: A type wrapped in the Type constructor. Also, there is
1257 nothing that can be done with type expressions, except for
1258 applying functions to them, so we can simply leave type
1259 arguments as they are.
1261 \startdesc{Dictionary arguments}
1262 In the core language, dictionary arguments are used to find
1263 operations operating on one of the type arguments (mostly for
1264 finding class methods). Since we will not actually evaluatie
1265 the function body for builtin functions and can generate
1266 code for builtin functions by just looking at the type
1267 arguments, these arguments can be ignored and left as they
1270 \startdesc{Type level arguments}
1271 Sometimes, we want to pass a value to a builtin function, but
1272 we need to know the value at compile time. Additionally, the
1273 value has an impact on the type of the function. This is
1274 encoded using type-level values, where the actual value of the
1275 argument is not important, but the type encodes some integer,
1276 for example. Since the value is not important, the actual form
1277 of the expression does not matter either and we can leave
1278 these arguments as they are.
1280 \startdesc{Other arguments}
1281 Technically, there is still a wide array of arguments that can
1282 be passed, but does not fall into any of the above categories.
1283 However, none of the supported builtin functions requires such
1284 an argument. This leaves use with passing unsupported types to
1285 a function, such as calling \lam{head} on a list of functions.
1287 In these cases, it would be impossible to generate hardware
1288 for such a function call anyway, so we can ignore these
1291 The only way to generate hardware for builtin functions with
1292 arguments like these, is to expand the function call into an
1293 equivalent core expression (\eg, expand map into a series of
1294 function applications). But for now, we choose to simply not
1295 support expressions like these.
1298 From the above, we can conclude that we can simply ignore these
1299 other unrepresentable arguments and focus on the first two
1303 \subsubsection{Argument simplification}
1304 This transform deals with arguments to functions that
1305 are of a runtime representable type. It ensures that they will all become
1306 references to global variables, or local signals in the resulting \small{VHDL}.
1308 TODO: It seems we can map an expression to a port, not only a signal.
1309 Perhaps this makes this transformation not needed?
1310 TODO: Say something about dataconstructors (without arguments, like True
1311 or False), which are variable references of a runtime representable
1312 type, but do not result in a signal.
1314 To reduce a complex expression to a simple variable reference, we create
1315 a new let expression around the application, which binds the complex
1316 expression to a new variable. The original function is then applied to
1321 -------------------- \lam{N} is of a representable type
1322 let x = N in M x \lam{N} is not a local variable reference
1330 let x = add a 1 in add x 1
1333 \transexample{Argument extraction}{from}{to}
1335 \subsubsection{Function extraction}
1336 This transform deals with function-typed arguments to builtin functions.
1337 Since these arguments cannot be propagated, we choose to extract them
1338 into a new global function instead.
1340 Any free variables occuring in the extracted arguments will become
1341 parameters to the new global function. The original argument is replaced
1342 with a reference to the new function, applied to any free variables from
1343 the original argument.
1345 This transformation is useful when applying higher order builtin functions
1346 like \hs{map} to a lambda abstraction, for example. In this case, the code
1347 that generates \small{VHDL} for \hs{map} only needs to handle top level functions and
1348 partial applications, not any other expression (such as lambda abstractions or
1349 even more complicated expressions).
1352 M N \lam{M} is a (partial aplication of a) builtin function.
1353 --------------------- \lam{f0 ... fn} = free local variables of \lam{N}
1354 M x f0 ... fn \lam{N :: a -> b}
1355 ~ \lam{N} is not a (partial application of) a top level function
1360 map (λa . add a b) xs
1374 \transexample{Function extraction}{from}{to}
1376 \subsubsection{Argument propagation}
1377 This transform deals with arguments to user-defined functions that are
1378 not representable at runtime. This means these arguments cannot be
1379 preserved in the final form and most be {\em propagated}.
1381 Propagation means to create a specialized version of the called
1382 function, with the propagated argument already filled in. As a simple
1383 example, in the following program:
1390 we could {\em propagate} the constant argument 1, with the following
1398 Special care must be taken when the to-be-propagated expression has any
1399 free variables. If this is the case, the original argument should not be
1400 removed alltogether, but replaced by all the free variables of the
1401 expression. In this way, the original expression can still be evaluated
1402 inside the new function. Also, this brings us closer to our goal: All
1403 these free variables will be simple variable references.
1405 To prevent us from propagating the same argument over and over, a simple
1406 local variable reference is not propagated (since is has exactly one
1407 free variable, itself, we would only replace that argument with itself).
1409 This shows that any free local variables that are not runtime representable
1410 cannot be brought into normal form by this transform. We rely on an
1411 inlining transformation to replace such a variable with an expression we
1412 can propagate again.
1417 x Y0 ... Yi ... Yn \lam{Yi} is not of a runtime representable type
1418 --------------------------------------------- \lam{Yi} is not a local variable reference
1419 x' y0 ... yi-1 f0 ... fm Yi+1 ... Yn \lam{f0 ... fm} = free local vars of \lam{Yi}
1421 x' = λy0 ... yi-1 f0 ... fm yi+1 ... yn .
1422 E y0 ... yi-1 Yi yi+1 ... yn
1428 \subsection{Cast propagation / simplification}
1429 This transform pushes casts down into the expression as far as possible. Since
1430 its exact role and need is not clear yet, this transformation is not yet
1433 \subsection{Return value simplification}
1434 This transformation ensures that the return value of a function is always a
1435 simple local variable reference.
1437 Currently implemented using lambda simplification, let simplification, and
1438 top simplification. Should change into something like the following, which
1439 works only on the result of a function instead of any subexpression. This is
1440 achieved by the contexts, like \lam{x = E}, though this is strictly not
1441 correct (you could read this as "if there is any function \lam{x} that binds
1442 \lam{E}, any \lam{E} can be transformed, while we only mean the \lam{E} that
1443 is bound by \lam{x}. This might need some extra notes or something).
1446 x = E \lam{E} is representable
1447 ~ \lam{E} is not a lambda abstraction
1448 E \lam{E} is not a let expression
1449 --------------------------- \lam{E} is not a local variable reference
1455 ~ \lam{E} is representable
1456 E \lam{E} is not a let expression
1457 --------------------------- \lam{E} is not a local variable reference
1462 x = λv0 ... λvn.let ... in E
1463 ~ \lam{E} is representable
1464 E \lam{E} is not a local variable reference
1465 ---------------------------
1474 x = let x = add 1 2 in x
1477 \transexample{Return value simplification}{from}{to}