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 \define[3]\transexample{
19 \placeexample[here]{#1}
20 \startcombination[2*1]
21 {\example{#2}}{Original program}
22 {\example{#3}}{Transformed program}
26 %\define[3]\transexampleh{
27 %% \placeexample[here]{#1}
28 %% \startcombination[1*2]
29 %% {\example{#2}}{Original program}
30 %% {\example{#3}}{Transformed program}
34 The first step in the core to \small{VHDL} translation process, is normalization. We
35 aim to bring the core description into a simpler form, which we can
36 subsequently translate into \small{VHDL} easily. This normal form is needed because
37 the full core language is more expressive than \small{VHDL} in some areas and because
38 core can describe expressions that do not have a direct hardware
41 TODO: Describe core properties not supported in \small{VHDL}, and describe how the
42 \small{VHDL} we want to generate should look like.
45 The transformations described here have a well-defined goal: To bring the
46 program in a well-defined form that is directly translatable to hardware,
47 while fully preserving the semantics of the program. We refer to this form as
48 the \emph{normal form} of the program. The formal definition of this normal
51 \placedefinition{}{A program is in \emph{normal form} if none of the
52 transformations from this chapter apply.}
54 Of course, this is an \quote{easy} definition of the normal form, since our
55 program will end up in normal form automatically. The more interesting part is
56 to see if this normal form actually has the properties we would like it to
59 But, before getting into more definitions and details about this normal form,
60 let's try to get a feeling for it first. The easiest way to do this is by
61 describing the things we want to not have in a normal form.
64 \item Any \emph{polymorphism} must be removed. When laying down hardware, we
65 can't generate any signals that can have multiple types. All types must be
66 completely known to generate hardware.
68 \item Any \emph{higher order} constructions must be removed. We can't
69 generate a hardware signal that contains a function, so all values,
70 arguments and returns values used must be first order.
72 \item Any complex \emph{nested scopes} must be removed. In the \small{VHDL}
73 description, every signal is in a single scope. Also, full expressions are
74 not supported everywhere (in particular port maps can only map signal names,
75 not expressions). To make the \small{VHDL} generation easy, all values must be bound
76 on the \quote{top level}.
79 TODO: Intermezzo: functions vs plain values
81 A very simple example of a program in normal form is given in
82 \in{example}[ex:MulSum]. As you can see, all arguments to the function (which
83 will become input ports in the final hardware) are at the top. This means that
84 the body of the final lambda abstraction is never a function, but always a
87 After the lambda abstractions, we see a single let expression, that binds two
88 variables (\lam{mul} and \lam{sum}). These variables will be signals in the
89 final hardware, bound to the output port of the \lam{*} and \lam{+}
92 The final line (the \quote{return value} of the function) selects the
93 \lam{sum} signal to be the output port of the function. This \quote{return
94 value} can always only be a variable reference, never a more complex
98 alu :: Bit -> Word -> Word -> Word
107 \startuseMPgraphic{MulSum}
108 save a, b, c, mul, add, sum;
111 newCircle.a(btex $a$ etex) "framed(false)";
112 newCircle.b(btex $b$ etex) "framed(false)";
113 newCircle.c(btex $c$ etex) "framed(false)";
114 newCircle.sum(btex $res$ etex) "framed(false)";
117 newCircle.mul(btex - etex);
118 newCircle.add(btex + etex);
120 a.c - b.c = (0cm, 2cm);
121 b.c - c.c = (0cm, 2cm);
122 add.c = c.c + (2cm, 0cm);
123 mul.c = midpoint(a.c, b.c) + (2cm, 0cm);
124 sum.c = add.c + (2cm, 0cm);
127 % Draw objects and lines
128 drawObj(a, b, c, mul, add, sum);
130 ncarc(a)(mul) "arcangle(15)";
131 ncarc(b)(mul) "arcangle(-15)";
137 \placeexample[here][ex:MulSum]{Simple architecture consisting of an adder and a
139 \startcombination[2*1]
140 {\typebufferlam{MulSum}}{Core description in normal form.}
141 {\boxedgraphic{MulSum}}{The architecture described by the normal form.}
144 The previous example described composing an architecture by calling other
145 functions (operators), resulting in a simple architecture with component and
146 connection. There is of course also some mechanism for choice in the normal
147 form. In a normal Core program, the \emph{case} expression can be used in a
148 few different ways to describe choice. In normal form, this is limited to a
151 \in{Example}[ex:AddSubAlu] shows an example describing a
152 simple \small{ALU}, which chooses between two operations based on an opcode
153 bit. The main structure is the same as in \in{example}[ex:MulSum], but this
154 time the \lam{res} variable is bound to a case expression. This case
155 expression scrutinizes the variable \lam{opcode} (and scrutinizing more
156 complex expressions is not supported). The case expression can select a
157 different variable based on the constructor of \lam{opcode}.
159 \startbuffer[AddSubAlu]
160 alu :: Bit -> Word -> Word -> Word
172 \startuseMPgraphic{AddSubAlu}
173 save opcode, a, b, add, sub, mux, res;
176 newCircle.opcode(btex $opcode$ etex) "framed(false)";
177 newCircle.a(btex $a$ etex) "framed(false)";
178 newCircle.b(btex $b$ etex) "framed(false)";
179 newCircle.res(btex $res$ etex) "framed(false)";
181 newCircle.add(btex + etex);
182 newCircle.sub(btex - etex);
185 opcode.c - a.c = (0cm, 2cm);
186 add.c - a.c = (4cm, 0cm);
187 sub.c - b.c = (4cm, 0cm);
188 a.c - b.c = (0cm, 3cm);
189 mux.c = midpoint(add.c, sub.c) + (1.5cm, 0cm);
190 res.c - mux.c = (1.5cm, 0cm);
193 % Draw objects and lines
194 drawObj(opcode, a, b, res, add, sub, mux);
196 ncline(a)(add) "posA(e)";
197 ncline(b)(sub) "posA(e)";
198 nccurve(a)(sub) "posA(e)", "angleA(0)";
199 nccurve(b)(add) "posA(e)", "angleA(0)";
200 nccurve(add)(mux) "posB(inpa)", "angleB(0)";
201 nccurve(sub)(mux) "posB(inpb)", "angleB(0)";
202 nccurve(opcode)(mux) "posB(n)", "angleA(0)", "angleB(-90)";
203 ncline(mux)(res) "posA(out)";
206 \placeexample[here][ex:AddSubAlu]{Simple \small{ALU} supporting two operations.}
207 \startcombination[2*1]
208 {\typebufferlam{AddSubAlu}}{Core description in normal form.}
209 {\boxedgraphic{AddSubAlu}}{The architecture described by the normal form.}
212 As a more complete example, consider \in{example}[ex:NormalComplete]. This
213 example contains everything that is supported in normal form, with the
214 exception of builtin higher order functions. The graphical version of the
215 architecture contains a slightly simplified version, since the state tuple
216 packing and unpacking have been left out. Instead, two seperate registers are
217 drawn. Also note that most synthesis tools will further optimize this
218 architecture by removing the multiplexers at the register input and replace
219 them with some logic in the clock inputs, but we want to show the architecture
220 as close to the description as possible.
222 \startbuffer[NormalComplete]
225 -> State (Word, Word)
226 -> (State (Word, Word), Word)
228 -- All arguments are an inital lambda
230 -- There are nested let expressions at top level
232 -- Unpack the state by coercion (\eg, cast from
233 -- State (Word, Word) to (Word, Word))
234 s = sp :: (Word, Word)
235 -- Extract both registers from the state
236 r1 = case s of (fst, snd) -> fst
237 r2 = case s of (fst, snd) -> snd
238 -- Calling some other user-defined function.
240 -- Conditional connections
252 -- pack the state by coercion (\eg, cast from
253 -- (Word, Word) to State (Word, Word))
254 sp' = s' :: State (Word, Word)
255 -- Pack our return value
262 \startuseMPgraphic{NormalComplete}
263 save a, d, r, foo, muxr, muxout, out;
266 newCircle.a(btex \lam{a} etex) "framed(false)";
267 newCircle.d(btex \lam{d} etex) "framed(false)";
268 newCircle.out(btex \lam{out} etex) "framed(false)";
270 %newCircle.add(btex + etex);
271 newBox.foo(btex \lam{foo} etex);
272 newReg.r1(btex $\lam{r1}$ etex) "dx(4mm)", "dy(6mm)";
273 newReg.r2(btex $\lam{r2}$ etex) "dx(4mm)", "dy(6mm)", "reflect(true)";
275 % Reflect over the vertical axis
276 reflectObj(muxr1)((0,0), (0,1));
279 rotateObj(muxout)(-90);
281 d.c = foo.c + (0cm, 1.5cm);
282 a.c = (xpart r2.c + 2cm, ypart d.c - 0.5cm);
283 foo.c = midpoint(muxr1.c, muxr2.c) + (0cm, 2cm);
284 muxr1.c = r1.c + (0cm, 2cm);
285 muxr2.c = r2.c + (0cm, 2cm);
286 r2.c = r1.c + (4cm, 0cm);
288 muxout.c = midpoint(r1.c, r2.c) - (0cm, 2cm);
289 out.c = muxout.c - (0cm, 1.5cm);
291 % % Draw objects and lines
292 drawObj(a, d, foo, r1, r2, muxr1, muxr2, muxout, out);
295 nccurve(foo)(muxr1) "angleA(-90)", "posB(inpa)", "angleB(180)";
296 nccurve(foo)(muxr2) "angleA(-90)", "posB(inpb)", "angleB(0)";
297 nccurve(muxr1)(r1) "posA(out)", "angleA(180)", "posB(d)", "angleB(0)";
298 nccurve(r1)(muxr1) "posA(out)", "angleA(0)", "posB(inpb)", "angleB(180)";
299 nccurve(muxr2)(r2) "posA(out)", "angleA(0)", "posB(d)", "angleB(180)";
300 nccurve(r2)(muxr2) "posA(out)", "angleA(180)", "posB(inpa)", "angleB(0)";
301 nccurve(r1)(muxout) "posA(out)", "angleA(0)", "posB(inpb)", "angleB(-90)";
302 nccurve(r2)(muxout) "posA(out)", "angleA(180)", "posB(inpa)", "angleB(-90)";
304 nccurve(a)(muxout) "angleA(-90)", "angleB(180)", "posB(sel)";
305 nccurve(a)(muxr1) "angleA(180)", "angleB(-90)", "posB(sel)";
306 nccurve(a)(muxr2) "angleA(180)", "angleB(-90)", "posB(sel)";
307 ncline(muxout)(out) "posA(out)";
310 \placeexample[here][ex:NormalComplete]{Simple architecture consisting of an adder and a
312 \startcombination[2*1]
313 {\typebufferlam{NormalComplete}}{Core description in normal form.}
314 {\boxedgraphic{NormalComplete}}{The architecture described by the normal form.}
317 \subsection{Normal form definition}
318 Now we have some intuition for the normal form, we can describe how we want
319 the normal form to look like in a slightly more formal manner. The following
320 EBNF-like description completely captures the intended structure (and
321 generates a subset of GHC's core format).
323 Some clauses have an expression listed in parentheses. These are conditions
324 that need to apply to the clause.
327 \italic{normal} = \italic{lambda}
328 \italic{lambda} = λvar.\italic{lambda} (representable(var))
330 \italic{toplet} = let \italic{binding} in \italic{toplet}
331 | letrec [\italic{binding}] in \italic{toplet}
332 | var (representable(varvar))
333 \italic{binding} = var = \italic{rhs} (representable(rhs))
334 -- State packing and unpacking by coercion
335 | var0 = var1 :: State ty (lvar(var1))
336 | var0 = var1 :: ty (var0 :: State ty) (lvar(var1))
337 \italic{rhs} = userapp
340 | case var of C a0 ... an -> ai (lvar(var))
342 | case var of (lvar(var))
343 DEFAULT -> var0 (lvar(var0))
344 C w0 ... wn -> resvar (\forall{}i, wi \neq resvar, lvar(resvar))
345 \italic{userapp} = \italic{userfunc}
346 | \italic{userapp} {userarg}
347 \italic{userfunc} = var (gvar(var))
348 \italic{userarg} = var (lvar(var))
349 \italic{builtinapp} = \italic{builtinfunc}
350 | \italic{builtinapp} \italic{builtinarg}
351 \italic{builtinfunc} = var (bvar(var))
352 \italic{builtinarg} = \italic{coreexpr}
355 -- TODO: Limit builtinarg further
357 -- TODO: There can still be other casts around (which the code can handle,
358 e.g., ignore), which still need to be documented here.
360 -- TODO: Note about the selector case. It just supports Bit and Bool
361 currently, perhaps it should be generalized in the normal form?
363 When looking at such a program from a hardware perspective, the top level
364 lambda's define the input ports. The value produced by the let expression is
365 the output port. Most function applications bound by the let expression
366 define a component instantiation, where the input and output ports are mapped
367 to local signals or arguments. Some of the others use a builtin
368 construction (\eg the \lam{case} statement) or call a builtin function
369 (\eg \lam{add} or \lam{sub}). For these, a hardcoded \small{VHDL} translation is
372 \section{Transformation notation}
373 To be able to concisely present transformations, we use a specific format to
374 them. It is a simple format, similar to one used in logic reasoning.
376 Such a transformation description looks like the following.
381 <original expression>
382 -------------------------- <expression conditions>
383 <transformed expresssion>
388 This format desribes a transformation that applies to \lam{original
389 expresssion} and transforms it into \lam{transformed expression}, assuming
390 that all conditions apply. In this format, there are a number of placeholders
391 in pointy brackets, most of which should be rather obvious in their meaning.
392 Nevertheless, we will more precisely specify their meaning below:
394 \startdesc{<original expression>} The expression pattern that will be matched
395 against (subexpressions of) the expression to be transformed. We call this a
396 pattern, because it can contain \emph{placeholders} (variables), which match
397 any expression or binder. Any such placeholder is said to be \emph{bound} to
398 the expression it matches. It is convention to use an uppercase latter (\eg
399 \lam{M} or \lam{E} to refer to any expression (including a simple variable
400 reference) and lowercase letters (\eg \lam{v} or \lam{b}) to refer to
401 (references to) binders.
403 For example, the pattern \lam{a + B} will match the expression
404 \lam{v + (2 * w)} (and bind \lam{a} to \lam{v} and \lam{B} to
405 \lam{(2 * 2)}), but not \lam{v + (2 * w)}.
408 \startdesc{<expression conditions>}
409 These are extra conditions on the expression that is matched. These
410 conditions can be used to further limit the cases in which the
411 transformation applies, in particular to prevent a transformation from
412 causing a loop with itself or another transformation.
414 Only if these if these conditions are \emph{all} true, this transformation
418 \startdesc{<context conditions>}
419 These are a number of extra conditions on the context of the function. In
420 particular, these conditions can require some other top level function to be
421 present, whose value matches the pattern given here. The format of each of
422 these conditions is: \lam{binder = <pattern>}.
424 Typically, the binder is some placeholder bound in the \lam{<original
425 expression>}, while the pattern contains some placeholders that are used in
426 the \lam{transformed expression}.
428 Only if a top level binder exists that matches each binder and pattern, this
429 transformation applies.
432 \startdesc{<transformed expression>}
433 This is the expression template that is the result of the transformation. If, looking
434 at the above three items, the transformation applies, the \lam{original
435 expression} is completely replaced with the \lam{<transformed expression>}.
436 We call this a template, because it can contain placeholders, referring to
437 any placeholder bound by the \lam{<original expression>} or the
438 \lam{<context conditions>}. The resulting expression will have those
439 placeholders replaced by the values bound to them.
441 Any binder (lowercase) placeholder that has no value bound to it yet will be
442 bound to (and replaced with) a fresh binder.
445 \startdesc{<context additions>}
446 These are templates for new functions to add to the context. This is a way
447 to have a transformation create new top level functiosn.
449 Each addition has the form \lam{binder = template}. As above, any
450 placeholder in the addition is replaced with the value bound to it, and any
451 binder placeholder that has no value bound to it yet will be bound to (and
452 replaced with) a fresh binder.
455 As an example, we'll look at η-abstraction:
459 -------------- \lam{E} does not occur on a function position in an application
460 λx.E x \lam{E} is not a lambda abstraction.
463 Consider the following function, which is a fairly obvious way to specify a
464 simple ALU (Note \at{example}[ex:AddSubAlu] is the normal form of this
468 alu :: Bit -> Word -> Word -> Word
469 alu = λopcode. case opcode of
474 There are a few subexpressions in this function to which we could possibly
475 apply the transformation. Since the pattern of the transformation is only
476 the placeholder \lam{E}, any expression will match that. Whether the
477 transformation applies to an expression is thus solely decided by the
478 conditions to the right of the transformation.
480 We will look at each expression in the function in a top down manner. The
481 first expression is the entire expression the function is bound to.
484 λopcode. case opcode of
489 As said, the expression pattern matches this. The type of this expression is
490 \lam{Bit -> Word -> Word -> Word}, which matches \lam{a -> b} (Note that in
491 this case \lam{a = Bit} and \lam{b = Word -> Word -> Word}).
493 Since this expression is at top level, it does not occur at a function
494 position of an application. However, The expression is a lambda abstraction,
495 so this transformation does not apply.
497 The next expression we could apply this transformation to, is the body of
498 the lambda abstraction:
506 The type of this expression is \lam{Word -> Word -> Word}, which again
507 matches \lam{a -> b}. The expression is the body of a lambda expression, so
508 it does not occur at a function position of an application. Finally, the
509 expression is not a lambda abstraction but a case expression, so all the
510 conditions match. There are no context conditions to match, so the
511 transformation applies.
513 By now, the placeholder \lam{E} is bound to the entire expression. The
514 placeholder \lam{x}, which occurs in the replacement template, is not bound
515 yet, so we need to generate a fresh binder for that. Let's use the binder
516 \lam{a}. This results in the following replacement expression:
524 Continuing with this expression, we see that the transformation does not
525 apply again (it is a lambda expression). Next we look at the body of this
534 Here, the transformation does apply, binding \lam{E} to the entire
535 expression and \lam{x} to the fresh binder \lam{b}, resulting in the
544 Again, the transformation does not apply to this lambda abstraction, so we
545 look at its body. For brevity, we'll put the case statement on one line from
549 (case opcode of Low -> (+); High -> (-)) a b
552 The type of this expression is \lam{Word}, so it does not match \lam{a -> b}
553 and the transformation does not apply. Next, we have two options for the
554 next expression to look at: The function position and argument position of
555 the application. The expression in the argument position is \lam{b}, which
556 has type \lam{Word}, so the transformation does not apply. The expression in
557 the function position is:
560 (case opcode of Low -> (+); High -> (-)) a
563 Obviously, the transformation does not apply here, since it occurs in
564 function position. In the same way the transformation does not apply to both
565 components of this expression (\lam{case opcode of Low -> (+); High -> (-)}
566 and \lam{a}), so we'll skip to the components of the case expression: The
567 scrutinee and both alternatives. Since the opcode is not a function, it does
568 not apply here, and we'll leave both alternatives as an exercise to the
569 reader. The final function, after all these transformations becomes:
572 alu :: Bit -> Word -> Word -> Word
573 alu = λopcode.λa.b. (case opcode of
574 Low -> λa1.λb1 (+) a1 b1
575 High -> λa2.λb2 (-) a2 b2) a b
578 In this case, the transformation does not apply anymore, though this might
579 not always be the case (e.g., the application of a transformation on a
580 subexpression might open up possibilities to apply the transformation
581 further up in the expression).
583 \subsection{Transformation application}
584 In this chapter we define a number of transformations, but how will we apply
585 these? As stated before, our normal form is reached as soon as no
586 transformation applies anymore. This means our application strategy is to
587 simply apply any transformation that applies, and continuing to do that with
588 the result of each transformation.
590 In particular, we define no particular order of transformations. Since
591 transformation order should not influence the resulting normal form (see TODO
592 ref), this leaves the implementation free to choose any application order that
593 results in an efficient implementation.
595 When applying a single transformation, we try to apply it to every (sub)expression
596 in a function, not just the top level function. This allows us to keep the
597 transformation descriptions concise and powerful.
599 \subsection{Definitions}
600 In the following sections, we will be using a number of functions and
601 notations, which we will define here.
603 \subsubsection{Other concepts}
604 A \emph{global variable} is any variable that is bound at the
605 top level of a program, or an external module. A \emph{local variable} is any
606 other variable (\eg, variables local to a function, which can be bound by
607 lambda abstractions, let expressions and pattern matches of case
608 alternatives). Note that this is a slightly different notion of global versus
609 local than what \small{GHC} uses internally.
610 \defref{global variable} \defref{local variable}
612 A \emph{hardware representable} (or just \emph{representable}) type or value
613 is (a value of) a type that we can generate a signal for in hardware. For
614 example, a bit, a vector of bits, a 32 bit unsigned word, etc. Types that are
615 not runtime representable notably include (but are not limited to): Types,
616 dictionaries, functions.
617 \defref{representable}
619 A \emph{builtin function} is a function supplied by the Cλash framework, whose
620 implementation is not valid Cλash. The implementation is of course valid
621 Haskell, for simulation, but it is not expressable in Cλash.
622 \defref{builtin function} \defref{user-defined function}
624 For these functions, Cλash has a \emph{builtin hardware translation}, so calls
625 to these functions can still be translated. These are functions like
626 \lam{map}, \lam{hwor} and \lam{length}.
628 A \emph{user-defined} function is a function for which we do have a Cλash
629 implementation available.
631 \subsubsection{Functions}
632 Here, we define a number of functions that can be used below to concisely
635 \refdef{global variable}\emph{gvar(expr)} is true when \emph{expr} is a variable that references a
636 global variable. It is false when it references a local variable.
638 \refdef{local variable}\emph{lvar(expr)} is the complement of \emph{gvar}; it is true when \emph{expr}
639 references a local variable, false when it references a global variable.
641 \refdef{representable}\emph{representable(expr)} or \emph{representable(var)} is true when
642 \emph{expr} or \emph{var} is \emph{representable}.
644 \subsection{Binder uniqueness}
645 A common problem in transformation systems, is binder uniqueness. When not
646 considering this problem, it is easy to create transformations that mix up
647 bindings and cause name collisions. Take for example, the following core
651 (λa.λb.λc. a * b * c) x c
654 By applying β-reduction (see below) once, we can simplify this expression to:
660 Now, we have replaced the \lam{a} binder with a reference to the \lam{x}
661 binder. No harm done here. But note that we see multiple occurences of the
662 \lam{c} binder. The first is a binding occurence, to which the second refers.
663 The last, however refers to \emph{another} instance of \lam{c}, which is
664 bound somewhere outside of this expression. Now, if we would apply beta
665 reduction without taking heed of binder uniqueness, we would get:
671 This is obviously not what was supposed to happen! The root of this problem is
672 the reuse of binders: Identical binders can be bound in different scopes, such
673 that only the inner one is \quote{visible} in the inner expression. In the example
674 above, the \lam{c} binder was bound outside of the expression and in the inner
675 lambda expression. Inside that lambda expression, only the inner \lam{c} is
678 There are a number of ways to solve this. \small{GHC} has isolated this
679 problem to their binder substitution code, which performs \emph{deshadowing}
680 during its expression traversal. This means that any binding that shadows
681 another binding on a higher level is replaced by a new binder that does not
682 shadow any other binding. This non-shadowing invariant is enough to prevent
683 binder uniqueness problems in \small{GHC}.
685 In our transformation system, maintaining this non-shadowing invariant is
686 a bit harder to do (mostly due to implementation issues, the prototype doesn't
687 use \small{GHC}'s subsitution code). Also, we can observe the following
691 \item Deshadowing does not guarantee overall uniqueness. For example, the
692 following (slightly contrived) expression shows the identifier \lam{x} bound in
693 two seperate places (and to different values), even though no shadowing
697 (let x = 1 in x) + (let x = 2 in x)
700 \item In our normal form (and the resulting \small{VHDL}), all binders
701 (signals) will end up in the same scope. To allow this, all binders within the
702 same function should be unique.
704 \item When we know that all binders in an expression are unique, moving around
705 or removing a subexpression will never cause any binder conflicts. If we have
706 some way to generate fresh binders, introducing new subexpressions will not
707 cause any problems either. The only way to cause conflicts is thus to
708 duplicate an existing subexpression.
711 Given the above, our prototype maintains a unique binder invariant. This
712 meanst that in any given moment during normalization, all binders \emph{within
713 a single function} must be unique. To achieve this, we apply the following
716 TODO: Define fresh binders and unique supplies
719 \item Before starting normalization, all binders in the function are made
720 unique. This is done by generating a fresh binder for every binder used. This
721 also replaces binders that did not pose any conflict, but it does ensure that
722 all binders within the function are generated by the same unique supply. See
723 (TODO: ref fresh binder).
724 \item Whenever a new binder must be generated, we generate a fresh binder that
725 is guaranteed to be different from \emph{all binders generated so far}. This
726 can thus never introduce duplication and will maintain the invariant.
727 \item Whenever (part of) an expression is duplicated (for example when
728 inlining), all binders in the expression are replaced with fresh binders
729 (using the same method as at the start of normalization). These fresh binders
730 can never introduce duplication, so this will maintain the invariant.
731 \item Whenever we move part of an expression around within the function, there
732 is no need to do anything special. There is obviously no way to introduce
733 duplication by moving expressions around. Since we know that each of the
734 binders is already unique, there is no way to introduce (incorrect) shadowing
738 \section{Transform passes}
739 In this section we describe the actual transforms. Here we're using
740 the core language in a notation that resembles lambda calculus.
742 Each of these transforms is meant to be applied to every (sub)expression
743 in a program, for as long as it applies. Only when none of the
744 transformations can be applied anymore, the program is in normal form (by
745 definition). We hope to be able to prove that this form will obey all of the
746 constraints defined above, but this has yet to happen (though it seems likely
749 Each of the transforms will be described informally first, explaining
750 the need for and goal of the transform. Then, a formal definition is
751 given, using a familiar syntax from the world of logic. Each transform
752 is specified as a number of conditions (above the horizontal line) and a
753 number of conclusions (below the horizontal line). The details of using
754 this notation are still a bit fuzzy, so comments are welcom.
756 \subsection{η-abstraction}
757 This transformation makes sure that all arguments of a function-typed
758 expression are named, by introducing lambda expressions. When combined with
759 β-reduction and function inlining below, all function-typed expressions should
760 be lambda abstractions or global identifiers.
764 -------------- \lam{E} is not the first argument of an application.
765 λx.E x \lam{E} is not a lambda abstraction.
766 \lam{x} is a variable that does not occur free in \lam{E}.
776 foo = λa.λx.(case a of
781 \transexample{η-abstraction}{from}{to}
783 \subsection{β-reduction}
784 β-reduction is a well known transformation from lambda calculus, where it is
785 the main reduction step. It reduces applications of labmda abstractions,
786 removing both the lambda abstraction and the application.
788 In our transformation system, this step helps to remove unwanted lambda
789 abstractions (basically all but the ones at the top level). Other
790 transformations (application propagation, non-representable inlining) make
791 sure that most lambda abstractions will eventually be reducable by
794 TODO: Define substitution syntax
811 \transexample{β-reduction}{from}{to}
813 \subsection{Application propagation}
814 This transformation is meant to propagate application expressions downwards
815 into expressions as far as possible. This allows partial applications inside
816 expressions to become fully applied and exposes new transformation
817 possibilities for other transformations (like β-reduction).
841 \transexample{Application propagation for a let expression}{from}{to}
869 \transexample{Application propagation for a case expression}{from}{to}
871 \subsection{Let derecursification}
872 This transformation is meant to make lets non-recursive whenever possible.
873 This might allow other optimizations to do their work better. TODO: Why is
876 \subsection{Let flattening}
877 This transformation puts nested lets in the same scope, by lifting the
878 binding(s) of the inner let into a new let around the outer let. Eventually,
879 this will cause all let bindings to appear in the same scope (they will all be
880 in scope for the function return value).
882 Note that this transformation does not try to be smart when faced with
883 recursive lets, it will just leave the lets recursive (possibly joining a
884 recursive and non-recursive let into a single recursive let). The let
885 rederecursification transformation will do this instead.
888 letnonrec x = (let bindings in M) in N
889 ------------------------------------------
890 let bindings in (letnonrec x = M) in N
896 x = (let bindings in M)
900 ------------------------------------------
919 b = let c = 3 in a + c
940 \transexample{Let flattening}{from}{to}
942 \subsection{Empty let removal}
943 This transformation is simple: It removes recursive lets that have no bindings
944 (which usually occurs when let derecursification removes the last binding from
953 \subsection{Simple let binding removal}
954 This transformation inlines simple let bindings (\eg a = b).
956 This transformation is not needed to get into normal form, but makes the
957 resulting \small{VHDL} a lot shorter.
983 \subsection{Unused let binding removal}
984 This transformation removes let bindings that are never used. Usually,
985 the desugarer introduces some unused let bindings.
987 This normalization pass should really be unneeded to get into normal form
988 (since ununsed bindings are not forbidden by the normal form), but in practice
989 the desugarer or simplifier emits some unused bindings that cannot be
990 normalized (e.g., calls to a \type{PatError} (TODO: Check this name)). Also,
991 this transformation makes the resulting \small{VHDL} a lot shorter.
995 ---------------------------- \lam{a} does not occur free in \lam{M}
1006 ---------------------------- \lam{a} does not occur free in \lam{M}
1014 \subsection{Non-representable binding inlining}
1015 This transform inlines let bindings that have a non-representable type. Since
1016 we can never generate a signal assignment for these bindings (we cannot
1017 declare a signal assignment with a non-representable type, for obvious
1018 reasons), we have no choice but to inline the binding to remove it.
1020 If the binding is non-representable because it is a lambda abstraction, it is
1021 likely that it will inlined into an application and β-reduction will remove
1022 the lambda abstraction and turn it into a representable expression at the
1023 inline site. The same holds for partial applications, which can be turned into
1024 full applications by inlining.
1026 Other cases of non-representable bindings we see in practice are primitive
1027 Haskell types. In most cases, these will not result in a valid normalized
1028 output, but then the input would have been invalid to start with. There is one
1029 exception to this: When a builtin function is applied to a non-representable
1030 expression, things might work out in some cases. For example, when you write a
1031 literal \hs{SizedInt} in Haskell, like \hs{1 :: SizedInt D8}, this results in
1032 the following core: \lam{fromInteger (smallInteger 10)}, where for example
1033 \lam{10 :: GHC.Prim.Int\#} and \lam{smallInteger 10 :: Integer} have
1034 non-representable types. TODO: This/these paragraph(s) should probably become a
1035 separate discussion somewhere else.
1038 letnonrec a = E in M
1039 -------------------------- \lam{E} has a non-representable type.
1050 -------------------------- \lam{E} has a non-representable type.
1070 x = fromInteger (smallInteger 10)
1072 (λa -> add a 1) (add 1 x)
1075 \transexample{Let flattening}{from}{to}
1077 \subsection{Compiler generated top level binding inlining}
1080 \subsection{Scrutinee simplification}
1081 This transform ensures that the scrutinee of a case expression is always
1082 a simple variable reference.
1087 ----------------- \lam{E} is not a local variable reference
1106 \transexample{Let flattening}{from}{to}
1109 \subsection{Case simplification}
1110 This transformation ensures that all case expressions become normal form. This
1111 means they will become one of:
1113 \item An extractor case with a single alternative that picks a single field
1114 from a datatype, \eg \lam{case x of (a, b) -> a}.
1115 \item A selector case with multiple alternatives and only wild binders, that
1116 makes a choice between expressions based on the constructor of another
1117 expression, \eg \lam{case x of Low -> a; High -> b}.
1122 C0 v0,0 ... v0,m -> E0
1124 Cn vn,0 ... vn,m -> En
1125 --------------------------------------------------- \forall i \forall j, 0 <= i <= n, 0 <= i < m (\lam{wi,j} is a wild (unused) binder)
1127 v0,0 = case x of C0 v0,0 .. v0,m -> v0,0
1129 v0,m = case x of C0 v0,0 .. v0,m -> v0,m
1132 vn,m = case x of Cn vn,0 .. vn,m -> vn,m
1136 C0 w0,0 ... w0,m -> x0
1138 Cn wn,0 ... wn,m -> xn
1141 TODO: This transformation specified like this is complicated and misses
1142 conditions to prevent looping with itself. Perhaps we should split it here for
1161 \transexample{Selector case simplification}{from}{to}
1169 b = case a of (,) b c -> b
1170 c = case a of (,) b c -> c
1177 \transexample{Extractor case simplification}{from}{to}
1179 \subsection{Case removal}
1180 This transform removes any case statements with a single alternative and
1183 These "useless" case statements are usually leftovers from case simplification
1184 on extractor case (see the previous example).
1189 ---------------------- \lam{\forall i, 0 <= i <= m} (\lam{vi} does not occur free in E)
1202 \transexample{Case removal}{from}{to}
1204 \subsection{Argument simplification}
1205 The transforms in this section deal with simplifying application
1206 arguments into normal form. The goal here is to:
1209 \item Make all arguments of user-defined functions (\eg, of which
1210 we have a function body) simple variable references of a runtime
1211 representable type. This is needed, since these applications will be turned
1212 into component instantiations.
1213 \item Make all arguments of builtin functions one of:
1215 \item A type argument.
1216 \item A dictionary argument.
1217 \item A type level expression.
1218 \item A variable reference of a runtime representable type.
1219 \item A variable reference or partial application of a function type.
1223 When looking at the arguments of a user-defined function, we can
1224 divide them into two categories:
1226 \item Arguments of a runtime representable type (\eg bits or vectors).
1228 These arguments can be preserved in the program, since they can
1229 be translated to input ports later on. However, since we can
1230 only connect signals to input ports, these arguments must be
1231 reduced to simple variables (for which signals will be
1232 produced). This is taken care of by the argument extraction
1234 \item Non-runtime representable typed arguments.
1236 These arguments cannot be preserved in the program, since we
1237 cannot represent them as input or output ports in the resulting
1238 \small{VHDL}. To remove them, we create a specialized version of the
1239 called function with these arguments filled in. This is done by
1240 the argument propagation transform.
1242 Typically, these arguments are type and dictionary arguments that are
1243 used to make functions polymorphic. By propagating these arguments, we
1244 are essentially doing the same which GHC does when it specializes
1245 functions: Creating multiple variants of the same function, one for
1246 each type for which it is used. Other common non-representable
1247 arguments are functions, e.g. when calling a higher order function
1248 with another function or a lambda abstraction as an argument.
1250 The reason for doing this is similar to the reasoning provided for
1251 the inlining of non-representable let bindings above. In fact, this
1252 argument propagation could be viewed as a form of cross-function
1256 TODO: Check the following itemization.
1258 When looking at the arguments of a builtin function, we can divide them
1262 \item Arguments of a runtime representable type.
1264 As we have seen with user-defined functions, these arguments can
1265 always be reduced to a simple variable reference, by the
1266 argument extraction transform. Performing this transform for
1267 builtin functions as well, means that the translation of builtin
1268 functions can be limited to signal references, instead of
1269 needing to support all possible expressions.
1271 \item Arguments of a function type.
1273 These arguments are functions passed to higher order builtins,
1274 like \lam{map} and \lam{foldl}. Since implementing these
1275 functions for arbitrary function-typed expressions (\eg, lambda
1276 expressions) is rather comlex, we reduce these arguments to
1277 (partial applications of) global functions.
1279 We can still support arbitrary expressions from the user code,
1280 by creating a new global function containing that expression.
1281 This way, we can simply replace the argument with a reference to
1282 that new function. However, since the expression can contain any
1283 number of free variables we also have to include partial
1284 applications in our normal form.
1286 This category of arguments is handled by the function extraction
1288 \item Other unrepresentable arguments.
1290 These arguments can take a few different forms:
1291 \startdesc{Type arguments}
1292 In the core language, type arguments can only take a single
1293 form: A type wrapped in the Type constructor. Also, there is
1294 nothing that can be done with type expressions, except for
1295 applying functions to them, so we can simply leave type
1296 arguments as they are.
1298 \startdesc{Dictionary arguments}
1299 In the core language, dictionary arguments are used to find
1300 operations operating on one of the type arguments (mostly for
1301 finding class methods). Since we will not actually evaluatie
1302 the function body for builtin functions and can generate
1303 code for builtin functions by just looking at the type
1304 arguments, these arguments can be ignored and left as they
1307 \startdesc{Type level arguments}
1308 Sometimes, we want to pass a value to a builtin function, but
1309 we need to know the value at compile time. Additionally, the
1310 value has an impact on the type of the function. This is
1311 encoded using type-level values, where the actual value of the
1312 argument is not important, but the type encodes some integer,
1313 for example. Since the value is not important, the actual form
1314 of the expression does not matter either and we can leave
1315 these arguments as they are.
1317 \startdesc{Other arguments}
1318 Technically, there is still a wide array of arguments that can
1319 be passed, but does not fall into any of the above categories.
1320 However, none of the supported builtin functions requires such
1321 an argument. This leaves use with passing unsupported types to
1322 a function, such as calling \lam{head} on a list of functions.
1324 In these cases, it would be impossible to generate hardware
1325 for such a function call anyway, so we can ignore these
1328 The only way to generate hardware for builtin functions with
1329 arguments like these, is to expand the function call into an
1330 equivalent core expression (\eg, expand map into a series of
1331 function applications). But for now, we choose to simply not
1332 support expressions like these.
1335 From the above, we can conclude that we can simply ignore these
1336 other unrepresentable arguments and focus on the first two
1340 \subsubsection{Argument simplification}
1341 This transform deals with arguments to functions that
1342 are of a runtime representable type. It ensures that they will all become
1343 references to global variables, or local signals in the resulting \small{VHDL}.
1345 TODO: It seems we can map an expression to a port, not only a signal.
1346 Perhaps this makes this transformation not needed?
1347 TODO: Say something about dataconstructors (without arguments, like True
1348 or False), which are variable references of a runtime representable
1349 type, but do not result in a signal.
1351 To reduce a complex expression to a simple variable reference, we create
1352 a new let expression around the application, which binds the complex
1353 expression to a new variable. The original function is then applied to
1358 -------------------- \lam{N} is of a representable type
1359 let x = N in M x \lam{N} is not a local variable reference
1367 let x = add a 1 in add x 1
1370 \transexample{Argument extraction}{from}{to}
1372 \subsubsection{Function extraction}
1373 This transform deals with function-typed arguments to builtin functions.
1374 Since these arguments cannot be propagated, we choose to extract them
1375 into a new global function instead.
1377 Any free variables occuring in the extracted arguments will become
1378 parameters to the new global function. The original argument is replaced
1379 with a reference to the new function, applied to any free variables from
1380 the original argument.
1382 This transformation is useful when applying higher order builtin functions
1383 like \hs{map} to a lambda abstraction, for example. In this case, the code
1384 that generates \small{VHDL} for \hs{map} only needs to handle top level functions and
1385 partial applications, not any other expression (such as lambda abstractions or
1386 even more complicated expressions).
1389 M N \lam{M} is a (partial aplication of a) builtin function.
1390 --------------------- \lam{f0 ... fn} = free local variables of \lam{N}
1391 M x f0 ... fn \lam{N :: a -> b}
1392 ~ \lam{N} is not a (partial application of) a top level function
1397 map (λa . add a b) xs
1411 \transexample{Function extraction}{from}{to}
1413 \subsubsection{Argument propagation}
1414 This transform deals with arguments to user-defined functions that are
1415 not representable at runtime. This means these arguments cannot be
1416 preserved in the final form and most be {\em propagated}.
1418 Propagation means to create a specialized version of the called
1419 function, with the propagated argument already filled in. As a simple
1420 example, in the following program:
1427 we could {\em propagate} the constant argument 1, with the following
1435 Special care must be taken when the to-be-propagated expression has any
1436 free variables. If this is the case, the original argument should not be
1437 removed alltogether, but replaced by all the free variables of the
1438 expression. In this way, the original expression can still be evaluated
1439 inside the new function. Also, this brings us closer to our goal: All
1440 these free variables will be simple variable references.
1442 To prevent us from propagating the same argument over and over, a simple
1443 local variable reference is not propagated (since is has exactly one
1444 free variable, itself, we would only replace that argument with itself).
1446 This shows that any free local variables that are not runtime representable
1447 cannot be brought into normal form by this transform. We rely on an
1448 inlining transformation to replace such a variable with an expression we
1449 can propagate again.
1454 x Y0 ... Yi ... Yn \lam{Yi} is not of a runtime representable type
1455 --------------------------------------------- \lam{Yi} is not a local variable reference
1456 x' y0 ... yi-1 f0 ... fm Yi+1 ... Yn \lam{f0 ... fm} = free local vars of \lam{Yi}
1458 x' = λy0 ... yi-1 f0 ... fm yi+1 ... yn .
1459 E y0 ... yi-1 Yi yi+1 ... yn
1465 \subsection{Cast propagation / simplification}
1466 This transform pushes casts down into the expression as far as possible. Since
1467 its exact role and need is not clear yet, this transformation is not yet
1470 \subsection{Return value simplification}
1471 This transformation ensures that the return value of a function is always a
1472 simple local variable reference.
1474 Currently implemented using lambda simplification, let simplification, and
1475 top simplification. Should change into something like the following, which
1476 works only on the result of a function instead of any subexpression. This is
1477 achieved by the contexts, like \lam{x = E}, though this is strictly not
1478 correct (you could read this as "if there is any function \lam{x} that binds
1479 \lam{E}, any \lam{E} can be transformed, while we only mean the \lam{E} that
1480 is bound by \lam{x}. This might need some extra notes or something).
1483 x = E \lam{E} is representable
1484 ~ \lam{E} is not a lambda abstraction
1485 E \lam{E} is not a let expression
1486 --------------------------- \lam{E} is not a local variable reference
1492 ~ \lam{E} is representable
1493 E \lam{E} is not a let expression
1494 --------------------------- \lam{E} is not a local variable reference
1499 x = λv0 ... λvn.let ... in E
1500 ~ \lam{E} is representable
1501 E \lam{E} is not a local variable reference
1502 ---------------------------
1511 x = let x = add 1 2 in x
1514 \transexample{Return value simplification}{from}{to}