1 \chapter[chap:normalization]{Normalization}
2 % A helper to print a single example in the half the page width. The example
3 % text should be in a buffer whose name is given in an argument.
5 % The align=right option really does left-alignment, but without the program
6 % will end up on a single line. The strut=no option prevents a bunch of empty
7 % space at the start of the frame.
9 \framed[offset=1mm,align=right,strut=no,background=box,frame=off]{
10 \setuptyping[option=LAM,style=sans,before=,after=,strip=auto]
12 \setuptyping[option=none,style=\tttf,strip=auto]
16 \define[4]\transexample{
17 \placeexample[here][ex:trans:#1]{#2}
18 \startcombination[2*1]
19 {\example{#3}}{Original program}
20 {\example{#4}}{Transformed program}
24 The first step in the core to \small{VHDL} translation process, is normalization. We
25 aim to bring the core description into a simpler form, which we can
26 subsequently translate into \small{VHDL} easily. This normal form is needed because
27 the full core language is more expressive than \small{VHDL} in some areas and because
28 core can describe expressions that do not have a direct hardware
32 The transformations described here have a well-defined goal: To bring the
33 program in a well-defined form that is directly translatable to hardware,
34 while fully preserving the semantics of the program. We refer to this form as
35 the \emph{normal form} of the program. The formal definition of this normal
38 \placedefinition{}{A program is in \emph{normal form} if none of the
39 transformations from this chapter apply.}
41 Of course, this is an \quote{easy} definition of the normal form, since our
42 program will end up in normal form automatically. The more interesting part is
43 to see if this normal form actually has the properties we would like it to
46 But, before getting into more definitions and details about this normal form,
47 let's try to get a feeling for it first. The easiest way to do this is by
48 describing the things we want to not have in a normal form.
51 \item Any \emph{polymorphism} must be removed. When laying down hardware, we
52 can't generate any signals that can have multiple types. All types must be
53 completely known to generate hardware.
55 \item Any \emph{higher order} constructions must be removed. We can't
56 generate a hardware signal that contains a function, so all values,
57 arguments and returns values used must be first order.
59 \item Any complex \emph{nested scopes} must be removed. In the \small{VHDL}
60 description, every signal is in a single scope. Also, full expressions are
61 not supported everywhere (in particular port maps can only map signal
62 names and constants, not complete expressions). To make the \small{VHDL}
63 generation easy, a separate binder must be bound to ever application or
67 \todo{Intermezzo: functions vs plain values}
69 A very simple example of a program in normal form is given in
70 \in{example}[ex:MulSum]. As you can see, all arguments to the function (which
71 will become input ports in the final hardware) are at the outer level.
72 This means that the body of the inner lambda abstraction is never a
73 function, but always a plain value.
75 As the body of the inner lambda abstraction, we see a single (recursive)
76 let expression, that binds two variables (\lam{mul} and \lam{sum}). These
77 variables will be signals in the final hardware, bound to the output port
78 of the \lam{*} and \lam{+} components.
80 The final line (the \quote{return value} of the function) selects the
81 \lam{sum} signal to be the output port of the function. This \quote{return
82 value} can always only be a variable reference, never a more complex
85 \todo{Add generated VHDL}
88 alu :: Bit -> Word -> Word -> Word
97 \startuseMPgraphic{MulSum}
98 save a, b, c, mul, add, sum;
101 newCircle.a(btex $a$ etex) "framed(false)";
102 newCircle.b(btex $b$ etex) "framed(false)";
103 newCircle.c(btex $c$ etex) "framed(false)";
104 newCircle.sum(btex $res$ etex) "framed(false)";
107 newCircle.mul(btex * etex);
108 newCircle.add(btex + etex);
110 a.c - b.c = (0cm, 2cm);
111 b.c - c.c = (0cm, 2cm);
112 add.c = c.c + (2cm, 0cm);
113 mul.c = midpoint(a.c, b.c) + (2cm, 0cm);
114 sum.c = add.c + (2cm, 0cm);
117 % Draw objects and lines
118 drawObj(a, b, c, mul, add, sum);
120 ncarc(a)(mul) "arcangle(15)";
121 ncarc(b)(mul) "arcangle(-15)";
127 \placeexample[here][ex:MulSum]{Simple architecture consisting of a
128 multiplier and a subtractor.}
129 \startcombination[2*1]
130 {\typebufferlam{MulSum}}{Core description in normal form.}
131 {\boxedgraphic{MulSum}}{The architecture described by the normal form.}
134 The previous example described composing an architecture by calling other
135 functions (operators), resulting in a simple architecture with components and
136 connections. There is of course also some mechanism for choice in the normal
137 form. In a normal Core program, the \emph{case} expression can be used in a
138 few different ways to describe choice. In normal form, this is limited to a
141 \in{Example}[ex:AddSubAlu] shows an example describing a
142 simple \small{ALU}, which chooses between two operations based on an opcode
143 bit. The main structure is similar to \in{example}[ex:MulSum], but this
144 time the \lam{res} variable is bound to a case expression. This case
145 expression scrutinizes the variable \lam{opcode} (and scrutinizing more
146 complex expressions is not supported). The case expression can select a
147 different variable based on the constructor of \lam{opcode}.
149 \startbuffer[AddSubAlu]
150 alu :: Bit -> Word -> Word -> Word
162 \startuseMPgraphic{AddSubAlu}
163 save opcode, a, b, add, sub, mux, res;
166 newCircle.opcode(btex $opcode$ etex) "framed(false)";
167 newCircle.a(btex $a$ etex) "framed(false)";
168 newCircle.b(btex $b$ etex) "framed(false)";
169 newCircle.res(btex $res$ etex) "framed(false)";
171 newCircle.add(btex + etex);
172 newCircle.sub(btex - etex);
175 opcode.c - a.c = (0cm, 2cm);
176 add.c - a.c = (4cm, 0cm);
177 sub.c - b.c = (4cm, 0cm);
178 a.c - b.c = (0cm, 3cm);
179 mux.c = midpoint(add.c, sub.c) + (1.5cm, 0cm);
180 res.c - mux.c = (1.5cm, 0cm);
183 % Draw objects and lines
184 drawObj(opcode, a, b, res, add, sub, mux);
186 ncline(a)(add) "posA(e)";
187 ncline(b)(sub) "posA(e)";
188 nccurve(a)(sub) "posA(e)", "angleA(0)";
189 nccurve(b)(add) "posA(e)", "angleA(0)";
190 nccurve(add)(mux) "posB(inpa)", "angleB(0)";
191 nccurve(sub)(mux) "posB(inpb)", "angleB(0)";
192 nccurve(opcode)(mux) "posB(n)", "angleA(0)", "angleB(-90)";
193 ncline(mux)(res) "posA(out)";
196 \placeexample[here][ex:AddSubAlu]{Simple \small{ALU} supporting two operations.}
197 \startcombination[2*1]
198 {\typebufferlam{AddSubAlu}}{Core description in normal form.}
199 {\boxedgraphic{AddSubAlu}}{The architecture described by the normal form.}
202 As a more complete example, consider \in{example}[ex:NormalComplete]. This
203 example contains everything that is supported in normal form, with the
204 exception of builtin higher order functions. The graphical version of the
205 architecture contains a slightly simplified version, since the state tuple
206 packing and unpacking have been left out. Instead, two seperate registers are
207 drawn. Also note that most synthesis tools will further optimize this
208 architecture by removing the multiplexers at the register input and
209 instead put some gates in front of the register's clock input, but we want
210 to show the architecture as close to the description as possible.
212 As you can see from the previous examples, the generation of the final
213 architecture from the normal form is straightforward. In each of the
214 examples, there is a direct match between the normal form structure,
215 the generated VHDL and the architecture shown in the images.
217 \startbuffer[NormalComplete]
220 -> State (Word, Word)
221 -> (State (Word, Word), Word)
223 -- All arguments are an inital lambda (address, data, packed state)
225 -- There are nested let expressions at top level
227 -- Unpack the state by coercion (\eg, cast from
228 -- State (Word, Word) to (Word, Word))
229 s = sp ▶ (Word, Word)
230 -- Extract both registers from the state
231 r1 = case s of (a, b) -> a
232 r2 = case s of (a, b) -> b
233 -- Calling some other user-defined function.
235 -- Conditional connections
247 -- pack the state by coercion (\eg, cast from
248 -- (Word, Word) to State (Word, Word))
249 sp' = s' ▶ State (Word, Word)
250 -- Pack our return value
257 \startuseMPgraphic{NormalComplete}
258 save a, d, r, foo, muxr, muxout, out;
261 newCircle.a(btex \lam{a} etex) "framed(false)";
262 newCircle.d(btex \lam{d} etex) "framed(false)";
263 newCircle.out(btex \lam{out} etex) "framed(false)";
265 %newCircle.add(btex + etex);
266 newBox.foo(btex \lam{foo} etex);
267 newReg.r1(btex $\lam{r1}$ etex) "dx(4mm)", "dy(6mm)";
268 newReg.r2(btex $\lam{r2}$ etex) "dx(4mm)", "dy(6mm)", "reflect(true)";
270 % Reflect over the vertical axis
271 reflectObj(muxr1)((0,0), (0,1));
274 rotateObj(muxout)(-90);
276 d.c = foo.c + (0cm, 1.5cm);
277 a.c = (xpart r2.c + 2cm, ypart d.c - 0.5cm);
278 foo.c = midpoint(muxr1.c, muxr2.c) + (0cm, 2cm);
279 muxr1.c = r1.c + (0cm, 2cm);
280 muxr2.c = r2.c + (0cm, 2cm);
281 r2.c = r1.c + (4cm, 0cm);
283 muxout.c = midpoint(r1.c, r2.c) - (0cm, 2cm);
284 out.c = muxout.c - (0cm, 1.5cm);
286 % % Draw objects and lines
287 drawObj(a, d, foo, r1, r2, muxr1, muxr2, muxout, out);
290 nccurve(foo)(muxr1) "angleA(-90)", "posB(inpa)", "angleB(180)";
291 nccurve(foo)(muxr2) "angleA(-90)", "posB(inpb)", "angleB(0)";
292 nccurve(muxr1)(r1) "posA(out)", "angleA(180)", "posB(d)", "angleB(0)";
293 nccurve(r1)(muxr1) "posA(out)", "angleA(0)", "posB(inpb)", "angleB(180)";
294 nccurve(muxr2)(r2) "posA(out)", "angleA(0)", "posB(d)", "angleB(180)";
295 nccurve(r2)(muxr2) "posA(out)", "angleA(180)", "posB(inpa)", "angleB(0)";
296 nccurve(r1)(muxout) "posA(out)", "angleA(0)", "posB(inpb)", "angleB(-90)";
297 nccurve(r2)(muxout) "posA(out)", "angleA(180)", "posB(inpa)", "angleB(-90)";
299 nccurve(a)(muxout) "angleA(-90)", "angleB(180)", "posB(sel)";
300 nccurve(a)(muxr1) "angleA(180)", "angleB(-90)", "posB(sel)";
301 nccurve(a)(muxr2) "angleA(180)", "angleB(-90)", "posB(sel)";
302 ncline(muxout)(out) "posA(out)";
305 \todo{Don't split registers in this image?}
306 \placeexample[here][ex:NormalComplete]{Simple architecture consisting of an adder and a
308 \startcombination[2*1]
309 {\typebufferlam{NormalComplete}}{Core description in normal form.}
310 {\boxedgraphic{NormalComplete}}{The architecture described by the normal form.}
315 \subsection[sec:normalization:intendednormalform]{Intended normal form definition}
316 Now we have some intuition for the normal form, we can describe how we want
317 the normal form to look like in a slightly more formal manner. The following
318 EBNF-like description completely captures the intended structure (and
319 generates a subset of GHC's core format).
321 Some clauses have an expression listed in parentheses. These are conditions
322 that need to apply to the clause.
324 \defref{intended normal form definition}
325 \todo{Fix indentation}
327 \italic{normal} := \italic{lambda}
328 \italic{lambda} := λvar.\italic{lambda} (representable(var))
330 \italic{toplet} := letrec [\italic{binding}...] in var (representable(var))
331 \italic{binding} := var = \italic{rhs} (representable(rhs))
332 -- State packing and unpacking by coercion
333 | var0 = var1 ▶ State ty (lvar(var1))
334 | var0 = var1 ▶ ty (var1 :: State ty ∧ lvar(var1))
335 \italic{rhs} := userapp
338 | case var of C a0 ... an -> ai (lvar(var))
340 | case var of (lvar(var))
341 [ DEFAULT -> var ] (lvar(var))
342 C0 w0,0 ... w0,n -> var0
344 Cm wm,0 ... wm,n -> varm (\forall{}i \forall{}j, wi,j \neq vari, lvar(vari))
345 \italic{userapp} := \italic{userfunc}
346 | \italic{userapp} {userarg}
347 \italic{userfunc} := var (gvar(var))
348 \italic{userarg} := var (lvar(var))
349 \italic{builtinapp} := \italic{builtinfunc}
350 | \italic{builtinapp} \italic{builtinarg}
351 \italic{builtinfunc} := var (bvar(var))
352 \italic{builtinarg} := var (representable(var) ∧ lvar(var))
353 | \italic{partapp} (partapp :: a -> b)
354 | \italic{coreexpr} (¬representable(coreexpr) ∧ ¬(coreexpr :: a -> b))
355 \italic{partapp} := \italic{userapp} | \italic{builtinapp}
358 \todo{There can still be other casts around (which the code can handle,
359 e.g., ignore), which still need to be documented here}
361 When looking at such a program from a hardware perspective, the top level
362 lambda's define the input ports. The variable reference in the body of
363 the recursive let expression is the output port. Most function
364 applications bound by the let expression define a component
365 instantiation, where the input and output ports are mapped to local
366 signals or arguments. Some of the others use a builtin construction (\eg
367 the \lam{case} expression) or call a builtin function (\eg \lam{+} or
368 \lam{map}). For these, a hardcoded \small{VHDL} translation is
371 \section[sec:normalization:transformation]{Transformation notation}
372 To be able to concisely present transformations, we use a specific format
373 for them. It is a simple format, similar to one used in logic reasoning.
375 Such a transformation description looks like the following.
380 <original expression>
381 -------------------------- <expression conditions>
382 <transformed expresssion>
387 This format desribes a transformation that applies to \lam{<original
388 expresssion>} and transforms it into \lam{<transformed expression>}, assuming
389 that all conditions apply. In this format, there are a number of placeholders
390 in pointy brackets, most of which should be rather obvious in their meaning.
391 Nevertheless, we will more precisely specify their meaning below:
393 \startdesc{<original expression>} The expression pattern that will be matched
394 against (subexpressions of) the expression to be transformed. We call this a
395 pattern, because it can contain \emph{placeholders} (variables), which match
396 any expression or binder. Any such placeholder is said to be \emph{bound} to
397 the expression it matches. It is convention to use an uppercase letter (\eg
398 \lam{M} or \lam{E}) to refer to any expression (including a simple variable
399 reference) and lowercase letters (\eg \lam{v} or \lam{b}) to refer to
400 (references to) binders.
402 For example, the pattern \lam{a + B} will match the expression
403 \lam{v + (2 * w)} (binding \lam{a} to \lam{v} and \lam{B} to
404 \lam{(2 * w)}), but not \lam{(2 * w) + v}.
407 \startdesc{<expression conditions>}
408 These are extra conditions on the expression that is matched. These
409 conditions can be used to further limit the cases in which the
410 transformation applies, commonly to prevent a transformation from
411 causing a loop with itself or another transformation.
413 Only if these conditions are \emph{all} true, the transformation
417 \startdesc{<context conditions>}
418 These are a number of extra conditions on the context of the function. In
419 particular, these conditions can require some (other) top level function to be
420 present, whose value matches the pattern given here. The format of each of
421 these conditions is: \lam{binder = <pattern>}.
423 Typically, the binder is some placeholder bound in the \lam{<original
424 expression>}, while the pattern contains some placeholders that are used in
425 the \lam{transformed expression}.
427 Only if a top level binder exists that matches each binder and pattern,
428 the transformation applies.
431 \startdesc{<transformed expression>}
432 This is the expression template that is the result of the transformation. If, looking
433 at the above three items, the transformation applies, the \lam{<original
434 expression>} is completely replaced with the \lam{<transformed expression>}.
435 We call this a template, because it can contain placeholders, referring to
436 any placeholder bound by the \lam{<original expression>} or the
437 \lam{<context conditions>}. The resulting expression will have those
438 placeholders replaced by the values bound to them.
440 Any binder (lowercase) placeholder that has no value bound to it yet will be
441 bound to (and replaced with) a fresh binder.
444 \startdesc{<context additions>}
445 These are templates for new functions to add to the context. This is a way
446 to have a transformation create new top level functions.
448 Each addition has the form \lam{binder = template}. As above, any
449 placeholder in the addition is replaced with the value bound to it, and any
450 binder placeholder that has no value bound to it yet will be bound to (and
451 replaced with) a fresh binder.
454 As an example, we'll look at η-abstraction:
458 -------------- \lam{E} does not occur on a function position in an application
459 λx.E x \lam{E} is not a lambda abstraction.
462 η-abstraction is a well known transformation from lambda calculus. What
463 this transformation does, is take any expression that has a function type
464 and turn it into a lambda expression (giving an explicit name to the
465 argument). There are some extra conditions that ensure that this
466 transformation does not apply infinitely (which are not necessarily part
467 of the conventional definition of η-abstraction).
469 Consider the following function, which is a fairly obvious way to specify a
470 simple ALU (Note that \in{example}[ex:AddSubAlu] shows the normal form of this
471 function). The parentheses around the \lam{+} and \lam{-} operators are
472 commonly used in Haskell to show that the operators are used as normal
473 functions, instead of \emph{infix} operators (\eg, the operators appear
474 before their arguments, instead of in between).
477 alu :: Bit -> Word -> Word -> Word
478 alu = λopcode. case opcode of
483 There are a few subexpressions in this function to which we could possibly
484 apply the transformation. Since the pattern of the transformation is only
485 the placeholder \lam{E}, any expression will match that. Whether the
486 transformation applies to an expression is thus solely decided by the
487 conditions to the right of the transformation.
489 We will look at each expression in the function in a top down manner. The
490 first expression is the entire expression the function is bound to.
493 λopcode. case opcode of
498 As said, the expression pattern matches this. The type of this expression is
499 \lam{Bit -> Word -> Word -> Word}, which matches \lam{a -> b} (Note that in
500 this case \lam{a = Bit} and \lam{b = Word -> Word -> Word}).
502 Since this expression is at top level, it does not occur at a function
503 position of an application. However, The expression is a lambda abstraction,
504 so this transformation does not apply.
506 The next expression we could apply this transformation to, is the body of
507 the lambda abstraction:
515 The type of this expression is \lam{Word -> Word -> Word}, which again
516 matches \lam{a -> b}. The expression is the body of a lambda expression, so
517 it does not occur at a function position of an application. Finally, the
518 expression is not a lambda abstraction but a case expression, so all the
519 conditions match. There are no context conditions to match, so the
520 transformation applies.
522 By now, the placeholder \lam{E} is bound to the entire expression. The
523 placeholder \lam{x}, which occurs in the replacement template, is not bound
524 yet, so we need to generate a fresh binder for that. Let's use the binder
525 \lam{a}. This results in the following replacement expression:
533 Continuing with this expression, we see that the transformation does not
534 apply again (it is a lambda expression). Next we look at the body of this
543 Here, the transformation does apply, binding \lam{E} to the entire
544 expression and \lam{x} to the fresh binder \lam{b}, resulting in the
553 Again, the transformation does not apply to this lambda abstraction, so we
554 look at its body. For brevity, we'll put the case expression on one line from
558 (case opcode of Low -> (+); High -> (-)) a b
561 The type of this expression is \lam{Word}, so it does not match \lam{a -> b}
562 and the transformation does not apply. Next, we have two options for the
563 next expression to look at: The function position and argument position of
564 the application. The expression in the argument position is \lam{b}, which
565 has type \lam{Word}, so the transformation does not apply. The expression in
566 the function position is:
569 (case opcode of Low -> (+); High -> (-)) a
572 Obviously, the transformation does not apply here, since it occurs in
573 function position (which makes the second condition false). In the same
574 way the transformation does not apply to both components of this
575 expression (\lam{case opcode of Low -> (+); High -> (-)} and \lam{a}), so
576 we'll skip to the components of the case expression: The scrutinee and
577 both alternatives. Since the opcode is not a function, it does not apply
580 The first alternative is \lam{(+)}. This expression has a function type
581 (the operator still needs two arguments). It does not occur in function
582 position of an application and it is not a lambda expression, so the
583 transformation applies.
585 We look at the \lam{<original expression>} pattern, which is \lam{E}.
586 This means we bind \lam{E} to \lam{(+)}. We then replace the expression
587 with the \lam{<transformed expression>}, replacing all occurences of
588 \lam{E} with \lam{(+)}. In the \lam{<transformed expression>}, the This gives us the replacement expression:
589 \lam{λx.(+) x} (A lambda expression binding \lam{x}, with a body that
590 applies the addition operator to \lam{x}).
592 The complete function then becomes:
594 (case opcode of Low -> λa1.(+) a1; High -> (-)) a
597 Now the transformation no longer applies to the complete first alternative
598 (since it is a lambda expression). It does not apply to the addition
599 operator again, since it is now in function position in an application. It
600 does, however, apply to the application of the addition operator, since
601 that is neither a lambda expression nor does it occur in function
602 position. This means after one more application of the transformation, the
606 (case opcode of Low -> λa1.λb1.(+) a1 b1; High -> (-)) a
609 The other alternative is left as an exercise to the reader. The final
610 function, after applying η-abstraction until it does no longer apply is:
613 alu :: Bit -> Word -> Word -> Word
614 alu = λopcode.λa.b. (case opcode of
615 Low -> λa1.λb1 (+) a1 b1
616 High -> λa2.λb2 (-) a2 b2) a b
619 \subsection{Transformation application}
620 In this chapter we define a number of transformations, but how will we apply
621 these? As stated before, our normal form is reached as soon as no
622 transformation applies anymore. This means our application strategy is to
623 simply apply any transformation that applies, and continuing to do that with
624 the result of each transformation.
626 In particular, we define no particular order of transformations. Since
627 transformation order should not influence the resulting normal form,
628 this leaves the implementation free to choose any application order that
629 results in an efficient implementation. Unfortunately this is not
630 entirely true for the current set of transformations. See
631 \in{section}[sec:normalization:non-determinism] for a discussion of this
634 When applying a single transformation, we try to apply it to every (sub)expression
635 in a function, not just the top level function body. This allows us to
636 keep the transformation descriptions concise and powerful.
638 \subsection{Definitions}
639 In the following sections, we will be using a number of functions and
640 notations, which we will define here.
642 \subsubsection{Concepts}
643 A \emph{global variable} is any variable (binder) that is bound at the
644 top level of a program, or an external module. A \emph{local variable} is any
645 other variable (\eg, variables local to a function, which can be bound by
646 lambda abstractions, let expressions and pattern matches of case
647 alternatives). Note that this is a slightly different notion of global versus
648 local than what \small{GHC} uses internally.
649 \defref{global variable} \defref{local variable}
651 A \emph{hardware representable} (or just \emph{representable}) type or value
652 is (a value of) a type that we can generate a signal for in hardware. For
653 example, a bit, a vector of bits, a 32 bit unsigned word, etc. Values that are
654 not runtime representable notably include (but are not limited to): Types,
655 dictionaries, functions.
656 \defref{representable}
658 A \emph{builtin function} is a function supplied by the Cλash framework, whose
659 implementation is not valid Cλash. The implementation is of course valid
660 Haskell, for simulation, but it is not expressable in Cλash.
661 \defref{builtin function} \defref{user-defined function}
663 For these functions, Cλash has a \emph{builtin hardware translation}, so calls
664 to these functions can still be translated. These are functions like
665 \lam{map}, \lam{hwor} and \lam{length}.
667 A \emph{user-defined} function is a function for which we do have a Cλash
668 implementation available.
670 \subsubsection{Predicates}
671 Here, we define a number of predicates that can be used below to concisely
672 specify conditions.\refdef{global variable}
674 \emph{gvar(expr)} is true when \emph{expr} is a variable that references a
675 global variable. It is false when it references a local variable.
677 \refdef{local variable}\emph{lvar(expr)} is the complement of \emph{gvar}; it is true when \emph{expr}
678 references a local variable, false when it references a global variable.
680 \refdef{representable}\emph{representable(expr)} or \emph{representable(var)} is true when
681 \emph{expr} or \emph{var} is \emph{representable}.
683 \subsection[sec:normalization:uniq]{Binder uniqueness}
684 A common problem in transformation systems, is binder uniqueness. When not
685 considering this problem, it is easy to create transformations that mix up
686 bindings and cause name collisions. Take for example, the following core
690 (λa.λb.λc. a * b * c) x c
693 By applying β-reduction (see \in{section}[sec:normalization:beta]) once,
694 we can simplify this expression to:
700 Now, we have replaced the \lam{a} binder with a reference to the \lam{x}
701 binder. No harm done here. But note that we see multiple occurences of the
702 \lam{c} binder. The first is a binding occurence, to which the second refers.
703 The last, however refers to \emph{another} instance of \lam{c}, which is
704 bound somewhere outside of this expression. Now, if we would apply beta
705 reduction without taking heed of binder uniqueness, we would get:
711 This is obviously not what was supposed to happen! The root of this problem is
712 the reuse of binders: Identical binders can be bound in different scopes, such
713 that only the inner one is \quote{visible} in the inner expression. In the example
714 above, the \lam{c} binder was bound outside of the expression and in the inner
715 lambda expression. Inside that lambda expression, only the inner \lam{c} is
718 There are a number of ways to solve this. \small{GHC} has isolated this
719 problem to their binder substitution code, which performs \emph{deshadowing}
720 during its expression traversal. This means that any binding that shadows
721 another binding on a higher level is replaced by a new binder that does not
722 shadow any other binding. This non-shadowing invariant is enough to prevent
723 binder uniqueness problems in \small{GHC}.
725 In our transformation system, maintaining this non-shadowing invariant is
726 a bit harder to do (mostly due to implementation issues, the prototype doesn't
727 use \small{GHC}'s subsitution code). Also, the following points can be
731 \item Deshadowing does not guarantee overall uniqueness. For example, the
732 following (slightly contrived) expression shows the identifier \lam{x} bound in
733 two seperate places (and to different values), even though no shadowing
737 (let x = 1 in x) + (let x = 2 in x)
740 \item In our normal form (and the resulting \small{VHDL}), all binders
741 (signals) within the same function (entity) will end up in the same
742 scope. To allow this, all binders within the same function should be
745 \item When we know that all binders in an expression are unique, moving around
746 or removing a subexpression will never cause any binder conflicts. If we have
747 some way to generate fresh binders, introducing new subexpressions will not
748 cause any problems either. The only way to cause conflicts is thus to
749 duplicate an existing subexpression.
752 Given the above, our prototype maintains a unique binder invariant. This
753 means that in any given moment during normalization, all binders \emph{within
754 a single function} must be unique. To achieve this, we apply the following
757 \todo{Define fresh binders and unique supplies}
760 \item Before starting normalization, all binders in the function are made
761 unique. This is done by generating a fresh binder for every binder used. This
762 also replaces binders that did not cause any conflict, but it does ensure that
763 all binders within the function are generated by the same unique supply.
764 \refdef{fresh binder}
765 \item Whenever a new binder must be generated, we generate a fresh binder that
766 is guaranteed to be different from \emph{all binders generated so far}. This
767 can thus never introduce duplication and will maintain the invariant.
768 \item Whenever (a part of) an expression is duplicated (for example when
769 inlining), all binders in the expression are replaced with fresh binders
770 (using the same method as at the start of normalization). These fresh binders
771 can never introduce duplication, so this will maintain the invariant.
772 \item Whenever we move part of an expression around within the function, there
773 is no need to do anything special. There is obviously no way to introduce
774 duplication by moving expressions around. Since we know that each of the
775 binders is already unique, there is no way to introduce (incorrect) shadowing
779 \section{Transform passes}
780 In this section we describe the actual transforms.
782 Each transformation will be described informally first, explaining
783 the need for and goal of the transformation. Then, we will formally define
784 the transformation using the syntax introduced in
785 \in{section}[sec:normalization:transformation].
787 \subsection{General cleanup}
788 These transformations are general cleanup transformations, that aim to
789 make expressions simpler. These transformations usually clean up the
790 mess left behind by other transformations or clean up expressions to
791 expose new transformation opportunities for other transformations.
793 Most of these transformations are standard optimizations in other
794 compilers as well. However, in our compiler, most of these are not just
795 optimizations, but they are required to get our program into intended
799 \defref{substitution notation}
800 \startframedtext[width=8cm,background=box,frame=no]
801 \startalignment[center]
802 {\tfa Substitution notation}
806 In some of the transformations in this chapter, we need to perform
807 substitution on an expression. Substitution means replacing every
808 occurence of some expression (usually a variable reference) with
811 There have been a lot of different notations used in literature for
812 specifying substitution. The notation that will be used in this report
819 This means expression \lam{E} with all occurences of \lam{A} replaced
824 \subsubsection[sec:normalization:beta]{β-reduction}
825 β-reduction is a well known transformation from lambda calculus, where it is
826 the main reduction step. It reduces applications of lambda abstractions,
827 removing both the lambda abstraction and the application.
829 In our transformation system, this step helps to remove unwanted lambda
830 abstractions (basically all but the ones at the top level). Other
831 transformations (application propagation, non-representable inlining) make
832 sure that most lambda abstractions will eventually be reducable by
835 Note that β-reduction also works on type lambda abstractions and type
836 applications as well. This means the substitution below also works on
837 type variables, in the case that the binder is a type variable and teh
838 expression applied to is a type.
855 \transexample{beta}{β-reduction}{from}{to}
865 \transexample{beta-type}{β-reduction for type abstractions}{from}{to}
867 \subsubsection{Empty let removal}
868 This transformation is simple: It removes recursive lets that have no bindings
869 (which usually occurs when unused let binding removal removes the last
872 Note that there is no need to define this transformation for
873 non-recursive lets, since they always contain exactly one binding.
883 \subsubsection[sec:normalization:simplelet]{Simple let binding removal}
884 This transformation inlines simple let bindings, that bind some
885 binder to some other binder instead of a more complex expression (\ie
888 This transformation is not needed to get an expression into intended
889 normal form (since these bindings are part of the intended normal
890 form), but makes the resulting \small{VHDL} a lot shorter.
892 \refdef{substitution notation}
902 ----------------------------- \lam{b} is a variable reference
903 letrec \lam{ai} ≠ \lam{b}
916 \subsubsection{Unused let binding removal}
917 This transformation removes let bindings that are never used.
918 Occasionally, \GHC's desugarer introduces some unused let bindings.
920 This normalization pass should really be unneeded to get into intended normal form
921 (since unused bindings are not forbidden by the normal form), but in practice
922 the desugarer or simplifier emits some unused bindings that cannot be
923 normalized (e.g., calls to a \type{PatError}\todo{Check this name}). Also,
924 this transformation makes the resulting \small{VHDL} a lot shorter.
926 \todo{Don't use old-style numerals in transformations}
935 M \lam{ai} does not occur free in \lam{M}
936 ---------------------------- \forall j, 0 ≤ j ≤ n, j ≠ i (\lam{ai} does not occur free in \lam{Ej})
950 \subsubsection{Cast propagation / simplification}
951 This transform pushes casts down into the expression as far as possible.
952 Since its exact role and need is not clear yet, this transformation is
955 \todo{Cast propagation}
957 \subsubsection{Top level binding inlining}
958 This transform takes simple top level bindings generated by the
959 \small{GHC} compiler. \small{GHC} sometimes generates very simple
960 \quote{wrapper} bindings, which are bound to just a variable
961 reference, or a partial application to constants or other variable
964 Note that this transformation is completely optional. It is not
965 required to get any function into intended normal form, but it does help making
966 the resulting VHDL output easier to read (since it removes a bunch of
967 components that are really boring).
969 This transform takes any top level binding generated by the compiler,
970 whose normalized form contains only a single let binding.
973 x = λa0 ... λan.let y = E in y
976 -------------------------------------- \lam{x} is generated by the compiler
977 λa0 ... λan.let y = E in y
981 (+) :: Word -> Word -> Word
982 (+) = GHC.Num.(+) @Word \$dNum
987 GHC.Num.(+) @ Alu.Word \$dNum a b
990 \transexample{toplevelinline}{Top level binding inlining}{from}{to}
992 \in{Example}[ex:trans:toplevelinline] shows a typical application of
993 the addition operator generated by \GHC. The type and dictionary
994 arguments used here are described in
995 \in{Section}[section:prototype:polymorphism].
997 Without this transformation, there would be a \lam{(+)} entity
998 in the \VHDL which would just add its inputs. This generates a
999 lot of overhead in the \VHDL, which is particularly annoying
1000 when browsing the generated RTL schematic (especially since most
1001 non-alphanumerics, like all characters in \lam{(+)}, are not
1002 allowed in \VHDL architecture names\footnote{Technically, it is
1003 allowed to use non-alphanumerics when using extended
1004 identifiers, but it seems that none of the tooling likes
1005 extended identifiers in filenames, so it effectively doesn't
1006 work.}, so the entity would be called \quote{w7aA7f} or
1007 something similarly unreadable and autogenerated).
1009 \subsection{Program structure}
1010 These transformations are aimed at normalizing the overall structure
1011 into the intended form. This means ensuring there is a lambda abstraction
1012 at the top for every argument (input port or current state), putting all
1013 of the other value definitions in let bindings and making the final
1014 return value a simple variable reference.
1016 \subsubsection[sec:normalization:eta]{η-abstraction}
1017 This transformation makes sure that all arguments of a function-typed
1018 expression are named, by introducing lambda expressions. When combined with
1019 β-reduction and non-representable binding inlining, all function-typed
1020 expressions should be lambda abstractions or global identifiers.
1024 -------------- \lam{E} is not the first argument of an application.
1025 λx.E x \lam{E} is not a lambda abstraction.
1026 \lam{x} is a variable that does not occur free in \lam{E}.
1036 foo = λa.λx.(case a of
1041 \transexample{eta}{η-abstraction}{from}{to}
1043 \subsubsection[sec:normalization:appprop]{Application propagation}
1044 This transformation is meant to propagate application expressions downwards
1045 into expressions as far as possible. This allows partial applications inside
1046 expressions to become fully applied and exposes new transformation
1047 opportunities for other transformations (like β-reduction and
1050 Since all binders in our expression are unique (see
1051 \in{section}[sec:normalization:uniq]), there is no risk that we will
1052 introduce unintended shadowing by moving an expression into a lower
1053 scope. Also, since only move expression into smaller scopes (down into
1054 our expression), there is no risk of moving a variable reference out
1055 of the scope in which it is defined.
1058 (letrec binds in E) M
1059 ------------------------
1079 \transexample{appproplet}{Application propagation for a let expression}{from}{to}
1107 \transexample{apppropcase}{Application propagation for a case expression}{from}{to}
1109 \subsubsection[sec:normalization:letrecurse]{Let recursification}
1110 This transformation makes all non-recursive lets recursive. In the
1111 end, we want a single recursive let in our normalized program, so all
1112 non-recursive lets can be converted. This also makes other
1113 transformations simpler: They can simply assume all lets are
1121 ------------------------------------------
1128 \subsubsection{Let flattening}
1129 This transformation puts nested lets in the same scope, by lifting the
1130 binding(s) of the inner let into the outer let. Eventually, this will
1131 cause all let bindings to appear in the same scope.
1133 This transformation only applies to recursive lets, since all
1134 non-recursive lets will be made recursive (see
1135 \in{section}[sec:normalization:letrecurse]).
1137 Since we are joining two scopes together, there is no risk of moving a
1138 variable reference out of the scope where it is defined.
1144 ai = (letrec bindings in M)
1149 ------------------------------------------
1184 \transexample{letflat}{Let flattening}{from}{to}
1186 \subsubsection{Return value simplification}
1187 This transformation ensures that the return value of a function is always a
1188 simple local variable reference.
1190 Currently implemented using lambda simplification, let simplification, and
1191 top simplification. Should change into something like the following, which
1192 works only on the result of a function instead of any subexpression. This is
1193 achieved by the contexts, like \lam{x = E}, though this is strictly not
1194 correct (you could read this as "if there is any function \lam{x} that binds
1195 \lam{E}, any \lam{E} can be transformed, while we only mean the \lam{E} that
1196 is bound by \lam{x}. This might need some extra notes or something).
1198 Note that the return value is not simplified if its not representable.
1199 Otherwise, this would cause a direct loop with the inlining of
1200 unrepresentable bindings. If the return value is not
1201 representable because it has a function type, η-abstraction should
1202 make sure that this transformation will eventually apply. If the value
1203 is not representable for other reasons, the function result itself is
1204 not representable, meaning this function is not translatable anyway.
1207 x = E \lam{E} is representable
1208 ~ \lam{E} is not a lambda abstraction
1209 E \lam{E} is not a let expression
1210 --------------------------- \lam{E} is not a local variable reference
1216 ~ \lam{E} is representable
1217 E \lam{E} is not a let expression
1218 --------------------------- \lam{E} is not a local variable reference
1223 x = λv0 ... λvn.let ... in E
1224 ~ \lam{E} is representable
1225 E \lam{E} is not a local variable reference
1226 -----------------------------
1235 x = letrec x = add 1 2 in x
1238 \transexample{retvalsimpl}{Return value simplification}{from}{to}
1240 \todo{More examples}
1242 \subsection[sec:normalization:argsimpl]{Representable arguments simplification}
1243 This section contains just a single transformation that deals with
1244 representable arguments in applications. Non-representable arguments are
1245 handled by the transformations in
1246 \in{section}[sec:normalization:nonrep].
1248 This transformation ensures that all representable arguments will become
1249 references to local variables. This ensures they will become references
1250 to local signals in the resulting \small{VHDL}, which is required due to
1251 limitations in the component instantiation code in \VHDL (one can only
1252 assign a signal or constant to an input port). By ensuring that all
1253 arguments are always simple variable references, we always have a signal
1254 available to map to the input ports.
1256 To reduce a complex expression to a simple variable reference, we create
1257 a new let expression around the application, which binds the complex
1258 expression to a new variable. The original function is then applied to
1261 \refdef{global variable}
1262 Note that references to \emph{global variables} (like a top level
1263 function without arguments, but also an argumentless dataconstructors
1264 like \lam{True}) are also simplified. Only local variables generate
1265 signals in the resulting architecture. Even though argumentless
1266 dataconstructors generate constants in generated \VHDL code and could be
1267 mapped to an input port directly, they are still simplified to make the
1268 normal form more regular.
1270 \refdef{representable}
1273 -------------------- \lam{N} is representable
1274 letrec x = N in M x \lam{N} is not a local variable reference
1276 \refdef{local variable}
1283 letrec x = add a 1 in add x 1
1286 \transexample{argsimpl}{Argument simplification}{from}{to}
1288 \subsection[sec:normalization:builtins]{Builtin functions}
1289 This section deals with (arguments to) builtin functions. In the
1290 intended normal form definition\refdef{intended normal form definition}
1291 we can see that there are three sorts of arguments a builtin function
1295 \item A representable local variable reference. This is the most
1296 common argument to any function. The argument simplification
1297 transformation described in \in{section}[sec:normalization:argsimpl]
1298 makes sure that \emph{any} representable argument to \emph{any}
1299 function (including builtin functions) is turned into a local variable
1301 \item (A partial application of) a top level function (either builtin on
1302 user-defined). The function extraction transformation described in
1303 this section takes care of turning every functiontyped argument into
1304 (a partial application of) a top level function.
1305 \item Any expression that is not representable and does not have a
1306 function type. Since these can be any expression, there is no
1307 transformation needed. Note that this category is exactly all
1308 expressions that are not transformed by the transformations for the
1309 previous two categories. This means that \emph{any} core expression
1310 that is used as an argument to a builtin function will be either
1311 transformed into one of the above categories, or end up in this
1312 categorie. In any case, the result is in normal form.
1315 As noted, the argument simplification will handle any representable
1316 arguments to a builtin function. The following transformation is needed
1317 to handle non-representable arguments with a function type, all other
1318 non-representable arguments don't need any special handling.
1320 \subsubsection[sec:normalization:funextract]{Function extraction}
1321 This transform deals with function-typed arguments to builtin
1323 Since builtin functions cannot be specialized (see
1324 \in{section}[sec:normalization:specialize]) to remove the arguments,
1325 these arguments are extracted into a new global function instead. In
1326 other words, we create a new top level function that has exactly the
1327 extracted argument as its body. This greatly simplifies the
1328 translation rules needed for builtin functions, since they only need
1329 to handle (partial applications of) top level functions.
1331 Any free variables occuring in the extracted arguments will become
1332 parameters to the new global function. The original argument is replaced
1333 with a reference to the new function, applied to any free variables from
1334 the original argument.
1336 This transformation is useful when applying higher order builtin functions
1337 like \hs{map} to a lambda abstraction, for example. In this case, the code
1338 that generates \small{VHDL} for \hs{map} only needs to handle top level functions and
1339 partial applications, not any other expression (such as lambda abstractions or
1340 even more complicated expressions).
1343 M N \lam{M} is (a partial aplication of) a builtin function.
1344 --------------------- \lam{f0 ... fn} are all free local variables of \lam{N}
1345 M (x f0 ... fn) \lam{N :: a -> b}
1346 ~ \lam{N} is not a (partial application of) a top level function
1351 addList = λb.λxs.map (λa . add a b) xs
1355 addList = λb.λxs.map (f b) xs
1360 \transexample{funextract}{Function extraction}{from}{to}
1362 Note that the function \lam{f} will still need normalization after
1365 \subsection{Case normalisation}
1366 \subsubsection{Scrutinee simplification}
1367 This transform ensures that the scrutinee of a case expression is always
1368 a simple variable reference.
1373 ----------------- \lam{E} is not a local variable reference
1392 \transexample{letflat}{Case normalisation}{from}{to}
1395 \subsubsection{Case simplification}
1396 This transformation ensures that all case expressions become normal form. This
1397 means they will become one of:
1399 \item An extractor case with a single alternative that picks a single field
1400 from a datatype, \eg \lam{case x of (a, b) -> a}.
1401 \item A selector case with multiple alternatives and only wild binders, that
1402 makes a choice between expressions based on the constructor of another
1403 expression, \eg \lam{case x of Low -> a; High -> b}.
1406 \defref{wild binder}
1409 C0 v0,0 ... v0,m -> E0
1411 Cn vn,0 ... vn,m -> En
1412 --------------------------------------------------- \forall i \forall j, 0 ≤ i ≤ n, 0 ≤ i < m (\lam{wi,j} is a wild (unused) binder)
1414 v0,0 = case E of C0 v0,0 .. v0,m -> v0,0
1416 v0,m = case E of C0 v0,0 .. v0,m -> v0,m
1418 vn,m = case E of Cn vn,0 .. vn,m -> vn,m
1424 C0 w0,0 ... w0,m -> x0
1426 Cn wn,0 ... wn,m -> xn
1428 \todo{Check the subscripts of this transformation}
1430 Note that this transformation applies to case expressions with any
1431 scrutinee. If the scrutinee is a complex expression, this might result
1432 in duplicate hardware. An extra condition to only apply this
1433 transformation when the scrutinee is already simple (effectively
1434 causing this transformation to be only applied after the scrutinee
1435 simplification transformation) might be in order.
1437 \fxnote{This transformation specified like this is complicated and misses
1438 conditions to prevent looping with itself. Perhaps it should be split here for
1457 \transexample{selcasesimpl}{Selector case simplification}{from}{to}
1465 b = case a of (,) b c -> b
1466 c = case a of (,) b c -> c
1473 \transexample{excasesimpl}{Extractor case simplification}{from}{to}
1475 \refdef{selector case}
1476 In \in{example}[ex:trans:excasesimpl] the case expression is expanded
1477 into multiple case expressions, including a pretty useless expression
1478 (that is neither a selector or extractor case). This case can be
1479 removed by the Case removal transformation in
1480 \in{section}[sec:transformation:caseremoval].
1482 \subsubsection[sec:transformation:caseremoval]{Case removal}
1483 This transform removes any case expression with a single alternative and
1486 These "useless" case expressions are usually leftovers from case simplification
1487 on extractor case (see the previous example).
1492 ---------------------- \lam{\forall i, 0 ≤ i ≤ m} (\lam{vi} does not occur free in E)
1505 \transexample{caserem}{Case removal}{from}{to}
1507 \subsection[sec:normalization:nonrep]{Removing unrepresentable values}
1508 The transformations in this section are aimed at making all the
1509 values used in our expression representable. There are two main
1510 transformations that are applied to \emph{all} unrepresentable let
1511 bindings and function arguments. These are meant to address three
1512 different kinds of unrepresentable values: Polymorphic values, higher
1513 order values and literals. The transformation are described generically:
1514 They apply to all non-representable values. However, non-representable
1515 values that don't fall into one of these three categories will be moved
1516 around by these transformations but are unlikely to completely
1517 disappear. They usually mean the program was not valid in the first
1518 place, because unsupported types were used (for example, a program using
1521 Each of these three categories will be detailed below, followed by the
1522 actual transformations.
1524 \subsubsection{Removing Polymorphism}
1525 As noted in \in{section}[sec:prototype:polymporphism],
1526 polymorphism is made explicit in Core through type and
1527 dictionary arguments. To remove the polymorphism from a
1528 function, we can simply specialize the polymorphic function for
1529 the particular type applied to it. The same goes for dictionary
1530 arguments. To remove polymorphism from let bound values, we
1531 simply inline the let bindings that have a polymorphic type,
1532 which should (eventually) make sure that the polymorphic
1533 expression is applied to a type and/or dictionary, which can
1534 then be removed by β-reduction (\in{section}[sec:normalization:beta]).
1536 Since both type and dictionary arguments are not representable,
1537 \refdef{representable}
1538 the non-representable argument specialization and
1539 non-representable let binding inlining transformations below
1540 take care of exactly this.
1542 There is one case where polymorphism cannot be completely
1543 removed: Builtin functions are still allowed to be polymorphic
1544 (Since we have no function body that we could properly
1545 specialize). However, the code that generates \VHDL for builtin
1546 functions knows how to handle this, so this is not a problem.
1548 \subsubsection{Defunctionalization}
1549 These transformations remove higher order expressions from our
1550 program, making all values first-order.
1552 Higher order values are always introduced by lambda abstractions, none
1553 of the other Core expression elements can introduce a function type.
1554 However, other expressions can \emph{have} a function type, when they
1555 have a lambda expression in their body.
1557 For example, the following expression is a higher order expression
1558 that is not a lambda expression itself:
1560 \refdef{id function}
1567 The reference to the \lam{id} function shows that we can introduce a
1568 higher order expression in our program without using a lambda
1569 expression directly. However, inside the definition of the \lam{id}
1570 function, we can be sure that a lambda expression is present.
1572 Looking closely at the definition of our normal form in
1573 \in{section}[sec:normalization:intendednormalform], we can see that
1574 there are three possibilities for higher order values to appear in our
1575 intended normal form:
1578 \item[item:toplambda] Lambda abstractions can appear at the highest level of a
1579 top level function. These lambda abstractions introduce the
1580 arguments (input ports / current state) of the function.
1581 \item[item:builtinarg] (Partial applications of) top level functions can appear as an
1582 argument to a builtin function.
1583 \item[item:completeapp] (Partial applications of) top level functions can appear in
1584 function position of an application. Since a partial application
1585 cannot appear anywhere else (except as builtin function arguments),
1586 all partial applications are applied, meaning that all applications
1587 will become complete applications. However, since application of
1588 arguments happens one by one, in the expression:
1592 the subexpression \lam{f 1} has a function type. But this is
1593 allowed, since it is inside a complete application.
1596 We will take a typical function with some higher order values as an
1597 example. The following function takes two arguments: a \lam{Bit} and a
1598 list of numbers. Depending on the first argument, each number in the
1599 list is doubled, or the list is returned unmodified. For the sake of
1600 the example, no polymorphism is shown. In reality, at least map would
1604 λy.let double = λx. x + x in
1610 This example shows a number of higher order values that we cannot
1611 translate to \VHDL directly. The \lam{double} binder bound in the let
1612 expression has a function type, as well as both of the alternatives of
1613 the case expression. The first alternative is a partial application of
1614 the \lam{map} builtin function, whereas the second alternative is a
1617 To reduce all higher order values to one of the above items, a number
1618 of transformations we've already seen are used. The η-abstraction
1619 transformation from \in{section}[sec:normalization:eta] ensures all
1620 function arguments are introduced by lambda abstraction on the highest
1621 level of a function. These lambda arguments are allowed because of
1622 \in{item}[item:toplambda] above. After η-abstraction, our example
1623 becomes a bit bigger:
1626 λy.λq.(let double = λx. x + x in
1633 η-abstraction also introduces extra applications (the application of
1634 the let expression to \lam{q} in the above example). These
1635 applications can then propagated down by the application propagation
1636 transformation (\in{section}[sec:normalization:appprop]). In our
1637 example, the \lam{q} and \lam{r} variable will be propagated into the
1638 let expression and then into the case expression:
1641 λy.λq.let double = λx. x + x in
1647 This propagation makes higher order values become applied (in
1648 particular both of the alternatives of the case now have a
1649 representable type). Completely applied top level functions (like the
1650 first alternative) are now no longer invalid (they fall under
1651 \in{item}[item:completeapp] above). (Completely) applied lambda
1652 abstractions can be removed by β-abstraction. For our example,
1653 applying β-abstraction results in the following:
1656 λy.λq.let double = λx. x + x in
1662 As you can see in our example, all of this moves applications towards
1663 the higher order values, but misses higher order functions bound by
1664 let expressions. The applications cannot be moved towards these values
1665 (since they can be used in multiple places), so the values will have
1666 to be moved towards the applications. This is achieved by inlining all
1667 higher order values bound by let applications, by the
1668 non-representable binding inlining transformation below. When applying
1669 it to our example, we get the following:
1673 Low -> map (λx. x + x) q
1677 We've nearly eliminated all unsupported higher order values from this
1678 expressions. The one that's remaining is the first argument to the
1679 \lam{map} function. Having higher order arguments to a builtin
1680 function like \lam{map} is allowed in the intended normal form, but
1681 only if the argument is a (partial application) of a top level
1682 function. This is easily done by introducing a new top level function
1683 and put the lambda abstraction inside. This is done by the function
1684 extraction transformation from
1685 \in{section}[sec:normalization:funextract].
1693 This also introduces a new function, that we have called \lam{func}:
1699 Note that this does not actually remove the lambda, but now it is a
1700 lambda at the highest level of a function, which is allowed in the
1701 intended normal form.
1703 There is one case that has not been discussed yet. What if the
1704 \lam{map} function in the example above was not a builtin function
1705 but a user-defined function? Then extracting the lambda expression
1706 into a new function would not be enough, since user-defined functions
1707 can never have higher order arguments. For example, the following
1708 expression shows an example:
1711 twice :: (Word -> Word) -> Word -> Word
1712 twice = λf.λa.f (f a)
1714 main = λa.app (λx. x + x) a
1717 This example shows a function \lam{twice} that takes a function as a
1718 first argument and applies that function twice to the second argument.
1719 Again, we've made the function monomorphic for clarity, even though
1720 this function would be a lot more useful if it was polymorphic. The
1721 function \lam{main} uses \lam{twice} to apply a lambda epression twice.
1723 When faced with a user defined function, a body is available for that
1724 function. This means we could create a specialized version of the
1725 function that only works for this particular higher order argument
1726 (\ie, we can just remove the argument and call the specialized
1727 function without the argument). This transformation is detailed below.
1728 Applying this transformation to the example gives:
1731 twice' :: Word -> Word
1732 twice' = λb.(λf.λa.f (f a)) (λx. x + x) b
1737 The \lam{main} function is now in normal form, since the only higher
1738 order value there is the top level lambda expression. The new
1739 \lam{twice'} function is a bit complex, but the entire original body of
1740 the original \lam{twice} function is wrapped in a lambda abstraction
1741 and applied to the argument we've specialized for (\lam{λx. x + x})
1742 and the other arguments. This complex expression can fortunately be
1743 effectively reduced by repeatedly applying β-reduction:
1746 twice' :: Word -> Word
1747 twice' = λb.(b + b) + (b + b)
1750 This example also shows that the resulting normal form might not be as
1751 efficient as we might hope it to be (it is calculating \lam{(b + b)}
1752 twice). This is discussed in more detail in
1753 \in{section}[sec:normalization:duplicatework].
1755 \subsubsection{Literals}
1756 There are a limited number of literals available in Haskell and Core.
1757 \refdef{enumerated types} When using (enumerating) algebraic
1758 datatypes, a literal is just a reference to the corresponding data
1759 constructor, which has a representable type (the algebraic datatype)
1760 and can be translated directly. This also holds for literals of the
1761 \hs{Bool} Haskell type, which is just an enumerated type.
1763 There is, however, a second type of literal that does not have a
1764 representable type: Integer literals. Cλash supports using integer
1765 literals for all three integer types supported (\hs{SizedWord},
1766 \hs{SizedInt} and \hs{RangedWord}). This is implemented using
1767 Haskell's \hs{Num} typeclass, which offers a \hs{fromInteger} method
1768 that converts any \hs{Integer} to the Cλash datatypes.
1770 When \GHC sees integer literals, it will automatically insert calls to
1771 the \hs{fromInteger} method in the resulting Core expression. For
1772 example, the following expression in Haskell creates a 32 bit unsigned
1773 word with the value 1. The explicit type signature is needed, since
1774 there is no context for \GHC to determine the type from otherwise.
1780 This Haskell code results in the following Core expression:
1783 fromInteger @(SizedWord D32) \$dNum (smallInteger 10)
1786 The literal 10 will have the type \lam{GHC.Prim.Int\#}, which is
1787 converted into an \lam{Integer} by \lam{smallInteger}. Finally, the
1788 \lam{fromInteger} function will finally convert this into a
1789 \lam{SizedWord D32}.
1791 Both the \lam{GHC.Prim.Int\#} and \lam{Integer} types are not
1792 representable, and cannot be translated directly. Fortunately, there
1793 is no need to translate them, since \lam{fromInteger} is a builtin
1794 function that knows how to handle these values. However, this does
1795 require that the \lam{fromInteger} function is directly applied to
1796 these non-representable literal values, otherwise errors will occur.
1797 For example, the following expression is not in the intended normal
1798 form, since one of the let bindings has an unrepresentable type
1802 let l = smallInteger 10 in fromInteger @(SizedWord D32) \$dNum l
1805 By inlining these let-bindings, we can ensure that unrepresentable
1806 literals bound by a let binding end up in an application of the
1807 appropriate builtin function, where they are allowed. Since it is
1808 possible that the application of that function is in a different
1809 function than the definition of the literal value, we will always need
1810 to specialize away any unrepresentable literals that are used as
1811 function arguments. The following two transformations do exactly this.
1813 \subsubsection{Non-representable binding inlining}
1814 This transform inlines let bindings that are bound to a
1815 non-representable value. Since we can never generate a signal
1816 assignment for these bindings (we cannot declare a signal assignment
1817 with a non-representable type, for obvious reasons), we have no choice
1818 but to inline the binding to remove it.
1820 As we have seen in the previous sections, inlining these bindings
1821 solves (part of) the polymorphism, higher order values and
1822 unrepresentable literals in an expression.
1824 \refdef{substitution notation}
1834 -------------------------- \lam{Ei} has a non-representable type.
1836 a0 = E0 [ai=>Ei] \vdots
1837 ai-1 = Ei-1 [ai=>Ei]
1838 ai+1 = Ei+1 [ai=>Ei]
1857 x = fromInteger (smallInteger 10)
1859 (λb -> add b 1) (add 1 x)
1862 \transexample{nonrepinline}{Nonrepresentable binding inlining}{from}{to}
1864 \subsubsection[sec:normalization:specialize]{Function specialization}
1865 This transform removes arguments to user-defined functions that are
1866 not representable at runtime. This is done by creating a
1867 \emph{specialized} version of the function that only works for one
1868 particular value of that argument (in other words, the argument can be
1871 Specialization means to create a specialized version of the called
1872 function, with one argument already filled in. As a simple example, in
1873 the following program (this is not actual Core, since it directly uses
1874 a literal with the unrepresentable type \lam{GHC.Prim.Int\#}).
1881 We could specialize the function \lam{f} against the literal argument
1882 1, with the following result:
1889 In some way, this transformation is similar to β-reduction, but it
1890 operates across function boundaries. It is also similar to
1891 non-representable let binding inlining above, since it sort of
1892 \quote{inlines} an expression into a called function.
1894 Special care must be taken when the argument has any free variables.
1895 If this is the case, the original argument should not be removed
1896 completely, but replaced by all the free variables of the expression.
1897 In this way, the original expression can still be evaluated inside the
1900 To prevent us from propagating the same argument over and over, a
1901 simple local variable reference is not propagated (since is has
1902 exactly one free variable, itself, we would only replace that argument
1905 This shows that any free local variables that are not runtime
1906 representable cannot be brought into normal form by this transform. We
1907 rely on an inlining or β-reduction transformation to replace such a
1908 variable with an expression we can propagate again.
1913 x Y0 ... Yi ... Yn \lam{Yi} is not representable
1914 --------------------------------------------- \lam{Yi} is not a local variable reference
1915 x' y0 ... yi-1 f0 ... fm Yi+1 ... Yn \lam{f0 ... fm} are all free local vars of \lam{Yi}
1916 ~ \lam{T0 ... Tn} are the types of \lam{Y0 ... Yn}
1917 x' = λ(y0 :: T0) ... λ(yi-1 :: Ty-1). λf0 ... λfm. λ(yi+1 :: Ty+1) ... λ(yn :: Tn).
1918 E y0 ... yi-1 Yi yi+1 ... yn
1921 This is a bit of a complex transformation. It transforms an
1922 application of the function \lam{x}, where one of the arguments
1923 (\lam{Y_i}) is not representable. A new
1924 function \lam{x'} is created that wraps the body of the old function.
1925 The body of the new function becomes a number of nested lambda
1926 abstractions, one for each of the original arguments that are left
1929 The ith argument is replaced with the free variables of
1930 \lam{Y_i}. Note that we reuse the same binders as those used in
1931 \lam{Y_i}, since we can then just use \lam{Y_i} inside the new
1932 function body and have all of the variables it uses be in scope.
1934 The argument that we are specializing for, \lam{Y_i}, is put inside
1935 the new function body. The old function body is applied to it. Since
1936 we use this new function only in place of an application with that
1937 particular argument \lam{Y_i}, behaviour should not change.
1939 Note that the types of the arguments of our new function are taken
1940 from the types of the \emph{actual} arguments (\lam{T0 ... Tn}). This
1941 means that any polymorphism in the arguments is removed, even when the
1942 corresponding explicit type lambda is not removed
1943 yet.\refdef{type lambda}
1945 \todo{Examples. Perhaps reference the previous sections}
1947 \section{Unsolved problems}
1948 The above system of transformations has been implemented in the prototype
1949 and seems to work well to compile simple and more complex examples of
1950 hardware descriptions. \todo{Ref christiaan?} However, this normalization
1951 system has not seen enough review and work to be complete and work for
1952 every Core expression that is supplied to it. A number of problems
1953 have already been identified and are discussed in this section.
1955 \subsection[sec:normalization:duplicatework]{Work duplication}
1956 A possible problem of β-reduction is that it could duplicate work.
1957 When the expression applied is not a simple variable reference, but
1958 requires calculation and the binder the lambda abstraction binds to
1959 is used more than once, more hardware might be generated than strictly
1962 As an example, consider the expression:
1968 When applying β-reduction to this expression, we get:
1974 which of course calculates \lam{(a * b)} twice.
1976 A possible solution to this would be to use the following alternative
1977 transformation, which is of course no longer normal β-reduction. The
1978 followin transformation has not been tested in the prototype, but is
1979 given here for future reference:
1987 This doesn't seem like much of an improvement, but it does get rid of
1988 the lambda expression (and the associated higher order value), while
1989 at the same time introducing a new let binding. Since the result of
1990 every application or case expression must be bound by a let expression
1991 in the intended normal form anyway, this is probably not a problem. If
1992 the argument happens to be a variable reference, then simple let
1993 binding removal (\in{section}[sec:normalization:simplelet]) will
1994 remove it, making the result identical to that of the original
1995 β-reduction transformation.
1997 When also applying argument simplification to the above example, we
1998 get the following expression:
2006 Looking at this, we could imagine an alternative approach: Create a
2007 transformation that removes let bindings that bind identical values.
2008 In the above expression, the \lam{y} and \lam{z} variables could be
2009 merged together, resulting in the more efficient expression:
2012 let y = (a * b) in y + y
2015 \subsection[sec:normalization:non-determinism]{Non-determinism}
2016 As an example, again consider the following expression:
2022 We can apply both β-reduction (\in{section}[sec:normalization:beta])
2023 as well as argument simplification
2024 (\in{section}[sec:normalization:argsimpl]) to this expression.
2026 When applying argument simplification first and then β-reduction, we
2027 get the following expression:
2030 let y = (a * b) in y + y
2033 When applying β-reduction first and then argument simplification, we
2034 get the following expression:
2042 As you can see, this is a different expression. This means that the
2043 order of expressions, does in fact change the resulting normal form,
2044 which is something that we would like to avoid. In this particular
2045 case one of the alternatives is even clearly more efficient, so we
2046 would of course like the more efficient form to be the normal form.
2048 For this particular problem, the solutions for duplication of work
2049 seem from the previous section seem to fix the determinism of our
2050 transformation system as well. However, it is likely that there are
2051 other occurences of this problem.
2054 We do not fully understand the use of cast expressions in Core, so
2055 there are probably expressions involving cast expressions that cannot
2056 be brought into intended normal form by this transformation system.
2058 The uses of casts in the core system should be investigated more and
2059 transformations will probably need updating to handle them in all
2062 \subsection[sec:normalization:stateproblems]{Normalization of stateful descriptions}
2063 Currently, the intended normal form definition\refdef{intended
2064 normal form definition} offers enough freedom to describe all
2065 valid stateful descriptions, but is not limiting enough. It is
2066 possible to write descriptions which are in intended normal
2067 form, but cannot be translated into \VHDL in a meaningful way
2068 (\eg, a function that swaps two substates in its result, or a
2069 function that changes a substate itself instead of passing it to
2072 It is now up to the programmer to not do anything funny with
2073 these state values, whereas the normalization just tries not to
2074 mess up the flow of state values. In practice, there are
2075 situations where a Core program that \emph{could} be a valid
2076 stateful description is not translateable by the prototype. This
2077 most often happens when statefulness is mixed with pattern
2078 matching, causing a state input to be unpacked multiple times or
2079 be unpacked and repacked only in some of the code paths.
2081 Without going into detail about the exact problems (of which
2082 there are probably more than have shown up so far), it seems
2083 unlikely that these problems can be solved entirely by just
2084 improving the \VHDL state generation in the final stage. The
2085 normalization stage seems the best place to apply the rewriting
2086 needed to support more complex stateful descriptions. This does
2087 of course mean that the intended normal form definition must be
2088 extended as well to be more specific about how state handling
2089 should look like in normal form.
2090 \in{Section}[sec:prototype:statelimits] already contains a
2091 tight description of the limitations on the use of state
2092 variables, which could be adapted into the intended normal form.
2094 \section[sec:normalization:properties]{Provable properties}
2095 When looking at the system of transformations outlined above, there are a
2096 number of questions that we can ask ourselves. The main question is of course:
2097 \quote{Does our system work as intended?}. We can split this question into a
2098 number of subquestions:
2101 \item[q:termination] Does our system \emph{terminate}? Since our system will
2102 keep running as long as transformations apply, there is an obvious risk that
2103 it will keep running indefinitely. This typically happens when one
2104 transformation produces a result that is transformed back to the original
2105 by another transformation, or when one or more transformations keep
2106 expanding some expression.
2107 \item[q:soundness] Is our system \emph{sound}? Since our transformations
2108 continuously modify the expression, there is an obvious risk that the final
2109 normal form will not be equivalent to the original program: Its meaning could
2111 \item[q:completeness] Is our system \emph{complete}? Since we have a complex
2112 system of transformations, there is an obvious risk that some expressions will
2113 not end up in our intended normal form, because we forgot some transformation.
2114 In other words: Does our transformation system result in our intended normal
2115 form for all possible inputs?
2116 \item[q:determinism] Is our system \emph{deterministic}? Since we have defined
2117 no particular order in which the transformation should be applied, there is an
2118 obvious risk that different transformation orderings will result in
2119 \emph{different} normal forms. They might still both be intended normal forms
2120 (if our system is \emph{complete}) and describe correct hardware (if our
2121 system is \emph{sound}), so this property is less important than the previous
2122 three: The translator would still function properly without it.
2125 Unfortunately, the final transformation system has only been
2126 developed in the final part of the research, leaving no more time
2127 for verifying these properties. In fact, it is likely that the
2128 current transformation system still violates some of these
2129 properties in some cases and should be improved (or extra conditions
2130 on the input hardware descriptions should be formulated).
2132 This is most likely the case with the completeness and determinism
2133 properties, perhaps als the termination property. The soundness
2134 property probably holds, since it is easier to manually verify (each
2135 transformation can be reviewed separately).
2137 Even though no complete proofs have been made, some ideas for
2138 possible proof strategies are shown below.
2140 \subsection{Graph representation}
2141 Before looking into how to prove these properties, we'll look at
2142 transformation systems from a graph perspective. We will first define
2143 the graph view and then illustrate it using a simple example from lambda
2144 calculus (which is a different system than the Cλash normalization
2145 system). The nodes of the graph are all possible Core expressions. The
2146 (directed) edges of the graph are transformations. When a transformation
2147 α applies to an expression \lam{A} to produce an expression \lam{B}, we
2148 add an edge from the node for \lam{A} to the node for \lam{B}, labeled
2151 \startuseMPgraphic{TransformGraph}
2155 newCircle.a(btex \lam{(λx.λy. (+) x y) 1} etex);
2156 newCircle.b(btex \lam{λy. (+) 1 y} etex);
2157 newCircle.c(btex \lam{(λx.(+) x) 1} etex);
2158 newCircle.d(btex \lam{(+) 1} etex);
2161 c.c = b.c + (4cm, 0cm);
2162 a.c = midpoint(b.c, c.c) + (0cm, 4cm);
2163 d.c = midpoint(b.c, c.c) - (0cm, 3cm);
2165 % β-conversion between a and b
2166 ncarc.a(a)(b) "name(bred)";
2167 ObjLabel.a(btex $\xrightarrow[normal]{}{β}$ etex) "labpathname(bred)", "labdir(rt)";
2168 ncarc.b(b)(a) "name(bexp)", "linestyle(dashed withdots)";
2169 ObjLabel.b(btex $\xleftarrow[normal]{}{β}$ etex) "labpathname(bexp)", "labdir(lft)";
2171 % η-conversion between a and c
2172 ncarc.a(a)(c) "name(ered)";
2173 ObjLabel.a(btex $\xrightarrow[normal]{}{η}$ etex) "labpathname(ered)", "labdir(rt)";
2174 ncarc.c(c)(a) "name(eexp)", "linestyle(dashed withdots)";
2175 ObjLabel.c(btex $\xleftarrow[normal]{}{η}$ etex) "labpathname(eexp)", "labdir(lft)";
2177 % η-conversion between b and d
2178 ncarc.b(b)(d) "name(ered)";
2179 ObjLabel.b(btex $\xrightarrow[normal]{}{η}$ etex) "labpathname(ered)", "labdir(rt)";
2180 ncarc.d(d)(b) "name(eexp)", "linestyle(dashed withdots)";
2181 ObjLabel.d(btex $\xleftarrow[normal]{}{η}$ etex) "labpathname(eexp)", "labdir(lft)";
2183 % β-conversion between c and d
2184 ncarc.c(c)(d) "name(bred)";
2185 ObjLabel.c(btex $\xrightarrow[normal]{}{β}$ etex) "labpathname(bred)", "labdir(rt)";
2186 ncarc.d(d)(c) "name(bexp)", "linestyle(dashed withdots)";
2187 ObjLabel.d(btex $\xleftarrow[normal]{}{β}$ etex) "labpathname(bexp)", "labdir(lft)";
2189 % Draw objects and lines
2190 drawObj(a, b, c, d);
2193 \placeexample[right][ex:TransformGraph]{Partial graph of a lambda calculus
2194 system with β and η reduction (solid lines) and expansion (dotted lines).}
2195 \boxedgraphic{TransformGraph}
2197 Of course the graph for Cλash is unbounded, since we can construct an
2198 infinite amount of Core expressions. Also, there might potentially be
2199 multiple edges between two given nodes (with different labels), though
2200 seems unlikely to actually happen in our system.
2202 See \in{example}[ex:TransformGraph] for the graph representation of a very
2203 simple lambda calculus that contains just the expressions \lam{(λx.λy. (+) x
2204 y) 1}, \lam{λy. (+) 1 y}, \lam{(λx.(+) x) 1} and \lam{(+) 1}. The
2205 transformation system consists of β-reduction and η-reduction (solid edges) or
2206 β-expansion and η-expansion (dotted edges).
2208 \todo{Define β-reduction and η-reduction?}
2210 Note that the normal form of such a system consists of the set of nodes
2211 (expressions) without outgoing edges, since those are the expression to which
2212 no transformation applies anymore. We call this set of nodes the \emph{normal
2213 set}. The set of nodes containing expressions in intended normal
2214 form \refdef{intended normal form} is called the \emph{intended
2217 From such a graph, we can derive some properties easily:
2219 \item A system will \emph{terminate} if there is no path of infinite length
2220 in the graph (this includes cycles, but can also happen without cycles).
2221 \item Soundness is not easily represented in the graph.
2222 \item A system is \emph{complete} if all of the nodes in the normal set have
2223 the intended normal form. The inverse (that all of the nodes outside of
2224 the normal set are \emph{not} in the intended normal form) is not
2225 strictly required. In other words, our normal set must be a
2226 subset of the intended normal form, but they do not need to be
2229 \item A system is deterministic if all paths starting at a particular
2230 node, which end in a node in the normal set, end at the same node.
2233 When looking at the \in{example}[ex:TransformGraph], we see that the system
2234 terminates for both the reduction and expansion systems (but note that, for
2235 expansion, this is only true because we've limited the possible
2236 expressions. In comlete lambda calculus, there would be a path from
2237 \lam{(λx.λy. (+) x y) 1} to \lam{(λx.λy.(λz.(+) z) x y) 1} to
2238 \lam{(λx.λy.(λz.(λq.(+) q) z) x y) 1} etc.)
2240 If we would consider the system with both expansion and reduction, there
2241 would no longer be termination either, since there would be cycles all
2244 The reduction and expansion systems have a normal set of containing just
2245 \lam{(+) 1} or \lam{(λx.λy. (+) x y) 1} respectively. Since all paths in
2246 either system end up in these normal forms, both systems are \emph{complete}.
2247 Also, since there is only one node in the normal set, it must obviously be
2248 \emph{deterministic} as well.
2250 \subsection{Termination}
2251 In general, proving termination of an arbitrary program is a very
2252 hard problem. \todo{Ref about arbitrary termination} Fortunately,
2253 we only have to prove termination for our specific transformation
2256 A common approach for these kinds of proofs is to associate a
2257 measure with each possible expression in our system. If we can
2258 show that each transformation strictly decreases this measure
2259 (\ie, the expression transformed to has a lower measure than the
2260 expression transformed from). \todo{ref about measure-based
2261 termination proofs / analysis}
2263 A good measure for a system consisting of just β-reduction would
2264 be the number of lambda expressions in the expression. Since every
2265 application of β-reduction removes a lambda abstraction (and there
2266 is always a bounded number of lambda abstractions in every
2267 expression) we can easily see that a transformation system with
2268 just β-reduction will always terminate.
2270 For our complete system, this measure would be fairly complex
2271 (probably the sum of a lot of things). Since the (conditions on)
2272 our transformations are pretty complex, we would need to include
2273 both simple things like the number of let expressions as well as
2274 more complex things like the number of case expressions that are
2275 not yet in normal form.
2277 No real attempt has been made at finding a suitable measure for
2280 \subsection{Soundness}
2281 Soundness is a property that can be proven for each transformation
2282 separately. Since our system only runs separate transformations
2283 sequentially, if each of our transformations leaves the
2284 \emph{meaning} of the expression unchanged, then the entire system
2285 will of course leave the meaning unchanged and is thus
2288 The current prototype has only been verified in an ad-hoc fashion
2289 by inspecting (the code for) each transformation. A more formal
2290 verification would be more appropriate.
2292 To be able to formally show that each transformation properly
2293 preserves the meaning of every expression, we require an exact
2294 definition of the \emph{meaning} of every expression, so we can
2295 compare them. Currently there seems to be no formal definition of
2296 the meaning or semantics of \GHC's core language, only informal
2297 descriptions are available.
2299 It should be possible to have a single formal definition of
2300 meaning for Core for both normal Core compilation by \GHC and for
2301 our compilation to \VHDL. The main difference seems to be that in
2302 hardware every expression is always evaluated, while in software
2303 it is only evaluated if needed, but it should be possible to
2304 assign a meaning to core expressions that assumes neither.
2306 Since each of the transformations can be applied to any
2307 subexpression as well, there is a constraint on our meaning
2308 definition: The meaning of an expression should depend only on the
2309 meaning of subexpressions, not on the expressions themselves. For
2310 example, the meaning of the application in \lam{f (let x = 4 in
2311 x)} should be the same as the meaning of the application in \lam{f
2312 4}, since the argument subexpression has the same meaning (though
2313 the actual expression is different).
2315 \subsection{Completeness}
2316 Proving completeness is probably not hard, but it could be a lot
2317 of work. We have seen above that to prove completeness, we must
2318 show that the normal set of our graph representation is a subset
2319 of the intended normal set.
2321 However, it is hard to systematically generate or reason about the
2322 normal set, since it is defined as any nodes to which no
2323 transformation applies. To determine this set, each transformation
2324 must be considered and when a transformation is added, the entire
2325 set should be re-evaluated. This means it is hard to show that
2326 each node in the normal set is also in the intended normal set.
2327 Reasoning about our intended normal set is easier, since we know
2328 how to generate it from its definition. \refdef{intended normal
2331 Fortunately, we can also prove the complement (which is
2332 equivalent, since $A \subseteq B \Leftrightarrow \overline{B}
2333 \subseteq \overline{A}$): Show that the set of nodes not in
2334 intended normal form is a subset of the set of nodes not in normal
2335 form. In other words, show that for every expression that is not
2336 in intended normal form, that there is at least one transformation
2337 that applies to it (since that means it is not in normal form
2338 either and since $A \subseteq C \Leftrightarrow \forall x (x \in A
2339 \rightarrow x \in C)$).
2341 By systematically reviewing the entire Core language definition
2342 along with the intended normal form definition (both of which have
2343 a similar structure), it should be possible to identify all
2344 possible (sets of) core expressions that are not in intended
2345 normal form and identify a transformation that applies to it.
2347 This approach is especially useful for proving completeness of our
2348 system, since if expressions exist to which none of the
2349 transformations apply (\ie if the system is not yet complete), it
2350 is immediately clear which expressions these are and adding
2351 (or modifying) transformations to fix this should be relatively
2354 As observed above, applying this approach is a lot of work, since
2355 we need to check every (set of) transformation(s) separately.
2357 \todo{Perhaps do a few steps of the proofs as proof-of-concept}
2359 % vim: set sw=2 sts=2 expandtab: