1 \chapter[chap:normalization]{Normalization}
2 % A helper to print a single example in the half the page width. The example
3 % text should be in a buffer whose name is given in an argument.
5 % The align=right option really does left-alignment, but without the program
6 % will end up on a single line. The strut=no option prevents a bunch of empty
7 % space at the start of the frame.
9 \framed[offset=1mm,align=right,strut=no,background=box,frame=off]{
10 \setuptyping[option=LAM,style=sans,before=,after=,strip=auto]
12 \setuptyping[option=none,style=\tttf,strip=auto]
16 \define[4]\transexample{
17 \placeexample[here][ex:trans:#1]{#2}
18 \startcombination[2*1]
19 {\example{#3}}{Original program}
20 {\example{#4}}{Transformed program}
24 The first step in the core to \small{VHDL} translation process, is normalization. We
25 aim to bring the core description into a simpler form, which we can
26 subsequently translate into \small{VHDL} easily. This normal form is needed because
27 the full core language is more expressive than \small{VHDL} in some areas and because
28 core can describe expressions that do not have a direct hardware
31 \todo{Describe core properties not supported in \VHDL, and describe how the
32 \VHDL we want to generate should look like.}
35 \todo{Refresh or refer to distinct hardware per application principle}
36 The transformations described here have a well-defined goal: To bring the
37 program in a well-defined form that is directly translatable to hardware,
38 while fully preserving the semantics of the program. We refer to this form as
39 the \emph{normal form} of the program. The formal definition of this normal
42 \placedefinition{}{A program is in \emph{normal form} if none of the
43 transformations from this chapter apply.}
45 Of course, this is an \quote{easy} definition of the normal form, since our
46 program will end up in normal form automatically. The more interesting part is
47 to see if this normal form actually has the properties we would like it to
50 But, before getting into more definitions and details about this normal form,
51 let's try to get a feeling for it first. The easiest way to do this is by
52 describing the things we want to not have in a normal form.
55 \item Any \emph{polymorphism} must be removed. When laying down hardware, we
56 can't generate any signals that can have multiple types. All types must be
57 completely known to generate hardware.
59 \item Any \emph{higher order} constructions must be removed. We can't
60 generate a hardware signal that contains a function, so all values,
61 arguments and returns values used must be first order.
63 \item Any complex \emph{nested scopes} must be removed. In the \small{VHDL}
64 description, every signal is in a single scope. Also, full expressions are
65 not supported everywhere (in particular port maps can only map signal
66 names and constants, not complete expressions). To make the \small{VHDL}
67 generation easy, a separate binder must be bound to ever application or
71 \todo{Intermezzo: functions vs plain values}
73 A very simple example of a program in normal form is given in
74 \in{example}[ex:MulSum]. As you can see, all arguments to the function (which
75 will become input ports in the final hardware) are at the outer level.
76 This means that the body of the inner lambda abstraction is never a
77 function, but always a plain value.
79 As the body of the inner lambda abstraction, we see a single (recursive)
80 let expression, that binds two variables (\lam{mul} and \lam{sum}). These
81 variables will be signals in the final hardware, bound to the output port
82 of the \lam{*} and \lam{+} components.
84 The final line (the \quote{return value} of the function) selects the
85 \lam{sum} signal to be the output port of the function. This \quote{return
86 value} can always only be a variable reference, never a more complex
89 \todo{Add generated VHDL}
92 alu :: Bit -> Word -> Word -> Word
101 \startuseMPgraphic{MulSum}
102 save a, b, c, mul, add, sum;
105 newCircle.a(btex $a$ etex) "framed(false)";
106 newCircle.b(btex $b$ etex) "framed(false)";
107 newCircle.c(btex $c$ etex) "framed(false)";
108 newCircle.sum(btex $res$ etex) "framed(false)";
111 newCircle.mul(btex * etex);
112 newCircle.add(btex + etex);
114 a.c - b.c = (0cm, 2cm);
115 b.c - c.c = (0cm, 2cm);
116 add.c = c.c + (2cm, 0cm);
117 mul.c = midpoint(a.c, b.c) + (2cm, 0cm);
118 sum.c = add.c + (2cm, 0cm);
121 % Draw objects and lines
122 drawObj(a, b, c, mul, add, sum);
124 ncarc(a)(mul) "arcangle(15)";
125 ncarc(b)(mul) "arcangle(-15)";
131 \placeexample[here][ex:MulSum]{Simple architecture consisting of a
132 multiplier and a subtractor.}
133 \startcombination[2*1]
134 {\typebufferlam{MulSum}}{Core description in normal form.}
135 {\boxedgraphic{MulSum}}{The architecture described by the normal form.}
138 The previous example described composing an architecture by calling other
139 functions (operators), resulting in a simple architecture with components and
140 connections. There is of course also some mechanism for choice in the normal
141 form. In a normal Core program, the \emph{case} expression can be used in a
142 few different ways to describe choice. In normal form, this is limited to a
145 \in{Example}[ex:AddSubAlu] shows an example describing a
146 simple \small{ALU}, which chooses between two operations based on an opcode
147 bit. The main structure is similar to \in{example}[ex:MulSum], but this
148 time the \lam{res} variable is bound to a case expression. This case
149 expression scrutinizes the variable \lam{opcode} (and scrutinizing more
150 complex expressions is not supported). The case expression can select a
151 different variable based on the constructor of \lam{opcode}.
153 \startbuffer[AddSubAlu]
154 alu :: Bit -> Word -> Word -> Word
166 \startuseMPgraphic{AddSubAlu}
167 save opcode, a, b, add, sub, mux, res;
170 newCircle.opcode(btex $opcode$ etex) "framed(false)";
171 newCircle.a(btex $a$ etex) "framed(false)";
172 newCircle.b(btex $b$ etex) "framed(false)";
173 newCircle.res(btex $res$ etex) "framed(false)";
175 newCircle.add(btex + etex);
176 newCircle.sub(btex - etex);
179 opcode.c - a.c = (0cm, 2cm);
180 add.c - a.c = (4cm, 0cm);
181 sub.c - b.c = (4cm, 0cm);
182 a.c - b.c = (0cm, 3cm);
183 mux.c = midpoint(add.c, sub.c) + (1.5cm, 0cm);
184 res.c - mux.c = (1.5cm, 0cm);
187 % Draw objects and lines
188 drawObj(opcode, a, b, res, add, sub, mux);
190 ncline(a)(add) "posA(e)";
191 ncline(b)(sub) "posA(e)";
192 nccurve(a)(sub) "posA(e)", "angleA(0)";
193 nccurve(b)(add) "posA(e)", "angleA(0)";
194 nccurve(add)(mux) "posB(inpa)", "angleB(0)";
195 nccurve(sub)(mux) "posB(inpb)", "angleB(0)";
196 nccurve(opcode)(mux) "posB(n)", "angleA(0)", "angleB(-90)";
197 ncline(mux)(res) "posA(out)";
200 \placeexample[here][ex:AddSubAlu]{Simple \small{ALU} supporting two operations.}
201 \startcombination[2*1]
202 {\typebufferlam{AddSubAlu}}{Core description in normal form.}
203 {\boxedgraphic{AddSubAlu}}{The architecture described by the normal form.}
206 As a more complete example, consider \in{example}[ex:NormalComplete]. This
207 example contains everything that is supported in normal form, with the
208 exception of builtin higher order functions. The graphical version of the
209 architecture contains a slightly simplified version, since the state tuple
210 packing and unpacking have been left out. Instead, two seperate registers are
211 drawn. Also note that most synthesis tools will further optimize this
212 architecture by removing the multiplexers at the register input and
213 instead put some gates in front of the register's clock input, but we want
214 to show the architecture as close to the description as possible.
216 As you can see from the previous examples, the generation of the final
217 architecture from the normal form is straightforward. In each of the
218 examples, there is a direct match between the normal form structure,
219 the generated VHDL and the architecture shown in the images.
221 \startbuffer[NormalComplete]
224 -> State (Word, Word)
225 -> (State (Word, Word), Word)
227 -- All arguments are an inital lambda (address, data, packed state)
229 -- There are nested let expressions at top level
231 -- Unpack the state by coercion (\eg, cast from
232 -- State (Word, Word) to (Word, Word))
233 s = sp ▶ (Word, Word)
234 -- Extract both registers from the state
235 r1 = case s of (a, b) -> a
236 r2 = case s of (a, b) -> b
237 -- Calling some other user-defined function.
239 -- Conditional connections
251 -- pack the state by coercion (\eg, cast from
252 -- (Word, Word) to State (Word, Word))
253 sp' = s' ▶ State (Word, Word)
254 -- Pack our return value
261 \startuseMPgraphic{NormalComplete}
262 save a, d, r, foo, muxr, muxout, out;
265 newCircle.a(btex \lam{a} etex) "framed(false)";
266 newCircle.d(btex \lam{d} etex) "framed(false)";
267 newCircle.out(btex \lam{out} etex) "framed(false)";
269 %newCircle.add(btex + etex);
270 newBox.foo(btex \lam{foo} etex);
271 newReg.r1(btex $\lam{r1}$ etex) "dx(4mm)", "dy(6mm)";
272 newReg.r2(btex $\lam{r2}$ etex) "dx(4mm)", "dy(6mm)", "reflect(true)";
274 % Reflect over the vertical axis
275 reflectObj(muxr1)((0,0), (0,1));
278 rotateObj(muxout)(-90);
280 d.c = foo.c + (0cm, 1.5cm);
281 a.c = (xpart r2.c + 2cm, ypart d.c - 0.5cm);
282 foo.c = midpoint(muxr1.c, muxr2.c) + (0cm, 2cm);
283 muxr1.c = r1.c + (0cm, 2cm);
284 muxr2.c = r2.c + (0cm, 2cm);
285 r2.c = r1.c + (4cm, 0cm);
287 muxout.c = midpoint(r1.c, r2.c) - (0cm, 2cm);
288 out.c = muxout.c - (0cm, 1.5cm);
290 % % Draw objects and lines
291 drawObj(a, d, foo, r1, r2, muxr1, muxr2, muxout, out);
294 nccurve(foo)(muxr1) "angleA(-90)", "posB(inpa)", "angleB(180)";
295 nccurve(foo)(muxr2) "angleA(-90)", "posB(inpb)", "angleB(0)";
296 nccurve(muxr1)(r1) "posA(out)", "angleA(180)", "posB(d)", "angleB(0)";
297 nccurve(r1)(muxr1) "posA(out)", "angleA(0)", "posB(inpb)", "angleB(180)";
298 nccurve(muxr2)(r2) "posA(out)", "angleA(0)", "posB(d)", "angleB(180)";
299 nccurve(r2)(muxr2) "posA(out)", "angleA(180)", "posB(inpa)", "angleB(0)";
300 nccurve(r1)(muxout) "posA(out)", "angleA(0)", "posB(inpb)", "angleB(-90)";
301 nccurve(r2)(muxout) "posA(out)", "angleA(180)", "posB(inpa)", "angleB(-90)";
303 nccurve(a)(muxout) "angleA(-90)", "angleB(180)", "posB(sel)";
304 nccurve(a)(muxr1) "angleA(180)", "angleB(-90)", "posB(sel)";
305 nccurve(a)(muxr2) "angleA(180)", "angleB(-90)", "posB(sel)";
306 ncline(muxout)(out) "posA(out)";
309 \todo{Don't split registers in this image?}
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.}
319 \subsection{Intended normal form definition}
320 Now we have some intuition for the normal form, we can describe how we want
321 the normal form to look like in a slightly more formal manner. The following
322 EBNF-like description completely captures the intended structure (and
323 generates a subset of GHC's core format).
325 Some clauses have an expression listed in parentheses. These are conditions
326 that need to apply to the clause.
328 \defref{intended normal form definition}
329 \todo{Fix indentation}
331 \italic{normal} = \italic{lambda}
332 \italic{lambda} = λvar.\italic{lambda} (representable(var))
334 \italic{toplet} = letrec [\italic{binding}...] in var (representable(varvar))
335 \italic{binding} = var = \italic{rhs} (representable(rhs))
336 -- State packing and unpacking by coercion
337 | var0 = var1 ▶ State ty (lvar(var1))
338 | var0 = var1 ▶ ty (var0 :: State ty) (lvar(var1))
339 \italic{rhs} = userapp
342 | case var of C a0 ... an -> ai (lvar(var))
344 | case var of (lvar(var))
345 DEFAULT -> var0 (lvar(var0))
346 C w0 ... wn -> resvar (\forall{}i, wi \neq resvar, lvar(resvar))
347 \italic{userapp} = \italic{userfunc}
348 | \italic{userapp} {userarg}
349 \italic{userfunc} = var (gvar(var))
350 \italic{userarg} = var (lvar(var))
351 \italic{builtinapp} = \italic{builtinfunc}
352 | \italic{builtinapp} \italic{builtinarg}
353 \italic{builtinfunc} = var (bvar(var))
354 \italic{builtinarg} = \italic{coreexpr}
357 \todo{Limit builtinarg further}
359 \todo{There can still be other casts around (which the code can handle,
360 e.g., ignore), which still need to be documented here}
362 \todo{Note about the selector case. It just supports Bit and Bool
363 currently, perhaps it should be generalized in the normal form? This is
366 When looking at such a program from a hardware perspective, the top level
367 lambda's define the input ports. The variable referenc in the body of
368 the recursive let expression is the output port. Most function
369 applications bound by the let expression define a component
370 instantiation, where the input and output ports are mapped to local
371 signals or arguments. Some of the others use a builtin construction (\eg
372 the \lam{case} expression) or call a builtin function (\eg \lam{+} or
373 \lam{map}). For these, a hardcoded \small{VHDL} translation is
376 \section[sec:normalization:transformation]{Transformation notation}
377 To be able to concisely present transformations, we use a specific format
378 for 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 letter (\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)} (binding \lam{a} to \lam{v} and \lam{B} to
409 \lam{(2 * w)}), but not \lam{(2 * w) + v}.
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, commonly to prevent a transformation from
416 causing a loop with itself or another transformation.
418 Only if these conditions are \emph{all} true, the 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,
433 the 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 functions.
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 η-abstraction is a well known transformation from lambda calculus. What
468 this transformation does, is take any expression that has a function type
469 and turn it into a lambda expression (giving an explicit name to the
470 argument). There are some extra conditions that ensure that this
471 transformation does not apply infinitely (which are not necessarily part
472 of the conventional definition of η-abstraction).
474 Consider the following function, which is a fairly obvious way to specify a
475 simple ALU (Note that \in{example}[ex:AddSubAlu] shows the normal form of this
476 function). The parentheses around the \lam{+} and \lam{-} operators are
477 commonly used in Haskell to show that the operators are used as normal
478 functions, instead of \emph{infix} operators (\eg, the operators appear
479 before their arguments, instead of in between).
482 alu :: Bit -> Word -> Word -> Word
483 alu = λopcode. case opcode of
488 There are a few subexpressions in this function to which we could possibly
489 apply the transformation. Since the pattern of the transformation is only
490 the placeholder \lam{E}, any expression will match that. Whether the
491 transformation applies to an expression is thus solely decided by the
492 conditions to the right of the transformation.
494 We will look at each expression in the function in a top down manner. The
495 first expression is the entire expression the function is bound to.
498 λopcode. case opcode of
503 As said, the expression pattern matches this. The type of this expression is
504 \lam{Bit -> Word -> Word -> Word}, which matches \lam{a -> b} (Note that in
505 this case \lam{a = Bit} and \lam{b = Word -> Word -> Word}).
507 Since this expression is at top level, it does not occur at a function
508 position of an application. However, The expression is a lambda abstraction,
509 so this transformation does not apply.
511 The next expression we could apply this transformation to, is the body of
512 the lambda abstraction:
520 The type of this expression is \lam{Word -> Word -> Word}, which again
521 matches \lam{a -> b}. The expression is the body of a lambda expression, so
522 it does not occur at a function position of an application. Finally, the
523 expression is not a lambda abstraction but a case expression, so all the
524 conditions match. There are no context conditions to match, so the
525 transformation applies.
527 By now, the placeholder \lam{E} is bound to the entire expression. The
528 placeholder \lam{x}, which occurs in the replacement template, is not bound
529 yet, so we need to generate a fresh binder for that. Let's use the binder
530 \lam{a}. This results in the following replacement expression:
538 Continuing with this expression, we see that the transformation does not
539 apply again (it is a lambda expression). Next we look at the body of this
548 Here, the transformation does apply, binding \lam{E} to the entire
549 expression and \lam{x} to the fresh binder \lam{b}, resulting in the
558 Again, the transformation does not apply to this lambda abstraction, so we
559 look at its body. For brevity, we'll put the case statement on one line from
563 (case opcode of Low -> (+); High -> (-)) a b
566 The type of this expression is \lam{Word}, so it does not match \lam{a -> b}
567 and the transformation does not apply. Next, we have two options for the
568 next expression to look at: The function position and argument position of
569 the application. The expression in the argument position is \lam{b}, which
570 has type \lam{Word}, so the transformation does not apply. The expression in
571 the function position is:
574 (case opcode of Low -> (+); High -> (-)) a
577 Obviously, the transformation does not apply here, since it occurs in
578 function position (which makes the second condition false). In the same
579 way the transformation does not apply to both components of this
580 expression (\lam{case opcode of Low -> (+); High -> (-)} and \lam{a}), so
581 we'll skip to the components of the case expression: The scrutinee and
582 both alternatives. Since the opcode is not a function, it does not apply
585 The first alternative is \lam{(+)}. This expression has a function type
586 (the operator still needs two arguments). It does not occur in function
587 position of an application and it is not a lambda expression, so the
588 transformation applies.
590 We look at the \lam{<original expression>} pattern, which is \lam{E}.
591 This means we bind \lam{E} to \lam{(+)}. We then replace the expression
592 with the \lam{<transformed expression>}, replacing all occurences of
593 \lam{E} with \lam{(+)}. In the \lam{<transformed expression>}, the This gives us the replacement expression:
594 \lam{λx.(+) x} (A lambda expression binding \lam{x}, with a body that
595 applies the addition operator to \lam{x}).
597 The complete function then becomes:
599 (case opcode of Low -> λa1.(+) a1; High -> (-)) a
602 Now the transformation no longer applies to the complete first alternative
603 (since it is a lambda expression). It does not apply to the addition
604 operator again, since it is now in function position in an application. It
605 does, however, apply to the application of the addition operator, since
606 that is neither a lambda expression nor does it occur in function
607 position. This means after one more application of the transformation, the
611 (case opcode of Low -> λa1.λb1.(+) a1 b1; High -> (-)) a
614 The other alternative is left as an exercise to the reader. The final
615 function, after applying η-abstraction until it does no longer apply is:
618 alu :: Bit -> Word -> Word -> Word
619 alu = λopcode.λa.b. (case opcode of
620 Low -> λa1.λb1 (+) a1 b1
621 High -> λa2.λb2 (-) a2 b2) a b
624 \subsection{Transformation application}
625 In this chapter we define a number of transformations, but how will we apply
626 these? As stated before, our normal form is reached as soon as no
627 transformation applies anymore. This means our application strategy is to
628 simply apply any transformation that applies, and continuing to do that with
629 the result of each transformation.
631 In particular, we define no particular order of transformations. Since
632 transformation order should not influence the resulting normal form,
633 \todo{This is not really true, but would like it to be...} this leaves
634 the implementation free to choose any application order that results in
635 an efficient implementation.
637 When applying a single transformation, we try to apply it to every (sub)expression
638 in a function, not just the top level function body. This allows us to
639 keep the transformation descriptions concise and powerful.
641 \subsection{Definitions}
642 In the following sections, we will be using a number of functions and
643 notations, which we will define here.
645 \todo{Define substitution (notation)}
647 \subsubsection{Concepts}
648 A \emph{global variable} is any variable (binder) that is bound at the
649 top level of a program, or an external module. A \emph{local variable} is any
650 other variable (\eg, variables local to a function, which can be bound by
651 lambda abstractions, let expressions and pattern matches of case
652 alternatives). Note that this is a slightly different notion of global versus
653 local than what \small{GHC} uses internally.
654 \defref{global variable} \defref{local variable}
656 A \emph{hardware representable} (or just \emph{representable}) type or value
657 is (a value of) a type that we can generate a signal for in hardware. For
658 example, a bit, a vector of bits, a 32 bit unsigned word, etc. Values that are
659 not runtime representable notably include (but are not limited to): Types,
660 dictionaries, functions.
661 \defref{representable}
663 A \emph{builtin function} is a function supplied by the Cλash framework, whose
664 implementation is not valid Cλash. The implementation is of course valid
665 Haskell, for simulation, but it is not expressable in Cλash.
666 \defref{builtin function} \defref{user-defined function}
668 For these functions, Cλash has a \emph{builtin hardware translation}, so calls
669 to these functions can still be translated. These are functions like
670 \lam{map}, \lam{hwor} and \lam{length}.
672 A \emph{user-defined} function is a function for which we do have a Cλash
673 implementation available.
675 \subsubsection{Predicates}
676 Here, we define a number of predicates that can be used below to concisely
677 specify conditions.\refdef{global variable}
679 \emph{gvar(expr)} is true when \emph{expr} is a variable that references a
680 global variable. It is false when it references a local variable.
682 \refdef{local variable}\emph{lvar(expr)} is the complement of \emph{gvar}; it is true when \emph{expr}
683 references a local variable, false when it references a global variable.
685 \refdef{representable}\emph{representable(expr)} or \emph{representable(var)} is true when
686 \emph{expr} or \emph{var} is \emph{representable}.
688 \subsection[sec:normalization:uniq]{Binder uniqueness}
689 A common problem in transformation systems, is binder uniqueness. When not
690 considering this problem, it is easy to create transformations that mix up
691 bindings and cause name collisions. Take for example, the following core
695 (λa.λb.λc. a * b * c) x c
698 By applying β-reduction (see \in{section}[sec:normalization:beta]) once,
699 we can simplify this expression to:
705 Now, we have replaced the \lam{a} binder with a reference to the \lam{x}
706 binder. No harm done here. But note that we see multiple occurences of the
707 \lam{c} binder. The first is a binding occurence, to which the second refers.
708 The last, however refers to \emph{another} instance of \lam{c}, which is
709 bound somewhere outside of this expression. Now, if we would apply beta
710 reduction without taking heed of binder uniqueness, we would get:
716 This is obviously not what was supposed to happen! The root of this problem is
717 the reuse of binders: Identical binders can be bound in different scopes, such
718 that only the inner one is \quote{visible} in the inner expression. In the example
719 above, the \lam{c} binder was bound outside of the expression and in the inner
720 lambda expression. Inside that lambda expression, only the inner \lam{c} is
723 There are a number of ways to solve this. \small{GHC} has isolated this
724 problem to their binder substitution code, which performs \emph{deshadowing}
725 during its expression traversal. This means that any binding that shadows
726 another binding on a higher level is replaced by a new binder that does not
727 shadow any other binding. This non-shadowing invariant is enough to prevent
728 binder uniqueness problems in \small{GHC}.
730 In our transformation system, maintaining this non-shadowing invariant is
731 a bit harder to do (mostly due to implementation issues, the prototype doesn't
732 use \small{GHC}'s subsitution code). Also, the following points can be
736 \item Deshadowing does not guarantee overall uniqueness. For example, the
737 following (slightly contrived) expression shows the identifier \lam{x} bound in
738 two seperate places (and to different values), even though no shadowing
742 (let x = 1 in x) + (let x = 2 in x)
745 \item In our normal form (and the resulting \small{VHDL}), all binders
746 (signals) within the same function (entity) will end up in the same
747 scope. To allow this, all binders within the same function should be
750 \item When we know that all binders in an expression are unique, moving around
751 or removing a subexpression will never cause any binder conflicts. If we have
752 some way to generate fresh binders, introducing new subexpressions will not
753 cause any problems either. The only way to cause conflicts is thus to
754 duplicate an existing subexpression.
757 Given the above, our prototype maintains a unique binder invariant. This
758 means that in any given moment during normalization, all binders \emph{within
759 a single function} must be unique. To achieve this, we apply the following
762 \todo{Define fresh binders and unique supplies}
765 \item Before starting normalization, all binders in the function are made
766 unique. This is done by generating a fresh binder for every binder used. This
767 also replaces binders that did not cause any conflict, but it does ensure that
768 all binders within the function are generated by the same unique supply.
769 \refdef{fresh binder}
770 \item Whenever a new binder must be generated, we generate a fresh binder that
771 is guaranteed to be different from \emph{all binders generated so far}. This
772 can thus never introduce duplication and will maintain the invariant.
773 \item Whenever (a part of) an expression is duplicated (for example when
774 inlining), all binders in the expression are replaced with fresh binders
775 (using the same method as at the start of normalization). These fresh binders
776 can never introduce duplication, so this will maintain the invariant.
777 \item Whenever we move part of an expression around within the function, there
778 is no need to do anything special. There is obviously no way to introduce
779 duplication by moving expressions around. Since we know that each of the
780 binders is already unique, there is no way to introduce (incorrect) shadowing
784 \section{Transform passes}
785 In this section we describe the actual transforms.
787 Each transformation will be described informally first, explaining
788 the need for and goal of the transformation. Then, we will formally define
789 the transformation using the syntax introduced in
790 \in{section}[sec:normalization:transformation].
792 \subsection{General cleanup}
793 These transformations are general cleanup transformations, that aim to
794 make expressions simpler. These transformations usually clean up the
795 mess left behind by other transformations or clean up expressions to
796 expose new transformation opportunities for other transformations.
798 Most of these transformations are standard optimizations in other
799 compilers as well. However, in our compiler, most of these are not just
800 optimizations, but they are required to get our program into intended
803 \subsubsection[sec:normalization:beta]{β-reduction}
804 \defref{beta-reduction}
805 β-reduction is a well known transformation from lambda calculus, where it is
806 the main reduction step. It reduces applications of lambda abstractions,
807 removing both the lambda abstraction and the application.
809 In our transformation system, this step helps to remove unwanted lambda
810 abstractions (basically all but the ones at the top level). Other
811 transformations (application propagation, non-representable inlining) make
812 sure that most lambda abstractions will eventually be reducable by
815 Note that β-reduction also works on type lambda abstractions and type
816 applications as well. This means the substitution below also works on
817 type variables, in the case that the binder is a type variable and teh
818 expression applied to is a type.
835 \transexample{beta}{β-reduction}{from}{to}
845 \transexample{beta-type}{β-reduction for type abstractions}{from}{to}
848 \subsubsection{Empty let removal}
849 This transformation is simple: It removes recursive lets that have no bindings
850 (which usually occurs when unused let binding removal removes the last
853 Note that there is no need to define this transformation for
854 non-recursive lets, since they always contain exactly one binding.
864 \subsubsection{Simple let binding removal}
865 This transformation inlines simple let bindings, that bind some
866 binder to some other binder instead of a more complex expression (\ie
869 This transformation is not needed to get an expression into intended
870 normal form (since these bindings are part of the intended normal
871 form), but makes the resulting \small{VHDL} a lot shorter.
882 ----------------------------- \lam{b} is a variable reference
883 letrec \lam{ai} ≠ \lam{b}
896 \subsubsection{Unused let binding removal}
897 This transformation removes let bindings that are never used.
898 Occasionally, \GHC's desugarer introduces some unused let bindings.
900 This normalization pass should really be unneeded to get into intended normal form
901 (since unused bindings are not forbidden by the normal form), but in practice
902 the desugarer or simplifier emits some unused bindings that cannot be
903 normalized (e.g., calls to a \type{PatError}\todo{Check this name}). Also,
904 this transformation makes the resulting \small{VHDL} a lot shorter.
906 \todo{Don't use old-style numerals in transformations}
915 M \lam{ai} does not occur free in \lam{M}
916 ---------------------------- \forall j, 0 ≤ j ≤ n, j ≠ i (\lam{ai} does not occur free in \lam{Ej})
930 \subsubsection{Cast propagation / simplification}
931 This transform pushes casts down into the expression as far as possible.
932 Since its exact role and need is not clear yet, this transformation is
935 \todo{Cast propagation}
937 \subsubsection{Top level binding inlining}
938 This transform takes simple top level bindings generated by the
939 \small{GHC} compiler. \small{GHC} sometimes generates very simple
940 \quote{wrapper} bindings, which are bound to just a variable
941 reference, or a partial application to constants or other variable
944 Note that this transformation is completely optional. It is not
945 required to get any function into intended normal form, but it does help making
946 the resulting VHDL output easier to read (since it removes a bunch of
947 components that are really boring).
949 This transform takes any top level binding generated by the compiler,
950 whose normalized form contains only a single let binding.
953 x = λa0 ... λan.let y = E in y
956 -------------------------------------- \lam{x} is generated by the compiler
957 λa0 ... λan.let y = E in y
961 (+) :: Word -> Word -> Word
962 (+) = GHC.Num.(+) @Word $dNum
967 GHC.Num.(+) @ Alu.Word $dNum a b
970 \transexample{toplevelinline}{Top level binding inlining}{from}{to}
972 \in{Example}[ex:trans:toplevelinline] shows a typical application of
973 the addition operator generated by \GHC. The type and dictionary
974 arguments used here are described in
975 \in{Section}[section:prototype:polymorphism].
977 Without this transformation, there would be a \lam{(+)} entity
978 in the \VHDL which would just add its inputs. This generates a
979 lot of overhead in the \VHDL, which is particularly annoying
980 when browsing the generated RTL schematic (especially since most
981 non-alphanumerics, like all characters in \lam{(+)}, are not
982 allowed in \VHDL architecture names\footnote{Technically, it is
983 allowed to use non-alphanumerics when using extended
984 identifiers, but it seems that none of the tooling likes
985 extended identifiers in filenames, so it effectively doesn't
986 work.}, so the entity would be called \quote{w7aA7f} or
987 something similarly unreadable and autogenerated).
989 \subsection{Program structure}
990 These transformations are aimed at normalizing the overall structure
991 into the intended form. This means ensuring there is a lambda abstraction
992 at the top for every argument (input port or current state), putting all
993 of the other value definitions in let bindings and making the final
994 return value a simple variable reference.
996 \subsubsection{η-abstraction}
997 This transformation makes sure that all arguments of a function-typed
998 expression are named, by introducing lambda expressions. When combined with
999 β-reduction and non-representable binding inlining, all function-typed
1000 expressions should be lambda abstractions or global identifiers.
1004 -------------- \lam{E} is not the first argument of an application.
1005 λx.E x \lam{E} is not a lambda abstraction.
1006 \lam{x} is a variable that does not occur free in \lam{E}.
1016 foo = λa.λx.(case a of
1021 \transexample{eta}{η-abstraction}{from}{to}
1023 \subsubsection{Application propagation}
1024 This transformation is meant to propagate application expressions downwards
1025 into expressions as far as possible. This allows partial applications inside
1026 expressions to become fully applied and exposes new transformation
1027 opportunities for other transformations (like β-reduction and
1030 Since all binders in our expression are unique (see
1031 \in{section}[sec:normalization:uniq]), there is no risk that we will
1032 introduce unintended shadowing by moving an expression into a lower
1033 scope. Also, since only move expression into smaller scopes (down into
1034 our expression), there is no risk of moving a variable reference out
1035 of the scope in which it is defined.
1038 (letrec binds in E) M
1039 ------------------------
1059 \transexample{appproplet}{Application propagation for a let expression}{from}{to}
1087 \transexample{apppropcase}{Application propagation for a case expression}{from}{to}
1089 \subsubsection{Let recursification}
1090 This transformation makes all non-recursive lets recursive. In the
1091 end, we want a single recursive let in our normalized program, so all
1092 non-recursive lets can be converted. This also makes other
1093 transformations simpler: They can simply assume all lets are
1101 ------------------------------------------
1108 \subsubsection{Let flattening}
1109 This transformation puts nested lets in the same scope, by lifting the
1110 binding(s) of the inner let into the outer let. Eventually, this will
1111 cause all let bindings to appear in the same scope.
1113 This transformation only applies to recursive lets, since all
1114 non-recursive lets will be made recursive (see
1115 \in{section}[sec:normalization:letrecurse]).
1117 Since we are joining two scopes together, there is no risk of moving a
1118 variable reference out of the scope where it is defined.
1124 ai = (letrec bindings in M)
1129 ------------------------------------------
1164 \transexample{letflat}{Let flattening}{from}{to}
1166 \subsubsection{Return value simplification}
1167 This transformation ensures that the return value of a function is always a
1168 simple local variable reference.
1170 Currently implemented using lambda simplification, let simplification, and
1171 top simplification. Should change into something like the following, which
1172 works only on the result of a function instead of any subexpression. This is
1173 achieved by the contexts, like \lam{x = E}, though this is strictly not
1174 correct (you could read this as "if there is any function \lam{x} that binds
1175 \lam{E}, any \lam{E} can be transformed, while we only mean the \lam{E} that
1176 is bound by \lam{x}. This might need some extra notes or something).
1178 Note that the return value is not simplified if its not representable.
1179 Otherwise, this would cause a direct loop with the inlining of
1180 unrepresentable bindings. If the return value is not
1181 representable because it has a function type, η-abstraction should
1182 make sure that this transformation will eventually apply. If the value
1183 is not representable for other reasons, the function result itself is
1184 not representable, meaning this function is not translatable anyway.
1187 x = E \lam{E} is representable
1188 ~ \lam{E} is not a lambda abstraction
1189 E \lam{E} is not a let expression
1190 --------------------------- \lam{E} is not a local variable reference
1196 ~ \lam{E} is representable
1197 E \lam{E} is not a let expression
1198 --------------------------- \lam{E} is not a local variable reference
1203 x = λv0 ... λvn.let ... in E
1204 ~ \lam{E} is representable
1205 E \lam{E} is not a local variable reference
1206 -----------------------------
1215 x = letrec x = add 1 2 in x
1218 \transexample{retvalsimpl}{Return value simplification}{from}{to}
1220 \todo{More examples}
1222 \subsection{Argument simplification}
1223 The transforms in this section deal with simplifying application
1224 arguments into normal form. The goal here is to:
1226 \todo{This section should only talk about representable arguments. Non
1227 representable arguments are treated by specialization.}
1230 \item Make all arguments of user-defined functions (\eg, of which
1231 we have a function body) simple variable references of a runtime
1232 representable type. This is needed, since these applications will be turned
1233 into component instantiations.
1234 \item Make all arguments of builtin functions one of:
1236 \item A type argument.
1237 \item A dictionary argument.
1238 \item A type level expression.
1239 \item A variable reference of a runtime representable type.
1240 \item A variable reference or partial application of a function type.
1244 When looking at the arguments of a user-defined function, we can
1245 divide them into two categories:
1247 \item Arguments of a runtime representable type (\eg bits or vectors).
1249 These arguments can be preserved in the program, since they can
1250 be translated to input ports later on. However, since we can
1251 only connect signals to input ports, these arguments must be
1252 reduced to simple variables (for which signals will be
1253 produced). This is taken care of by the argument extraction
1255 \item Non-runtime representable typed arguments. \todo{Move this
1256 bullet to specialization}
1258 These arguments cannot be preserved in the program, since we
1259 cannot represent them as input or output ports in the resulting
1260 \small{VHDL}. To remove them, we create a specialized version of the
1261 called function with these arguments filled in. This is done by
1262 the argument propagation transform.
1264 Typically, these arguments are type and dictionary arguments that are
1265 used to make functions polymorphic. By propagating these arguments, we
1266 are essentially doing the same which GHC does when it specializes
1267 functions: Creating multiple variants of the same function, one for
1268 each type for which it is used. Other common non-representable
1269 arguments are functions, e.g. when calling a higher order function
1270 with another function or a lambda abstraction as an argument.
1272 The reason for doing this is similar to the reasoning provided for
1273 the inlining of non-representable let bindings above. In fact, this
1274 argument propagation could be viewed as a form of cross-function
1278 \todo{Move this itemization into a new section about builtin functions}
1279 When looking at the arguments of a builtin function, we can divide them
1283 \item Arguments of a runtime representable type.
1285 As we have seen with user-defined functions, these arguments can
1286 always be reduced to a simple variable reference, by the
1287 argument extraction transform. Performing this transform for
1288 builtin functions as well, means that the translation of builtin
1289 functions can be limited to signal references, instead of
1290 needing to support all possible expressions.
1292 \item Arguments of a function type.
1294 These arguments are functions passed to higher order builtins,
1295 like \lam{map} and \lam{foldl}. Since implementing these
1296 functions for arbitrary function-typed expressions (\eg, lambda
1297 expressions) is rather comlex, we reduce these arguments to
1298 (partial applications of) global functions.
1300 We can still support arbitrary expressions from the user code,
1301 by creating a new global function containing that expression.
1302 This way, we can simply replace the argument with a reference to
1303 that new function. However, since the expression can contain any
1304 number of free variables we also have to include partial
1305 applications in our normal form.
1307 This category of arguments is handled by the function extraction
1309 \item Other unrepresentable arguments.
1311 These arguments can take a few different forms:
1312 \startdesc{Type arguments}
1313 In the core language, type arguments can only take a single
1314 form: A type wrapped in the Type constructor. Also, there is
1315 nothing that can be done with type expressions, except for
1316 applying functions to them, so we can simply leave type
1317 arguments as they are.
1319 \startdesc{Dictionary arguments}
1320 In the core language, dictionary arguments are used to find
1321 operations operating on one of the type arguments (mostly for
1322 finding class methods). Since we will not actually evaluatie
1323 the function body for builtin functions and can generate
1324 code for builtin functions by just looking at the type
1325 arguments, these arguments can be ignored and left as they
1328 \startdesc{Type level arguments}
1329 Sometimes, we want to pass a value to a builtin function, but
1330 we need to know the value at compile time. Additionally, the
1331 value has an impact on the type of the function. This is
1332 encoded using type-level values, where the actual value of the
1333 argument is not important, but the type encodes some integer,
1334 for example. Since the value is not important, the actual form
1335 of the expression does not matter either and we can leave
1336 these arguments as they are.
1338 \startdesc{Other arguments}
1339 Technically, there is still a wide array of arguments that can
1340 be passed, but does not fall into any of the above categories.
1341 However, none of the supported builtin functions requires such
1342 an argument. This leaves use with passing unsupported types to
1343 a function, such as calling \lam{head} on a list of functions.
1345 In these cases, it would be impossible to generate hardware
1346 for such a function call anyway, so we can ignore these
1349 The only way to generate hardware for builtin functions with
1350 arguments like these, is to expand the function call into an
1351 equivalent core expression (\eg, expand map into a series of
1352 function applications). But for now, we choose to simply not
1353 support expressions like these.
1356 From the above, we can conclude that we can simply ignore these
1357 other unrepresentable arguments and focus on the first two
1361 \subsubsection{Argument simplification}
1362 This transform deals with arguments to functions that
1363 are of a runtime representable type. It ensures that they will all become
1364 references to global variables, or local signals in the resulting
1365 \small{VHDL}, which is required due to limitations in the component
1366 instantiation code in \VHDL (one can only assign a signal or constant
1367 to an input port). By ensuring that all arguments are always simple
1368 variable references, we always have a signal available to assign to
1371 \todo{Say something about dataconstructors (without arguments, like True
1372 or False), which are variable references of a runtime representable
1373 type, but do not result in a signal.}
1375 To reduce a complex expression to a simple variable reference, we create
1376 a new let expression around the application, which binds the complex
1377 expression to a new variable. The original function is then applied to
1380 Note that a reference to a \emph{global variable} (like a top level
1381 function without arguments, but also an argumentless dataconstructors
1382 like \lam{True}) is also simplified. Only local variables generate
1383 signals in the resulting architecture.
1385 \refdef{representable}
1388 -------------------- \lam{N} is representable
1389 letrec x = N in M x \lam{N} is not a local variable reference
1391 \refdef{local variable}
1398 letrec x = add a 1 in add x 1
1401 \transexample{argextract}{Argument extraction}{from}{to}
1403 \subsubsection[sec:normalization:funextract]{Function extraction}
1404 \todo{Move to section about builtin functions}
1405 This transform deals with function-typed arguments to builtin
1406 functions. Since builtin functions cannot be specialized to remove
1407 the arguments, we choose to extract these arguments into a new global
1408 function instead. This greatly simplifies the translation rules needed
1409 for builtin functions. \todo{Should we talk about these? Reference
1412 Any free variables occuring in the extracted arguments will become
1413 parameters to the new global function. The original argument is replaced
1414 with a reference to the new function, applied to any free variables from
1415 the original argument.
1417 This transformation is useful when applying higher order builtin functions
1418 like \hs{map} to a lambda abstraction, for example. In this case, the code
1419 that generates \small{VHDL} for \hs{map} only needs to handle top level functions and
1420 partial applications, not any other expression (such as lambda abstractions or
1421 even more complicated expressions).
1424 M N \lam{M} is (a partial aplication of) a builtin function.
1425 --------------------- \lam{f0 ... fn} are all free local variables of \lam{N}
1426 M (x f0 ... fn) \lam{N :: a -> b}
1427 ~ \lam{N} is not a (partial application of) a top level function
1431 \todo{Split this example}
1433 map (λa . add a b) xs
1447 \transexample{funextract}{Function extraction}{from}{to}
1449 Note that \lam{x0} and {x1} will still need normalization after this.
1451 \todo{Fill the gap left by moving argument propagation away}
1453 \subsection{Case normalisation}
1454 \subsubsection{Scrutinee simplification}
1455 This transform ensures that the scrutinee of a case expression is always
1456 a simple variable reference.
1461 ----------------- \lam{E} is not a local variable reference
1480 \transexample{letflat}{Let flattening}{from}{to}
1483 \subsubsection{Case simplification}
1484 This transformation ensures that all case expressions become normal form. This
1485 means they will become one of:
1487 \item An extractor case with a single alternative that picks a single field
1488 from a datatype, \eg \lam{case x of (a, b) -> a}.
1489 \item A selector case with multiple alternatives and only wild binders, that
1490 makes a choice between expressions based on the constructor of another
1491 expression, \eg \lam{case x of Low -> a; High -> b}.
1494 \defref{wild binder}
1497 C0 v0,0 ... v0,m -> E0
1499 Cn vn,0 ... vn,m -> En
1500 --------------------------------------------------- \forall i \forall j, 0 ≤ i ≤ n, 0 ≤ i < m (\lam{wi,j} is a wild (unused) binder)
1502 v0,0 = case E of C0 v0,0 .. v0,m -> v0,0
1504 v0,m = case E of C0 v0,0 .. v0,m -> v0,m
1506 vn,m = case E of Cn vn,0 .. vn,m -> vn,m
1512 C0 w0,0 ... w0,m -> x0
1514 Cn wn,0 ... wn,m -> xn
1516 \todo{Check the subscripts of this transformation}
1518 Note that this transformation applies to case statements with any
1519 scrutinee. If the scrutinee is a complex expression, this might result
1520 in duplicate hardware. An extra condition to only apply this
1521 transformation when the scrutinee is already simple (effectively
1522 causing this transformation to be only applied after the scrutinee
1523 simplification transformation) might be in order.
1525 \fxnote{This transformation specified like this is complicated and misses
1526 conditions to prevent looping with itself. Perhaps it should be split here for
1545 \transexample{selcasesimpl}{Selector case simplification}{from}{to}
1553 b = case a of (,) b c -> b
1554 c = case a of (,) b c -> c
1561 \transexample{excasesimpl}{Extractor case simplification}{from}{to}
1563 \refdef{selector case}
1564 In \in{example}[ex:trans:excasesimpl] the case expression is expanded
1565 into multiple case expressions, including a pretty useless expression
1566 (that is neither a selector or extractor case). This case can be
1567 removed by the Case removal transformation in
1568 \in{section}[sec:transformation:caseremoval].
1570 \subsubsection[sec:transformation:caseremoval]{Case removal}
1571 This transform removes any case statements with a single alternative and
1574 These "useless" case statements are usually leftovers from case simplification
1575 on extractor case (see the previous example).
1580 ---------------------- \lam{\forall i, 0 ≤ i ≤ m} (\lam{vi} does not occur free in E)
1593 \transexample{caserem}{Case removal}{from}{to}
1595 \subsection{Removing unrepresentable values}
1596 The transformations in this section are aimed at making all the
1597 values used in our expression representable. There are two main
1598 transformations that are applied to \emph{all} unrepresentable let
1599 bindings and function arguments, but these are really meant to
1600 address three different kinds of unrepresentable values:
1601 Polymorphic values, higher order values and literals. Each of these
1602 will be detailed below, followed by the actual transformations.
1604 \subsubsection{Removing Polymorphism}
1605 As noted in \in{section}[sec:prototype:polymporphism],
1606 polymorphism is made explicit in Core through type and
1607 dictionary arguments. To remove the polymorphism from a
1608 function, we can simply specialize the polymorphic function for
1609 the particular type applied to it. The same goes for dictionary
1610 arguments. To remove polymorphism from let bound values, we
1611 simply inline the let bindings that have a polymorphic type,
1612 which should (eventually) make sure that the polymorphic
1613 expression is applied to a type and/or dictionary, which can
1614 \refdef{beta-reduction}
1615 then be removed by β-reduction.
1617 Since both type and dictionary arguments are not representable,
1618 \refdef{representable}
1619 the non-representable argument specialization and
1620 non-representable let binding inlining transformations below
1621 take care of exactly this.
1623 There is one case where polymorphism cannot be completely
1624 removed: Builtin functions are still allowed to be polymorphic
1625 (Since we have no function body that we could properly
1626 specialize). However, the code that generates \VHDL for builtin
1627 functions knows how to handle this, so this is not a problem.
1629 \subsubsection{Defunctionalization}
1630 These transformations remove higher order expressions from our
1631 program, making all values first-order.
1633 \todo{Finish this section}
1635 There is one case where higher order values cannot be completely
1636 removed: Builtin functions are still allowed to have higher
1637 order arguments (Since we have no function body that we could
1638 properly specialize). These are limited to (partial applications
1639 of) top level functions, however, which is handled by the
1640 top-level function extraction (see
1641 \in{section}[sec:normalization:funextract]). However, the code
1642 that generates \VHDL for builtin functions knows how to handle
1643 these, so this is not a problem.
1645 \subsubsection{Literals}
1646 \todo{Fill this section}
1648 \subsubsection{Non-representable binding inlining}
1649 \todo{Move this section into a new section (together with
1651 This transform inlines let bindings that are bound to a
1652 non-representable value. Since we can never generate a signal
1653 assignment for these bindings (we cannot declare a signal assignment
1654 with a non-representable type, for obvious reasons), we have no choice
1655 but to inline the binding to remove it.
1657 If the binding is non-representable because it is a lambda abstraction, it is
1658 likely that it will inlined into an application and β-reduction will remove
1659 the lambda abstraction and turn it into a representable expression at the
1660 inline site. The same holds for partial applications, which can be turned into
1661 full applications by inlining.
1663 Other cases of non-representable bindings we see in practice are primitive
1664 Haskell types. In most cases, these will not result in a valid normalized
1665 output, but then the input would have been invalid to start with. There is one
1666 exception to this: When a builtin function is applied to a non-representable
1667 expression, things might work out in some cases. For example, when you write a
1668 literal \hs{SizedInt} in Haskell, like \hs{1 :: SizedInt D8}, this results in
1669 the following core: \lam{fromInteger (smallInteger 10)}, where for example
1670 \lam{10 :: GHC.Prim.Int\#} and \lam{smallInteger 10 :: Integer} have
1671 non-representable types. \todo{Expand on this. This/these paragraph(s)
1672 should probably become a separate discussion somewhere else}
1674 \todo{Can this duplicate work? -- For polymorphism probably, for
1675 higher order expressions only if they are inlined before they
1676 are themselves normalized.}
1687 -------------------------- \lam{Ei} has a non-representable type.
1689 a0 = E0 [ai=>Ei] \vdots
1690 ai-1 = Ei-1 [ai=>Ei]
1691 ai+1 = Ei+1 [ai=>Ei]
1710 x = fromInteger (smallInteger 10)
1712 (λb -> add b 1) (add 1 x)
1715 \transexample{nonrepinline}{Nonrepresentable binding inlining}{from}{to}
1717 \subsubsection{Argument propagation}
1718 \todo{Rename this section to specialization}
1720 This transform deals with arguments to user-defined functions that are
1721 not representable at runtime. This means these arguments cannot be
1722 preserved in the final form and most be {\em propagated}.
1724 Propagation means to create a specialized version of the called
1725 function, with the propagated argument already filled in. As a simple
1726 example, in the following program:
1733 We could {\em propagate} the constant argument 1, with the following
1741 Special care must be taken when the to-be-propagated expression has any
1742 free variables. If this is the case, the original argument should not be
1743 removed completely, but replaced by all the free variables of the
1744 expression. In this way, the original expression can still be evaluated
1745 inside the new function. Also, this brings us closer to our goal: All
1746 these free variables will be simple variable references.
1748 To prevent us from propagating the same argument over and over, a simple
1749 local variable reference is not propagated (since is has exactly one
1750 free variable, itself, we would only replace that argument with itself).
1752 This shows that any free local variables that are not runtime representable
1753 cannot be brought into normal form by this transform. We rely on an
1754 inlining transformation to replace such a variable with an expression we
1755 can propagate again.
1760 x Y0 ... Yi ... Yn \lam{Yi} is not of a runtime representable type
1761 --------------------------------------------- \lam{Yi} is not a local variable reference
1762 x' y0 ... yi-1 f0 ... fm Yi+1 ... Yn \lam{f0 ... fm} are all free local vars of \lam{Yi}
1764 x' = λy0 ... λyi-1. λf0 ... λfm. λyi+1 ... λyn .
1765 E y0 ... yi-1 Yi yi+1 ... yn
1768 \todo{Describe what the formal specification means}
1769 \todo{Note that we don't change the sepcialised function body, only
1771 \todo{This does not take care of updating the types of y0 ...
1772 yn. The code uses the types of Y0 ... Yn for this, regardless of
1773 whether the type arguments were properly propagated...}
1780 \section[sec:normalization:properties]{Provable properties}
1781 When looking at the system of transformations outlined above, there are a
1782 number of questions that we can ask ourselves. The main question is of course:
1783 \quote{Does our system work as intended?}. We can split this question into a
1784 number of subquestions:
1787 \item[q:termination] Does our system \emph{terminate}? Since our system will
1788 keep running as long as transformations apply, there is an obvious risk that
1789 it will keep running indefinitely. This typically happens when one
1790 transformation produces a result that is transformed back to the original
1791 by another transformation, or when one or more transformations keep
1792 expanding some expression.
1793 \item[q:soundness] Is our system \emph{sound}? Since our transformations
1794 continuously modify the expression, there is an obvious risk that the final
1795 normal form will not be equivalent to the original program: Its meaning could
1797 \item[q:completeness] Is our system \emph{complete}? Since we have a complex
1798 system of transformations, there is an obvious risk that some expressions will
1799 not end up in our intended normal form, because we forgot some transformation.
1800 In other words: Does our transformation system result in our intended normal
1801 form for all possible inputs?
1802 \item[q:determinism] Is our system \emph{deterministic}? Since we have defined
1803 no particular order in which the transformation should be applied, there is an
1804 obvious risk that different transformation orderings will result in
1805 \emph{different} normal forms. They might still both be intended normal forms
1806 (if our system is \emph{complete}) and describe correct hardware (if our
1807 system is \emph{sound}), so this property is less important than the previous
1808 three: The translator would still function properly without it.
1811 Unfortunately, the final transformation system has only been
1812 developed in the final part of the research, leaving no more time
1813 for verifying these properties. In fact, it is likely that the
1814 current transformation system still violates some of these
1815 properties in some cases and should be improved (or extra conditions
1816 on the input hardware descriptions should be formulated).
1818 This is most likely the case with the completeness and determinism
1819 properties, perhaps als the termination property. The soundness
1820 property probably holds, since it is easier to manually verify (each
1821 transformation can be reviewed separately).
1823 Even though no complete proofs have been made, some ideas for
1824 possible proof strategies are shown below.
1826 \subsection{Graph representation}
1827 Before looking into how to prove these properties, we'll look at our
1828 transformation system from a graph perspective. The nodes of the graph are
1829 all possible Core expressions. The (directed) edges of the graph are
1830 transformations. When a transformation α applies to an expression \lam{A} to
1831 produce an expression \lam{B}, we add an edge from the node for \lam{A} to the
1832 node for \lam{B}, labeled α.
1834 \startuseMPgraphic{TransformGraph}
1838 newCircle.a(btex \lam{(λx.λy. (+) x y) 1} etex);
1839 newCircle.b(btex \lam{λy. (+) 1 y} etex);
1840 newCircle.c(btex \lam{(λx.(+) x) 1} etex);
1841 newCircle.d(btex \lam{(+) 1} etex);
1844 c.c = b.c + (4cm, 0cm);
1845 a.c = midpoint(b.c, c.c) + (0cm, 4cm);
1846 d.c = midpoint(b.c, c.c) - (0cm, 3cm);
1848 % β-conversion between a and b
1849 ncarc.a(a)(b) "name(bred)";
1850 ObjLabel.a(btex $\xrightarrow[normal]{}{β}$ etex) "labpathname(bred)", "labdir(rt)";
1851 ncarc.b(b)(a) "name(bexp)", "linestyle(dashed withdots)";
1852 ObjLabel.b(btex $\xleftarrow[normal]{}{β}$ etex) "labpathname(bexp)", "labdir(lft)";
1854 % η-conversion between a and c
1855 ncarc.a(a)(c) "name(ered)";
1856 ObjLabel.a(btex $\xrightarrow[normal]{}{η}$ etex) "labpathname(ered)", "labdir(rt)";
1857 ncarc.c(c)(a) "name(eexp)", "linestyle(dashed withdots)";
1858 ObjLabel.c(btex $\xleftarrow[normal]{}{η}$ etex) "labpathname(eexp)", "labdir(lft)";
1860 % η-conversion between b and d
1861 ncarc.b(b)(d) "name(ered)";
1862 ObjLabel.b(btex $\xrightarrow[normal]{}{η}$ etex) "labpathname(ered)", "labdir(rt)";
1863 ncarc.d(d)(b) "name(eexp)", "linestyle(dashed withdots)";
1864 ObjLabel.d(btex $\xleftarrow[normal]{}{η}$ etex) "labpathname(eexp)", "labdir(lft)";
1866 % β-conversion between c and d
1867 ncarc.c(c)(d) "name(bred)";
1868 ObjLabel.c(btex $\xrightarrow[normal]{}{β}$ etex) "labpathname(bred)", "labdir(rt)";
1869 ncarc.d(d)(c) "name(bexp)", "linestyle(dashed withdots)";
1870 ObjLabel.d(btex $\xleftarrow[normal]{}{β}$ etex) "labpathname(bexp)", "labdir(lft)";
1872 % Draw objects and lines
1873 drawObj(a, b, c, d);
1876 \placeexample[right][ex:TransformGraph]{Partial graph of a lambda calculus
1877 system with β and η reduction (solid lines) and expansion (dotted lines).}
1878 \boxedgraphic{TransformGraph}
1880 Of course our graph is unbounded, since we can construct an infinite amount of
1881 Core expressions. Also, there might potentially be multiple edges between two
1882 given nodes (with different labels), though seems unlikely to actually happen
1885 See \in{example}[ex:TransformGraph] for the graph representation of a very
1886 simple lambda calculus that contains just the expressions \lam{(λx.λy. (+) x
1887 y) 1}, \lam{λy. (+) 1 y}, \lam{(λx.(+) x) 1} and \lam{(+) 1}. The
1888 transformation system consists of β-reduction and η-reduction (solid edges) or
1889 β-expansion and η-expansion (dotted edges).
1891 \todo{Define β-reduction and η-reduction?}
1893 Note that the normal form of such a system consists of the set of nodes
1894 (expressions) without outgoing edges, since those are the expression to which
1895 no transformation applies anymore. We call this set of nodes the \emph{normal
1896 set}. The set of nodes containing expressions in intended normal
1897 form \refdef{intended normal form} is called the \emph{intended
1900 From such a graph, we can derive some properties easily:
1902 \item A system will \emph{terminate} if there is no path of infinite length
1903 in the graph (this includes cycles, but can also happen without cycles).
1904 \item Soundness is not easily represented in the graph.
1905 \item A system is \emph{complete} if all of the nodes in the normal set have
1906 the intended normal form. The inverse (that all of the nodes outside of
1907 the normal set are \emph{not} in the intended normal form) is not
1908 strictly required. In other words, our normal set must be a
1909 subset of the intended normal form, but they do not need to be
1912 \item A system is deterministic if all paths starting at a particular
1913 node, which end in a node in the normal set, end at the same node.
1916 When looking at the \in{example}[ex:TransformGraph], we see that the system
1917 terminates for both the reduction and expansion systems (but note that, for
1918 expansion, this is only true because we've limited the possible
1919 expressions. In comlete lambda calculus, there would be a path from
1920 \lam{(λx.λy. (+) x y) 1} to \lam{(λx.λy.(λz.(+) z) x y) 1} to
1921 \lam{(λx.λy.(λz.(λq.(+) q) z) x y) 1} etc.)
1923 If we would consider the system with both expansion and reduction, there
1924 would no longer be termination either, since there would be cycles all
1927 The reduction and expansion systems have a normal set of containing just
1928 \lam{(+) 1} or \lam{(λx.λy. (+) x y) 1} respectively. Since all paths in
1929 either system end up in these normal forms, both systems are \emph{complete}.
1930 Also, since there is only one node in the normal set, it must obviously be
1931 \emph{deterministic} as well.
1933 \todo{Add content to these sections}
1934 \subsection{Termination}
1935 In general, proving termination of an arbitrary program is a very
1936 hard problem. \todo{Ref about arbitrary termination} Fortunately,
1937 we only have to prove termination for our specific transformation
1940 A common approach for these kinds of proofs is to associate a
1941 measure with each possible expression in our system. If we can
1942 show that each transformation strictly decreases this measure
1943 (\ie, the expression transformed to has a lower measure than the
1944 expression transformed from). \todo{ref about measure-based
1945 termination proofs / analysis}
1947 A good measure for a system consisting of just β-reduction would
1948 be the number of lambda expressions in the expression. Since every
1949 application of β-reduction removes a lambda abstraction (and there
1950 is always a bounded number of lambda abstractions in every
1951 expression) we can easily see that a transformation system with
1952 just β-reduction will always terminate.
1954 For our complete system, this measure would be fairly complex
1955 (probably the sum of a lot of things). Since the (conditions on)
1956 our transformations are pretty complex, we would need to include
1957 both simple things like the number of let expressions as well as
1958 more complex things like the number of case expressions that are
1959 not yet in normal form.
1961 No real attempt has been made at finding a suitable measure for
1964 \subsection{Soundness}
1965 Soundness is a property that can be proven for each transformation
1966 separately. Since our system only runs separate transformations
1967 sequentially, if each of our transformations leaves the
1968 \emph{meaning} of the expression unchanged, then the entire system
1969 will of course leave the meaning unchanged and is thus
1972 The current prototype has only been verified in an ad-hoc fashion
1973 by inspecting (the code for) each transformation. A more formal
1974 verification would be more appropriate.
1976 To be able to formally show that each transformation properly
1977 preserves the meaning of every expression, we require an exact
1978 definition of the \emph{meaning} of every expression, so we can
1979 compare them. Currently there seems to be no formal definition of
1980 the meaning or semantics of \GHC's core language, only informal
1981 descriptions are available.
1983 It should be possible to have a single formal definition of
1984 meaning for Core for both normal Core compilation by \GHC and for
1985 our compilation to \VHDL. The main difference seems to be that in
1986 hardware every expression is always evaluated, while in software
1987 it is only evaluated if needed, but it should be possible to
1988 assign a meaning to core expressions that assumes neither.
1990 Since each of the transformations can be applied to any
1991 subexpression as well, there is a constraint on our meaning
1992 definition: The meaning of an expression should depend only on the
1993 meaning of subexpressions, not on the expressions themselves. For
1994 example, the meaning of the application in \lam{f (let x = 4 in
1995 x)} should be the same as the meaning of the application in \lam{f
1996 4}, since the argument subexpression has the same meaning (though
1997 the actual expression is different).
1999 \subsection{Completeness}
2000 Proving completeness is probably not hard, but it could be a lot
2001 of work. We have seen above that to prove completeness, we must
2002 show that the normal set of our graph representation is a subset
2003 of the intended normal set.
2005 However, it is hard to systematically generate or reason about the
2006 normal set, since it is defined as any nodes to which no
2007 transformation applies. To determine this set, each transformation
2008 must be considered and when a transformation is added, the entire
2009 set should be re-evaluated. This means it is hard to show that
2010 each node in the normal set is also in the intended normal set.
2011 Reasoning about our intended normal set is easier, since we know
2012 how to generate it from its definition. \refdef{intended normal
2015 Fortunately, we can also prove the complement (which is
2016 equivalent, since $A \subseteq B \Leftrightarrow \overline{B}
2017 \subseteq \overline{A}$): Show that the set of nodes not in
2018 intended normal form is a subset of the set of nodes not in normal
2019 form. In other words, show that for every expression that is not
2020 in intended normal form, that there is at least one transformation
2021 that applies to it (since that means it is not in normal form
2022 either and since $A \subseteq C \Leftrightarrow \forall x (x \in A
2023 \rightarrow x \in C)$).
2025 By systematically reviewing the entire Core language definition
2026 along with the intended normal form definition (both of which have
2027 a similar structure), it should be possible to identify all
2028 possible (sets of) core expressions that are not in intended
2029 normal form and identify a transformation that applies to it.
2031 This approach is especially useful for proving completeness of our
2032 system, since if expressions exist to which none of the
2033 transformations apply (\ie if the system is not yet complete), it
2034 is immediately clear which expressions these are and adding
2035 (or modifying) transformations to fix this should be relatively
2038 As observed above, applying this approach is a lot of work, since
2039 we need to check every (set of) transformation(s) separately.
2041 \todo{Perhaps do a few steps of the proofs as proof-of-concept}
2043 % vim: set sw=2 sts=2 expandtab: