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
28 areas (higher order expressions, limited polymorphism using type
29 classes, etc.) and because core can describe expressions that do not
30 have a direct hardware interpretation.
33 The transformations described here have a well-defined goal: To bring the
34 program in a well-defined form that is directly translatable to
35 \VHDL, while fully preserving the semantics of the program. We refer
36 to this form as the \emph{normal form} of the program. The formal
37 definition of this normal form is quite simple:
39 \placedefinition{}{\startboxed A program is in \emph{normal form} if none of the
40 transformations from this chapter apply.\stopboxed}
42 Of course, this is an \quote{easy} definition of the normal form, since our
43 program will end up in normal form automatically. The more interesting part is
44 to see if this normal form actually has the properties we would like it to
47 But, before getting into more definitions and details about this normal form,
48 let's try to get a feeling for it first. The easiest way to do this is by
49 describing the things we want to not have in a normal form.
52 \item Any \emph{polymorphism} must be removed. When laying down hardware, we
53 can't generate any signals that can have multiple types. All types must be
54 completely known to generate hardware.
56 \item All \emph{higher order} constructions must be removed. We can't
57 generate a hardware signal that contains a function, so all values,
58 arguments and return values used must be first order.
60 \item All complex \emph{nested scopes} must be removed. In the \small{VHDL}
61 description, every signal is in a single scope. Also, full expressions are
62 not supported everywhere (in particular port maps can only map signal
63 names and constants, not complete expressions). To make the \small{VHDL}
64 generation easy, a separate binder must be bound to ever application or
68 \todo{Intermezzo: functions vs plain values}
70 A very simple example of a program in normal form is given in
71 \in{example}[ex:MulSum]. As you can see, all arguments to the function (which
72 will become input ports in the generated \VHDL) are at the outer level.
73 This means that the body of the inner lambda abstraction is never a
74 function, but always a plain value.
76 As the body of the inner lambda abstraction, we see a single (recursive)
77 let expression, that binds two variables (\lam{mul} and \lam{sum}). These
78 variables will be signals in the generated \VHDL, bound to the output port
79 of the \lam{*} and \lam{+} components.
81 The final line (the \quote{return value} of the function) selects the
82 \lam{sum} signal to be the output port of the function. This \quote{return
83 value} can always only be a variable reference, never a more complex
86 \todo{Add generated VHDL}
89 alu :: Bit -> Word -> Word -> Word
98 \startuseMPgraphic{MulSum}
99 save a, b, c, mul, add, sum;
102 newCircle.a(btex $a$ etex) "framed(false)";
103 newCircle.b(btex $b$ etex) "framed(false)";
104 newCircle.c(btex $c$ etex) "framed(false)";
105 newCircle.sum(btex $sum$ etex) "framed(false)";
108 newCircle.mul(btex * etex);
109 newCircle.add(btex + etex);
111 a.c - b.c = (0cm, 2cm);
112 b.c - c.c = (0cm, 2cm);
113 add.c = c.c + (2cm, 0cm);
114 mul.c = midpoint(a.c, b.c) + (2cm, 0cm);
115 sum.c = add.c + (2cm, 0cm);
118 % Draw objects and lines
119 drawObj(a, b, c, mul, add, sum);
121 ncarc(a)(mul) "arcangle(15)";
122 ncarc(b)(mul) "arcangle(-15)";
128 \placeexample[here][ex:MulSum]{Simple architecture consisting of a
129 multiplier and a subtractor.}
130 \startcombination[2*1]
131 {\typebufferlam{MulSum}}{Core description in normal form.}
132 {\boxedgraphic{MulSum}}{The architecture described by the normal form.}
135 \in{Example}[ex:MulSum] showed a function that just applied two
136 other functions (multiplication and addition), resulting in a simple
137 architecture with two components and some connections. There is of
138 course also some mechanism for choice in the normal form. In a
139 normal Core program, the \emph{case} expression can be used in a few
140 different ways to describe choice. In normal form, this is limited
141 to a very specific form.
143 \in{Example}[ex:AddSubAlu] shows an example describing a
144 simple \small{ALU}, which chooses between two operations based on an opcode
145 bit. The main structure is similar to \in{example}[ex:MulSum], but this
146 time the \lam{res} variable is bound to a case expression. This case
147 expression scrutinizes the variable \lam{opcode} (and scrutinizing more
148 complex expressions is not supported). The case expression can select a
149 different variable based on the constructor of \lam{opcode}.
150 \refdef{case expression}
152 \startbuffer[AddSubAlu]
153 alu :: Bit -> Word -> Word -> Word
165 \startuseMPgraphic{AddSubAlu}
166 save opcode, a, b, add, sub, mux, res;
169 newCircle.opcode(btex $opcode$ etex) "framed(false)";
170 newCircle.a(btex $a$ etex) "framed(false)";
171 newCircle.b(btex $b$ etex) "framed(false)";
172 newCircle.res(btex $res$ etex) "framed(false)";
174 newCircle.add(btex + etex);
175 newCircle.sub(btex - etex);
178 opcode.c - a.c = (0cm, 2cm);
179 add.c - a.c = (4cm, 0cm);
180 sub.c - b.c = (4cm, 0cm);
181 a.c - b.c = (0cm, 3cm);
182 mux.c = midpoint(add.c, sub.c) + (1.5cm, 0cm);
183 res.c - mux.c = (1.5cm, 0cm);
186 % Draw objects and lines
187 drawObj(opcode, a, b, res, add, sub, mux);
189 ncline(a)(add) "posA(e)";
190 ncline(b)(sub) "posA(e)";
191 nccurve(a)(sub) "posA(e)", "angleA(0)";
192 nccurve(b)(add) "posA(e)", "angleA(0)";
193 nccurve(add)(mux) "posB(inpa)", "angleB(0)";
194 nccurve(sub)(mux) "posB(inpb)", "angleB(0)";
195 nccurve(opcode)(mux) "posB(n)", "angleA(0)", "angleB(-90)";
196 ncline(mux)(res) "posA(out)";
199 \placeexample[here][ex:AddSubAlu]{Simple \small{ALU} supporting two operations.}
200 \startcombination[2*1]
201 {\typebufferlam{AddSubAlu}}{Core description in normal form.}
202 {\boxedgraphic{AddSubAlu}}{The architecture described by the normal form.}
205 As a more complete example, consider
206 \in{example}[ex:NormalComplete]. This example shows everything that
207 is allowed in normal form, except for builtin higher order functions
208 (like \lam{map}). The graphical version of the architecture contains
209 a slightly simplified version, since the state tuple packing and
210 unpacking have been left out. Instead, two separate registers are
211 drawn. Also note that most synthesis tools will further optimize
212 this architecture by removing the multiplexers at the register input
213 and instead put some gates in front of the register's clock input,
214 but we want to show the architecture as close to the description as
217 As you can see from the previous examples, the generation of the final
218 architecture from the normal form is straightforward. In each of the
219 examples, there is a direct match between the normal form structure,
220 the generated VHDL and the architecture shown in the images.
222 \startbuffer[NormalComplete]
225 -> State (Word, Word)
226 -> (State (Word, Word), Word)
228 -- All arguments are an inital lambda (address, data, packed state)
230 -- There are nested let expressions at top level
232 -- Unpack the state by coercion (\eg, cast from
233 -- State (Word, Word) to (Word, Word))
234 s = sp ▶ (Word, Word)
235 -- Extract both registers from the state
236 r1 = case s of (a, b) -> a
237 r2 = case s of (a, b) -> b
238 -- Calling some other user-defined function.
240 -- Conditional connections
252 -- pack the state by coercion (\eg, cast from
253 -- (Word, Word) to State (Word, Word))
254 sp' = s' ▶ State (Word, Word)
255 -- Pack our return value
262 \startuseMPgraphic{NormalComplete}
263 save a, d, r, foo, muxr, muxout, out;
266 newCircle.a(btex \lam{a} etex) "framed(false)";
267 newCircle.d(btex \lam{d} etex) "framed(false)";
268 newCircle.out(btex \lam{out} etex) "framed(false)";
270 %newCircle.add(btex + etex);
271 newBox.foo(btex \lam{foo} etex);
272 newReg.r1(btex $\lam{r1}$ etex) "dx(4mm)", "dy(6mm)";
273 newReg.r2(btex $\lam{r2}$ etex) "dx(4mm)", "dy(6mm)", "reflect(true)";
275 % Reflect over the vertical axis
276 reflectObj(muxr1)((0,0), (0,1));
279 rotateObj(muxout)(-90);
281 d.c = foo.c + (0cm, 1.5cm);
282 a.c = (xpart r2.c + 2cm, ypart d.c - 0.5cm);
283 foo.c = midpoint(muxr1.c, muxr2.c) + (0cm, 2cm);
284 muxr1.c = r1.c + (0cm, 2cm);
285 muxr2.c = r2.c + (0cm, 2cm);
286 r2.c = r1.c + (4cm, 0cm);
288 muxout.c = midpoint(r1.c, r2.c) - (0cm, 2cm);
289 out.c = muxout.c - (0cm, 1.5cm);
291 % % Draw objects and lines
292 drawObj(a, d, foo, r1, r2, muxr1, muxr2, muxout, out);
295 nccurve(foo)(muxr1) "angleA(-90)", "posB(inpa)", "angleB(180)";
296 nccurve(foo)(muxr2) "angleA(-90)", "posB(inpb)", "angleB(0)";
297 nccurve(muxr1)(r1) "posA(out)", "angleA(180)", "posB(d)", "angleB(0)";
298 nccurve(r1)(muxr1) "posA(out)", "angleA(0)", "posB(inpb)", "angleB(180)";
299 nccurve(muxr2)(r2) "posA(out)", "angleA(0)", "posB(d)", "angleB(180)";
300 nccurve(r2)(muxr2) "posA(out)", "angleA(180)", "posB(inpa)", "angleB(0)";
301 nccurve(r1)(muxout) "posA(out)", "angleA(0)", "posB(inpb)", "angleB(-90)";
302 nccurve(r2)(muxout) "posA(out)", "angleA(180)", "posB(inpa)", "angleB(-90)";
304 nccurve(a)(muxout) "angleA(-90)", "angleB(180)", "posB(sel)";
305 nccurve(a)(muxr1) "angleA(180)", "angleB(-90)", "posB(sel)";
306 nccurve(a)(muxr2) "angleA(180)", "angleB(-90)", "posB(sel)";
307 ncline(muxout)(out) "posA(out)";
310 \todo{Don't split registers in this image?}
311 \placeexample[here][ex:NormalComplete]{Simple architecture consisting of an adder and a
313 \startcombination[2*1]
314 {\typebufferlam{NormalComplete}}{Core description in normal form.}
315 {\boxedgraphic{NormalComplete}}{The architecture described by the normal form.}
320 \subsection[sec:normalization:intendednormalform]{Intended normal form definition}
321 Now we have some intuition for the normal form, we can describe how we want
322 the normal form to look like in a slightly more formal manner. The following
323 EBNF-like description captures most of the intended structure (and
324 generates a subset of GHC's core format).
326 There are two things missing: Cast expressions are sometimes
327 allowed by the prototype, but not specified here and the below
328 definition allows uses of state that cannot be translated to \VHDL
329 properly. These two problems are discussed in
330 \in{section}[sec:normalization:castproblems] and
331 \in{section}[sec:normalization:stateproblems] respectively.
333 Some clauses have an expression listed behind them in parentheses.
334 These are conditions that need to apply to the clause. The
335 predicates used there (\lam{lvar()}, \lam{representable()},
336 \lam{gvar()}) will be defined in
337 \in{section}[sec:normalization:predicates].
339 An expression is in normal form if it matches the first
340 definition, \emph{normal}.
342 \todo{Fix indentation}
343 \startbuffer[IntendedNormal]
344 \italic{normal} := \italic{lambda}
345 \italic{lambda} := λvar.\italic{lambda} (representable(var))
347 \italic{toplet} := letrec [\italic{binding}...] in var (representable(var))
348 \italic{binding} := var = \italic{rhs} (representable(rhs))
349 -- State packing and unpacking by coercion
350 | var0 = var1 ▶ State ty (lvar(var1))
351 | var0 = var1 ▶ ty (var1 :: State ty ∧ lvar(var1))
352 \italic{rhs} := \italic{userapp}
353 | \italic{builtinapp}
355 | case var of C a0 ... an -> ai (lvar(var))
357 | case var of (lvar(var))
358 [ DEFAULT -> var ] (lvar(var))
359 C0 w0,0 ... w0,n -> var0
361 Cm wm,0 ... wm,n -> varm (\forall{}i \forall{}j, wi,j \neq vari, lvar(vari))
362 \italic{userapp} := \italic{userfunc}
363 | \italic{userapp} {userarg}
364 \italic{userfunc} := var (gvar(var))
365 \italic{userarg} := var (lvar(var))
366 \italic{builtinapp} := \italic{builtinfunc}
367 | \italic{builtinapp} \italic{builtinarg}
368 \italic{builtinfunc} := var (bvar(var))
369 \italic{builtinarg} := var (representable(var) ∧ lvar(var))
370 | \italic{partapp} (partapp :: a -> b)
371 | \italic{coreexpr} (¬representable(coreexpr) ∧ ¬(coreexpr :: a -> b))
372 \italic{partapp} := \italic{userapp}
373 | \italic{builtinapp}
376 \placedefinition[][def:IntendedNormal]{Definition of the intended nnormal form using an \small{EBNF}-like syntax.}
377 {\defref{intended normal form definition}
378 \typebufferlam{IntendedNormal}}
380 When looking at such a program from a hardware perspective, the
381 top level lambda abstractions define the input ports. Lambda
382 abstractions cannot appear anywhere else. The variable reference
383 in the body of the recursive let expression is the output port.
384 Most function applications bound by the let expression define a
385 component instantiation, where the input and output ports are
386 mapped to local signals or arguments. Some of the others use a
387 builtin construction (\eg the \lam{case} expression) or call a
388 builtin function (\eg \lam{+} or \lam{map}). For these, a
389 hardcoded \small{VHDL} translation is available.
391 \section[sec:normalization:transformation]{Transformation notation}
392 To be able to concisely present transformations, we use a specific format
393 for them. It is a simple format, similar to one used in logic reasoning.
395 Such a transformation description looks like the following.
400 <original expression>
401 -------------------------- <expression conditions>
402 <transformed expresssion>
407 This format desribes a transformation that applies to \lam{<original
408 expresssion>} and transforms it into \lam{<transformed expression>}, assuming
409 that all conditions are satisfied. In this format, there are a number of placeholders
410 in pointy brackets, most of which should be rather obvious in their meaning.
411 Nevertheless, we will more precisely specify their meaning below:
413 \startdesc{<original expression>} The expression pattern that will be matched
414 against (subexpressions of) the expression to be transformed. We call this a
415 pattern, because it can contain \emph{placeholders} (variables), which match
416 any expression or binder. Any such placeholder is said to be \emph{bound} to
417 the expression it matches. It is convention to use an uppercase letter (\eg
418 \lam{M} or \lam{E}) to refer to any expression (including a simple variable
419 reference) and lowercase letters (\eg \lam{v} or \lam{b}) to refer to
420 (references to) binders.
422 For example, the pattern \lam{a + B} will match the expression
423 \lam{v + (2 * w)} (binding \lam{a} to \lam{v} and \lam{B} to
424 \lam{(2 * w)}), but not \lam{(2 * w) + v}.
427 \startdesc{<expression conditions>}
428 These are extra conditions on the expression that is matched. These
429 conditions can be used to further limit the cases in which the
430 transformation applies, commonly to prevent a transformation from
431 causing a loop with itself or another transformation.
433 Only if these conditions are \emph{all} satisfied, the transformation
437 \startdesc{<context conditions>}
438 These are a number of extra conditions on the context of the function. In
439 particular, these conditions can require some (other) top level function to be
440 present, whose value matches the pattern given here. The format of each of
441 these conditions is: \lam{binder = <pattern>}.
443 Typically, the binder is some placeholder bound in the \lam{<original
444 expression>}, while the pattern contains some placeholders that are used in
445 the \lam{transformed expression}.
447 Only if a top level binder exists that matches each binder and pattern,
448 the transformation applies.
451 \startdesc{<transformed expression>}
452 This is the expression template that is the result of the transformation. If, looking
453 at the above three items, the transformation applies, the \lam{<original
454 expression>} is completely replaced by the \lam{<transformed expression>}.
455 We call this a template, because it can contain placeholders, referring to
456 any placeholder bound by the \lam{<original expression>} or the
457 \lam{<context conditions>}. The resulting expression will have those
458 placeholders replaced by the values bound to them.
460 Any binder (lowercase) placeholder that has no value bound to it yet will be
461 bound to (and replaced with) a fresh binder.
464 \startdesc{<context additions>}
465 These are templates for new functions to be added to the context.
466 This is a way to let a transformation create new top level
469 Each addition has the form \lam{binder = template}. As above, any
470 placeholder in the addition is replaced with the value bound to it, and any
471 binder placeholder that has no value bound to it yet will be bound to (and
472 replaced with) a fresh binder.
475 To understand this notation better, the step by step application of
476 the η-abstraction transformation to a simple \small{ALU} will be
477 shown. Consider η-abstraction, described using above notation as
482 -------------- \lam{E} does not occur on a function position in an application
483 λx.E x \lam{E} is not a lambda abstraction.
486 η-abstraction is a well known transformation from lambda calculus. What
487 this transformation does, is take any expression that has a function type
488 and turn it into a lambda expression (giving an explicit name to the
489 argument). There are some extra conditions that ensure that this
490 transformation does not apply infinitely (which are not necessarily part
491 of the conventional definition of η-abstraction).
493 Consider the following function, in Core notation, which is a fairly obvious way to specify a
494 simple \small{ALU} (Note that it is not yet in normal form, but
495 \in{example}[ex:AddSubAlu] shows the normal form of this function).
496 The parentheses around the \lam{+} and \lam{-} operators are
497 commonly used in Haskell to show that the operators are used as
498 normal functions, instead of \emph{infix} operators (\eg, the
499 operators appear before their arguments, instead of in between).
502 alu :: Bit -> Word -> Word -> Word
503 alu = λopcode. case opcode of
508 There are a few subexpressions in this function to which we could possibly
509 apply the transformation. Since the pattern of the transformation is only
510 the placeholder \lam{E}, any expression will match that. Whether the
511 transformation applies to an expression is thus solely decided by the
512 conditions to the right of the transformation.
514 We will look at each expression in the function in a top down manner. The
515 first expression is the entire expression the function is bound to.
518 λopcode. case opcode of
523 As said, the expression pattern matches this. The type of this expression is
524 \lam{Bit -> Word -> Word -> Word}, which matches \lam{a -> b} (Note that in
525 this case \lam{a = Bit} and \lam{b = Word -> Word -> Word}).
527 Since this expression is at top level, it does not occur at a function
528 position of an application. However, The expression is a lambda abstraction,
529 so this transformation does not apply.
531 The next expression we could apply this transformation to, is the body of
532 the lambda abstraction:
540 The type of this expression is \lam{Word -> Word -> Word}, which again
541 matches \lam{a -> b}. The expression is the body of a lambda expression, so
542 it does not occur at a function position of an application. Finally, the
543 expression is not a lambda abstraction but a case expression, so all the
544 conditions match. There are no context conditions to match, so the
545 transformation applies.
547 By now, the placeholder \lam{E} is bound to the entire expression. The
548 placeholder \lam{x}, which occurs in the replacement template, is not bound
549 yet, so we need to generate a fresh binder for that. Let's use the binder
550 \lam{a}. This results in the following replacement expression:
558 Continuing with this expression, we see that the transformation does not
559 apply again (it is a lambda expression). Next we look at the body of this
568 Here, the transformation does apply, binding \lam{E} to the entire
569 expression (which has type \lam{Word -> Word}) and binding \lam{x}
570 to the fresh binder \lam{b}, resulting in the replacement:
578 The transformation does not apply to this lambda abstraction, so we
579 look at its body. For brevity, we'll put the case expression on one line from
583 (case opcode of Low -> (+); High -> (-)) a b
586 The type of this expression is \lam{Word}, so it does not match \lam{a -> b}
587 and the transformation does not apply. Next, we have two options for the
588 next expression to look at: The function position and argument position of
589 the application. The expression in the argument position is \lam{b}, which
590 has type \lam{Word}, so the transformation does not apply. The expression in
591 the function position is:
594 (case opcode of Low -> (+); High -> (-)) a
597 Obviously, the transformation does not apply here, since it occurs in
598 function position (which makes the second condition false). In the same
599 way the transformation does not apply to both components of this
600 expression (\lam{case opcode of Low -> (+); High -> (-)} and \lam{a}), so
601 we'll skip to the components of the case expression: The scrutinee and
602 both alternatives. Since the opcode is not a function, it does not apply
605 The first alternative is \lam{(+)}. This expression has a function type
606 (the operator still needs two arguments). It does not occur in function
607 position of an application and it is not a lambda expression, so the
608 transformation applies.
610 We look at the \lam{<original expression>} pattern, which is \lam{E}.
611 This means we bind \lam{E} to \lam{(+)}. We then replace the expression
612 with the \lam{<transformed expression>}, replacing all occurences of
613 \lam{E} with \lam{(+)}. In the \lam{<transformed expression>}, the This gives us the replacement expression:
614 \lam{λx.(+) x} (A lambda expression binding \lam{x}, with a body that
615 applies the addition operator to \lam{x}).
617 The complete function then becomes:
619 (case opcode of Low -> λa1.(+) a1; High -> (-)) a
622 Now the transformation no longer applies to the complete first alternative
623 (since it is a lambda expression). It does not apply to the addition
624 operator again, since it is now in function position in an application. It
625 does, however, apply to the application of the addition operator, since
626 that is neither a lambda expression nor does it occur in function
627 position. This means after one more application of the transformation, the
631 (case opcode of Low -> λa1.λb1.(+) a1 b1; High -> (-)) a
634 The other alternative is left as an exercise to the reader. The final
635 function, after applying η-abstraction until it does no longer apply is:
638 alu :: Bit -> Word -> Word -> Word
639 alu = λopcode.λa.b. (case opcode of
640 Low -> λa1.λb1 (+) a1 b1
641 High -> λa2.λb2 (-) a2 b2) a b
644 \subsection{Transformation application}
645 In this chapter we define a number of transformations, but how will we apply
646 these? As stated before, our normal form is reached as soon as no
647 transformation applies anymore. This means our application strategy is to
648 simply apply any transformation that applies, and continuing to do that with
649 the result of each transformation.
651 In particular, we define no particular order of transformations. Since
652 transformation order should not influence the resulting normal form,
653 this leaves the implementation free to choose any application order that
654 results in an efficient implementation. Unfortunately this is not
655 entirely true for the current set of transformations. See
656 \in{section}[sec:normalization:non-determinism] for a discussion of this
659 When applying a single transformation, we try to apply it to every (sub)expression
660 in a function, not just the top level function body. This allows us to
661 keep the transformation descriptions concise and powerful.
663 \subsection{Definitions}
664 A \emph{global variable} is any variable (binder) that is bound at the
665 top level of a program, or an external module. A \emph{local variable} is any
666 other variable (\eg, variables local to a function, which can be bound by
667 lambda abstractions, let expressions and pattern matches of case
668 alternatives). This is a slightly different notion of global versus
669 local than what \small{GHC} uses internally, but for our purposes
670 the distinction \GHC makes is not useful.
671 \defref{global variable} \defref{local variable}
673 A \emph{hardware representable} (or just \emph{representable}) type or value
674 is (a value of) a type that we can generate a signal for in hardware. For
675 example, a bit, a vector of bits, a 32 bit unsigned word, etc. Values that are
676 not runtime representable notably include (but are not limited to): Types,
677 dictionaries, functions.
678 \defref{representable}
680 A \emph{builtin function} is a function supplied by the Cλash framework, whose
681 implementation is not valid Cλash. The implementation is of course valid
682 Haskell, for simulation, but it is not expressable in Cλash.
683 \defref{builtin function} \defref{user-defined function}
685 For these functions, Cλash has a \emph{builtin hardware translation}, so calls
686 to these functions can still be translated. These are functions like
687 \lam{map}, \lam{hwor} and \lam{length}.
689 A \emph{user-defined} function is a function for which we do have a Cλash
690 implementation available.
692 \subsubsection[sec:normalization:predicates]{Predicates}
693 Here, we define a number of predicates that can be used below to concisely
696 \emph{gvar(expr)} is true when \emph{expr} is a variable that references a
697 global variable. It is false when it references a local variable.
699 \emph{lvar(expr)} is the complement of \emph{gvar}; it is true when \emph{expr}
700 references a local variable, false when it references a global variable.
702 \emph{representable(expr)} is true when \emph{expr} is \emph{representable}.
704 \subsection[sec:normalization:uniq]{Binder uniqueness}
705 A common problem in transformation systems, is binder uniqueness. When not
706 considering this problem, it is easy to create transformations that mix up
707 bindings and cause name collisions. Take for example, the following core
711 (λa.λb.λc. a * b * c) x c
714 By applying β-reduction (see \in{section}[sec:normalization:beta]) once,
715 we can simplify this expression to:
721 Now, we have replaced the \lam{a} binder with a reference to the \lam{x}
722 binder. No harm done here. But note that we see multiple occurences of the
723 \lam{c} binder. The first is a binding occurence, to which the second refers.
724 The last, however refers to \emph{another} instance of \lam{c}, which is
725 bound somewhere outside of this expression. Now, if we would apply beta
726 reduction without taking heed of binder uniqueness, we would get:
732 This is obviously not what was supposed to happen! The root of this problem is
733 the reuse of binders: Identical binders can be bound in different,
734 but overlapping scopes. Any variable reference in those
735 overlapping scopes then refers to the variable bound in the inner
736 (smallest) scope. There is not way to refer to the variable in the
737 outer scope. This effect is usually referred to as
738 \emph{shadowing}: When a binder is bound in a scope where the
739 binder already had a value, the inner binding is said to
740 \emph{shadow} the outer binding. In the example above, the \lam{c}
741 binder was bound outside of the expression and in the inner lambda
742 expression. Inside that lambda expression, only the inner \lam{c}
745 There are a number of ways to solve this. \small{GHC} has isolated this
746 problem to their binder substitution code, which performs \emph{deshadowing}
747 during its expression traversal. This means that any binding that shadows
748 another binding on a higher level is replaced by a new binder that does not
749 shadow any other binding. This non-shadowing invariant is enough to prevent
750 binder uniqueness problems in \small{GHC}.
752 In our transformation system, maintaining this non-shadowing invariant is
753 a bit harder to do (mostly due to implementation issues, the prototype doesn't
754 use \small{GHC}'s subsitution code). Also, the following points can be
758 \item Deshadowing does not guarantee overall uniqueness. For example, the
759 following (slightly contrived) expression shows the identifier \lam{x} bound in
760 two seperate places (and to different values), even though no shadowing
764 (let x = 1 in x) + (let x = 2 in x)
767 \item In our normal form (and the resulting \small{VHDL}), all binders
768 (signals) within the same function (entity) will end up in the same
769 scope. To allow this, all binders within the same function should be
772 \item When we know that all binders in an expression are unique, moving around
773 or removing a subexpression will never cause any binder conflicts. If we have
774 some way to generate fresh binders, introducing new subexpressions will not
775 cause any problems either. The only way to cause conflicts is thus to
776 duplicate an existing subexpression.
779 Given the above, our prototype maintains a unique binder invariant. This
780 means that in any given moment during normalization, all binders \emph{within
781 a single function} must be unique. To achieve this, we apply the following
784 \todo{Define fresh binders and unique supplies}
787 \item Before starting normalization, all binders in the function are made
788 unique. This is done by generating a fresh binder for every binder used. This
789 also replaces binders that did not cause any conflict, but it does ensure that
790 all binders within the function are generated by the same unique supply.
791 \refdef{fresh binder}
792 \item Whenever a new binder must be generated, we generate a fresh binder that
793 is guaranteed to be different from \emph{all binders generated so far}. This
794 can thus never introduce duplication and will maintain the invariant.
795 \item Whenever (a part of) an expression is duplicated (for example when
796 inlining), all binders in the expression are replaced with fresh binders
797 (using the same method as at the start of normalization). These fresh binders
798 can never introduce duplication, so this will maintain the invariant.
799 \item Whenever we move part of an expression around within the function, there
800 is no need to do anything special. There is obviously no way to introduce
801 duplication by moving expressions around. Since we know that each of the
802 binders is already unique, there is no way to introduce (incorrect) shadowing
806 \section{Transform passes}
807 In this section we describe the actual transforms.
809 Each transformation will be described informally first, explaining
810 the need for and goal of the transformation. Then, we will formally define
811 the transformation using the syntax introduced in
812 \in{section}[sec:normalization:transformation].
814 \subsection{General cleanup}
815 These transformations are general cleanup transformations, that aim to
816 make expressions simpler. These transformations usually clean up the
817 mess left behind by other transformations or clean up expressions to
818 expose new transformation opportunities for other transformations.
820 Most of these transformations are standard optimizations in other
821 compilers as well. However, in our compiler, most of these are not just
822 optimizations, but they are required to get our program into intended
826 \defref{substitution notation}
827 \startframedtext[width=8cm,background=box,frame=no]
828 \startalignment[center]
829 {\tfa Substitution notation}
833 In some of the transformations in this chapter, we need to perform
834 substitution on an expression. Substitution means replacing every
835 occurence of some expression (usually a variable reference) with
838 There have been a lot of different notations used in literature for
839 specifying substitution. The notation that will be used in this report
846 This means expression \lam{E} with all occurences of \lam{A} replaced
851 \subsubsection[sec:normalization:beta]{β-reduction}
852 β-reduction is a well known transformation from lambda calculus, where it is
853 the main reduction step. It reduces applications of lambda abstractions,
854 removing both the lambda abstraction and the application.
856 In our transformation system, this step helps to remove unwanted lambda
857 abstractions (basically all but the ones at the top level). Other
858 transformations (application propagation, non-representable inlining) make
859 sure that most lambda abstractions will eventually be reducable by
862 Note that β-reduction also works on type lambda abstractions and type
863 applications as well. This means the substitution below also works on
864 type variables, in the case that the binder is a type variable and teh
865 expression applied to is a type.
882 \transexample{beta}{β-reduction}{from}{to}
892 \transexample{beta-type}{β-reduction for type abstractions}{from}{to}
894 \subsubsection{Empty let removal}
895 This transformation is simple: It removes recursive lets that have no bindings
896 (which usually occurs when unused let binding removal removes the last
899 Note that there is no need to define this transformation for
900 non-recursive lets, since they always contain exactly one binding.
910 \subsubsection[sec:normalization:simplelet]{Simple let binding removal}
911 This transformation inlines simple let bindings, that bind some
912 binder to some other binder instead of a more complex expression (\ie
915 This transformation is not needed to get an expression into intended
916 normal form (since these bindings are part of the intended normal
917 form), but makes the resulting \small{VHDL} a lot shorter.
919 \refdef{substitution notation}
929 ----------------------------- \lam{b} is a variable reference
930 letrec \lam{ai} ≠ \lam{b}
943 \subsubsection{Unused let binding removal}
944 This transformation removes let bindings that are never used.
945 Occasionally, \GHC's desugarer introduces some unused let bindings.
947 This normalization pass should really be not be necessary to get
948 into intended normal form (since the intended normal form
949 definition \refdef{intended normal form definition} does not
950 require that every binding is used), but in practice the
951 desugarer or simplifier emits some bindings that cannot be
952 normalized (e.g., calls to a
953 \hs{Control.Exception.Base.patError}) but are not used anywhere
954 either. To prevent the \VHDL generation from breaking on these
955 artifacts, this transformation removes them.
957 \todo{Don't use old-style numerals in transformations}
966 M \lam{ai} does not occur free in \lam{M}
967 ---------------------------- \forall j, 0 ≤ j ≤ n, j ≠ i (\lam{ai} does not occur free in \lam{Ej})
981 \subsubsection{Cast propagation / simplification}
982 This transform pushes casts down into the expression as far as
983 possible. This transformation has been added to make a few
984 specific corner cases work, but it is not clear yet if this
985 transformation handles cast expressions completely or in the
986 right way. See \in{section}[sec:normalization:castproblems].
990 -------------------------
1000 -------------------------
1007 \subsubsection{Top level binding inlining}
1008 \refdef{top level binding}
1009 This transform takes simple top level bindings generated by the
1010 \small{GHC} compiler. \small{GHC} sometimes generates very simple
1011 \quote{wrapper} bindings, which are bound to just a variable
1012 reference, or contain just a (partial) function appliation with
1013 the type and dictionary arguments filled in (such as the
1014 \lam{(+)} in the example below).
1016 Note that this transformation is completely optional. It is not
1017 required to get any function into intended normal form, but it does help making
1018 the resulting VHDL output easier to read (since it removes a bunch of
1019 components that are really boring).
1021 This transform takes any top level binding generated by \GHC,
1022 whose normalized form contains only a single let binding.
1025 x = λa0 ... λan.let y = E in y
1028 -------------------------------------- \lam{x} is generated by the compiler
1029 λa0 ... λan.let y = E in y
1033 (+) :: Word -> Word -> Word
1034 (+) = GHC.Num.(+) @Word \$dNum
1039 GHC.Num.(+) @ Alu.Word \$dNum a b
1042 \transexample{toplevelinline}{Top level binding inlining}{from}{to}
1044 \in{Example}[ex:trans:toplevelinline] shows a typical application of
1045 the addition operator generated by \GHC. The type and dictionary
1046 arguments used here are described in
1047 \in{Section}[section:prototype:polymorphism].
1049 Without this transformation, there would be a \lam{(+)} entity
1050 in the \VHDL which would just add its inputs. This generates a
1051 lot of overhead in the \VHDL, which is particularly annoying
1052 when browsing the generated RTL schematic (especially since most
1053 non-alphanumerics, like all characters in \lam{(+)}, are not
1054 allowed in \VHDL architecture names\footnote{Technically, it is
1055 allowed to use non-alphanumerics when using extended
1056 identifiers, but it seems that none of the tooling likes
1057 extended identifiers in filenames, so it effectively doesn't
1058 work.}, so the entity would be called \quote{w7aA7f} or
1059 something similarly unreadable and autogenerated).
1061 \subsection{Program structure}
1062 These transformations are aimed at normalizing the overall structure
1063 into the intended form. This means ensuring there is a lambda abstraction
1064 at the top for every argument (input port or current state), putting all
1065 of the other value definitions in let bindings and making the final
1066 return value a simple variable reference.
1068 \subsubsection[sec:normalization:eta]{η-abstraction}
1069 This transformation makes sure that all arguments of a function-typed
1070 expression are named, by introducing lambda expressions. When combined with
1071 β-reduction and non-representable binding inlining, all function-typed
1072 expressions should be lambda abstractions or global identifiers.
1076 -------------- \lam{E} does not occur on a function position in an application
1077 λx.E x \lam{E} is not a lambda abstraction.
1087 foo = λa.λx.(case a of
1092 \transexample{eta}{η-abstraction}{from}{to}
1094 \subsubsection[sec:normalization:appprop]{Application propagation}
1095 This transformation is meant to propagate application expressions downwards
1096 into expressions as far as possible. This allows partial applications inside
1097 expressions to become fully applied and exposes new transformation
1098 opportunities for other transformations (like β-reduction and
1101 Since all binders in our expression are unique (see
1102 \in{section}[sec:normalization:uniq]), there is no risk that we will
1103 introduce unintended shadowing by moving an expression into a lower
1104 scope. Also, since only move expression into smaller scopes (down into
1105 our expression), there is no risk of moving a variable reference out
1106 of the scope in which it is defined.
1109 (letrec binds in E) M
1110 ------------------------
1130 \transexample{appproplet}{Application propagation for a let expression}{from}{to}
1158 \transexample{apppropcase}{Application propagation for a case expression}{from}{to}
1160 \subsubsection[sec:normalization:letrecurse]{Let recursification}
1161 This transformation makes all non-recursive lets recursive. In the
1162 end, we want a single recursive let in our normalized program, so all
1163 non-recursive lets can be converted. This also makes other
1164 transformations simpler: They can simply assume all lets are
1172 ------------------------------------------
1179 \subsubsection{Let flattening}
1180 This transformation puts nested lets in the same scope, by lifting the
1181 binding(s) of the inner let into the outer let. Eventually, this will
1182 cause all let bindings to appear in the same scope.
1184 This transformation only applies to recursive lets, since all
1185 non-recursive lets will be made recursive (see
1186 \in{section}[sec:normalization:letrecurse]).
1188 Since we are joining two scopes together, there is no risk of moving a
1189 variable reference out of the scope where it is defined.
1195 ai = (letrec bindings in M)
1200 ------------------------------------------
1235 \transexample{letflat}{Let flattening}{from}{to}
1237 \subsubsection{Return value simplification}
1238 This transformation ensures that the return value of a function is always a
1239 simple local variable reference.
1241 This transformation only applies to the entire body of a
1242 function instead of any subexpression in a function. This is
1243 achieved by the contexts, like \lam{x = E}, though this is
1244 strictly not correct (you could read this as "if there is any
1245 function \lam{x} that binds \lam{E}, any \lam{E} can be
1246 transformed, while we only mean the \lam{E} that is bound by
1249 Note that the return value is not simplified if its not
1250 representable. Otherwise, this would cause a direct loop with
1251 the inlining of unrepresentable bindings. If the return value is
1252 not representable because it has a function type, η-abstraction
1253 should make sure that this transformation will eventually apply.
1254 If the value is not representable for other reasons, the
1255 function result itself is not representable, meaning this
1256 function is not translatable anyway.
1259 x = E \lam{E} is representable
1260 ~ \lam{E} is not a lambda abstraction
1261 E \lam{E} is not a let expression
1262 --------------------------- \lam{E} is not a local variable reference
1268 ~ \lam{E} is representable
1269 E \lam{E} is not a let expression
1270 --------------------------- \lam{E} is not a local variable reference
1275 x = λv0 ... λvn.let ... in E
1276 ~ \lam{E} is representable
1277 E \lam{E} is not a local variable reference
1278 -----------------------------
1287 x = letrec x = add 1 2 in x
1290 \transexample{retvalsimpl}{Return value simplification}{from}{to}
1292 \todo{More examples}
1294 \subsection[sec:normalization:argsimpl]{Representable arguments simplification}
1295 This section contains just a single transformation that deals with
1296 representable arguments in applications. Non-representable arguments are
1297 handled by the transformations in
1298 \in{section}[sec:normalization:nonrep].
1300 This transformation ensures that all representable arguments will become
1301 references to local variables. This ensures they will become references
1302 to local signals in the resulting \small{VHDL}, which is required due to
1303 limitations in the component instantiation code in \VHDL (one can only
1304 assign a signal or constant to an input port). By ensuring that all
1305 arguments are always simple variable references, we always have a signal
1306 available to map to the input ports.
1308 To reduce a complex expression to a simple variable reference, we create
1309 a new let expression around the application, which binds the complex
1310 expression to a new variable. The original function is then applied to
1313 \refdef{global variable}
1314 Note that references to \emph{global variables} (like a top level
1315 function without arguments, but also an argumentless dataconstructors
1316 like \lam{True}) are also simplified. Only local variables generate
1317 signals in the resulting architecture. Even though argumentless
1318 dataconstructors generate constants in generated \VHDL code and could be
1319 mapped to an input port directly, they are still simplified to make the
1320 normal form more regular.
1322 \refdef{representable}
1325 -------------------- \lam{N} is representable
1326 letrec x = N in M x \lam{N} is not a local variable reference
1328 \refdef{local variable}
1335 letrec x = add a 1 in add x 1
1338 \transexample{argsimpl}{Argument simplification}{from}{to}
1340 \subsection[sec:normalization:builtins]{Builtin functions}
1341 This section deals with (arguments to) builtin functions. In the
1342 intended normal form definition\refdef{intended normal form definition}
1343 we can see that there are three sorts of arguments a builtin function
1347 \item A representable local variable reference. This is the most
1348 common argument to any function. The argument simplification
1349 transformation described in \in{section}[sec:normalization:argsimpl]
1350 makes sure that \emph{any} representable argument to \emph{any}
1351 function (including builtin functions) is turned into a local variable
1353 \item (A partial application of) a top level function (either builtin on
1354 user-defined). The function extraction transformation described in
1355 this section takes care of turning every functiontyped argument into
1356 (a partial application of) a top level function.
1357 \item Any expression that is not representable and does not have a
1358 function type. Since these can be any expression, there is no
1359 transformation needed. Note that this category is exactly all
1360 expressions that are not transformed by the transformations for the
1361 previous two categories. This means that \emph{any} core expression
1362 that is used as an argument to a builtin function will be either
1363 transformed into one of the above categories, or end up in this
1364 categorie. In any case, the result is in normal form.
1367 As noted, the argument simplification will handle any representable
1368 arguments to a builtin function. The following transformation is needed
1369 to handle non-representable arguments with a function type, all other
1370 non-representable arguments don't need any special handling.
1372 \subsubsection[sec:normalization:funextract]{Function extraction}
1373 This transform deals with function-typed arguments to builtin
1375 Since builtin functions cannot be specialized (see
1376 \in{section}[sec:normalization:specialize]) to remove the arguments,
1377 these arguments are extracted into a new global function instead. In
1378 other words, we create a new top level function that has exactly the
1379 extracted argument as its body. This greatly simplifies the
1380 translation rules needed for builtin functions, since they only need
1381 to handle (partial applications of) top level functions.
1383 Any free variables occuring in the extracted arguments will become
1384 parameters to the new global function. The original argument is replaced
1385 with a reference to the new function, applied to any free variables from
1386 the original argument.
1388 This transformation is useful when applying higher order builtin functions
1389 like \hs{map} to a lambda abstraction, for example. In this case, the code
1390 that generates \small{VHDL} for \hs{map} only needs to handle top level functions and
1391 partial applications, not any other expression (such as lambda abstractions or
1392 even more complicated expressions).
1395 M N \lam{M} is (a partial aplication of) a builtin function.
1396 --------------------- \lam{f0 ... fn} are all free local variables of \lam{N}
1397 M (x f0 ... fn) \lam{N :: a -> b}
1398 ~ \lam{N} is not a (partial application of) a top level function
1403 addList = λb.λxs.map (λa . add a b) xs
1407 addList = λb.λxs.map (f b) xs
1412 \transexample{funextract}{Function extraction}{from}{to}
1414 Note that the function \lam{f} will still need normalization after
1417 \subsection{Case normalisation}
1418 \subsubsection{Scrutinee simplification}
1419 This transform ensures that the scrutinee of a case expression is always
1420 a simple variable reference.
1425 ----------------- \lam{E} is not a local variable reference
1444 \transexample{letflat}{Case normalisation}{from}{to}
1447 \subsubsection{Case normalization}
1448 This transformation ensures that all case expressions get a form
1449 that is allowed by the intended normal form. This means they
1450 will become one of: \refdef{intended normal form definition}
1452 \item An extractor case with a single alternative that picks a field
1453 from a datatype, \eg \lam{case x of (a, b) -> a}.
1454 \item A selector case with multiple alternatives and only wild binders, that
1455 makes a choice between expressions based on the constructor of another
1456 expression, \eg \lam{case x of Low -> a; High -> b}.
1459 For an arbitrary case, that has \lam{n} alternatives, with
1460 \lam{m} binders in each alternatives, this will result in \lam{m
1461 * n} extractor case expression to get at each variable, \lam{n}
1462 let bindings for each of the alternatives' value and a single
1463 selector case to select the right value out of these.
1465 Technically, the defintion of this transformation would require
1466 that the constructor for every alternative has exactly the same
1467 amount (\lam{m}) of arguments, but of course this transformation
1468 also applies when this is not the case.
1472 C0 v0,0 ... v0,m -> E0
1474 Cn vn,0 ... vn,m -> En
1475 --------------------------------------------------- \forall i \forall j, 0 ≤ i ≤ n, 0 ≤ i < m (\lam{wi,j} is a wild (unused) binder)
1476 letrec The case expression is not an extractor case
1477 v0,0 = case E of C0 x0,0 .. x0,m -> x0,0 The case expression is not a selector case
1479 v0,m = case E of C0 x0,0 .. x0,m -> x0,m
1481 vn,m = case E of Cn xn,0 .. xn,m -> xn,m
1487 C0 w0,0 ... w0,m -> y0
1489 Cn wn,0 ... wn,m -> yn
1492 \refdef{wild binder}
1493 Note that this transformation applies to case expressions with any
1494 scrutinee. If the scrutinee is a complex expression, this might
1495 result in duplication of work (hardware). An extra condition to
1496 only apply this transformation when the scrutinee is already
1497 simple (effectively causing this transformation to be only
1498 applied after the scrutinee simplification transformation) might
1517 \transexample{selcasesimpl}{Selector case simplification}{from}{to}
1525 b = case a of (,) b c -> b
1526 c = case a of (,) b c -> c
1533 \transexample{excasesimpl}{Extractor case simplification}{from}{to}
1535 \refdef{selector case}
1536 In \in{example}[ex:trans:excasesimpl] the case expression is expanded
1537 into multiple case expressions, including a pretty useless expression
1538 (that is neither a selector or extractor case). This case can be
1539 removed by the Case removal transformation in
1540 \in{section}[sec:transformation:caseremoval].
1542 \subsubsection[sec:transformation:caseremoval]{Case removal}
1543 This transform removes any case expression with a single alternative and
1544 only wild binders.\refdef{wild binder}
1546 These "useless" case expressions are usually leftovers from case simplification
1547 on extractor case (see the previous example).
1552 ---------------------- \lam{\forall i, 0 ≤ i ≤ m} (\lam{vi} does not occur free in E)
1565 \transexample{caserem}{Case removal}{from}{to}
1567 \subsection[sec:normalization:nonrep]{Removing unrepresentable values}
1568 The transformations in this section are aimed at making all the
1569 values used in our expression representable. There are two main
1570 transformations that are applied to \emph{all} unrepresentable let
1571 bindings and function arguments. These are meant to address three
1572 different kinds of unrepresentable values: Polymorphic values, higher
1573 order values and literals. The transformation are described generically:
1574 They apply to all non-representable values. However, non-representable
1575 values that don't fall into one of these three categories will be moved
1576 around by these transformations but are unlikely to completely
1577 disappear. They usually mean the program was not valid in the first
1578 place, because unsupported types were used (for example, a program using
1581 Each of these three categories will be detailed below, followed by the
1582 actual transformations.
1584 \subsubsection{Removing Polymorphism}
1585 As noted in \in{section}[sec:prototype:polymporphism],
1586 polymorphism is made explicit in Core through type and
1587 dictionary arguments. To remove the polymorphism from a
1588 function, we can simply specialize the polymorphic function for
1589 the particular type applied to it. The same goes for dictionary
1590 arguments. To remove polymorphism from let bound values, we
1591 simply inline the let bindings that have a polymorphic type,
1592 which should (eventually) make sure that the polymorphic
1593 expression is applied to a type and/or dictionary, which can
1594 then be removed by β-reduction (\in{section}[sec:normalization:beta]).
1596 Since both type and dictionary arguments are not representable,
1597 \refdef{representable}
1598 the non-representable argument specialization and
1599 non-representable let binding inlining transformations below
1600 take care of exactly this.
1602 There is one case where polymorphism cannot be completely
1603 removed: Builtin functions are still allowed to be polymorphic
1604 (Since we have no function body that we could properly
1605 specialize). However, the code that generates \VHDL for builtin
1606 functions knows how to handle this, so this is not a problem.
1608 \subsubsection{Defunctionalization}
1609 These transformations remove higher order expressions from our
1610 program, making all values first-order.
1612 Higher order values are always introduced by lambda abstractions, none
1613 of the other Core expression elements can introduce a function type.
1614 However, other expressions can \emph{have} a function type, when they
1615 have a lambda expression in their body.
1617 For example, the following expression is a higher order expression
1618 that is not a lambda expression itself:
1620 \refdef{id function}
1627 The reference to the \lam{id} function shows that we can introduce a
1628 higher order expression in our program without using a lambda
1629 expression directly. However, inside the definition of the \lam{id}
1630 function, we can be sure that a lambda expression is present.
1632 Looking closely at the definition of our normal form in
1633 \in{section}[sec:normalization:intendednormalform], we can see that
1634 there are three possibilities for higher order values to appear in our
1635 intended normal form:
1638 \item[item:toplambda] Lambda abstractions can appear at the highest level of a
1639 top level function. These lambda abstractions introduce the
1640 arguments (input ports / current state) of the function.
1641 \item[item:builtinarg] (Partial applications of) top level functions can appear as an
1642 argument to a builtin function.
1643 \item[item:completeapp] (Partial applications of) top level functions can appear in
1644 function position of an application. Since a partial application
1645 cannot appear anywhere else (except as builtin function arguments),
1646 all partial applications are applied, meaning that all applications
1647 will become complete applications. However, since application of
1648 arguments happens one by one, in the expression:
1652 the subexpression \lam{f 1} has a function type. But this is
1653 allowed, since it is inside a complete application.
1656 We will take a typical function with some higher order values as an
1657 example. The following function takes two arguments: a \lam{Bit} and a
1658 list of numbers. Depending on the first argument, each number in the
1659 list is doubled, or the list is returned unmodified. For the sake of
1660 the example, no polymorphism is shown. In reality, at least map would
1664 λy.let double = λx. x + x in
1670 This example shows a number of higher order values that we cannot
1671 translate to \VHDL directly. The \lam{double} binder bound in the let
1672 expression has a function type, as well as both of the alternatives of
1673 the case expression. The first alternative is a partial application of
1674 the \lam{map} builtin function, whereas the second alternative is a
1677 To reduce all higher order values to one of the above items, a number
1678 of transformations we've already seen are used. The η-abstraction
1679 transformation from \in{section}[sec:normalization:eta] ensures all
1680 function arguments are introduced by lambda abstraction on the highest
1681 level of a function. These lambda arguments are allowed because of
1682 \in{item}[item:toplambda] above. After η-abstraction, our example
1683 becomes a bit bigger:
1686 λy.λq.(let double = λx. x + x in
1693 η-abstraction also introduces extra applications (the application of
1694 the let expression to \lam{q} in the above example). These
1695 applications can then propagated down by the application propagation
1696 transformation (\in{section}[sec:normalization:appprop]). In our
1697 example, the \lam{q} and \lam{r} variable will be propagated into the
1698 let expression and then into the case expression:
1701 λy.λq.let double = λx. x + x in
1707 This propagation makes higher order values become applied (in
1708 particular both of the alternatives of the case now have a
1709 representable type). Completely applied top level functions (like the
1710 first alternative) are now no longer invalid (they fall under
1711 \in{item}[item:completeapp] above). (Completely) applied lambda
1712 abstractions can be removed by β-abstraction. For our example,
1713 applying β-abstraction results in the following:
1716 λy.λq.let double = λx. x + x in
1722 As you can see in our example, all of this moves applications towards
1723 the higher order values, but misses higher order functions bound by
1724 let expressions. The applications cannot be moved towards these values
1725 (since they can be used in multiple places), so the values will have
1726 to be moved towards the applications. This is achieved by inlining all
1727 higher order values bound by let applications, by the
1728 non-representable binding inlining transformation below. When applying
1729 it to our example, we get the following:
1733 Low -> map (λx. x + x) q
1737 We've nearly eliminated all unsupported higher order values from this
1738 expressions. The one that's remaining is the first argument to the
1739 \lam{map} function. Having higher order arguments to a builtin
1740 function like \lam{map} is allowed in the intended normal form, but
1741 only if the argument is a (partial application) of a top level
1742 function. This is easily done by introducing a new top level function
1743 and put the lambda abstraction inside. This is done by the function
1744 extraction transformation from
1745 \in{section}[sec:normalization:funextract].
1753 This also introduces a new function, that we have called \lam{func}:
1759 Note that this does not actually remove the lambda, but now it is a
1760 lambda at the highest level of a function, which is allowed in the
1761 intended normal form.
1763 There is one case that has not been discussed yet. What if the
1764 \lam{map} function in the example above was not a builtin function
1765 but a user-defined function? Then extracting the lambda expression
1766 into a new function would not be enough, since user-defined functions
1767 can never have higher order arguments. For example, the following
1768 expression shows an example:
1771 twice :: (Word -> Word) -> Word -> Word
1772 twice = λf.λa.f (f a)
1774 main = λa.app (λx. x + x) a
1777 This example shows a function \lam{twice} that takes a function as a
1778 first argument and applies that function twice to the second argument.
1779 Again, we've made the function monomorphic for clarity, even though
1780 this function would be a lot more useful if it was polymorphic. The
1781 function \lam{main} uses \lam{twice} to apply a lambda epression twice.
1783 When faced with a user defined function, a body is available for that
1784 function. This means we could create a specialized version of the
1785 function that only works for this particular higher order argument
1786 (\ie, we can just remove the argument and call the specialized
1787 function without the argument). This transformation is detailed below.
1788 Applying this transformation to the example gives:
1791 twice' :: Word -> Word
1792 twice' = λb.(λf.λa.f (f a)) (λx. x + x) b
1797 The \lam{main} function is now in normal form, since the only higher
1798 order value there is the top level lambda expression. The new
1799 \lam{twice'} function is a bit complex, but the entire original body of
1800 the original \lam{twice} function is wrapped in a lambda abstraction
1801 and applied to the argument we've specialized for (\lam{λx. x + x})
1802 and the other arguments. This complex expression can fortunately be
1803 effectively reduced by repeatedly applying β-reduction:
1806 twice' :: Word -> Word
1807 twice' = λb.(b + b) + (b + b)
1810 This example also shows that the resulting normal form might not be as
1811 efficient as we might hope it to be (it is calculating \lam{(b + b)}
1812 twice). This is discussed in more detail in
1813 \in{section}[sec:normalization:duplicatework].
1815 \subsubsection{Literals}
1816 There are a limited number of literals available in Haskell and Core.
1817 \refdef{enumerated types} When using (enumerating) algebraic
1818 datatypes, a literal is just a reference to the corresponding data
1819 constructor, which has a representable type (the algebraic datatype)
1820 and can be translated directly. This also holds for literals of the
1821 \hs{Bool} Haskell type, which is just an enumerated type.
1823 There is, however, a second type of literal that does not have a
1824 representable type: Integer literals. Cλash supports using integer
1825 literals for all three integer types supported (\hs{SizedWord},
1826 \hs{SizedInt} and \hs{RangedWord}). This is implemented using
1827 Haskell's \hs{Num} typeclass, which offers a \hs{fromInteger} method
1828 that converts any \hs{Integer} to the Cλash datatypes.
1830 When \GHC sees integer literals, it will automatically insert calls to
1831 the \hs{fromInteger} method in the resulting Core expression. For
1832 example, the following expression in Haskell creates a 32 bit unsigned
1833 word with the value 1. The explicit type signature is needed, since
1834 there is no context for \GHC to determine the type from otherwise.
1840 This Haskell code results in the following Core expression:
1843 fromInteger @(SizedWord D32) \$dNum (smallInteger 10)
1846 The literal 10 will have the type \lam{GHC.Prim.Int\#}, which is
1847 converted into an \lam{Integer} by \lam{smallInteger}. Finally, the
1848 \lam{fromInteger} function will finally convert this into a
1849 \lam{SizedWord D32}.
1851 Both the \lam{GHC.Prim.Int\#} and \lam{Integer} types are not
1852 representable, and cannot be translated directly. Fortunately, there
1853 is no need to translate them, since \lam{fromInteger} is a builtin
1854 function that knows how to handle these values. However, this does
1855 require that the \lam{fromInteger} function is directly applied to
1856 these non-representable literal values, otherwise errors will occur.
1857 For example, the following expression is not in the intended normal
1858 form, since one of the let bindings has an unrepresentable type
1862 let l = smallInteger 10 in fromInteger @(SizedWord D32) \$dNum l
1865 By inlining these let-bindings, we can ensure that unrepresentable
1866 literals bound by a let binding end up in an application of the
1867 appropriate builtin function, where they are allowed. Since it is
1868 possible that the application of that function is in a different
1869 function than the definition of the literal value, we will always need
1870 to specialize away any unrepresentable literals that are used as
1871 function arguments. The following two transformations do exactly this.
1873 \subsubsection{Non-representable binding inlining}
1874 This transform inlines let bindings that are bound to a
1875 non-representable value. Since we can never generate a signal
1876 assignment for these bindings (we cannot declare a signal assignment
1877 with a non-representable type, for obvious reasons), we have no choice
1878 but to inline the binding to remove it.
1880 As we have seen in the previous sections, inlining these bindings
1881 solves (part of) the polymorphism, higher order values and
1882 unrepresentable literals in an expression.
1884 \refdef{substitution notation}
1894 -------------------------- \lam{Ei} has a non-representable type.
1896 a0 = E0 [ai=>Ei] \vdots
1897 ai-1 = Ei-1 [ai=>Ei]
1898 ai+1 = Ei+1 [ai=>Ei]
1917 x = fromInteger (smallInteger 10)
1919 (λb -> add b 1) (add 1 x)
1922 \transexample{nonrepinline}{Nonrepresentable binding inlining}{from}{to}
1924 \subsubsection[sec:normalization:specialize]{Function specialization}
1925 This transform removes arguments to user-defined functions that are
1926 not representable at runtime. This is done by creating a
1927 \emph{specialized} version of the function that only works for one
1928 particular value of that argument (in other words, the argument can be
1931 Specialization means to create a specialized version of the called
1932 function, with one argument already filled in. As a simple example, in
1933 the following program (this is not actual Core, since it directly uses
1934 a literal with the unrepresentable type \lam{GHC.Prim.Int\#}).
1941 We could specialize the function \lam{f} against the literal argument
1942 1, with the following result:
1949 In some way, this transformation is similar to β-reduction, but it
1950 operates across function boundaries. It is also similar to
1951 non-representable let binding inlining above, since it sort of
1952 \quote{inlines} an expression into a called function.
1954 Special care must be taken when the argument has any free variables.
1955 If this is the case, the original argument should not be removed
1956 completely, but replaced by all the free variables of the expression.
1957 In this way, the original expression can still be evaluated inside the
1960 To prevent us from propagating the same argument over and over, a
1961 simple local variable reference is not propagated (since is has
1962 exactly one free variable, itself, we would only replace that argument
1965 This shows that any free local variables that are not runtime
1966 representable cannot be brought into normal form by this transform. We
1967 rely on an inlining or β-reduction transformation to replace such a
1968 variable with an expression we can propagate again.
1973 x Y0 ... Yi ... Yn \lam{Yi} is not representable
1974 --------------------------------------------- \lam{Yi} is not a local variable reference
1975 x' Y0 ... Yi-1 f0 ... fm Yi+1 ... Yn \lam{f0 ... fm} are all free local vars of \lam{Yi}
1976 ~ \lam{T0 ... Tn} are the types of \lam{Y0 ... Yn}
1977 x' = λ(y0 :: T0) ... λ(yi-1 :: Ty-1).
1979 λ(yi+1 :: Ty+1) ... λ(yn :: Tn).
1980 E y0 ... yi-1 Yi yi+1 ... yn
1983 This is a bit of a complex transformation. It transforms an
1984 application of the function \lam{x}, where one of the arguments
1985 (\lam{Y_i}) is not representable. A new
1986 function \lam{x'} is created that wraps the body of the old function.
1987 The body of the new function becomes a number of nested lambda
1988 abstractions, one for each of the original arguments that are left
1991 The ith argument is replaced with the free variables of
1992 \lam{Y_i}. Note that we reuse the same binders as those used in
1993 \lam{Y_i}, since we can then just use \lam{Y_i} inside the new
1994 function body and have all of the variables it uses be in scope.
1996 The argument that we are specializing for, \lam{Y_i}, is put inside
1997 the new function body. The old function body is applied to it. Since
1998 we use this new function only in place of an application with that
1999 particular argument \lam{Y_i}, behaviour should not change.
2001 Note that the types of the arguments of our new function are taken
2002 from the types of the \emph{actual} arguments (\lam{T0 ... Tn}). This
2003 means that any polymorphism in the arguments is removed, even when the
2004 corresponding explicit type lambda is not removed
2007 \todo{Examples. Perhaps reference the previous sections}
2009 \section{Unsolved problems}
2010 The above system of transformations has been implemented in the prototype
2011 and seems to work well to compile simple and more complex examples of
2012 hardware descriptions. \todo{Ref christiaan?} However, this normalization
2013 system has not seen enough review and work to be complete and work for
2014 every Core expression that is supplied to it. A number of problems
2015 have already been identified and are discussed in this section.
2017 \subsection[sec:normalization:duplicatework]{Work duplication}
2018 A possible problem of β-reduction is that it could duplicate work.
2019 When the expression applied is not a simple variable reference, but
2020 requires calculation and the binder the lambda abstraction binds to
2021 is used more than once, more hardware might be generated than strictly
2024 As an example, consider the expression:
2030 When applying β-reduction to this expression, we get:
2036 which of course calculates \lam{(a * b)} twice.
2038 A possible solution to this would be to use the following alternative
2039 transformation, which is of course no longer normal β-reduction. The
2040 followin transformation has not been tested in the prototype, but is
2041 given here for future reference:
2049 This doesn't seem like much of an improvement, but it does get rid of
2050 the lambda expression (and the associated higher order value), while
2051 at the same time introducing a new let binding. Since the result of
2052 every application or case expression must be bound by a let expression
2053 in the intended normal form anyway, this is probably not a problem. If
2054 the argument happens to be a variable reference, then simple let
2055 binding removal (\in{section}[sec:normalization:simplelet]) will
2056 remove it, making the result identical to that of the original
2057 β-reduction transformation.
2059 When also applying argument simplification to the above example, we
2060 get the following expression:
2068 Looking at this, we could imagine an alternative approach: Create a
2069 transformation that removes let bindings that bind identical values.
2070 In the above expression, the \lam{y} and \lam{z} variables could be
2071 merged together, resulting in the more efficient expression:
2074 let y = (a * b) in y + y
2077 \subsection[sec:normalization:non-determinism]{Non-determinism}
2078 As an example, again consider the following expression:
2084 We can apply both β-reduction (\in{section}[sec:normalization:beta])
2085 as well as argument simplification
2086 (\in{section}[sec:normalization:argsimpl]) to this expression.
2088 When applying argument simplification first and then β-reduction, we
2089 get the following expression:
2092 let y = (a * b) in y + y
2095 When applying β-reduction first and then argument simplification, we
2096 get the following expression:
2104 As you can see, this is a different expression. This means that the
2105 order of expressions, does in fact change the resulting normal form,
2106 which is something that we would like to avoid. In this particular
2107 case one of the alternatives is even clearly more efficient, so we
2108 would of course like the more efficient form to be the normal form.
2110 For this particular problem, the solutions for duplication of work
2111 seem from the previous section seem to fix the determinism of our
2112 transformation system as well. However, it is likely that there are
2113 other occurences of this problem.
2115 \subsection[sec:normalization:castproblems]{Casts}
2116 We do not fully understand the use of cast expressions in Core, so
2117 there are probably expressions involving cast expressions that cannot
2118 be brought into intended normal form by this transformation system.
2120 The uses of casts in the core system should be investigated more and
2121 transformations will probably need updating to handle them in all
2124 \subsection[sec:normalization:stateproblems]{Normalization of stateful descriptions}
2125 Currently, the intended normal form definition\refdef{intended
2126 normal form definition} offers enough freedom to describe all
2127 valid stateful descriptions, but is not limiting enough. It is
2128 possible to write descriptions which are in intended normal
2129 form, but cannot be translated into \VHDL in a meaningful way
2130 (\eg, a function that swaps two substates in its result, or a
2131 function that changes a substate itself instead of passing it to
2134 It is now up to the programmer to not do anything funny with
2135 these state values, whereas the normalization just tries not to
2136 mess up the flow of state values. In practice, there are
2137 situations where a Core program that \emph{could} be a valid
2138 stateful description is not translateable by the prototype. This
2139 most often happens when statefulness is mixed with pattern
2140 matching, causing a state input to be unpacked multiple times or
2141 be unpacked and repacked only in some of the code paths.
2143 Without going into detail about the exact problems (of which
2144 there are probably more than have shown up so far), it seems
2145 unlikely that these problems can be solved entirely by just
2146 improving the \VHDL state generation in the final stage. The
2147 normalization stage seems the best place to apply the rewriting
2148 needed to support more complex stateful descriptions. This does
2149 of course mean that the intended normal form definition must be
2150 extended as well to be more specific about how state handling
2151 should look like in normal form.
2152 \in{Section}[sec:prototype:statelimits] already contains a
2153 tight description of the limitations on the use of state
2154 variables, which could be adapted into the intended normal form.
2156 \section[sec:normalization:properties]{Provable properties}
2157 When looking at the system of transformations outlined above, there are a
2158 number of questions that we can ask ourselves. The main question is of course:
2159 \quote{Does our system work as intended?}. We can split this question into a
2160 number of subquestions:
2163 \item[q:termination] Does our system \emph{terminate}? Since our system will
2164 keep running as long as transformations apply, there is an obvious risk that
2165 it will keep running indefinitely. This typically happens when one
2166 transformation produces a result that is transformed back to the original
2167 by another transformation, or when one or more transformations keep
2168 expanding some expression.
2169 \item[q:soundness] Is our system \emph{sound}? Since our transformations
2170 continuously modify the expression, there is an obvious risk that the final
2171 normal form will not be equivalent to the original program: Its meaning could
2173 \item[q:completeness] Is our system \emph{complete}? Since we have a complex
2174 system of transformations, there is an obvious risk that some expressions will
2175 not end up in our intended normal form, because we forgot some transformation.
2176 In other words: Does our transformation system result in our intended normal
2177 form for all possible inputs?
2178 \item[q:determinism] Is our system \emph{deterministic}? Since we have defined
2179 no particular order in which the transformation should be applied, there is an
2180 obvious risk that different transformation orderings will result in
2181 \emph{different} normal forms. They might still both be intended normal forms
2182 (if our system is \emph{complete}) and describe correct hardware (if our
2183 system is \emph{sound}), so this property is less important than the previous
2184 three: The translator would still function properly without it.
2187 Unfortunately, the final transformation system has only been
2188 developed in the final part of the research, leaving no more time
2189 for verifying these properties. In fact, it is likely that the
2190 current transformation system still violates some of these
2191 properties in some cases and should be improved (or extra conditions
2192 on the input hardware descriptions should be formulated).
2194 This is most likely the case with the completeness and determinism
2195 properties, perhaps als the termination property. The soundness
2196 property probably holds, since it is easier to manually verify (each
2197 transformation can be reviewed separately).
2199 Even though no complete proofs have been made, some ideas for
2200 possible proof strategies are shown below.
2202 \subsection{Graph representation}
2203 Before looking into how to prove these properties, we'll look at
2204 transformation systems from a graph perspective. We will first define
2205 the graph view and then illustrate it using a simple example from lambda
2206 calculus (which is a different system than the Cλash normalization
2207 system). The nodes of the graph are all possible Core expressions. The
2208 (directed) edges of the graph are transformations. When a transformation
2209 α applies to an expression \lam{A} to produce an expression \lam{B}, we
2210 add an edge from the node for \lam{A} to the node for \lam{B}, labeled
2213 \startuseMPgraphic{TransformGraph}
2217 newCircle.a(btex \lam{(λx.λy. (+) x y) 1} etex);
2218 newCircle.b(btex \lam{λy. (+) 1 y} etex);
2219 newCircle.c(btex \lam{(λx.(+) x) 1} etex);
2220 newCircle.d(btex \lam{(+) 1} etex);
2223 c.c = b.c + (4cm, 0cm);
2224 a.c = midpoint(b.c, c.c) + (0cm, 4cm);
2225 d.c = midpoint(b.c, c.c) - (0cm, 3cm);
2227 % β-conversion between a and b
2228 ncarc.a(a)(b) "name(bred)";
2229 ObjLabel.a(btex $\xrightarrow[normal]{}{β}$ etex) "labpathname(bred)", "labdir(rt)";
2230 ncarc.b(b)(a) "name(bexp)", "linestyle(dashed withdots)";
2231 ObjLabel.b(btex $\xleftarrow[normal]{}{β}$ etex) "labpathname(bexp)", "labdir(lft)";
2233 % η-conversion between a and c
2234 ncarc.a(a)(c) "name(ered)";
2235 ObjLabel.a(btex $\xrightarrow[normal]{}{η}$ etex) "labpathname(ered)", "labdir(rt)";
2236 ncarc.c(c)(a) "name(eexp)", "linestyle(dashed withdots)";
2237 ObjLabel.c(btex $\xleftarrow[normal]{}{η}$ etex) "labpathname(eexp)", "labdir(lft)";
2239 % η-conversion between b and d
2240 ncarc.b(b)(d) "name(ered)";
2241 ObjLabel.b(btex $\xrightarrow[normal]{}{η}$ etex) "labpathname(ered)", "labdir(rt)";
2242 ncarc.d(d)(b) "name(eexp)", "linestyle(dashed withdots)";
2243 ObjLabel.d(btex $\xleftarrow[normal]{}{η}$ etex) "labpathname(eexp)", "labdir(lft)";
2245 % β-conversion between c and d
2246 ncarc.c(c)(d) "name(bred)";
2247 ObjLabel.c(btex $\xrightarrow[normal]{}{β}$ etex) "labpathname(bred)", "labdir(rt)";
2248 ncarc.d(d)(c) "name(bexp)", "linestyle(dashed withdots)";
2249 ObjLabel.d(btex $\xleftarrow[normal]{}{β}$ etex) "labpathname(bexp)", "labdir(lft)";
2251 % Draw objects and lines
2252 drawObj(a, b, c, d);
2255 \placeexample[right][ex:TransformGraph]{Partial graph of a lambda calculus
2256 system with β and η reduction (solid lines) and expansion (dotted lines).}
2257 \boxedgraphic{TransformGraph}
2259 Of course the graph for Cλash is unbounded, since we can construct an
2260 infinite amount of Core expressions. Also, there might potentially be
2261 multiple edges between two given nodes (with different labels), though
2262 seems unlikely to actually happen in our system.
2264 See \in{example}[ex:TransformGraph] for the graph representation of a very
2265 simple lambda calculus that contains just the expressions \lam{(λx.λy. (+) x
2266 y) 1}, \lam{λy. (+) 1 y}, \lam{(λx.(+) x) 1} and \lam{(+) 1}. The
2267 transformation system consists of β-reduction and η-reduction (solid edges) or
2268 β-expansion and η-expansion (dotted edges).
2270 \todo{Define β-reduction and η-reduction?}
2272 Note that the normal form of such a system consists of the set of nodes
2273 (expressions) without outgoing edges, since those are the expressions to which
2274 no transformation applies anymore. We call this set of nodes the \emph{normal
2275 set}. The set of nodes containing expressions in intended normal
2276 form \refdef{intended normal form} is called the \emph{intended
2279 From such a graph, we can derive some properties easily:
2281 \item A system will \emph{terminate} if there is no path of infinite length
2282 in the graph (this includes cycles, but can also happen without cycles).
2283 \item Soundness is not easily represented in the graph.
2284 \item A system is \emph{complete} if all of the nodes in the normal set have
2285 the intended normal form. The inverse (that all of the nodes outside of
2286 the normal set are \emph{not} in the intended normal form) is not
2287 strictly required. In other words, our normal set must be a
2288 subset of the intended normal form, but they do not need to be
2291 \item A system is deterministic if all paths starting at a particular
2292 node, which end in a node in the normal set, end at the same node.
2295 When looking at the \in{example}[ex:TransformGraph], we see that the system
2296 terminates for both the reduction and expansion systems (but note that, for
2297 expansion, this is only true because we've limited the possible
2298 expressions. In comlete lambda calculus, there would be a path from
2299 \lam{(λx.λy. (+) x y) 1} to \lam{(λx.λy.(λz.(+) z) x y) 1} to
2300 \lam{(λx.λy.(λz.(λq.(+) q) z) x y) 1} etc.)
2302 If we would consider the system with both expansion and reduction, there
2303 would no longer be termination either, since there would be cycles all
2306 The reduction and expansion systems have a normal set of containing just
2307 \lam{(+) 1} or \lam{(λx.λy. (+) x y) 1} respectively. Since all paths in
2308 either system end up in these normal forms, both systems are \emph{complete}.
2309 Also, since there is only one node in the normal set, it must obviously be
2310 \emph{deterministic} as well.
2312 \subsection{Termination}
2313 In general, proving termination of an arbitrary program is a very
2314 hard problem. \todo{Ref about arbitrary termination} Fortunately,
2315 we only have to prove termination for our specific transformation
2318 A common approach for these kinds of proofs is to associate a
2319 measure with each possible expression in our system. If we can
2320 show that each transformation strictly decreases this measure
2321 (\ie, the expression transformed to has a lower measure than the
2322 expression transformed from). \todo{ref about measure-based
2323 termination proofs / analysis}
2325 A good measure for a system consisting of just β-reduction would
2326 be the number of lambda expressions in the expression. Since every
2327 application of β-reduction removes a lambda abstraction (and there
2328 is always a bounded number of lambda abstractions in every
2329 expression) we can easily see that a transformation system with
2330 just β-reduction will always terminate.
2332 For our complete system, this measure would be fairly complex
2333 (probably the sum of a lot of things). Since the (conditions on)
2334 our transformations are pretty complex, we would need to include
2335 both simple things like the number of let expressions as well as
2336 more complex things like the number of case expressions that are
2337 not yet in normal form.
2339 No real attempt has been made at finding a suitable measure for
2342 \subsection{Soundness}
2343 Soundness is a property that can be proven for each transformation
2344 separately. Since our system only runs separate transformations
2345 sequentially, if each of our transformations leaves the
2346 \emph{meaning} of the expression unchanged, then the entire system
2347 will of course leave the meaning unchanged and is thus
2350 The current prototype has only been verified in an ad-hoc fashion
2351 by inspecting (the code for) each transformation. A more formal
2352 verification would be more appropriate.
2354 To be able to formally show that each transformation properly
2355 preserves the meaning of every expression, we require an exact
2356 definition of the \emph{meaning} of every expression, so we can
2357 compare them. A definition of the operational semantics of \GHC's Core
2358 language is available \cite[sulzmann07], but this does not seem
2359 sufficient for our goals (but it is a good start).
2361 It should be possible to have a single formal definition of
2362 meaning for Core for both normal Core compilation by \GHC and for
2363 our compilation to \VHDL. The main difference seems to be that in
2364 hardware every expression is always evaluated, while in software
2365 it is only evaluated if needed, but it should be possible to
2366 assign a meaning to core expressions that assumes neither.
2368 Since each of the transformations can be applied to any
2369 subexpression as well, there is a constraint on our meaning
2370 definition: The meaning of an expression should depend only on the
2371 meaning of subexpressions, not on the expressions themselves. For
2372 example, the meaning of the application in \lam{f (let x = 4 in
2373 x)} should be the same as the meaning of the application in \lam{f
2374 4}, since the argument subexpression has the same meaning (though
2375 the actual expression is different).
2377 \subsection{Completeness}
2378 Proving completeness is probably not hard, but it could be a lot
2379 of work. We have seen above that to prove completeness, we must
2380 show that the normal set of our graph representation is a subset
2381 of the intended normal set.
2383 However, it is hard to systematically generate or reason about the
2384 normal set, since it is defined as any nodes to which no
2385 transformation applies. To determine this set, each transformation
2386 must be considered and when a transformation is added, the entire
2387 set should be re-evaluated. This means it is hard to show that
2388 each node in the normal set is also in the intended normal set.
2389 Reasoning about our intended normal set is easier, since we know
2390 how to generate it from its definition. \refdef{intended normal
2393 Fortunately, we can also prove the complement (which is
2394 equivalent, since $A \subseteq B \Leftrightarrow \overline{B}
2395 \subseteq \overline{A}$): Show that the set of nodes not in
2396 intended normal form is a subset of the set of nodes not in normal
2397 form. In other words, show that for every expression that is not
2398 in intended normal form, that there is at least one transformation
2399 that applies to it (since that means it is not in normal form
2400 either and since $A \subseteq C \Leftrightarrow \forall x (x \in A
2401 \rightarrow x \in C)$).
2403 By systematically reviewing the entire Core language definition
2404 along with the intended normal form definition (both of which have
2405 a similar structure), it should be possible to identify all
2406 possible (sets of) core expressions that are not in intended
2407 normal form and identify a transformation that applies to it.
2409 This approach is especially useful for proving completeness of our
2410 system, since if expressions exist to which none of the
2411 transformations apply (\ie if the system is not yet complete), it
2412 is immediately clear which expressions these are and adding
2413 (or modifying) transformations to fix this should be relatively
2416 As observed above, applying this approach is a lot of work, since
2417 we need to check every (set of) transformation(s) separately.
2419 \todo{Perhaps do a few steps of the proofs as proof-of-concept}
2421 % vim: set sw=2 sts=2 expandtab: