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 \section{Transformation notation}
377 To be able to concisely present transformations, we use a specific format to
378 them. It is a simple format, similar to one used in logic reasoning.
380 Such a transformation description looks like the following.
385 <original expression>
386 -------------------------- <expression conditions>
387 <transformed expresssion>
392 This format desribes a transformation that applies to \lam{original
393 expresssion} and transforms it into \lam{transformed expression}, assuming
394 that all conditions apply. In this format, there are a number of placeholders
395 in pointy brackets, most of which should be rather obvious in their meaning.
396 Nevertheless, we will more precisely specify their meaning below:
398 \startdesc{<original expression>} The expression pattern that will be matched
399 against (subexpressions of) the expression to be transformed. We call this a
400 pattern, because it can contain \emph{placeholders} (variables), which match
401 any expression or binder. Any such placeholder is said to be \emph{bound} to
402 the expression it matches. It is convention to use an uppercase latter (\eg
403 \lam{M} or \lam{E} to refer to any expression (including a simple variable
404 reference) and lowercase letters (\eg \lam{v} or \lam{b}) to refer to
405 (references to) binders.
407 For example, the pattern \lam{a + B} will match the expression
408 \lam{v + (2 * w)} (and bind \lam{a} to \lam{v} and \lam{B} to
409 \lam{(2 * 2)}), but not \lam{v + (2 * w)}.
412 \startdesc{<expression conditions>}
413 These are extra conditions on the expression that is matched. These
414 conditions can be used to further limit the cases in which the
415 transformation applies, in particular to prevent a transformation from
416 causing a loop with itself or another transformation.
418 Only if these if these conditions are \emph{all} true, this transformation
422 \startdesc{<context conditions>}
423 These are a number of extra conditions on the context of the function. In
424 particular, these conditions can require some other top level function to be
425 present, whose value matches the pattern given here. The format of each of
426 these conditions is: \lam{binder = <pattern>}.
428 Typically, the binder is some placeholder bound in the \lam{<original
429 expression>}, while the pattern contains some placeholders that are used in
430 the \lam{transformed expression}.
432 Only if a top level binder exists that matches each binder and pattern, this
433 transformation applies.
436 \startdesc{<transformed expression>}
437 This is the expression template that is the result of the transformation. If, looking
438 at the above three items, the transformation applies, the \lam{original
439 expression} is completely replaced with the \lam{<transformed expression>}.
440 We call this a template, because it can contain placeholders, referring to
441 any placeholder bound by the \lam{<original expression>} or the
442 \lam{<context conditions>}. The resulting expression will have those
443 placeholders replaced by the values bound to them.
445 Any binder (lowercase) placeholder that has no value bound to it yet will be
446 bound to (and replaced with) a fresh binder.
449 \startdesc{<context additions>}
450 These are templates for new functions to add to the context. This is a way
451 to have a transformation create new top level functiosn.
453 Each addition has the form \lam{binder = template}. As above, any
454 placeholder in the addition is replaced with the value bound to it, and any
455 binder placeholder that has no value bound to it yet will be bound to (and
456 replaced with) a fresh binder.
459 As an example, we'll look at η-abstraction:
463 -------------- \lam{E} does not occur on a function position in an application
464 λx.E x \lam{E} is not a lambda abstraction.
467 Consider the following function, which is a fairly obvious way to specify a
468 simple ALU (Note \at{example}[ex:AddSubAlu] is the normal form of this
472 alu :: Bit -> Word -> Word -> Word
473 alu = λopcode. case opcode of
478 There are a few subexpressions in this function to which we could possibly
479 apply the transformation. Since the pattern of the transformation is only
480 the placeholder \lam{E}, any expression will match that. Whether the
481 transformation applies to an expression is thus solely decided by the
482 conditions to the right of the transformation.
484 We will look at each expression in the function in a top down manner. The
485 first expression is the entire expression the function is bound to.
488 λopcode. case opcode of
493 As said, the expression pattern matches this. The type of this expression is
494 \lam{Bit -> Word -> Word -> Word}, which matches \lam{a -> b} (Note that in
495 this case \lam{a = Bit} and \lam{b = Word -> Word -> Word}).
497 Since this expression is at top level, it does not occur at a function
498 position of an application. However, The expression is a lambda abstraction,
499 so this transformation does not apply.
501 The next expression we could apply this transformation to, is the body of
502 the lambda abstraction:
510 The type of this expression is \lam{Word -> Word -> Word}, which again
511 matches \lam{a -> b}. The expression is the body of a lambda expression, so
512 it does not occur at a function position of an application. Finally, the
513 expression is not a lambda abstraction but a case expression, so all the
514 conditions match. There are no context conditions to match, so the
515 transformation applies.
517 By now, the placeholder \lam{E} is bound to the entire expression. The
518 placeholder \lam{x}, which occurs in the replacement template, is not bound
519 yet, so we need to generate a fresh binder for that. Let's use the binder
520 \lam{a}. This results in the following replacement expression:
528 Continuing with this expression, we see that the transformation does not
529 apply again (it is a lambda expression). Next we look at the body of this
538 Here, the transformation does apply, binding \lam{E} to the entire
539 expression and \lam{x} to the fresh binder \lam{b}, resulting in the
548 Again, the transformation does not apply to this lambda abstraction, so we
549 look at its body. For brevity, we'll put the case statement on one line from
553 (case opcode of Low -> (+); High -> (-)) a b
556 The type of this expression is \lam{Word}, so it does not match \lam{a -> b}
557 and the transformation does not apply. Next, we have two options for the
558 next expression to look at: The function position and argument position of
559 the application. The expression in the argument position is \lam{b}, which
560 has type \lam{Word}, so the transformation does not apply. The expression in
561 the function position is:
564 (case opcode of Low -> (+); High -> (-)) a
567 Obviously, the transformation does not apply here, since it occurs in
568 function position. In the same way the transformation does not apply to both
569 components of this expression (\lam{case opcode of Low -> (+); High -> (-)}
570 and \lam{a}), so we'll skip to the components of the case expression: The
571 scrutinee and both alternatives. Since the opcode is not a function, it does
572 not apply here, and we'll leave both alternatives as an exercise to the
573 reader. The final function, after all these transformations becomes:
576 alu :: Bit -> Word -> Word -> Word
577 alu = λopcode.λa.b. (case opcode of
578 Low -> λa1.λb1 (+) a1 b1
579 High -> λa2.λb2 (-) a2 b2) a b
582 In this case, the transformation does not apply anymore, though this might
583 not always be the case (e.g., the application of a transformation on a
584 subexpression might open up possibilities to apply the transformation
585 further up in the expression).
587 \subsection{Transformation application}
588 In this chapter we define a number of transformations, but how will we apply
589 these? As stated before, our normal form is reached as soon as no
590 transformation applies anymore. This means our application strategy is to
591 simply apply any transformation that applies, and continuing to do that with
592 the result of each transformation.
594 In particular, we define no particular order of transformations. Since
595 transformation order should not influence the resulting normal form (see TODO
596 ref), this leaves the implementation free to choose any application order that
597 results in an efficient implementation.
599 When applying a single transformation, we try to apply it to every (sub)expression
600 in a function, not just the top level function. This allows us to keep the
601 transformation descriptions concise and powerful.
603 \subsection{Definitions}
604 In the following sections, we will be using a number of functions and
605 notations, which we will define here.
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 \emph{local variable} is any
610 other variable (\eg, variables local to a function, which can be bound by
611 lambda abstractions, let expressions and pattern matches of case
612 alternatives). Note that this is a slightly different notion of global versus
613 local than what \small{GHC} uses internally.
614 \defref{global variable} \defref{local variable}
616 A \emph{hardware representable} (or just \emph{representable}) type or value
617 is (a value of) a type that we can generate a signal for in hardware. For
618 example, a bit, a vector of bits, a 32 bit unsigned word, etc. Types that are
619 not runtime representable notably include (but are not limited to): Types,
620 dictionaries, functions.
621 \defref{representable}
623 A \emph{builtin function} is a function supplied by the Cλash framework, whose
624 implementation is not valid Cλash. The implementation is of course valid
625 Haskell, for simulation, but it is not expressable in Cλash.
626 \defref{builtin function} \defref{user-defined function}
628 For these functions, Cλash has a \emph{builtin hardware translation}, so calls
629 to these functions can still be translated. These are functions like
630 \lam{map}, \lam{hwor} and \lam{length}.
632 A \emph{user-defined} function is a function for which we do have a Cλash
633 implementation available.
635 \subsubsection{Functions}
636 Here, we define a number of functions that can be used below to concisely
639 \refdef{global variable}\emph{gvar(expr)} is true when \emph{expr} is a variable that references a
640 global variable. It is false when it references a local variable.
642 \refdef{local variable}\emph{lvar(expr)} is the complement of \emph{gvar}; it is true when \emph{expr}
643 references a local variable, false when it references a global variable.
645 \refdef{representable}\emph{representable(expr)} or \emph{representable(var)} is true when
646 \emph{expr} or \emph{var} is \emph{representable}.
648 \subsection{Binder uniqueness}
649 A common problem in transformation systems, is binder uniqueness. When not
650 considering this problem, it is easy to create transformations that mix up
651 bindings and cause name collisions. Take for example, the following core
655 (λa.λb.λc. a * b * c) x c
658 By applying β-reduction (see below) once, we can simplify this expression to:
664 Now, we have replaced the \lam{a} binder with a reference to the \lam{x}
665 binder. No harm done here. But note that we see multiple occurences of the
666 \lam{c} binder. The first is a binding occurence, to which the second refers.
667 The last, however refers to \emph{another} instance of \lam{c}, which is
668 bound somewhere outside of this expression. Now, if we would apply beta
669 reduction without taking heed of binder uniqueness, we would get:
675 This is obviously not what was supposed to happen! The root of this problem is
676 the reuse of binders: Identical binders can be bound in different scopes, such
677 that only the inner one is \quote{visible} in the inner expression. In the example
678 above, the \lam{c} binder was bound outside of the expression and in the inner
679 lambda expression. Inside that lambda expression, only the inner \lam{c} is
682 There are a number of ways to solve this. \small{GHC} has isolated this
683 problem to their binder substitution code, which performs \emph{deshadowing}
684 during its expression traversal. This means that any binding that shadows
685 another binding on a higher level is replaced by a new binder that does not
686 shadow any other binding. This non-shadowing invariant is enough to prevent
687 binder uniqueness problems in \small{GHC}.
689 In our transformation system, maintaining this non-shadowing invariant is
690 a bit harder to do (mostly due to implementation issues, the prototype doesn't
691 use \small{GHC}'s subsitution code). Also, we can observe the following
695 \item Deshadowing does not guarantee overall uniqueness. For example, the
696 following (slightly contrived) expression shows the identifier \lam{x} bound in
697 two seperate places (and to different values), even though no shadowing
701 (let x = 1 in x) + (let x = 2 in x)
704 \item In our normal form (and the resulting \small{VHDL}), all binders
705 (signals) will end up in the same scope. To allow this, all binders within the
706 same function should be unique.
708 \item When we know that all binders in an expression are unique, moving around
709 or removing a subexpression will never cause any binder conflicts. If we have
710 some way to generate fresh binders, introducing new subexpressions will not
711 cause any problems either. The only way to cause conflicts is thus to
712 duplicate an existing subexpression.
715 Given the above, our prototype maintains a unique binder invariant. This
716 meanst that in any given moment during normalization, all binders \emph{within
717 a single function} must be unique. To achieve this, we apply the following
720 TODO: Define fresh binders and unique supplies
723 \item Before starting normalization, all binders in the function are made
724 unique. This is done by generating a fresh binder for every binder used. This
725 also replaces binders that did not pose any conflict, but it does ensure that
726 all binders within the function are generated by the same unique supply. See
727 (TODO: ref fresh binder).
728 \item Whenever a new binder must be generated, we generate a fresh binder that
729 is guaranteed to be different from \emph{all binders generated so far}. This
730 can thus never introduce duplication and will maintain the invariant.
731 \item Whenever (part of) an expression is duplicated (for example when
732 inlining), all binders in the expression are replaced with fresh binders
733 (using the same method as at the start of normalization). These fresh binders
734 can never introduce duplication, so this will maintain the invariant.
735 \item Whenever we move part of an expression around within the function, there
736 is no need to do anything special. There is obviously no way to introduce
737 duplication by moving expressions around. Since we know that each of the
738 binders is already unique, there is no way to introduce (incorrect) shadowing
742 \section{Transform passes}
743 In this section we describe the actual transforms. Here we're using
744 the core language in a notation that resembles lambda calculus.
746 Each of these transforms is meant to be applied to every (sub)expression
747 in a program, for as long as it applies. Only when none of the
748 transformations can be applied anymore, the program is in normal form (by
749 definition). We hope to be able to prove that this form will obey all of the
750 constraints defined above, but this has yet to happen (though it seems likely
753 Each of the transforms will be described informally first, explaining
754 the need for and goal of the transform. Then, a formal definition is
755 given, using a familiar syntax from the world of logic. Each transform
756 is specified as a number of conditions (above the horizontal line) and a
757 number of conclusions (below the horizontal line). The details of using
758 this notation are still a bit fuzzy, so comments are welcom.
760 \subsection{η-abstraction}
761 This transformation makes sure that all arguments of a function-typed
762 expression are named, by introducing lambda expressions. When combined with
763 β-reduction and function inlining below, all function-typed expressions should
764 be lambda abstractions or global identifiers.
768 -------------- \lam{E} is not the first argument of an application.
769 λx.E x \lam{E} is not a lambda abstraction.
770 \lam{x} is a variable that does not occur free in \lam{E}.
780 foo = λa.λx.(case a of
785 \transexample{η-abstraction}{from}{to}
787 \subsection{β-reduction}
788 β-reduction is a well known transformation from lambda calculus, where it is
789 the main reduction step. It reduces applications of labmda abstractions,
790 removing both the lambda abstraction and the application.
792 In our transformation system, this step helps to remove unwanted lambda
793 abstractions (basically all but the ones at the top level). Other
794 transformations (application propagation, non-representable inlining) make
795 sure that most lambda abstractions will eventually be reducable by
798 TODO: Define substitution syntax
815 \transexample{β-reduction}{from}{to}
817 \subsection{Application propagation}
818 This transformation is meant to propagate application expressions downwards
819 into expressions as far as possible. This allows partial applications inside
820 expressions to become fully applied and exposes new transformation
821 possibilities for other transformations (like β-reduction).
845 \transexample{Application propagation for a let expression}{from}{to}
873 \transexample{Application propagation for a case expression}{from}{to}
875 \subsection{Let derecursification}
876 This transformation is meant to make lets non-recursive whenever possible.
877 This might allow other optimizations to do their work better. TODO: Why is
880 \subsection{Let flattening}
881 This transformation puts nested lets in the same scope, by lifting the
882 binding(s) of the inner let into a new let around the outer let. Eventually,
883 this will cause all let bindings to appear in the same scope (they will all be
884 in scope for the function return value).
886 Note that this transformation does not try to be smart when faced with
887 recursive lets, it will just leave the lets recursive (possibly joining a
888 recursive and non-recursive let into a single recursive let). The let
889 rederecursification transformation will do this instead.
892 letnonrec x = (let bindings in M) in N
893 ------------------------------------------
894 let bindings in (letnonrec x = M) in N
900 x = (let bindings in M)
904 ------------------------------------------
923 b = let c = 3 in a + c
944 \transexample{Let flattening}{from}{to}
946 \subsection{Empty let removal}
947 This transformation is simple: It removes recursive lets that have no bindings
948 (which usually occurs when let derecursification removes the last binding from
957 \subsection{Simple let binding removal}
958 This transformation inlines simple let bindings (\eg a = b).
960 This transformation is not needed to get into normal form, but makes the
961 resulting \small{VHDL} a lot shorter.
987 \subsection{Unused let binding removal}
988 This transformation removes let bindings that are never used. Usually,
989 the desugarer introduces some unused let bindings.
991 This normalization pass should really be unneeded to get into normal form
992 (since ununsed bindings are not forbidden by the normal form), but in practice
993 the desugarer or simplifier emits some unused bindings that cannot be
994 normalized (e.g., calls to a \type{PatError} (TODO: Check this name)). Also,
995 this transformation makes the resulting \small{VHDL} a lot shorter.
999 ---------------------------- \lam{a} does not occur free in \lam{M}
1010 ---------------------------- \lam{a} does not occur free in \lam{M}
1018 \subsection{Non-representable binding inlining}
1019 This transform inlines let bindings that have a non-representable type. Since
1020 we can never generate a signal assignment for these bindings (we cannot
1021 declare a signal assignment with a non-representable type, for obvious
1022 reasons), we have no choice but to inline the binding to remove it.
1024 If the binding is non-representable because it is a lambda abstraction, it is
1025 likely that it will inlined into an application and β-reduction will remove
1026 the lambda abstraction and turn it into a representable expression at the
1027 inline site. The same holds for partial applications, which can be turned into
1028 full applications by inlining.
1030 Other cases of non-representable bindings we see in practice are primitive
1031 Haskell types. In most cases, these will not result in a valid normalized
1032 output, but then the input would have been invalid to start with. There is one
1033 exception to this: When a builtin function is applied to a non-representable
1034 expression, things might work out in some cases. For example, when you write a
1035 literal \hs{SizedInt} in Haskell, like \hs{1 :: SizedInt D8}, this results in
1036 the following core: \lam{fromInteger (smallInteger 10)}, where for example
1037 \lam{10 :: GHC.Prim.Int\#} and \lam{smallInteger 10 :: Integer} have
1038 non-representable types. TODO: This/these paragraph(s) should probably become a
1039 separate discussion somewhere else.
1042 letnonrec a = E in M
1043 -------------------------- \lam{E} has a non-representable type.
1054 -------------------------- \lam{E} has a non-representable type.
1074 x = fromInteger (smallInteger 10)
1076 (λa -> add a 1) (add 1 x)
1079 \transexample{Let flattening}{from}{to}
1081 \subsection{Compiler generated top level binding inlining}
1084 \subsection{Scrutinee simplification}
1085 This transform ensures that the scrutinee of a case expression is always
1086 a simple variable reference.
1091 ----------------- \lam{E} is not a local variable reference
1110 \transexample{Let flattening}{from}{to}
1113 \subsection{Case simplification}
1114 This transformation ensures that all case expressions become normal form. This
1115 means they will become one of:
1117 \item An extractor case with a single alternative that picks a single field
1118 from a datatype, \eg \lam{case x of (a, b) -> a}.
1119 \item A selector case with multiple alternatives and only wild binders, that
1120 makes a choice between expressions based on the constructor of another
1121 expression, \eg \lam{case x of Low -> a; High -> b}.
1126 C0 v0,0 ... v0,m -> E0
1128 Cn vn,0 ... vn,m -> En
1129 --------------------------------------------------- \forall i \forall j, 0 <= i <= n, 0 <= i < m (\lam{wi,j} is a wild (unused) binder)
1131 v0,0 = case x of C0 v0,0 .. v0,m -> v0,0
1133 v0,m = case x of C0 v0,0 .. v0,m -> v0,m
1136 vn,m = case x of Cn vn,0 .. vn,m -> vn,m
1140 C0 w0,0 ... w0,m -> x0
1142 Cn wn,0 ... wn,m -> xn
1145 TODO: This transformation specified like this is complicated and misses
1146 conditions to prevent looping with itself. Perhaps we should split it here for
1165 \transexample{Selector case simplification}{from}{to}
1173 b = case a of (,) b c -> b
1174 c = case a of (,) b c -> c
1181 \transexample{Extractor case simplification}{from}{to}
1183 \subsection{Case removal}
1184 This transform removes any case statements with a single alternative and
1187 These "useless" case statements are usually leftovers from case simplification
1188 on extractor case (see the previous example).
1193 ---------------------- \lam{\forall i, 0 <= i <= m} (\lam{vi} does not occur free in E)
1206 \transexample{Case removal}{from}{to}
1208 \subsection{Argument simplification}
1209 The transforms in this section deal with simplifying application
1210 arguments into normal form. The goal here is to:
1213 \item Make all arguments of user-defined functions (\eg, of which
1214 we have a function body) simple variable references of a runtime
1215 representable type. This is needed, since these applications will be turned
1216 into component instantiations.
1217 \item Make all arguments of builtin functions one of:
1219 \item A type argument.
1220 \item A dictionary argument.
1221 \item A type level expression.
1222 \item A variable reference of a runtime representable type.
1223 \item A variable reference or partial application of a function type.
1227 When looking at the arguments of a user-defined function, we can
1228 divide them into two categories:
1230 \item Arguments of a runtime representable type (\eg bits or vectors).
1232 These arguments can be preserved in the program, since they can
1233 be translated to input ports later on. However, since we can
1234 only connect signals to input ports, these arguments must be
1235 reduced to simple variables (for which signals will be
1236 produced). This is taken care of by the argument extraction
1238 \item Non-runtime representable typed arguments.
1240 These arguments cannot be preserved in the program, since we
1241 cannot represent them as input or output ports in the resulting
1242 \small{VHDL}. To remove them, we create a specialized version of the
1243 called function with these arguments filled in. This is done by
1244 the argument propagation transform.
1246 Typically, these arguments are type and dictionary arguments that are
1247 used to make functions polymorphic. By propagating these arguments, we
1248 are essentially doing the same which GHC does when it specializes
1249 functions: Creating multiple variants of the same function, one for
1250 each type for which it is used. Other common non-representable
1251 arguments are functions, e.g. when calling a higher order function
1252 with another function or a lambda abstraction as an argument.
1254 The reason for doing this is similar to the reasoning provided for
1255 the inlining of non-representable let bindings above. In fact, this
1256 argument propagation could be viewed as a form of cross-function
1260 TODO: Check the following itemization.
1262 When looking at the arguments of a builtin function, we can divide them
1266 \item Arguments of a runtime representable type.
1268 As we have seen with user-defined functions, these arguments can
1269 always be reduced to a simple variable reference, by the
1270 argument extraction transform. Performing this transform for
1271 builtin functions as well, means that the translation of builtin
1272 functions can be limited to signal references, instead of
1273 needing to support all possible expressions.
1275 \item Arguments of a function type.
1277 These arguments are functions passed to higher order builtins,
1278 like \lam{map} and \lam{foldl}. Since implementing these
1279 functions for arbitrary function-typed expressions (\eg, lambda
1280 expressions) is rather comlex, we reduce these arguments to
1281 (partial applications of) global functions.
1283 We can still support arbitrary expressions from the user code,
1284 by creating a new global function containing that expression.
1285 This way, we can simply replace the argument with a reference to
1286 that new function. However, since the expression can contain any
1287 number of free variables we also have to include partial
1288 applications in our normal form.
1290 This category of arguments is handled by the function extraction
1292 \item Other unrepresentable arguments.
1294 These arguments can take a few different forms:
1295 \startdesc{Type arguments}
1296 In the core language, type arguments can only take a single
1297 form: A type wrapped in the Type constructor. Also, there is
1298 nothing that can be done with type expressions, except for
1299 applying functions to them, so we can simply leave type
1300 arguments as they are.
1302 \startdesc{Dictionary arguments}
1303 In the core language, dictionary arguments are used to find
1304 operations operating on one of the type arguments (mostly for
1305 finding class methods). Since we will not actually evaluatie
1306 the function body for builtin functions and can generate
1307 code for builtin functions by just looking at the type
1308 arguments, these arguments can be ignored and left as they
1311 \startdesc{Type level arguments}
1312 Sometimes, we want to pass a value to a builtin function, but
1313 we need to know the value at compile time. Additionally, the
1314 value has an impact on the type of the function. This is
1315 encoded using type-level values, where the actual value of the
1316 argument is not important, but the type encodes some integer,
1317 for example. Since the value is not important, the actual form
1318 of the expression does not matter either and we can leave
1319 these arguments as they are.
1321 \startdesc{Other arguments}
1322 Technically, there is still a wide array of arguments that can
1323 be passed, but does not fall into any of the above categories.
1324 However, none of the supported builtin functions requires such
1325 an argument. This leaves use with passing unsupported types to
1326 a function, such as calling \lam{head} on a list of functions.
1328 In these cases, it would be impossible to generate hardware
1329 for such a function call anyway, so we can ignore these
1332 The only way to generate hardware for builtin functions with
1333 arguments like these, is to expand the function call into an
1334 equivalent core expression (\eg, expand map into a series of
1335 function applications). But for now, we choose to simply not
1336 support expressions like these.
1339 From the above, we can conclude that we can simply ignore these
1340 other unrepresentable arguments and focus on the first two
1344 \subsubsection{Argument simplification}
1345 This transform deals with arguments to functions that
1346 are of a runtime representable type. It ensures that they will all become
1347 references to global variables, or local signals in the resulting \small{VHDL}.
1349 TODO: It seems we can map an expression to a port, not only a signal.
1350 Perhaps this makes this transformation not needed?
1351 TODO: Say something about dataconstructors (without arguments, like True
1352 or False), which are variable references of a runtime representable
1353 type, but do not result in a signal.
1355 To reduce a complex expression to a simple variable reference, we create
1356 a new let expression around the application, which binds the complex
1357 expression to a new variable. The original function is then applied to
1362 -------------------- \lam{N} is of a representable type
1363 let x = N in M x \lam{N} is not a local variable reference
1371 let x = add a 1 in add x 1
1374 \transexample{Argument extraction}{from}{to}
1376 \subsubsection{Function extraction}
1377 This transform deals with function-typed arguments to builtin functions.
1378 Since these arguments cannot be propagated, we choose to extract them
1379 into a new global function instead.
1381 Any free variables occuring in the extracted arguments will become
1382 parameters to the new global function. The original argument is replaced
1383 with a reference to the new function, applied to any free variables from
1384 the original argument.
1386 This transformation is useful when applying higher order builtin functions
1387 like \hs{map} to a lambda abstraction, for example. In this case, the code
1388 that generates \small{VHDL} for \hs{map} only needs to handle top level functions and
1389 partial applications, not any other expression (such as lambda abstractions or
1390 even more complicated expressions).
1393 M N \lam{M} is a (partial aplication of a) builtin function.
1394 --------------------- \lam{f0 ... fn} = free local variables of \lam{N}
1395 M x f0 ... fn \lam{N :: a -> b}
1396 ~ \lam{N} is not a (partial application of) a top level function
1401 map (λa . add a b) xs
1415 \transexample{Function extraction}{from}{to}
1417 \subsubsection{Argument propagation}
1418 This transform deals with arguments to user-defined functions that are
1419 not representable at runtime. This means these arguments cannot be
1420 preserved in the final form and most be {\em propagated}.
1422 Propagation means to create a specialized version of the called
1423 function, with the propagated argument already filled in. As a simple
1424 example, in the following program:
1431 we could {\em propagate} the constant argument 1, with the following
1439 Special care must be taken when the to-be-propagated expression has any
1440 free variables. If this is the case, the original argument should not be
1441 removed alltogether, but replaced by all the free variables of the
1442 expression. In this way, the original expression can still be evaluated
1443 inside the new function. Also, this brings us closer to our goal: All
1444 these free variables will be simple variable references.
1446 To prevent us from propagating the same argument over and over, a simple
1447 local variable reference is not propagated (since is has exactly one
1448 free variable, itself, we would only replace that argument with itself).
1450 This shows that any free local variables that are not runtime representable
1451 cannot be brought into normal form by this transform. We rely on an
1452 inlining transformation to replace such a variable with an expression we
1453 can propagate again.
1458 x Y0 ... Yi ... Yn \lam{Yi} is not of a runtime representable type
1459 --------------------------------------------- \lam{Yi} is not a local variable reference
1460 x' y0 ... yi-1 f0 ... fm Yi+1 ... Yn \lam{f0 ... fm} = free local vars of \lam{Yi}
1462 x' = λy0 ... yi-1 f0 ... fm yi+1 ... yn .
1463 E y0 ... yi-1 Yi yi+1 ... yn
1469 \subsection{Cast propagation / simplification}
1470 This transform pushes casts down into the expression as far as possible. Since
1471 its exact role and need is not clear yet, this transformation is not yet
1474 \subsection{Return value simplification}
1475 This transformation ensures that the return value of a function is always a
1476 simple local variable reference.
1478 Currently implemented using lambda simplification, let simplification, and
1479 top simplification. Should change into something like the following, which
1480 works only on the result of a function instead of any subexpression. This is
1481 achieved by the contexts, like \lam{x = E}, though this is strictly not
1482 correct (you could read this as "if there is any function \lam{x} that binds
1483 \lam{E}, any \lam{E} can be transformed, while we only mean the \lam{E} that
1484 is bound by \lam{x}. This might need some extra notes or something).
1487 x = E \lam{E} is representable
1488 ~ \lam{E} is not a lambda abstraction
1489 E \lam{E} is not a let expression
1490 --------------------------- \lam{E} is not a local variable reference
1496 ~ \lam{E} is representable
1497 E \lam{E} is not a let expression
1498 --------------------------- \lam{E} is not a local variable reference
1503 x = λv0 ... λvn.let ... in E
1504 ~ \lam{E} is representable
1505 E \lam{E} is not a local variable reference
1506 ---------------------------
1515 x = let x = add 1 2 in x
1518 \transexample{Return value simplification}{from}{to}