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
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9 \framed[offset=1mm,align=right,strut=no,background=box,frame=off]{
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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
48 form, let us try to get a feeling for it first. The easiest way to do this
49 is by describing the things that are unwanted in the intended normal form.
52 \item Any \emph{polymorphism} must be removed. When laying down hardware, we
53 cannot 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 cannot
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 built-in 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{built-infunc} := var (bvar(var))
369 \italic{built-inarg} := 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 built-in construction (\eg the \lam{case} expression) or call a
388 built-in 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 expression>
407 This format describes a transformation that applies to \lam{<original
408 expression>} 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, which is a common transformation from
478 labmda calculus, described using above notation as follows:
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 us 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 will 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 will 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{built-in 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{built-in function} \defref{user-defined function}
685 For these functions, Cλash has a \emph{built-in 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
754 does not use \small{GHC}'s subsitution code). Also, the following points
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{Unused let binding removal}
895 This transformation removes let bindings that are never used.
896 Occasionally, \GHC's desugarer introduces some unused let bindings.
898 This normalization pass should really be not be necessary to get
899 into intended normal form (since the intended normal form
900 definition \refdef{intended normal form definition} does not
901 require that every binding is used), but in practice the
902 desugarer or simplifier emits some bindings that cannot be
903 normalized (e.g., calls to a
904 \hs{Control.Exception.Base.patError}) but are not used anywhere
905 either. To prevent the \VHDL generation from breaking on these
906 artifacts, this transformation removes them.
908 \todo{Do not use old-style numerals in transformations}
917 M \lam{ai} does not occur free in \lam{M}
918 ---------------------------- \lam{\forall j, 0 ≤ j ≤ n, j ≠ i} (\lam{ai} does not occur free in \lam{Ej})
944 \transexample{unusedlet}{Unused let binding removal}{from}{to}
946 \subsubsection{Empty let removal}
947 This transformation is simple: It removes recursive lets that have no bindings
948 (which usually occurs when unused let binding removal removes the last
951 Note that there is no need to define this transformation for
952 non-recursive lets, since they always contain exactly one binding.
971 \transexample{emptylet}{Empty let removal}{from}{to}
973 \subsubsection[sec:normalization:simplelet]{Simple let binding removal}
974 This transformation inlines simple let bindings, that bind some
975 binder to some other binder instead of a more complex expression (\ie
978 This transformation is not needed to get an expression into intended
979 normal form (since these bindings are part of the intended normal
980 form), but makes the resulting \small{VHDL} a lot shorter.
982 \refdef{substitution notation}
992 ----------------------------- \lam{b} is a variable reference
993 letrec \lam{ai} ≠ \lam{b}
1006 \subsubsection{Cast propagation / simplification}
1007 This transform pushes casts down into the expression as far as
1008 possible. This transformation has been added to make a few
1009 specific corner cases work, but it is not clear yet if this
1010 transformation handles cast expressions completely or in the
1011 right way. See \in{section}[sec:normalization:castproblems].
1014 (let binds in E) ▶ T
1015 -------------------------
1016 let binds in (E ▶ T)
1025 -------------------------
1032 \subsubsection{Top level binding inlining}
1033 \refdef{top level binding}
1034 This transform takes simple top level bindings generated by the
1035 \small{GHC} compiler. \small{GHC} sometimes generates very simple
1036 \quote{wrapper} bindings, which are bound to just a variable
1037 reference, or contain just a (partial) function appliation with
1038 the type and dictionary arguments filled in (such as the
1039 \lam{(+)} in the example below).
1041 Note that this transformation is completely optional. It is not
1042 required to get any function into intended normal form, but it does help making
1043 the resulting VHDL output easier to read (since it removes components
1044 that do not add any real structure, but do hide away operations and
1045 cause extra clutter).
1047 This transform takes any top level binding generated by \GHC,
1048 whose normalized form contains only a single let binding.
1051 x = λa0 ... λan.let y = E in y
1054 -------------------------------------- \lam{x} is generated by the compiler
1055 λa0 ... λan.let y = E in y
1059 (+) :: Word -> Word -> Word
1060 (+) = GHC.Num.(+) @Word \$dNum
1065 GHC.Num.(+) @ Alu.Word \$dNum a b
1068 \transexample{toplevelinline}{Top level binding inlining}{from}{to}
1070 \in{Example}[ex:trans:toplevelinline] shows a typical application of
1071 the addition operator generated by \GHC. The type and dictionary
1072 arguments used here are described in
1073 \in{Section}[section:prototype:polymorphism].
1075 Without this transformation, there would be a \lam{(+)} entity
1076 in the \VHDL which would just add its inputs. This generates a
1077 lot of overhead in the \VHDL, which is particularly annoying
1078 when browsing the generated RTL schematic (especially since most
1079 non-alphanumerics, like all characters in \lam{(+)}, are not
1080 allowed in \VHDL architecture names\footnote{Technically, it is
1081 allowed to use non-alphanumerics when using extended
1082 identifiers, but it seems that none of the tooling likes
1083 extended identifiers in filenames, so it effectively does not
1084 work.}, so the entity would be called \quote{w7aA7f} or
1085 something similarly meaningless and autogenerated).
1087 \subsection{Program structure}
1088 These transformations are aimed at normalizing the overall structure
1089 into the intended form. This means ensuring there is a lambda abstraction
1090 at the top for every argument (input port or current state), putting all
1091 of the other value definitions in let bindings and making the final
1092 return value a simple variable reference.
1094 \subsubsection[sec:normalization:eta]{η-abstraction}
1095 This transformation makes sure that all arguments of a function-typed
1096 expression are named, by introducing lambda expressions. When combined with
1097 β-reduction and non-representable binding inlining, all function-typed
1098 expressions should be lambda abstractions or global identifiers.
1102 -------------- \lam{E} does not occur on a function position in an application
1103 λx.E x \lam{E} is not a lambda abstraction.
1113 foo = λa.λx.(case a of
1118 \transexample{eta}{η-abstraction}{from}{to}
1120 \subsubsection[sec:normalization:appprop]{Application propagation}
1121 This transformation is meant to propagate application expressions downwards
1122 into expressions as far as possible. This allows partial applications inside
1123 expressions to become fully applied and exposes new transformation
1124 opportunities for other transformations (like β-reduction and
1127 Since all binders in our expression are unique (see
1128 \in{section}[sec:normalization:uniq]), there is no risk that we will
1129 introduce unintended shadowing by moving an expression into a lower
1130 scope. Also, since only move expression into smaller scopes (down into
1131 our expression), there is no risk of moving a variable reference out
1132 of the scope in which it is defined.
1135 (letrec binds in E) M
1136 ------------------------
1156 \transexample{appproplet}{Application propagation for a let expression}{from}{to}
1184 \transexample{apppropcase}{Application propagation for a case expression}{from}{to}
1186 \subsubsection[sec:normalization:letrecurse]{Let recursification}
1187 This transformation makes all non-recursive lets recursive. In the
1188 end, we want a single recursive let in our normalized program, so all
1189 non-recursive lets can be converted. This also makes other
1190 transformations simpler: They only need to be specified for recursive
1191 let expressions (and simply will not apply to non-recursive let
1192 expressions until this transformation has been applied).
1199 ------------------------------------------
1206 \subsubsection{Let flattening}
1207 This transformation puts nested lets in the same scope, by lifting the
1208 binding(s) of the inner let into the outer let. Eventually, this will
1209 cause all let bindings to appear in the same scope.
1211 This transformation only applies to recursive lets, since all
1212 non-recursive lets will be made recursive (see
1213 \in{section}[sec:normalization:letrecurse]).
1215 Since we are joining two scopes together, there is no risk of moving a
1216 variable reference out of the scope where it is defined.
1222 ai = (letrec bindings in M)
1227 ------------------------------------------
1262 \transexample{letflat}{Let flattening}{from}{to}
1264 \subsubsection{Return value simplification}
1265 This transformation ensures that the return value of a function is always a
1266 simple local variable reference.
1268 This transformation only applies to the entire body of a
1269 function instead of any subexpression in a function. This is
1270 achieved by the contexts, like \lam{x = E}, though this is
1271 strictly not correct (you could read this as "if there is any
1272 function \lam{x} that binds \lam{E}, any \lam{E} can be
1273 transformed, while we only mean the \lam{E} that is bound by
1276 Note that the return value is not simplified if its not
1277 representable. Otherwise, this would cause a direct loop with
1278 the inlining of unrepresentable bindings. If the return value is
1279 not representable because it has a function type, η-abstraction
1280 should make sure that this transformation will eventually apply.
1281 If the value is not representable for other reasons, the
1282 function result itself is not representable, meaning this
1283 function is not translatable anyway.
1286 x = E \lam{E} is representable
1287 ~ \lam{E} is not a lambda abstraction
1288 E \lam{E} is not a let expression
1289 --------------------------- \lam{E} is not a local variable reference
1295 ~ \lam{E} is representable
1296 E \lam{E} is not a let expression
1297 --------------------------- \lam{E} is not a local variable reference
1302 x = λv0 ... λvn.let ... in E
1303 ~ \lam{E} is representable
1304 E \lam{E} is not a local variable reference
1305 -----------------------------
1314 x = letrec x = add 1 2 in x
1317 \transexample{retvalsimpl}{Return value simplification}{from}{to}
1319 \todo{More examples}
1321 \subsection[sec:normalization:argsimpl]{Representable arguments simplification}
1322 This section contains just a single transformation that deals with
1323 representable arguments in applications. Non-representable arguments are
1324 handled by the transformations in
1325 \in{section}[sec:normalization:nonrep].
1327 This transformation ensures that all representable arguments will become
1328 references to local variables. This ensures they will become references
1329 to local signals in the resulting \small{VHDL}, which is required due to
1330 limitations in the component instantiation code in \VHDL (one can only
1331 assign a signal or constant to an input port). By ensuring that all
1332 arguments are always simple variable references, we always have a signal
1333 available to map to the input ports.
1335 To reduce a complex expression to a simple variable reference, we create
1336 a new let expression around the application, which binds the complex
1337 expression to a new variable. The original function is then applied to
1340 \refdef{global variable}
1341 Note that references to \emph{global variables} (like a top level
1342 function without arguments, but also an argumentless dataconstructors
1343 like \lam{True}) are also simplified. Only local variables generate
1344 signals in the resulting architecture. Even though argumentless
1345 dataconstructors generate constants in generated \VHDL code and could be
1346 mapped to an input port directly, they are still simplified to make the
1347 normal form more regular.
1349 \refdef{representable}
1352 -------------------- \lam{N} is representable
1353 letrec x = N in M x \lam{N} is not a local variable reference
1355 \refdef{local variable}
1362 letrec x = add a 1 in add x 1
1365 \transexample{argsimpl}{Argument simplification}{from}{to}
1367 \subsection[sec:normalization:built-ins]{Built-in functions}
1368 This section deals with (arguments to) built-in functions. In the
1369 intended normal form definition\refdef{intended normal form definition}
1370 we can see that there are three sorts of arguments a built-in function
1374 \item A representable local variable reference. This is the most
1375 common argument to any function. The argument simplification
1376 transformation described in \in{section}[sec:normalization:argsimpl]
1377 makes sure that \emph{any} representable argument to \emph{any}
1378 function (including built-in functions) is turned into a local variable
1380 \item (A partial application of) a top level function (either built-in on
1381 user-defined). The function extraction transformation described in
1382 this section takes care of turning every functiontyped argument into
1383 (a partial application of) a top level function.
1384 \item Any expression that is not representable and does not have a
1385 function type. Since these can be any expression, there is no
1386 transformation needed. Note that this category is exactly all
1387 expressions that are not transformed by the transformations for the
1388 previous two categories. This means that \emph{any} core expression
1389 that is used as an argument to a built-in function will be either
1390 transformed into one of the above categories, or end up in this
1391 categorie. In any case, the result is in normal form.
1394 As noted, the argument simplification will handle any representable
1395 arguments to a built-in function. The following transformation is needed
1396 to handle non-representable arguments with a function type, all other
1397 non-representable arguments do not need any special handling.
1399 \subsubsection[sec:normalization:funextract]{Function extraction}
1400 This transform deals with function-typed arguments to built-in
1402 Since built-in functions cannot be specialized (see
1403 \in{section}[sec:normalization:specialize]) to remove the arguments,
1404 these arguments are extracted into a new global function instead. In
1405 other words, we create a new top level function that has exactly the
1406 extracted argument as its body. This greatly simplifies the
1407 translation rules needed for built-in functions, since they only need
1408 to handle (partial applications of) top level functions.
1410 Any free variables occuring in the extracted arguments will become
1411 parameters to the new global function. The original argument is replaced
1412 with a reference to the new function, applied to any free variables from
1413 the original argument.
1415 This transformation is useful when applying higher-order built-in functions
1416 like \hs{map} to a lambda abstraction, for example. In this case, the code
1417 that generates \small{VHDL} for \hs{map} only needs to handle top level functions and
1418 partial applications, not any other expression (such as lambda abstractions or
1419 even more complicated expressions).
1422 M N \lam{M} is (a partial aplication of) a built-in function.
1423 --------------------- \lam{f0 ... fn} are all free local variables of \lam{N}
1424 M (x f0 ... fn) \lam{N :: a -> b}
1425 ~ \lam{N} is not a (partial application of) a top level function
1430 addList = λb.λxs.map (λa . add a b) xs
1434 addList = λb.λxs.map (f b) xs
1439 \transexample{funextract}{Function extraction}{from}{to}
1441 Note that the function \lam{f} will still need normalization after
1444 \subsection{Case normalisation}
1445 \subsubsection{Scrutinee simplification}
1446 This transform ensures that the scrutinee of a case expression is always
1447 a simple variable reference.
1452 ----------------- \lam{E} is not a local variable reference
1471 \transexample{letflat}{Case normalisation}{from}{to}
1474 \subsubsection{Case normalization}
1475 This transformation ensures that all case expressions get a form
1476 that is allowed by the intended normal form. This means they
1477 will become one of: \refdef{intended normal form definition}
1479 \item An extractor case with a single alternative that picks a field
1480 from a datatype, \eg \lam{case x of (a, b) -> a}.
1481 \item A selector case with multiple alternatives and only wild binders, that
1482 makes a choice between expressions based on the constructor of another
1483 expression, \eg \lam{case x of Low -> a; High -> b}.
1486 For an arbitrary case, that has \lam{n} alternatives, with
1487 \lam{m} binders in each alternatives, this will result in \lam{m
1488 * n} extractor case expression to get at each variable, \lam{n}
1489 let bindings for each of the alternatives' value and a single
1490 selector case to select the right value out of these.
1492 Technically, the defintion of this transformation would require
1493 that the constructor for every alternative has exactly the same
1494 amount (\lam{m}) of arguments, but of course this transformation
1495 also applies when this is not the case.
1499 C0 v0,0 ... v0,m -> E0
1501 Cn vn,0 ... vn,m -> En
1502 --------------------------------------------------- \lam{\forall i \forall j, 0 ≤ i ≤ n, 0 ≤ i < m} (\lam{wi,j} is a wild (unused) binder)
1503 letrec The case expression is not an extractor case
1504 v0,0 = case E of C0 x0,0 .. x0,m -> x0,0 The case expression is not a selector case
1506 v0,m = case E of C0 x0,0 .. x0,m -> x0,m
1508 vn,m = case E of Cn xn,0 .. xn,m -> xn,m
1514 C0 w0,0 ... w0,m -> y0
1516 Cn wn,0 ... wn,m -> yn
1519 \refdef{wild binder}
1520 Note that this transformation applies to case expressions with any
1521 scrutinee. If the scrutinee is a complex expression, this might
1522 result in duplication of work (hardware). An extra condition to
1523 only apply this transformation when the scrutinee is already
1524 simple (effectively causing this transformation to be only
1525 applied after the scrutinee simplification transformation) might
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 expression with a single alternative and
1571 only wild binders.\refdef{wild binder}
1573 These "useless" case expressions 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[sec:normalization:nonrep]{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. These are meant to address three
1599 different kinds of unrepresentable values: Polymorphic values,
1600 higher-order values and literals. The transformation are described
1601 generically: They apply to all non-representable values. However,
1602 non-representable values that do not fall into one of these three
1603 categories will be moved around by these transformations but are
1604 unlikely to completely disappear. They usually mean the program was not
1605 valid in the first place, because unsupported types were used (for
1606 example, a program using strings).
1608 Each of these three categories will be detailed below, followed by the
1609 actual transformations.
1611 \subsubsection{Removing Polymorphism}
1612 As noted in \in{section}[sec:prototype:polymporphism],
1613 polymorphism is made explicit in Core through type and
1614 dictionary arguments. To remove the polymorphism from a
1615 function, we can simply specialize the polymorphic function for
1616 the particular type applied to it. The same goes for dictionary
1617 arguments. To remove polymorphism from let bound values, we
1618 simply inline the let bindings that have a polymorphic type,
1619 which should (eventually) make sure that the polymorphic
1620 expression is applied to a type and/or dictionary, which can
1621 then be removed by β-reduction (\in{section}[sec:normalization:beta]).
1623 Since both type and dictionary arguments are not representable,
1624 \refdef{representable}
1625 the non-representable argument specialization and
1626 non-representable let binding inlining transformations below
1627 take care of exactly this.
1629 There is one case where polymorphism cannot be completely
1630 removed: Built-in functions are still allowed to be polymorphic
1631 (Since we have no function body that we could properly
1632 specialize). However, the code that generates \VHDL for built-in
1633 functions knows how to handle this, so this is not a problem.
1635 \subsubsection[sec:normalization:defunctionalization]{Defunctionalization}
1636 These transformations remove higher-order expressions from our
1637 program, making all values first-order.
1639 Higher order values are always introduced by lambda abstractions, none
1640 of the other Core expression elements can introduce a function type.
1641 However, other expressions can \emph{have} a function type, when they
1642 have a lambda expression in their body.
1644 For example, the following expression is a higher-order expression
1645 that is not a lambda expression itself:
1647 \refdef{id function}
1654 The reference to the \lam{id} function shows that we can introduce a
1655 higher-order expression in our program without using a lambda
1656 expression directly. However, inside the definition of the \lam{id}
1657 function, we can be sure that a lambda expression is present.
1659 Looking closely at the definition of our normal form in
1660 \in{section}[sec:normalization:intendednormalform], we can see that
1661 there are three possibilities for higher-order values to appear in our
1662 intended normal form:
1665 \item[item:toplambda] Lambda abstractions can appear at the highest level of a
1666 top level function. These lambda abstractions introduce the
1667 arguments (input ports / current state) of the function.
1668 \item[item:built-inarg] (Partial applications of) top level functions can appear as an
1669 argument to a built-in function.
1670 \item[item:completeapp] (Partial applications of) top level functions can appear in
1671 function position of an application. Since a partial application
1672 cannot appear anywhere else (except as built-in function arguments),
1673 all partial applications are applied, meaning that all applications
1674 will become complete applications. However, since application of
1675 arguments happens one by one, in the expression:
1679 the subexpression \lam{f 1} has a function type. But this is
1680 allowed, since it is inside a complete application.
1683 We will take a typical function with some higher-order values as an
1684 example. The following function takes two arguments: a \lam{Bit} and a
1685 list of numbers. Depending on the first argument, each number in the
1686 list is doubled, or the list is returned unmodified. For the sake of
1687 the example, no polymorphism is shown. In reality, at least map would
1691 λy.let double = λx. x + x in
1697 This example shows a number of higher-order values that we cannot
1698 translate to \VHDL directly. The \lam{double} binder bound in the let
1699 expression has a function type, as well as both of the alternatives of
1700 the case expression. The first alternative is a partial application of
1701 the \lam{map} built-in function, whereas the second alternative is a
1704 To reduce all higher-order values to one of the above items, a number
1705 of transformations we have already seen are used. The η-abstraction
1706 transformation from \in{section}[sec:normalization:eta] ensures all
1707 function arguments are introduced by lambda abstraction on the highest
1708 level of a function. These lambda arguments are allowed because of
1709 \in{item}[item:toplambda] above. After η-abstraction, our example
1710 becomes a bit bigger:
1713 λy.λq.(let double = λx. x + x in
1720 η-abstraction also introduces extra applications (the application of
1721 the let expression to \lam{q} in the above example). These
1722 applications can then propagated down by the application propagation
1723 transformation (\in{section}[sec:normalization:appprop]). In our
1724 example, the \lam{q} and \lam{r} variable will be propagated into the
1725 let expression and then into the case expression:
1728 λy.λq.let double = λx. x + x in
1734 This propagation makes higher-order values become applied (in
1735 particular both of the alternatives of the case now have a
1736 representable type). Completely applied top level functions (like the
1737 first alternative) are now no longer invalid (they fall under
1738 \in{item}[item:completeapp] above). (Completely) applied lambda
1739 abstractions can be removed by β-abstraction. For our example,
1740 applying β-abstraction results in the following:
1743 λy.λq.let double = λx. x + x in
1749 As you can see in our example, all of this moves applications towards
1750 the higher-order values, but misses higher-order functions bound by
1751 let expressions. The applications cannot be moved towards these values
1752 (since they can be used in multiple places), so the values will have
1753 to be moved towards the applications. This is achieved by inlining all
1754 higher-order values bound by let applications, by the
1755 non-representable binding inlining transformation below. When applying
1756 it to our example, we get the following:
1760 Low -> map (λx. x + x) q
1764 We have nearly eliminated all unsupported higher-order values from this
1765 expressions. The one that is remaining is the first argument to the
1766 \lam{map} function. Having higher-order arguments to a built-in
1767 function like \lam{map} is allowed in the intended normal form, but
1768 only if the argument is a (partial application) of a top level
1769 function. This is easily done by introducing a new top level function
1770 and put the lambda abstraction inside. This is done by the function
1771 extraction transformation from
1772 \in{section}[sec:normalization:funextract].
1780 This also introduces a new function, that we have called \lam{func}:
1786 Note that this does not actually remove the lambda, but now it is a
1787 lambda at the highest level of a function, which is allowed in the
1788 intended normal form.
1790 There is one case that has not been discussed yet. What if the
1791 \lam{map} function in the example above was not a built-in function
1792 but a user-defined function? Then extracting the lambda expression
1793 into a new function would not be enough, since user-defined functions
1794 can never have higher-order arguments. For example, the following
1795 expression shows an example:
1798 twice :: (Word -> Word) -> Word -> Word
1799 twice = λf.λa.f (f a)
1801 main = λa.app (λx. x + x) a
1804 This example shows a function \lam{twice} that takes a function as a
1805 first argument and applies that function twice to the second argument.
1806 Again, we have made the function monomorphic for clarity, even though
1807 this function would be a lot more useful if it was polymorphic. The
1808 function \lam{main} uses \lam{twice} to apply a lambda epression twice.
1810 When faced with a user defined function, a body is available for that
1811 function. This means we could create a specialized version of the
1812 function that only works for this particular higher-order argument
1813 (\ie, we can just remove the argument and call the specialized
1814 function without the argument). This transformation is detailed below.
1815 Applying this transformation to the example gives:
1818 twice' :: Word -> Word
1819 twice' = λb.(λf.λa.f (f a)) (λx. x + x) b
1824 The \lam{main} function is now in normal form, since the only
1825 higher-order value there is the top level lambda expression. The new
1826 \lam{twice'} function is a bit complex, but the entire original body
1827 of the original \lam{twice} function is wrapped in a lambda
1828 abstraction and applied to the argument we have specialized for
1829 (\lam{λx. x + x}) and the other arguments. This complex expression can
1830 fortunately be effectively reduced by repeatedly applying β-reduction:
1833 twice' :: Word -> Word
1834 twice' = λb.(b + b) + (b + b)
1837 This example also shows that the resulting normal form might not be as
1838 efficient as we might hope it to be (it is calculating \lam{(b + b)}
1839 twice). This is discussed in more detail in
1840 \in{section}[sec:normalization:duplicatework].
1842 \subsubsection{Literals}
1843 There are a limited number of literals available in Haskell and Core.
1844 \refdef{enumerated types} When using (enumerating) algebraic
1845 datatypes, a literal is just a reference to the corresponding data
1846 constructor, which has a representable type (the algebraic datatype)
1847 and can be translated directly. This also holds for literals of the
1848 \hs{Bool} Haskell type, which is just an enumerated type.
1850 There is, however, a second type of literal that does not have a
1851 representable type: Integer literals. Cλash supports using integer
1852 literals for all three integer types supported (\hs{SizedWord},
1853 \hs{SizedInt} and \hs{RangedWord}). This is implemented using
1854 Haskell's \hs{Num} type class, which offers a \hs{fromInteger} method
1855 that converts any \hs{Integer} to the Cλash datatypes.
1857 When \GHC sees integer literals, it will automatically insert calls to
1858 the \hs{fromInteger} method in the resulting Core expression. For
1859 example, the following expression in Haskell creates a 32 bit unsigned
1860 word with the value 1. The explicit type signature is needed, since
1861 there is no context for \GHC to determine the type from otherwise.
1867 This Haskell code results in the following Core expression:
1870 fromInteger @(SizedWord D32) \$dNum (smallInteger 10)
1873 The literal 10 will have the type \lam{GHC.Prim.Int\#}, which is
1874 converted into an \lam{Integer} by \lam{smallInteger}. Finally, the
1875 \lam{fromInteger} function will finally convert this into a
1876 \lam{SizedWord D32}.
1878 Both the \lam{GHC.Prim.Int\#} and \lam{Integer} types are not
1879 representable, and cannot be translated directly. Fortunately, there
1880 is no need to translate them, since \lam{fromInteger} is a built-in
1881 function that knows how to handle these values. However, this does
1882 require that the \lam{fromInteger} function is directly applied to
1883 these non-representable literal values, otherwise errors will occur.
1884 For example, the following expression is not in the intended normal
1885 form, since one of the let bindings has an unrepresentable type
1889 let l = smallInteger 10 in fromInteger @(SizedWord D32) \$dNum l
1892 By inlining these let-bindings, we can ensure that unrepresentable
1893 literals bound by a let binding end up in an application of the
1894 appropriate built-in function, where they are allowed. Since it is
1895 possible that the application of that function is in a different
1896 function than the definition of the literal value, we will always need
1897 to specialize away any unrepresentable literals that are used as
1898 function arguments. The following two transformations do exactly this.
1900 \subsubsection{Non-representable binding inlining}
1901 This transform inlines let bindings that are bound to a
1902 non-representable value. Since we can never generate a signal
1903 assignment for these bindings (we cannot declare a signal assignment
1904 with a non-representable type, for obvious reasons), we have no choice
1905 but to inline the binding to remove it.
1907 As we have seen in the previous sections, inlining these bindings
1908 solves (part of) the polymorphism, higher-order values and
1909 unrepresentable literals in an expression.
1911 \refdef{substitution notation}
1921 -------------------------- \lam{Ei} has a non-representable type.
1923 a0 = E0 [ai=>Ei] \vdots
1924 ai-1 = Ei-1 [ai=>Ei]
1925 ai+1 = Ei+1 [ai=>Ei]
1944 x = fromInteger (smallInteger 10)
1946 (λb -> add b 1) (add 1 x)
1949 \transexample{nonrepinline}{Nonrepresentable binding inlining}{from}{to}
1951 \subsubsection[sec:normalization:specialize]{Function specialization}
1952 This transform removes arguments to user-defined functions that are
1953 not representable at runtime. This is done by creating a
1954 \emph{specialized} version of the function that only works for one
1955 particular value of that argument (in other words, the argument can be
1958 Specialization means to create a specialized version of the called
1959 function, with one argument already filled in. As a simple example, in
1960 the following program (this is not actual Core, since it directly uses
1961 a literal with the unrepresentable type \lam{GHC.Prim.Int\#}).
1968 We could specialize the function \lam{f} against the literal argument
1969 1, with the following result:
1976 In some way, this transformation is similar to β-reduction, but it
1977 operates across function boundaries. It is also similar to
1978 non-representable let binding inlining above, since it sort of
1979 \quote{inlines} an expression into a called function.
1981 Special care must be taken when the argument has any free variables.
1982 If this is the case, the original argument should not be removed
1983 completely, but replaced by all the free variables of the expression.
1984 In this way, the original expression can still be evaluated inside the
1987 To prevent us from propagating the same argument over and over, a
1988 simple local variable reference is not propagated (since is has
1989 exactly one free variable, itself, we would only replace that argument
1992 This shows that any free local variables that are not runtime
1993 representable cannot be brought into normal form by this transform. We
1994 rely on an inlining or β-reduction transformation to replace such a
1995 variable with an expression we can propagate again.
2000 x Y0 ... Yi ... Yn \lam{Yi} is not representable
2001 --------------------------------------------- \lam{Yi} is not a local variable reference
2002 x' Y0 ... Yi-1 f0 ... fm Yi+1 ... Yn \lam{f0 ... fm} are all free local vars of \lam{Yi}
2003 ~ \lam{T0 ... Tn} are the types of \lam{Y0 ... Yn}
2004 x' = λ(y0 :: T0) ... λ(yi-1 :: Ty-1).
2006 λ(yi+1 :: Ty+1) ... λ(yn :: Tn).
2007 E y0 ... yi-1 Yi yi+1 ... yn
2010 This is a bit of a complex transformation. It transforms an
2011 application of the function \lam{x}, where one of the arguments
2012 (\lam{Y_i}) is not representable. A new
2013 function \lam{x'} is created that wraps the body of the old function.
2014 The body of the new function becomes a number of nested lambda
2015 abstractions, one for each of the original arguments that are left
2018 The ith argument is replaced with the free variables of
2019 \lam{Y_i}. Note that we reuse the same binders as those used in
2020 \lam{Y_i}, since we can then just use \lam{Y_i} inside the new
2021 function body and have all of the variables it uses be in scope.
2023 The argument that we are specializing for, \lam{Y_i}, is put inside
2024 the new function body. The old function body is applied to it. Since
2025 we use this new function only in place of an application with that
2026 particular argument \lam{Y_i}, behaviour should not change.
2028 Note that the types of the arguments of our new function are taken
2029 from the types of the \emph{actual} arguments (\lam{T0 ... Tn}). This
2030 means that any polymorphism in the arguments is removed, even when the
2031 corresponding explicit type lambda is not removed
2034 \todo{Examples. Perhaps reference the previous sections}
2036 \section{Unsolved problems}
2037 The above system of transformations has been implemented in the prototype
2038 and seems to work well to compile simple and more complex examples of
2039 hardware descriptions. \todo{Ref christiaan?} However, this normalization
2040 system has not seen enough review and work to be complete and work for
2041 every Core expression that is supplied to it. A number of problems
2042 have already been identified and are discussed in this section.
2044 \subsection[sec:normalization:duplicatework]{Work duplication}
2045 A possible problem of β-reduction is that it could duplicate work.
2046 When the expression applied is not a simple variable reference, but
2047 requires calculation and the binder the lambda abstraction binds to
2048 is used more than once, more hardware might be generated than strictly
2051 As an example, consider the expression:
2057 When applying β-reduction to this expression, we get:
2063 which of course calculates \lam{(a * b)} twice.
2065 A possible solution to this would be to use the following alternative
2066 transformation, which is of course no longer normal β-reduction. The
2067 followin transformation has not been tested in the prototype, but is
2068 given here for future reference:
2076 This does not seem like much of an improvement, but it does get rid of
2077 the lambda expression (and the associated higher-order value), while
2078 at the same time introducing a new let binding. Since the result of
2079 every application or case expression must be bound by a let expression
2080 in the intended normal form anyway, this is probably not a problem. If
2081 the argument happens to be a variable reference, then simple let
2082 binding removal (\in{section}[sec:normalization:simplelet]) will
2083 remove it, making the result identical to that of the original
2084 β-reduction transformation.
2086 When also applying argument simplification to the above example, we
2087 get the following expression:
2095 Looking at this, we could imagine an alternative approach: Create a
2096 transformation that removes let bindings that bind identical values.
2097 In the above expression, the \lam{y} and \lam{z} variables could be
2098 merged together, resulting in the more efficient expression:
2101 let y = (a * b) in y + y
2104 \subsection[sec:normalization:non-determinism]{Non-determinism}
2105 As an example, again consider the following expression:
2111 We can apply both β-reduction (\in{section}[sec:normalization:beta])
2112 as well as argument simplification
2113 (\in{section}[sec:normalization:argsimpl]) to this expression.
2115 When applying argument simplification first and then β-reduction, we
2116 get the following expression:
2119 let y = (a * b) in y + y
2122 When applying β-reduction first and then argument simplification, we
2123 get the following expression:
2131 As you can see, this is a different expression. This means that the
2132 order of expressions, does in fact change the resulting normal form,
2133 which is something that we would like to avoid. In this particular
2134 case one of the alternatives is even clearly more efficient, so we
2135 would of course like the more efficient form to be the normal form.
2137 For this particular problem, the solutions for duplication of work
2138 seem from the previous section seem to fix the determinism of our
2139 transformation system as well. However, it is likely that there are
2140 other occurences of this problem.
2142 \subsection[sec:normalization:castproblems]{Casts}
2143 We do not fully understand the use of cast expressions in Core, so
2144 there are probably expressions involving cast expressions that cannot
2145 be brought into intended normal form by this transformation system.
2147 The uses of casts in the core system should be investigated more and
2148 transformations will probably need updating to handle them in all
2151 \subsection[sec:normalization:stateproblems]{Normalization of stateful descriptions}
2152 Currently, the intended normal form definition\refdef{intended
2153 normal form definition} offers enough freedom to describe all
2154 valid stateful descriptions, but is not limiting enough. It is
2155 possible to write descriptions which are in intended normal
2156 form, but cannot be translated into \VHDL in a meaningful way
2157 (\eg, a function that swaps two substates in its result, or a
2158 function that changes a substate itself instead of passing it to
2161 It is now up to the programmer to not do anything funny with
2162 these state values, whereas the normalization just tries not to
2163 mess up the flow of state values. In practice, there are
2164 situations where a Core program that \emph{could} be a valid
2165 stateful description is not translateable by the prototype. This
2166 most often happens when statefulness is mixed with pattern
2167 matching, causing a state input to be unpacked multiple times or
2168 be unpacked and repacked only in some of the code paths.
2170 Without going into detail about the exact problems (of which
2171 there are probably more than have shown up so far), it seems
2172 unlikely that these problems can be solved entirely by just
2173 improving the \VHDL state generation in the final stage. The
2174 normalization stage seems the best place to apply the rewriting
2175 needed to support more complex stateful descriptions. This does
2176 of course mean that the intended normal form definition must be
2177 extended as well to be more specific about how state handling
2178 should look like in normal form.
2179 \in{Section}[sec:prototype:statelimits] already contains a
2180 tight description of the limitations on the use of state
2181 variables, which could be adapted into the intended normal form.
2183 \section[sec:normalization:properties]{Provable properties}
2184 When looking at the system of transformations outlined above, there are a
2185 number of questions that we can ask ourselves. The main question is of course:
2186 \quote{Does our system work as intended?}. We can split this question into a
2187 number of subquestions:
2190 \item[q:termination] Does our system \emph{terminate}? Since our system will
2191 keep running as long as transformations apply, there is an obvious risk that
2192 it will keep running indefinitely. This typically happens when one
2193 transformation produces a result that is transformed back to the original
2194 by another transformation, or when one or more transformations keep
2195 expanding some expression.
2196 \item[q:soundness] Is our system \emph{sound}? Since our transformations
2197 continuously modify the expression, there is an obvious risk that the final
2198 normal form will not be equivalent to the original program: Its meaning could
2200 \item[q:completeness] Is our system \emph{complete}? Since we have a complex
2201 system of transformations, there is an obvious risk that some expressions will
2202 not end up in our intended normal form, because we forgot some transformation.
2203 In other words: Does our transformation system result in our intended normal
2204 form for all possible inputs?
2205 \item[q:determinism] Is our system \emph{deterministic}? Since we have defined
2206 no particular order in which the transformation should be applied, there is an
2207 obvious risk that different transformation orderings will result in
2208 \emph{different} normal forms. They might still both be intended normal forms
2209 (if our system is \emph{complete}) and describe correct hardware (if our
2210 system is \emph{sound}), so this property is less important than the previous
2211 three: The translator would still function properly without it.
2214 Unfortunately, the final transformation system has only been
2215 developed in the final part of the research, leaving no more time
2216 for verifying these properties. In fact, it is likely that the
2217 current transformation system still violates some of these
2218 properties in some cases and should be improved (or extra conditions
2219 on the input hardware descriptions should be formulated).
2221 This is most likely the case with the completeness and determinism
2222 properties, perhaps als the termination property. The soundness
2223 property probably holds, since it is easier to manually verify (each
2224 transformation can be reviewed separately).
2226 Even though no complete proofs have been made, some ideas for
2227 possible proof strategies are shown below.
2229 \subsection{Graph representation}
2230 Before looking into how to prove these properties, we will look at
2231 transformation systems from a graph perspective. We will first define
2232 the graph view and then illustrate it using a simple example from lambda
2233 calculus (which is a different system than the Cλash normalization
2234 system). The nodes of the graph are all possible Core expressions. The
2235 (directed) edges of the graph are transformations. When a transformation
2236 α applies to an expression \lam{A} to produce an expression \lam{B}, we
2237 add an edge from the node for \lam{A} to the node for \lam{B}, labeled
2240 \startuseMPgraphic{TransformGraph}
2244 newCircle.a(btex \lam{(λx.λy. (+) x y) 1} etex);
2245 newCircle.b(btex \lam{λy. (+) 1 y} etex);
2246 newCircle.c(btex \lam{(λx.(+) x) 1} etex);
2247 newCircle.d(btex \lam{(+) 1} etex);
2250 c.c = b.c + (4cm, 0cm);
2251 a.c = midpoint(b.c, c.c) + (0cm, 4cm);
2252 d.c = midpoint(b.c, c.c) - (0cm, 3cm);
2254 % β-conversion between a and b
2255 ncarc.a(a)(b) "name(bred)";
2256 ObjLabel.a(btex $\xrightarrow[normal]{}{β}$ etex) "labpathname(bred)", "labdir(rt)";
2257 ncarc.b(b)(a) "name(bexp)", "linestyle(dashed withdots)";
2258 ObjLabel.b(btex $\xleftarrow[normal]{}{β}$ etex) "labpathname(bexp)", "labdir(lft)";
2260 % η-conversion between a and c
2261 ncarc.a(a)(c) "name(ered)";
2262 ObjLabel.a(btex $\xrightarrow[normal]{}{η}$ etex) "labpathname(ered)", "labdir(rt)";
2263 ncarc.c(c)(a) "name(eexp)", "linestyle(dashed withdots)";
2264 ObjLabel.c(btex $\xleftarrow[normal]{}{η}$ etex) "labpathname(eexp)", "labdir(lft)";
2266 % η-conversion between b and d
2267 ncarc.b(b)(d) "name(ered)";
2268 ObjLabel.b(btex $\xrightarrow[normal]{}{η}$ etex) "labpathname(ered)", "labdir(rt)";
2269 ncarc.d(d)(b) "name(eexp)", "linestyle(dashed withdots)";
2270 ObjLabel.d(btex $\xleftarrow[normal]{}{η}$ etex) "labpathname(eexp)", "labdir(lft)";
2272 % β-conversion between c and d
2273 ncarc.c(c)(d) "name(bred)";
2274 ObjLabel.c(btex $\xrightarrow[normal]{}{β}$ etex) "labpathname(bred)", "labdir(rt)";
2275 ncarc.d(d)(c) "name(bexp)", "linestyle(dashed withdots)";
2276 ObjLabel.d(btex $\xleftarrow[normal]{}{β}$ etex) "labpathname(bexp)", "labdir(lft)";
2278 % Draw objects and lines
2279 drawObj(a, b, c, d);
2282 \placeexample[right][ex:TransformGraph]{Partial graph of a lambda calculus
2283 system with β and η reduction (solid lines) and expansion (dotted lines).}
2284 \boxedgraphic{TransformGraph}
2286 Of course the graph for Cλash is unbounded, since we can construct an
2287 infinite amount of Core expressions. Also, there might potentially be
2288 multiple edges between two given nodes (with different labels), though
2289 this seems unlikely to actually happen in our system.
2291 See \in{example}[ex:TransformGraph] for the graph representation of a very
2292 simple lambda calculus that contains just the expressions \lam{(λx.λy. (+) x
2293 y) 1}, \lam{λy. (+) 1 y}, \lam{(λx.(+) x) 1} and \lam{(+) 1}. The
2294 transformation system consists of β-reduction and η-reduction (solid edges) or
2295 β-expansion and η-expansion (dotted edges).
2297 \todo{Define β-reduction and η-reduction?}
2299 Note that the normal form of such a system consists of the set of nodes
2300 (expressions) without outgoing edges, since those are the expressions to which
2301 no transformation applies anymore. We call this set of nodes the \emph{normal
2302 set}. The set of nodes containing expressions in intended normal
2303 form \refdef{intended normal form} is called the \emph{intended
2306 From such a graph, we can derive some properties easily:
2308 \item A system will \emph{terminate} if there is no walk (sequence of
2309 edges, or transformations) of infinite length in the graph (this
2310 includes cycles, but can also happen without cycles).
2311 \item Soundness is not easily represented in the graph.
2312 \item A system is \emph{complete} if all of the nodes in the normal set have
2313 the intended normal form. The inverse (that all of the nodes outside of
2314 the normal set are \emph{not} in the intended normal form) is not
2315 strictly required. In other words, our normal set must be a
2316 subset of the intended normal form, but they do not need to be
2319 \item A system is deterministic if all paths starting at a particular
2320 node, which end in a node in the normal set, end at the same node.
2323 When looking at the \in{example}[ex:TransformGraph], we see that the system
2324 terminates for both the reduction and expansion systems (but note that, for
2325 expansion, this is only true because we have limited the possible
2326 expressions. In complete lambda calculus, there would be a path from
2327 \lam{(λx.λy. (+) x y) 1} to \lam{(λx.λy.(λz.(+) z) x y) 1} to
2328 \lam{(λx.λy.(λz.(λq.(+) q) z) x y) 1} etc.)
2330 If we would consider the system with both expansion and reduction, there
2331 would no longer be termination either, since there would be cycles all
2334 The reduction and expansion systems have a normal set of containing just
2335 \lam{(+) 1} or \lam{(λx.λy. (+) x y) 1} respectively. Since all paths in
2336 either system end up in these normal forms, both systems are \emph{complete}.
2337 Also, since there is only one node in the normal set, it must obviously be
2338 \emph{deterministic} as well.
2340 \subsection{Termination}
2341 In general, proving termination of an arbitrary program is a very
2342 hard problem. \todo{Ref about arbitrary termination} Fortunately,
2343 we only have to prove termination for our specific transformation
2346 A common approach for these kinds of proofs is to associate a
2347 measure with each possible expression in our system. If we can
2348 show that each transformation strictly decreases this measure
2349 (\ie, the expression transformed to has a lower measure than the
2350 expression transformed from). \todo{ref about measure-based
2351 termination proofs / analysis}
2353 A good measure for a system consisting of just β-reduction would
2354 be the number of lambda expressions in the expression. Since every
2355 application of β-reduction removes a lambda abstraction (and there
2356 is always a bounded number of lambda abstractions in every
2357 expression) we can easily see that a transformation system with
2358 just β-reduction will always terminate.
2360 For our complete system, this measure would be fairly complex
2361 (probably the sum of a lot of things). Since the (conditions on)
2362 our transformations are pretty complex, we would need to include
2363 both simple things like the number of let expressions as well as
2364 more complex things like the number of case expressions that are
2365 not yet in normal form.
2367 No real attempt has been made at finding a suitable measure for
2370 \subsection{Soundness}
2371 Soundness is a property that can be proven for each transformation
2372 separately. Since our system only runs separate transformations
2373 sequentially, if each of our transformations leaves the
2374 \emph{meaning} of the expression unchanged, then the entire system
2375 will of course leave the meaning unchanged and is thus
2378 The current prototype has only been verified in an ad-hoc fashion
2379 by inspecting (the code for) each transformation. A more formal
2380 verification would be more appropriate.
2382 To be able to formally show that each transformation properly
2383 preserves the meaning of every expression, we require an exact
2384 definition of the \emph{meaning} of every expression, so we can
2385 compare them. A definition of the operational semantics of \GHC's Core
2386 language is available \cite[sulzmann07], but this does not seem
2387 sufficient for our goals (but it is a good start).
2389 It should be possible to have a single formal definition of
2390 meaning for Core for both normal Core compilation by \GHC and for
2391 our compilation to \VHDL. The main difference seems to be that in
2392 hardware every expression is always evaluated, while in software
2393 it is only evaluated if needed, but it should be possible to
2394 assign a meaning to core expressions that assumes neither.
2396 Since each of the transformations can be applied to any
2397 subexpression as well, there is a constraint on our meaning
2398 definition: The meaning of an expression should depend only on the
2399 meaning of subexpressions, not on the expressions themselves. For
2400 example, the meaning of the application in \lam{f (let x = 4 in
2401 x)} should be the same as the meaning of the application in \lam{f
2402 4}, since the argument subexpression has the same meaning (though
2403 the actual expression is different).
2405 \subsection{Completeness}
2406 Proving completeness is probably not hard, but it could be a lot
2407 of work. We have seen above that to prove completeness, we must
2408 show that the normal set of our graph representation is a subset
2409 of the intended normal set.
2411 However, it is hard to systematically generate or reason about the
2412 normal set, since it is defined as any nodes to which no
2413 transformation applies. To determine this set, each transformation
2414 must be considered and when a transformation is added, the entire
2415 set should be re-evaluated. This means it is hard to show that
2416 each node in the normal set is also in the intended normal set.
2417 Reasoning about our intended normal set is easier, since we know
2418 how to generate it from its definition. \refdef{intended normal
2421 Fortunately, we can also prove the complement (which is
2422 equivalent, since $A \subseteq B \Leftrightarrow \overline{B}
2423 \subseteq \overline{A}$): Show that the set of nodes not in
2424 intended normal form is a subset of the set of nodes not in normal
2425 form. In other words, show that for every expression that is not
2426 in intended normal form, that there is at least one transformation
2427 that applies to it (since that means it is not in normal form
2428 either and since $A \subseteq C \Leftrightarrow \forall x (x \in A
2429 \rightarrow x \in C)$).
2431 By systematically reviewing the entire Core language definition
2432 along with the intended normal form definition (both of which have
2433 a similar structure), it should be possible to identify all
2434 possible (sets of) core expressions that are not in intended
2435 normal form and identify a transformation that applies to it.
2437 This approach is especially useful for proving completeness of our
2438 system, since if expressions exist to which none of the
2439 transformations apply (\ie if the system is not yet complete), it
2440 is immediately clear which expressions these are and adding
2441 (or modifying) transformations to fix this should be relatively
2444 As observed above, applying this approach is a lot of work, since
2445 we need to check every (set of) transformation(s) separately.
2447 \todo{Perhaps do a few steps of the proofs as proof-of-concept}
2449 % vim: set sw=2 sts=2 expandtab: