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[sec:normalization:intendednormalform]{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(var))
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 (var1 :: 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 reference 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}
847 \subsubsection{Empty let removal}
848 This transformation is simple: It removes recursive lets that have no bindings
849 (which usually occurs when unused let binding removal removes the last
852 Note that there is no need to define this transformation for
853 non-recursive lets, since they always contain exactly one binding.
863 \subsubsection[sec:normalization:simplelet]{Simple let binding removal}
864 This transformation inlines simple let bindings, that bind some
865 binder to some other binder instead of a more complex expression (\ie
868 This transformation is not needed to get an expression into intended
869 normal form (since these bindings are part of the intended normal
870 form), but makes the resulting \small{VHDL} a lot shorter.
881 ----------------------------- \lam{b} is a variable reference
882 letrec \lam{ai} ≠ \lam{b}
895 \subsubsection{Unused let binding removal}
896 This transformation removes let bindings that are never used.
897 Occasionally, \GHC's desugarer introduces some unused let bindings.
899 This normalization pass should really be unneeded to get into intended normal form
900 (since unused bindings are not forbidden by the normal form), but in practice
901 the desugarer or simplifier emits some unused bindings that cannot be
902 normalized (e.g., calls to a \type{PatError}\todo{Check this name}). Also,
903 this transformation makes the resulting \small{VHDL} a lot shorter.
905 \todo{Don't use old-style numerals in transformations}
914 M \lam{ai} does not occur free in \lam{M}
915 ---------------------------- \forall j, 0 ≤ j ≤ n, j ≠ i (\lam{ai} does not occur free in \lam{Ej})
929 \subsubsection{Cast propagation / simplification}
930 This transform pushes casts down into the expression as far as possible.
931 Since its exact role and need is not clear yet, this transformation is
934 \todo{Cast propagation}
936 \subsubsection{Top level binding inlining}
937 This transform takes simple top level bindings generated by the
938 \small{GHC} compiler. \small{GHC} sometimes generates very simple
939 \quote{wrapper} bindings, which are bound to just a variable
940 reference, or a partial application to constants or other variable
943 Note that this transformation is completely optional. It is not
944 required to get any function into intended normal form, but it does help making
945 the resulting VHDL output easier to read (since it removes a bunch of
946 components that are really boring).
948 This transform takes any top level binding generated by the compiler,
949 whose normalized form contains only a single let binding.
952 x = λa0 ... λan.let y = E in y
955 -------------------------------------- \lam{x} is generated by the compiler
956 λa0 ... λan.let y = E in y
960 (+) :: Word -> Word -> Word
961 (+) = GHC.Num.(+) @Word \$dNum
966 GHC.Num.(+) @ Alu.Word \$dNum a b
969 \transexample{toplevelinline}{Top level binding inlining}{from}{to}
971 \in{Example}[ex:trans:toplevelinline] shows a typical application of
972 the addition operator generated by \GHC. The type and dictionary
973 arguments used here are described in
974 \in{Section}[section:prototype:polymorphism].
976 Without this transformation, there would be a \lam{(+)} entity
977 in the \VHDL which would just add its inputs. This generates a
978 lot of overhead in the \VHDL, which is particularly annoying
979 when browsing the generated RTL schematic (especially since most
980 non-alphanumerics, like all characters in \lam{(+)}, are not
981 allowed in \VHDL architecture names\footnote{Technically, it is
982 allowed to use non-alphanumerics when using extended
983 identifiers, but it seems that none of the tooling likes
984 extended identifiers in filenames, so it effectively doesn't
985 work.}, so the entity would be called \quote{w7aA7f} or
986 something similarly unreadable and autogenerated).
988 \subsection{Program structure}
989 These transformations are aimed at normalizing the overall structure
990 into the intended form. This means ensuring there is a lambda abstraction
991 at the top for every argument (input port or current state), putting all
992 of the other value definitions in let bindings and making the final
993 return value a simple variable reference.
995 \subsubsection[sec:normalization:eta]{η-abstraction}
996 This transformation makes sure that all arguments of a function-typed
997 expression are named, by introducing lambda expressions. When combined with
998 β-reduction and non-representable binding inlining, all function-typed
999 expressions should be lambda abstractions or global identifiers.
1003 -------------- \lam{E} is not the first argument of an application.
1004 λx.E x \lam{E} is not a lambda abstraction.
1005 \lam{x} is a variable that does not occur free in \lam{E}.
1015 foo = λa.λx.(case a of
1020 \transexample{eta}{η-abstraction}{from}{to}
1022 \subsubsection[sec:normalization:appprop]{Application propagation}
1023 This transformation is meant to propagate application expressions downwards
1024 into expressions as far as possible. This allows partial applications inside
1025 expressions to become fully applied and exposes new transformation
1026 opportunities for other transformations (like β-reduction and
1029 Since all binders in our expression are unique (see
1030 \in{section}[sec:normalization:uniq]), there is no risk that we will
1031 introduce unintended shadowing by moving an expression into a lower
1032 scope. Also, since only move expression into smaller scopes (down into
1033 our expression), there is no risk of moving a variable reference out
1034 of the scope in which it is defined.
1037 (letrec binds in E) M
1038 ------------------------
1058 \transexample{appproplet}{Application propagation for a let expression}{from}{to}
1086 \transexample{apppropcase}{Application propagation for a case expression}{from}{to}
1088 \subsubsection{Let recursification}
1089 This transformation makes all non-recursive lets recursive. In the
1090 end, we want a single recursive let in our normalized program, so all
1091 non-recursive lets can be converted. This also makes other
1092 transformations simpler: They can simply assume all lets are
1100 ------------------------------------------
1107 \subsubsection{Let flattening}
1108 This transformation puts nested lets in the same scope, by lifting the
1109 binding(s) of the inner let into the outer let. Eventually, this will
1110 cause all let bindings to appear in the same scope.
1112 This transformation only applies to recursive lets, since all
1113 non-recursive lets will be made recursive (see
1114 \in{section}[sec:normalization:letrecurse]).
1116 Since we are joining two scopes together, there is no risk of moving a
1117 variable reference out of the scope where it is defined.
1123 ai = (letrec bindings in M)
1128 ------------------------------------------
1163 \transexample{letflat}{Let flattening}{from}{to}
1165 \subsubsection{Return value simplification}
1166 This transformation ensures that the return value of a function is always a
1167 simple local variable reference.
1169 Currently implemented using lambda simplification, let simplification, and
1170 top simplification. Should change into something like the following, which
1171 works only on the result of a function instead of any subexpression. This is
1172 achieved by the contexts, like \lam{x = E}, though this is strictly not
1173 correct (you could read this as "if there is any function \lam{x} that binds
1174 \lam{E}, any \lam{E} can be transformed, while we only mean the \lam{E} that
1175 is bound by \lam{x}. This might need some extra notes or something).
1177 Note that the return value is not simplified if its not representable.
1178 Otherwise, this would cause a direct loop with the inlining of
1179 unrepresentable bindings. If the return value is not
1180 representable because it has a function type, η-abstraction should
1181 make sure that this transformation will eventually apply. If the value
1182 is not representable for other reasons, the function result itself is
1183 not representable, meaning this function is not translatable anyway.
1186 x = E \lam{E} is representable
1187 ~ \lam{E} is not a lambda abstraction
1188 E \lam{E} is not a let expression
1189 --------------------------- \lam{E} is not a local variable reference
1195 ~ \lam{E} is representable
1196 E \lam{E} is not a let expression
1197 --------------------------- \lam{E} is not a local variable reference
1202 x = λv0 ... λvn.let ... in E
1203 ~ \lam{E} is representable
1204 E \lam{E} is not a local variable reference
1205 -----------------------------
1214 x = letrec x = add 1 2 in x
1217 \transexample{retvalsimpl}{Return value simplification}{from}{to}
1219 \todo{More examples}
1221 \subsection{Argument simplification}
1222 The transforms in this section deal with simplifying application
1223 arguments into normal form. The goal here is to:
1225 \todo{This section should only talk about representable arguments. Non
1226 representable arguments are treated by specialization.}
1229 \item Make all arguments of user-defined functions (\eg, of which
1230 we have a function body) simple variable references of a runtime
1231 representable type. This is needed, since these applications will be turned
1232 into component instantiations.
1233 \item Make all arguments of builtin functions one of:
1235 \item A type argument.
1236 \item A dictionary argument.
1237 \item A type level expression.
1238 \item A variable reference of a runtime representable type.
1239 \item A variable reference or partial application of a function type.
1243 When looking at the arguments of a user-defined function, we can
1244 divide them into two categories:
1246 \item Arguments of a runtime representable type (\eg bits or vectors).
1248 These arguments can be preserved in the program, since they can
1249 be translated to input ports later on. However, since we can
1250 only connect signals to input ports, these arguments must be
1251 reduced to simple variables (for which signals will be
1252 produced). This is taken care of by the argument extraction
1254 \item Non-runtime representable typed arguments. \todo{Move this
1255 bullet to specialization}
1257 These arguments cannot be preserved in the program, since we
1258 cannot represent them as input or output ports in the resulting
1259 \small{VHDL}. To remove them, we create a specialized version of the
1260 called function with these arguments filled in. This is done by
1261 the argument propagation transform.
1263 Typically, these arguments are type and dictionary arguments that are
1264 used to make functions polymorphic. By propagating these arguments, we
1265 are essentially doing the same which GHC does when it specializes
1266 functions: Creating multiple variants of the same function, one for
1267 each type for which it is used. Other common non-representable
1268 arguments are functions, e.g. when calling a higher order function
1269 with another function or a lambda abstraction as an argument.
1271 The reason for doing this is similar to the reasoning provided for
1272 the inlining of non-representable let bindings above. In fact, this
1273 argument propagation could be viewed as a form of cross-function
1277 \todo{Move this itemization into a new section about builtin functions}
1278 When looking at the arguments of a builtin function, we can divide them
1282 \item Arguments of a runtime representable type.
1284 As we have seen with user-defined functions, these arguments can
1285 always be reduced to a simple variable reference, by the
1286 argument extraction transform. Performing this transform for
1287 builtin functions as well, means that the translation of builtin
1288 functions can be limited to signal references, instead of
1289 needing to support all possible expressions.
1291 \item Arguments of a function type.
1293 These arguments are functions passed to higher order builtins,
1294 like \lam{map} and \lam{foldl}. Since implementing these
1295 functions for arbitrary function-typed expressions (\eg, lambda
1296 expressions) is rather comlex, we reduce these arguments to
1297 (partial applications of) global functions.
1299 We can still support arbitrary expressions from the user code,
1300 by creating a new global function containing that expression.
1301 This way, we can simply replace the argument with a reference to
1302 that new function. However, since the expression can contain any
1303 number of free variables we also have to include partial
1304 applications in our normal form.
1306 This category of arguments is handled by the function extraction
1308 \item Other unrepresentable arguments.
1310 These arguments can take a few different forms:
1311 \startdesc{Type arguments}
1312 In the core language, type arguments can only take a single
1313 form: A type wrapped in the Type constructor. Also, there is
1314 nothing that can be done with type expressions, except for
1315 applying functions to them, so we can simply leave type
1316 arguments as they are.
1318 \startdesc{Dictionary arguments}
1319 In the core language, dictionary arguments are used to find
1320 operations operating on one of the type arguments (mostly for
1321 finding class methods). Since we will not actually evaluatie
1322 the function body for builtin functions and can generate
1323 code for builtin functions by just looking at the type
1324 arguments, these arguments can be ignored and left as they
1327 \startdesc{Type level arguments}
1328 Sometimes, we want to pass a value to a builtin function, but
1329 we need to know the value at compile time. Additionally, the
1330 value has an impact on the type of the function. This is
1331 encoded using type-level values, where the actual value of the
1332 argument is not important, but the type encodes some integer,
1333 for example. Since the value is not important, the actual form
1334 of the expression does not matter either and we can leave
1335 these arguments as they are.
1337 \startdesc{Other arguments}
1338 Technically, there is still a wide array of arguments that can
1339 be passed, but does not fall into any of the above categories.
1340 However, none of the supported builtin functions requires such
1341 an argument. This leaves use with passing unsupported types to
1342 a function, such as calling \lam{head} on a list of functions.
1344 In these cases, it would be impossible to generate hardware
1345 for such a function call anyway, so we can ignore these
1348 The only way to generate hardware for builtin functions with
1349 arguments like these, is to expand the function call into an
1350 equivalent core expression (\eg, expand map into a series of
1351 function applications). But for now, we choose to simply not
1352 support expressions like these.
1355 From the above, we can conclude that we can simply ignore these
1356 other unrepresentable arguments and focus on the first two
1360 \subsubsection[sec:normalization:argsimpl]{Argument simplification}
1361 This transform deals with arguments to functions that
1362 are of a runtime representable type. It ensures that they will all become
1363 references to global variables, or local signals in the resulting
1364 \small{VHDL}, which is required due to limitations in the component
1365 instantiation code in \VHDL (one can only assign a signal or constant
1366 to an input port). By ensuring that all arguments are always simple
1367 variable references, we always have a signal available to assign to
1370 \todo{Say something about dataconstructors (without arguments, like True
1371 or False), which are variable references of a runtime representable
1372 type, but do not result in a signal.}
1374 To reduce a complex expression to a simple variable reference, we create
1375 a new let expression around the application, which binds the complex
1376 expression to a new variable. The original function is then applied to
1379 Note that a reference to a \emph{global variable} (like a top level
1380 function without arguments, but also an argumentless dataconstructors
1381 like \lam{True}) is also simplified. Only local variables generate
1382 signals in the resulting architecture.
1384 \refdef{representable}
1387 -------------------- \lam{N} is representable
1388 letrec x = N in M x \lam{N} is not a local variable reference
1390 \refdef{local variable}
1397 letrec x = add a 1 in add x 1
1400 \transexample{argextract}{Argument extraction}{from}{to}
1402 \subsubsection[sec:normalization:funextract]{Function extraction}
1403 \todo{Move to section about builtin functions}
1404 This transform deals with function-typed arguments to builtin
1405 functions. Since builtin functions cannot be specialized to remove
1406 the arguments, we choose to extract these arguments into a new global
1407 function instead. This greatly simplifies the translation rules needed
1408 for builtin functions. \todo{Should we talk about these? Reference
1411 Any free variables occuring in the extracted arguments will become
1412 parameters to the new global function. The original argument is replaced
1413 with a reference to the new function, applied to any free variables from
1414 the original argument.
1416 This transformation is useful when applying higher order builtin functions
1417 like \hs{map} to a lambda abstraction, for example. In this case, the code
1418 that generates \small{VHDL} for \hs{map} only needs to handle top level functions and
1419 partial applications, not any other expression (such as lambda abstractions or
1420 even more complicated expressions).
1423 M N \lam{M} is (a partial aplication of) a builtin function.
1424 --------------------- \lam{f0 ... fn} are all free local variables of \lam{N}
1425 M (x f0 ... fn) \lam{N :: a -> b}
1426 ~ \lam{N} is not a (partial application of) a top level function
1430 \todo{Split this example}
1432 map (λa . add a b) xs
1446 \transexample{funextract}{Function extraction}{from}{to}
1448 Note that \lam{x0} and {x1} will still need normalization after this.
1450 \todo{Fill the gap left by moving argument propagation away}
1452 \subsection{Case normalisation}
1453 \subsubsection{Scrutinee simplification}
1454 This transform ensures that the scrutinee of a case expression is always
1455 a simple variable reference.
1460 ----------------- \lam{E} is not a local variable reference
1479 \transexample{letflat}{Let flattening}{from}{to}
1482 \subsubsection{Case simplification}
1483 This transformation ensures that all case expressions become normal form. This
1484 means they will become one of:
1486 \item An extractor case with a single alternative that picks a single field
1487 from a datatype, \eg \lam{case x of (a, b) -> a}.
1488 \item A selector case with multiple alternatives and only wild binders, that
1489 makes a choice between expressions based on the constructor of another
1490 expression, \eg \lam{case x of Low -> a; High -> b}.
1493 \defref{wild binder}
1496 C0 v0,0 ... v0,m -> E0
1498 Cn vn,0 ... vn,m -> En
1499 --------------------------------------------------- \forall i \forall j, 0 ≤ i ≤ n, 0 ≤ i < m (\lam{wi,j} is a wild (unused) binder)
1501 v0,0 = case E of C0 v0,0 .. v0,m -> v0,0
1503 v0,m = case E of C0 v0,0 .. v0,m -> v0,m
1505 vn,m = case E of Cn vn,0 .. vn,m -> vn,m
1511 C0 w0,0 ... w0,m -> x0
1513 Cn wn,0 ... wn,m -> xn
1515 \todo{Check the subscripts of this transformation}
1517 Note that this transformation applies to case statements with any
1518 scrutinee. If the scrutinee is a complex expression, this might result
1519 in duplicate hardware. An extra condition to only apply this
1520 transformation when the scrutinee is already simple (effectively
1521 causing this transformation to be only applied after the scrutinee
1522 simplification transformation) might be in order.
1524 \fxnote{This transformation specified like this is complicated and misses
1525 conditions to prevent looping with itself. Perhaps it should be split here for
1544 \transexample{selcasesimpl}{Selector case simplification}{from}{to}
1552 b = case a of (,) b c -> b
1553 c = case a of (,) b c -> c
1560 \transexample{excasesimpl}{Extractor case simplification}{from}{to}
1562 \refdef{selector case}
1563 In \in{example}[ex:trans:excasesimpl] the case expression is expanded
1564 into multiple case expressions, including a pretty useless expression
1565 (that is neither a selector or extractor case). This case can be
1566 removed by the Case removal transformation in
1567 \in{section}[sec:transformation:caseremoval].
1569 \subsubsection[sec:transformation:caseremoval]{Case removal}
1570 This transform removes any case statements with a single alternative and
1573 These "useless" case statements are usually leftovers from case simplification
1574 on extractor case (see the previous example).
1579 ---------------------- \lam{\forall i, 0 ≤ i ≤ m} (\lam{vi} does not occur free in E)
1592 \transexample{caserem}{Case removal}{from}{to}
1594 \subsection{Removing unrepresentable values}
1595 The transformations in this section are aimed at making all the
1596 values used in our expression representable. There are two main
1597 transformations that are applied to \emph{all} unrepresentable let
1598 bindings and function arguments, but these are really meant to
1599 address three different kinds of unrepresentable values:
1600 Polymorphic values, higher order values and literals. Each of these
1601 will be detailed below, followed by the actual transformations.
1603 \subsubsection{Removing Polymorphism}
1604 As noted in \in{section}[sec:prototype:polymporphism],
1605 polymorphism is made explicit in Core through type and
1606 dictionary arguments. To remove the polymorphism from a
1607 function, we can simply specialize the polymorphic function for
1608 the particular type applied to it. The same goes for dictionary
1609 arguments. To remove polymorphism from let bound values, we
1610 simply inline the let bindings that have a polymorphic type,
1611 which should (eventually) make sure that the polymorphic
1612 expression is applied to a type and/or dictionary, which can
1613 \refdef{beta-reduction}
1614 then be removed by β-reduction.
1616 Since both type and dictionary arguments are not representable,
1617 \refdef{representable}
1618 the non-representable argument specialization and
1619 non-representable let binding inlining transformations below
1620 take care of exactly this.
1622 There is one case where polymorphism cannot be completely
1623 removed: Builtin functions are still allowed to be polymorphic
1624 (Since we have no function body that we could properly
1625 specialize). However, the code that generates \VHDL for builtin
1626 functions knows how to handle this, so this is not a problem.
1628 \subsubsection{Defunctionalization}
1629 These transformations remove higher order expressions from our
1630 program, making all values first-order.
1632 Higher order values are always introduced by lambda abstractions, none
1633 of the other Core expression elements can introduce a function type.
1634 However, other expressions can \emph{have} a function type, when they
1635 have a lambda expression in their body.
1637 For example, the following expression is a higher order expression
1638 that is not a lambda expression itself:
1640 \refdef{id function}
1647 The reference to the \lam{id} function shows that we can introduce a
1648 higher order expression in our program without using a lambda
1649 expression directly. However, inside the definition of the \lam{id}
1650 function, we can be sure that a lambda expression is present.
1652 Looking closely at the definition of our normal form in
1653 \in{section}[sec:normalization:intendednormalform], we can see that
1654 there are three possibilities for higher order values to appear in our
1655 intended normal form:
1658 \item[item:toplambda] Lambda abstractions can appear at the highest level of a
1659 top level function. These lambda abstractions introduce the
1660 arguments (input ports / current state) of the function.
1661 \item[item:builtinarg] (Partial applications of) top level functions can appear as an
1662 argument to a builtin function.
1663 \item[item:completeapp] (Partial applications of) top level functions can appear in
1664 function position of an application. Since a partial application
1665 cannot appear anywhere else (except as builtin function arguments),
1666 all partial applications are applied, meaning that all applications
1667 will become complete applications. However, since application of
1668 arguments happens one by one, in the expression:
1672 the subexpression \lam{f 1} has a function type. But this is
1673 allowed, since it is inside a complete application.
1676 We will take a typical function with some higher order values as an
1677 example. The following function takes two arguments: a \lam{Bit} and a
1678 list of numbers. Depending on the first argument, each number in the
1679 list is doubled, or the list is returned unmodified. For the sake of
1680 the example, no polymorphism is shown. In reality, at least map would
1684 λy.let double = λx. x + x in
1690 This example shows a number of higher order values that we cannot
1691 translate to \VHDL directly. The \lam{double} binder bound in the let
1692 expression has a function type, as well as both of the alternatives of
1693 the case expression. The first alternative is a partial application of
1694 the \lam{map} builtin function, whereas the second alternative is a
1697 To reduce all higher order values to one of the above items, a number
1698 of transformations we've already seen are used. The η-abstraction
1699 transformation from \in{section}[sec:normalization:eta] ensures all
1700 function arguments are introduced by lambda abstraction on the highest
1701 level of a function. These lambda arguments are allowed because of
1702 \in{item}[item:toplambda] above. After η-abstraction, our example
1703 becomes a bit bigger:
1706 λy.λq.(let double = λx. x + x in
1713 η-abstraction also introduces extra applications (the application of
1714 the let expression to \lam{q} in the above example). These
1715 applications can then propagated down by the application propagation
1716 transformation (\in{section}[sec:normalization:approp]). In our
1717 example, the \lam{q} and \lam{r} variable will be propagated into the
1718 let expression and then into the case expression:
1721 λy.λq.let double = λx. x + x in
1727 This propagation makes higher order values become applied (in
1728 particular both of the alternatives of the case now have a
1729 representable type. Completely applied top level functions (like the
1730 first alternative) are now no longer invalid (they fall under
1731 \in{item}[item:completeapp] above). (Completely) applied lambda
1732 abstractions can be removed by β-abstraction. For our example,
1733 applying β-abstraction results in the following:
1736 λy.λq.let double = λx. x + x in
1742 As you can see in our example, all of this moves applications towards
1743 the higher order values, but misses higher order functions bound by
1744 let expressions. The applications cannot be moved towards these values
1745 (since they can be used in multiple places), so the values will have
1746 to be moved towards the applications. This is achieved by inlining all
1747 higher order values bound by let applications, by the
1748 non-representable binding inlining transformation below. When applying
1749 it to our example, we get the following:
1753 Low -> map (λx. x + x) q
1757 We've nearly eliminated all unsupported higher order values from this
1758 expressions. The one that's remaining is the first argument to the
1759 \lam{map} function. Having higher order arguments to a builtin
1760 function like \lam{map} is allowed in the intended normal form, but
1761 only if the argument is a (partial application) of a top level
1762 function. This is easily done by introducing a new top level function
1763 and put the lambda abstraction inside. This is done by the function
1764 extraction transformation from
1765 \in{section}[sec:normalization:funextract].
1773 This also introduces a new function, that we have called \lam{func}:
1779 Note that this does not actually remove the lambda, but now it is a
1780 lambda at the highest level of a function, which is allowed in the
1781 intended normal form.
1783 There is one case that has not been discussed yet. What if the
1784 \lam{map} function in the example above was not a builtin function
1785 but a user-defined function? Then extracting the lambda expression
1786 into a new function would not be enough, since user-defined functions
1787 can never have higher order arguments. For example, the following
1788 expression shows an example:
1791 app2 :: (Word -> Word) -> Word -> Word
1792 app2 = λf.λa.f (f a)
1794 main = λa.app (λx. x + x) a
1797 This example shows a function \lam{app2} that takes a function as a
1798 first argument and applies that function twice to the second argument.
1799 Again, we've made the function monomorphic for clarity, even though
1800 this function would be a lot more useful if it was polymorphic. The
1801 function \lam{main} uses \lam{app2} to apply a lambda epression twice.
1803 When faced with a user defined function, a body is available for that
1804 function. This means we could create a specialized version of the
1805 function that only works for this particular higher order argument
1806 (\ie, we can just remove the argument and call the specialized
1807 function without the argument). This transformation is detailed below.
1808 Applying this transformation to the example gives:
1811 app2' :: Word -> Word
1812 app2' = λb.(λf.λa.f (f a)) (λx. x + x) b
1817 The \lam{main} function is now in normal form, since the only higher
1818 order value there is the top level lambda expression. The new
1819 \lam{app2'} function is a bit complex, but the entire original body of
1820 the original \lam{app2} function is wrapped in a lambda abstraction
1821 and applied to the argument we've specialized for (\lam{λx. x + x})
1822 and the other arguments. This complex expression can fortunately be
1823 effectively reduced by repeatedly applying β-reduction:
1826 app2' :: Word -> Word
1827 app2' = λb.(b + b) + (b + b)
1830 This example also shows that the resulting normal form might not be as
1831 efficient as we might hope it to be (it is calculating \lam{(b + b)}
1832 twice). This is discussed in more detail in
1833 \in{section}[sec:normalization:duplicatework].
1835 \subsubsection{Literals}
1836 There are a limited number of literals available in Haskell and Core.
1837 \refdef{enumerated types} When using (enumerating) algebraic
1838 datatypes, a literal is just a reference to the corresponding data
1839 constructor, which has a representable type (the algebraic datatype)
1840 and can be translated directly. This also holds for literals of the
1841 \hs{Bool} Haskell type, which is just an enumerated type.
1843 There is, however, a second type of literal that does not have a
1844 representable type: Integer literals. Cλash supports using integer
1845 literals for all three integer types supported (\hs{SizedWord},
1846 \hs{SizedInt} and \hs{RangedWord}). This is implemented using
1847 Haskell's \hs{Num} typeclass, which offers a \hs{fromInteger} method
1848 that converts any \hs{Integer} to the Cλash datatypes.
1850 When \GHC sees integer literals, it will automatically insert calls to
1851 the \hs{fromInteger} method in the resulting Core expression. For
1852 example, the following expression in Haskell creates a 32 bit unsigned
1853 word with the value 1. The explicit type signature is needed, since
1854 there is no context for \GHC to determine the type from otherwise.
1860 This Haskell code results in the following Core expression:
1863 fromInteger @(SizedWord D32) \$dNum (smallInteger 10)
1866 The literal 10 will have the type \lam{GHC.Prim.Int\#}, which is
1867 converted into an \lam{Integer} by \lam{smallInteger}. Finally, the
1868 \lam{fromInteger} function will finally convert this into a
1869 \lam{SizedWord D32}.
1871 Both the \lam{GHC.Prim.Int\#} and \lam{Integer} types are not
1872 representable, and cannot be translated directly. Fortunately, there
1873 is no need to translate them, since \lam{fromInteger} is a builtin
1874 function that knows how to handle these values. However, this does
1875 require that the \lam{fromInteger} function is directly applied to
1876 these non-representable literal values, otherwise errors will occur.
1877 For example, the following expression is not in the intended normal
1878 form, since one of the let bindings has an unrepresentable type
1882 let l = smallInteger 10 in fromInteger @(SizedWord D32) \$dNum l
1885 By inlining these let-bindings, we can ensure that unrepresentable
1886 literals bound by a let binding end up in an application of the
1887 appropriate builtin function, where they are allowed. Since it is
1888 possible that the application of that function is in a different
1889 function than the definition of the literal value, we will always need
1890 to specialize away any unrepresentable literals that are used as
1891 function arguments. The following two transformations do exactly this.
1893 \subsubsection{Non-representable binding inlining}
1894 This transform inlines let bindings that are bound to a
1895 non-representable value. Since we can never generate a signal
1896 assignment for these bindings (we cannot declare a signal assignment
1897 with a non-representable type, for obvious reasons), we have no choice
1898 but to inline the binding to remove it.
1900 As we have seen in the previous sections, inlining these bindings
1901 solves (part of) the polymorphism, higher order values and
1902 unrepresentable literals in an expression.
1913 -------------------------- \lam{Ei} has a non-representable type.
1915 a0 = E0 [ai=>Ei] \vdots
1916 ai-1 = Ei-1 [ai=>Ei]
1917 ai+1 = Ei+1 [ai=>Ei]
1936 x = fromInteger (smallInteger 10)
1938 (λb -> add b 1) (add 1 x)
1941 \transexample{nonrepinline}{Nonrepresentable binding inlining}{from}{to}
1943 \subsubsection{Function specialization}
1944 This transform removes arguments to user-defined functions that are
1945 not representable at runtime. This is done by creating a
1946 \emph{specialized} version of the function that only works for one
1947 particular value of that argument (in other words, the argument can be
1950 Specialization means to create a specialized version of the called
1951 function, with one argument already filled in. As a simple example, in
1952 the following program (this is not actual Core, since it directly uses
1953 a literal with the unrepresentable type \lam{GHC.Prim.Int\#}).
1960 We could specialize the function \lam{f} against the literal argument
1961 1, with the following result:
1968 In some way, this transformation is similar to β-reduction, but it
1969 operates across function boundaries. It is also similar to
1970 non-representable let binding inlining above, since it sort of
1971 \quote{inlines} an expression into a called function.
1973 Special care must be taken when the argument has any free variables.
1974 If this is the case, the original argument should not be removed
1975 completely, but replaced by all the free variables of the expression.
1976 In this way, the original expression can still be evaluated inside the
1979 To prevent us from propagating the same argument over and over, a
1980 simple local variable reference is not propagated (since is has
1981 exactly one free variable, itself, we would only replace that argument
1984 This shows that any free local variables that are not runtime
1985 representable cannot be brought into normal form by this transform. We
1986 rely on an inlining or β-reduction transformation to replace such a
1987 variable with an expression we can propagate again.
1992 x Y0 ... Yi ... Yn \lam{Yi} is not representable
1993 --------------------------------------------- \lam{Yi} is not a local variable reference
1994 x' y0 ... yi-1 f0 ... fm Yi+1 ... Yn \lam{f0 ... fm} are all free local vars of \lam{Yi}
1995 ~ \lam{T0 ... Tn} are the types of \lam{Y0 ... Yn}
1996 x' = λ(y0 :: T0) ... λ(yi-1 :: Ty-1). λf0 ... λfm. λ(yi+1 :: Ty+1) ... λ(yn :: Tn).
1997 E y0 ... yi-1 Yi yi+1 ... yn
2000 This is a bit of a complex transformation. It transforms an
2001 application of the function \lam{x}, where one of the arguments
2002 (\lam{Y_i}) is not representable. A new
2003 function \lam{x'} is created that wraps the body of the old function.
2004 The body of the new function becomes a number of nested lambda
2005 abstractions, one for each of the original arguments that are left
2008 The ith argument is replaced with the free variables of
2009 \lam{Y_i}. Note that we reuse the same binders as those used in
2010 \lam{Y_i}, since we can then just use \lam{Y_i} inside the new
2011 function body and have all of the variables it uses be in scope.
2013 The argument that we are specializing for, \lam{Y_i}, is put inside
2014 the new function body. The old function body is applied to it. Since
2015 we use this new function only in place of an application with that
2016 particular argument \lam{Y_i}, behaviour should not change.
2018 Note that the types of the arguments of our new function are taken
2019 from the types of the \emph{actual} arguments (\lam{T0 ... Tn}). This
2020 means that any polymorphism in the arguments is removed, even when the
2021 corresponding explicit type lambda is not removed
2022 yet.\refdef{type lambda}
2024 \todo{Examples. Perhaps reference the previous sections}
2027 \section{Unsolved problems}
2028 The above system of transformations has been implemented in the prototype
2029 and seems to work well to compile simple and more complex examples of
2030 hardware descriptions. \todo{Ref christiaan?} However, this normalization
2031 system has not seen enough review and work to be complete and work for
2032 every Core expression that is supplied to it. A number of problems
2033 have already been identified and are discussed in this section.
2035 \subsection[sec:normalization:duplicatework]{Work duplication}
2036 A possible problem of β-reduction is that it could duplicate work.
2037 When the expression applied is not a simple variable reference, but
2038 requires calculation and the binder the lambda abstraction binds to
2039 is used more than once, more hardware might be generated than strictly
2042 As an example, consider the expression:
2048 When applying β-reduction to this expression, we get:
2054 which of course calculates \lam{(a * b)} twice.
2056 A possible solution to this would be to use the following alternative
2057 transformation, which is of course no longer normal β-reduction. The
2058 followin transformation has not been tested in the prototype, but is
2059 given here for future reference:
2067 This doesn't seem like much of an improvement, but it does get rid of
2068 the lambda expression (and the associated higher order value), while
2069 at the same time introducing a new let binding. Since the result of
2070 every application or case expression must be bound by a let expression
2071 in the intended normal form anyway, this is probably not a problem. If
2072 the argument happens to be a variable reference, then simple let
2073 binding removal (\in{section}[sec:normalization:simplelet]) will
2074 remove it, making the result identical to that of the original
2075 β-reduction transformation.
2077 When also applying argument simplification to the above example, we
2078 get the following expression:
2086 Looking at this, we could imagine an alternative approach: Create a
2087 transformation that removes let bindings that bind identical values.
2088 In the above expression, the \lam{y} and \lam{z} variables could be
2089 merged together, resulting in the more efficient expression:
2092 let y = (a * b) in y + y
2095 \subsection{Non-determinism}
2096 As an example, again consider the following expression:
2102 We can apply both β-reduction (\in{section}[sec:normalization:beta])
2103 as well as argument simplification
2104 (\in{section}[sec:normalization:argsimpl]) to this expression.
2106 When applying argument simplification first and then β-reduction, we
2107 get the following expression:
2110 let y = (a * b) in y + y
2113 When applying β-reduction first and then argument simplification, we
2114 get the following expression:
2122 As you can see, this is a different expression. This means that the
2123 order of expressions, does in fact change the resulting normal form,
2124 which is something that we would like to avoid. In this particular
2125 case one of the alternatives is even clearly more efficient, so we
2126 would of course like the more efficient form to be the normal form.
2128 For this particular problem, the solutions for duplication of work
2129 seem from the previous section seem to fix the determinism of our
2130 transformation system as well. However, it is likely that there are
2131 other occurences of this problem.
2134 We do not fully understand the use of cast expressions in Core, so
2135 there are probably expressions involving cast expressions that cannot
2136 be brought into intended normal form by this transformation system.
2138 The uses of casts in the core system should be investigated more and
2139 transformations will probably need updating to handle them in all
2142 \section[sec:normalization:properties]{Provable properties}
2143 When looking at the system of transformations outlined above, there are a
2144 number of questions that we can ask ourselves. The main question is of course:
2145 \quote{Does our system work as intended?}. We can split this question into a
2146 number of subquestions:
2149 \item[q:termination] Does our system \emph{terminate}? Since our system will
2150 keep running as long as transformations apply, there is an obvious risk that
2151 it will keep running indefinitely. This typically happens when one
2152 transformation produces a result that is transformed back to the original
2153 by another transformation, or when one or more transformations keep
2154 expanding some expression.
2155 \item[q:soundness] Is our system \emph{sound}? Since our transformations
2156 continuously modify the expression, there is an obvious risk that the final
2157 normal form will not be equivalent to the original program: Its meaning could
2159 \item[q:completeness] Is our system \emph{complete}? Since we have a complex
2160 system of transformations, there is an obvious risk that some expressions will
2161 not end up in our intended normal form, because we forgot some transformation.
2162 In other words: Does our transformation system result in our intended normal
2163 form for all possible inputs?
2164 \item[q:determinism] Is our system \emph{deterministic}? Since we have defined
2165 no particular order in which the transformation should be applied, there is an
2166 obvious risk that different transformation orderings will result in
2167 \emph{different} normal forms. They might still both be intended normal forms
2168 (if our system is \emph{complete}) and describe correct hardware (if our
2169 system is \emph{sound}), so this property is less important than the previous
2170 three: The translator would still function properly without it.
2173 Unfortunately, the final transformation system has only been
2174 developed in the final part of the research, leaving no more time
2175 for verifying these properties. In fact, it is likely that the
2176 current transformation system still violates some of these
2177 properties in some cases and should be improved (or extra conditions
2178 on the input hardware descriptions should be formulated).
2180 This is most likely the case with the completeness and determinism
2181 properties, perhaps als the termination property. The soundness
2182 property probably holds, since it is easier to manually verify (each
2183 transformation can be reviewed separately).
2185 Even though no complete proofs have been made, some ideas for
2186 possible proof strategies are shown below.
2188 \subsection{Graph representation}
2189 Before looking into how to prove these properties, we'll look at our
2190 transformation system from a graph perspective. The nodes of the graph are
2191 all possible Core expressions. The (directed) edges of the graph are
2192 transformations. When a transformation α applies to an expression \lam{A} to
2193 produce an expression \lam{B}, we add an edge from the node for \lam{A} to the
2194 node for \lam{B}, labeled α.
2196 \startuseMPgraphic{TransformGraph}
2200 newCircle.a(btex \lam{(λx.λy. (+) x y) 1} etex);
2201 newCircle.b(btex \lam{λy. (+) 1 y} etex);
2202 newCircle.c(btex \lam{(λx.(+) x) 1} etex);
2203 newCircle.d(btex \lam{(+) 1} etex);
2206 c.c = b.c + (4cm, 0cm);
2207 a.c = midpoint(b.c, c.c) + (0cm, 4cm);
2208 d.c = midpoint(b.c, c.c) - (0cm, 3cm);
2210 % β-conversion between a and b
2211 ncarc.a(a)(b) "name(bred)";
2212 ObjLabel.a(btex $\xrightarrow[normal]{}{β}$ etex) "labpathname(bred)", "labdir(rt)";
2213 ncarc.b(b)(a) "name(bexp)", "linestyle(dashed withdots)";
2214 ObjLabel.b(btex $\xleftarrow[normal]{}{β}$ etex) "labpathname(bexp)", "labdir(lft)";
2216 % η-conversion between a and c
2217 ncarc.a(a)(c) "name(ered)";
2218 ObjLabel.a(btex $\xrightarrow[normal]{}{η}$ etex) "labpathname(ered)", "labdir(rt)";
2219 ncarc.c(c)(a) "name(eexp)", "linestyle(dashed withdots)";
2220 ObjLabel.c(btex $\xleftarrow[normal]{}{η}$ etex) "labpathname(eexp)", "labdir(lft)";
2222 % η-conversion between b and d
2223 ncarc.b(b)(d) "name(ered)";
2224 ObjLabel.b(btex $\xrightarrow[normal]{}{η}$ etex) "labpathname(ered)", "labdir(rt)";
2225 ncarc.d(d)(b) "name(eexp)", "linestyle(dashed withdots)";
2226 ObjLabel.d(btex $\xleftarrow[normal]{}{η}$ etex) "labpathname(eexp)", "labdir(lft)";
2228 % β-conversion between c and d
2229 ncarc.c(c)(d) "name(bred)";
2230 ObjLabel.c(btex $\xrightarrow[normal]{}{β}$ etex) "labpathname(bred)", "labdir(rt)";
2231 ncarc.d(d)(c) "name(bexp)", "linestyle(dashed withdots)";
2232 ObjLabel.d(btex $\xleftarrow[normal]{}{β}$ etex) "labpathname(bexp)", "labdir(lft)";
2234 % Draw objects and lines
2235 drawObj(a, b, c, d);
2238 \placeexample[right][ex:TransformGraph]{Partial graph of a lambda calculus
2239 system with β and η reduction (solid lines) and expansion (dotted lines).}
2240 \boxedgraphic{TransformGraph}
2242 Of course our graph is unbounded, since we can construct an infinite amount of
2243 Core expressions. Also, there might potentially be multiple edges between two
2244 given nodes (with different labels), though seems unlikely to actually happen
2247 See \in{example}[ex:TransformGraph] for the graph representation of a very
2248 simple lambda calculus that contains just the expressions \lam{(λx.λy. (+) x
2249 y) 1}, \lam{λy. (+) 1 y}, \lam{(λx.(+) x) 1} and \lam{(+) 1}. The
2250 transformation system consists of β-reduction and η-reduction (solid edges) or
2251 β-expansion and η-expansion (dotted edges).
2253 \todo{Define β-reduction and η-reduction?}
2255 Note that the normal form of such a system consists of the set of nodes
2256 (expressions) without outgoing edges, since those are the expression to which
2257 no transformation applies anymore. We call this set of nodes the \emph{normal
2258 set}. The set of nodes containing expressions in intended normal
2259 form \refdef{intended normal form} is called the \emph{intended
2262 From such a graph, we can derive some properties easily:
2264 \item A system will \emph{terminate} if there is no path of infinite length
2265 in the graph (this includes cycles, but can also happen without cycles).
2266 \item Soundness is not easily represented in the graph.
2267 \item A system is \emph{complete} if all of the nodes in the normal set have
2268 the intended normal form. The inverse (that all of the nodes outside of
2269 the normal set are \emph{not} in the intended normal form) is not
2270 strictly required. In other words, our normal set must be a
2271 subset of the intended normal form, but they do not need to be
2274 \item A system is deterministic if all paths starting at a particular
2275 node, which end in a node in the normal set, end at the same node.
2278 When looking at the \in{example}[ex:TransformGraph], we see that the system
2279 terminates for both the reduction and expansion systems (but note that, for
2280 expansion, this is only true because we've limited the possible
2281 expressions. In comlete lambda calculus, there would be a path from
2282 \lam{(λx.λy. (+) x y) 1} to \lam{(λx.λy.(λz.(+) z) x y) 1} to
2283 \lam{(λx.λy.(λz.(λq.(+) q) z) x y) 1} etc.)
2285 If we would consider the system with both expansion and reduction, there
2286 would no longer be termination either, since there would be cycles all
2289 The reduction and expansion systems have a normal set of containing just
2290 \lam{(+) 1} or \lam{(λx.λy. (+) x y) 1} respectively. Since all paths in
2291 either system end up in these normal forms, both systems are \emph{complete}.
2292 Also, since there is only one node in the normal set, it must obviously be
2293 \emph{deterministic} as well.
2295 \todo{Add content to these sections}
2296 \subsection{Termination}
2297 In general, proving termination of an arbitrary program is a very
2298 hard problem. \todo{Ref about arbitrary termination} Fortunately,
2299 we only have to prove termination for our specific transformation
2302 A common approach for these kinds of proofs is to associate a
2303 measure with each possible expression in our system. If we can
2304 show that each transformation strictly decreases this measure
2305 (\ie, the expression transformed to has a lower measure than the
2306 expression transformed from). \todo{ref about measure-based
2307 termination proofs / analysis}
2309 A good measure for a system consisting of just β-reduction would
2310 be the number of lambda expressions in the expression. Since every
2311 application of β-reduction removes a lambda abstraction (and there
2312 is always a bounded number of lambda abstractions in every
2313 expression) we can easily see that a transformation system with
2314 just β-reduction will always terminate.
2316 For our complete system, this measure would be fairly complex
2317 (probably the sum of a lot of things). Since the (conditions on)
2318 our transformations are pretty complex, we would need to include
2319 both simple things like the number of let expressions as well as
2320 more complex things like the number of case expressions that are
2321 not yet in normal form.
2323 No real attempt has been made at finding a suitable measure for
2326 \subsection{Soundness}
2327 Soundness is a property that can be proven for each transformation
2328 separately. Since our system only runs separate transformations
2329 sequentially, if each of our transformations leaves the
2330 \emph{meaning} of the expression unchanged, then the entire system
2331 will of course leave the meaning unchanged and is thus
2334 The current prototype has only been verified in an ad-hoc fashion
2335 by inspecting (the code for) each transformation. A more formal
2336 verification would be more appropriate.
2338 To be able to formally show that each transformation properly
2339 preserves the meaning of every expression, we require an exact
2340 definition of the \emph{meaning} of every expression, so we can
2341 compare them. Currently there seems to be no formal definition of
2342 the meaning or semantics of \GHC's core language, only informal
2343 descriptions are available.
2345 It should be possible to have a single formal definition of
2346 meaning for Core for both normal Core compilation by \GHC and for
2347 our compilation to \VHDL. The main difference seems to be that in
2348 hardware every expression is always evaluated, while in software
2349 it is only evaluated if needed, but it should be possible to
2350 assign a meaning to core expressions that assumes neither.
2352 Since each of the transformations can be applied to any
2353 subexpression as well, there is a constraint on our meaning
2354 definition: The meaning of an expression should depend only on the
2355 meaning of subexpressions, not on the expressions themselves. For
2356 example, the meaning of the application in \lam{f (let x = 4 in
2357 x)} should be the same as the meaning of the application in \lam{f
2358 4}, since the argument subexpression has the same meaning (though
2359 the actual expression is different).
2361 \subsection{Completeness}
2362 Proving completeness is probably not hard, but it could be a lot
2363 of work. We have seen above that to prove completeness, we must
2364 show that the normal set of our graph representation is a subset
2365 of the intended normal set.
2367 However, it is hard to systematically generate or reason about the
2368 normal set, since it is defined as any nodes to which no
2369 transformation applies. To determine this set, each transformation
2370 must be considered and when a transformation is added, the entire
2371 set should be re-evaluated. This means it is hard to show that
2372 each node in the normal set is also in the intended normal set.
2373 Reasoning about our intended normal set is easier, since we know
2374 how to generate it from its definition. \refdef{intended normal
2377 Fortunately, we can also prove the complement (which is
2378 equivalent, since $A \subseteq B \Leftrightarrow \overline{B}
2379 \subseteq \overline{A}$): Show that the set of nodes not in
2380 intended normal form is a subset of the set of nodes not in normal
2381 form. In other words, show that for every expression that is not
2382 in intended normal form, that there is at least one transformation
2383 that applies to it (since that means it is not in normal form
2384 either and since $A \subseteq C \Leftrightarrow \forall x (x \in A
2385 \rightarrow x \in C)$).
2387 By systematically reviewing the entire Core language definition
2388 along with the intended normal form definition (both of which have
2389 a similar structure), it should be possible to identify all
2390 possible (sets of) core expressions that are not in intended
2391 normal form and identify a transformation that applies to it.
2393 This approach is especially useful for proving completeness of our
2394 system, since if expressions exist to which none of the
2395 transformations apply (\ie if the system is not yet complete), it
2396 is immediately clear which expressions these are and adding
2397 (or modifying) transformations to fix this should be relatively
2400 As observed above, applying this approach is a lot of work, since
2401 we need to check every (set of) transformation(s) separately.
2403 \todo{Perhaps do a few steps of the proofs as proof-of-concept}
2405 % vim: set sw=2 sts=2 expandtab: