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[force]{}{\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
69 alu :: Bit -> Word -> Word -> Word
78 \startuseMPgraphic{MulSum}
79 save a, b, c, mul, add, sum;
82 newCircle.a(btex $a$ etex) "framed(false)";
83 newCircle.b(btex $b$ etex) "framed(false)";
84 newCircle.c(btex $c$ etex) "framed(false)";
85 newCircle.sum(btex $sum$ etex) "framed(false)";
88 newCircle.mul(btex * etex);
89 newCircle.add(btex + etex);
91 a.c - b.c = (0cm, 2cm);
92 b.c - c.c = (0cm, 2cm);
93 add.c = c.c + (2cm, 0cm);
94 mul.c = midpoint(a.c, b.c) + (2cm, 0cm);
95 sum.c = add.c + (2cm, 0cm);
98 % Draw objects and lines
99 drawObj(a, b, c, mul, add, sum);
101 ncarc(a)(mul) "arcangle(15)";
102 ncarc(b)(mul) "arcangle(-15)";
108 \placeexample[][ex:MulSum]{Simple architecture consisting of a
109 multiplier and a subtractor.}
110 \startcombination[2*1]
111 {\typebufferlam{MulSum}}{Core description in normal form.}
112 {\boxedgraphic{MulSum}}{The architecture described by the normal form.}
115 \todo{Intermezzo: functions vs plain values}
117 A very simple example of a program in normal form is given in
118 \in{example}[ex:MulSum]. As you can see, all arguments to the function (which
119 will become input ports in the generated \VHDL) are at the outer level.
120 This means that the body of the inner lambda abstraction is never a
121 function, but always a plain value.
123 As the body of the inner lambda abstraction, we see a single (recursive)
124 let expression, that binds two variables (\lam{mul} and \lam{sum}). These
125 variables will be signals in the generated \VHDL, bound to the output port
126 of the \lam{*} and \lam{+} components.
128 The final line (the \quote{return value} of the function) selects the
129 \lam{sum} signal to be the output port of the function. This \quote{return
130 value} can always only be a variable reference, never a more complex
133 \todo{Add generated VHDL}
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[][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. Most synthesis tools will further optimize this architecture by
212 removing the multiplexers at the register input and instead use the write
213 enable port of the register (when it is available), but we want to show
214 the architecture as close to the description as possible.
216 As you can see from the previous examples, the generation of the final
217 architecture from the normal form is straightforward. In each of the
218 examples, there is a direct match between the normal form structure,
219 the generated VHDL and the architecture shown in the images.
221 \startbuffer[NormalComplete]
224 -> State (Word, Word)
225 -> (State (Word, Word), Word)
227 -- All arguments are an inital lambda (address, data, packed state)
229 -- There are nested let expressions at top level
231 -- Unpack the state by coercion (\eg, cast from
232 -- State (Word, Word) to (Word, Word))
233 s = sp ▶ (Word, Word)
234 -- Extract both registers from the state
235 r1 = case s of (a, b) -> a
236 r2 = case s of (a, b) -> b
237 -- Calling some other user-defined function.
239 -- Conditional connections
251 -- pack the state by coercion (\eg, cast from
252 -- (Word, Word) to State (Word, Word))
253 sp' = s' ▶ State (Word, Word)
254 -- Pack our return value
261 \startuseMPgraphic{NormalComplete}
262 save a, d, r, foo, muxr, muxout, out;
265 newCircle.a(btex \lam{a} etex) "framed(false)";
266 newCircle.d(btex \lam{d} etex) "framed(false)";
267 newCircle.out(btex \lam{out} etex) "framed(false)";
269 %newCircle.add(btex + etex);
270 newBox.foo(btex \lam{foo} etex);
271 newReg.r1(btex $\lam{r1}$ etex) "dx(4mm)", "dy(6mm)";
272 newReg.r2(btex $\lam{r2}$ etex) "dx(4mm)", "dy(6mm)", "reflect(true)";
274 % Reflect over the vertical axis
275 reflectObj(muxr1)((0,0), (0,1));
278 rotateObj(muxout)(-90);
280 d.c = foo.c + (0cm, 1.5cm);
281 a.c = (xpart r2.c + 2cm, ypart d.c - 0.5cm);
282 foo.c = midpoint(muxr1.c, muxr2.c) + (0cm, 2cm);
283 muxr1.c = r1.c + (0cm, 2cm);
284 muxr2.c = r2.c + (0cm, 2cm);
285 r2.c = r1.c + (4cm, 0cm);
287 muxout.c = midpoint(r1.c, r2.c) - (0cm, 2cm);
288 out.c = muxout.c - (0cm, 1.5cm);
290 % % Draw objects and lines
291 drawObj(a, d, foo, r1, r2, muxr1, muxr2, muxout, out);
294 nccurve(foo)(muxr1) "angleA(-90)", "posB(inpa)", "angleB(180)";
295 nccurve(foo)(muxr2) "angleA(-90)", "posB(inpb)", "angleB(0)";
296 nccurve(muxr1)(r1) "posA(out)", "angleA(180)", "posB(d)", "angleB(0)";
297 nccurve(r1)(muxr1) "posA(out)", "angleA(0)", "posB(inpb)", "angleB(180)";
298 nccurve(muxr2)(r2) "posA(out)", "angleA(0)", "posB(d)", "angleB(180)";
299 nccurve(r2)(muxr2) "posA(out)", "angleA(180)", "posB(inpa)", "angleB(0)";
300 nccurve(r1)(muxout) "posA(out)", "angleA(0)", "posB(inpb)", "angleB(-90)";
301 nccurve(r2)(muxout) "posA(out)", "angleA(180)", "posB(inpa)", "angleB(-90)";
303 nccurve(a)(muxout) "angleA(-90)", "angleB(180)", "posB(sel)";
304 nccurve(a)(muxr1) "angleA(180)", "angleB(-90)", "posB(sel)";
305 nccurve(a)(muxr2) "angleA(180)", "angleB(-90)", "posB(sel)";
306 ncline(muxout)(out) "posA(out)";
309 \todo{Don't split registers in this image?}
310 \placeexample[][ex:NormalComplete]{Simple architecture consisting of an adder and a
312 \startcombination[2*1]
313 {\typebufferlam{NormalComplete}}{Core description in normal form.}
314 {\boxedgraphic{NormalComplete}}{The architecture described by the normal form.}
319 \subsection[sec:normalization:intendednormalform]{Intended normal form definition}
320 Now we have some intuition for the normal form, we can describe how we want
321 the normal form to look like in a slightly more formal manner. The
322 EBNF-like description in \in{definition}[def:IntendedNormal] captures
323 most of the intended structure (and generates a subset of \GHC's core
326 There are two things missing from this definition: cast expressions are
327 sometimes 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 top
381 level lambda abstractions (\italic{lambda}) define the input ports.
382 Lambda abstractions cannot appear anywhere else. The variable reference
383 in the body of the recursive let expression (\italic{toplet}) is the
384 output port. Most binders bound by the let expression define a
385 component instantiation (\italic{userapp}), where the input and output
386 ports are mapped to local signals (\italic{userarg}). Some of the others
387 use a built-in construction (\eg\ the \lam{case} expression) or call a
388 built-in function (\italic{builtinapp}) such as \lam{+} or \lam{map}.
389 For these, a 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
681 framework, whose implementation is not used to generate \VHDL. This is
682 either because it is no valid Cλash (like most list functions that need
683 recursion) or because a Cλash implementation would be unwanted (for the
684 addition operator, for example, we would rather use the \VHDL addition
685 operator to let the synthesis tool decide what kind of adder to use
686 instead of explicitly describing one in Cλash). \defref{built-in
689 These are functions like \lam{map}, \lam{hwor}, \lam{+} and \lam{length}.
691 For these functions, Cλash has a \emph{built-in hardware translation},
692 so calls to these functions can still be translated. Built-in functions
693 must have a valid Haskell implementation, of course, to allow
696 A \emph{user-defined} function is a function for which no built-in
697 translation is available and whose definition will thus need to be
698 translated to Cλash. \defref{user-defined function}
700 \subsubsection[sec:normalization:predicates]{Predicates}
701 Here, we define a number of predicates that can be used below to concisely
704 \emph{gvar(expr)} is true when \emph{expr} is a variable that references a
705 global variable. It is false when it references a local variable.
707 \emph{lvar(expr)} is the complement of \emph{gvar}; it is true when \emph{expr}
708 references a local variable, false when it references a global variable.
710 \emph{representable(expr)} is true when \emph{expr} is \emph{representable}.
712 \subsection[sec:normalization:uniq]{Binder uniqueness}
713 A common problem in transformation systems, is binder uniqueness. When not
714 considering this problem, it is easy to create transformations that mix up
715 bindings and cause name collisions. Take for example, the following core
719 (λa.λb.λc. a * b * c) x c
722 By applying β-reduction (see \in{section}[sec:normalization:beta]) once,
723 we can simplify this expression to:
729 Now, we have replaced the \lam{a} binder with a reference to the \lam{x}
730 binder. No harm done here. But note that we see multiple occurences of the
731 \lam{c} binder. The first is a binding occurence, to which the second refers.
732 The last, however refers to \emph{another} instance of \lam{c}, which is
733 bound somewhere outside of this expression. Now, if we would apply beta
734 reduction without taking heed of binder uniqueness, we would get:
740 This is obviously not what was supposed to happen! The root of this problem is
741 the reuse of binders: identical binders can be bound in different,
742 but overlapping scopes. Any variable reference in those
743 overlapping scopes then refers to the variable bound in the inner
744 (smallest) scope. There is not way to refer to the variable in the
745 outer scope. This effect is usually referred to as
746 \emph{shadowing}: when a binder is bound in a scope where the
747 binder already had a value, the inner binding is said to
748 \emph{shadow} the outer binding. In the example above, the \lam{c}
749 binder was bound outside of the expression and in the inner lambda
750 expression. Inside that lambda expression, only the inner \lam{c}
753 There are a number of ways to solve this. \small{GHC} has isolated this
754 problem to their binder substitution code, which performs \emph{deshadowing}
755 during its expression traversal. This means that any binding that shadows
756 another binding on a higher level is replaced by a new binder that does not
757 shadow any other binding. This non-shadowing invariant is enough to prevent
758 binder uniqueness problems in \small{GHC}.
760 In our transformation system, maintaining this non-shadowing invariant is
761 a bit harder to do (mostly due to implementation issues, the prototype
762 does not use \small{GHC}'s subsitution code). Also, the following points
766 \item Deshadowing does not guarantee overall uniqueness. For example, the
767 following (slightly contrived) expression shows the identifier \lam{x} bound in
768 two seperate places (and to different values), even though no shadowing
772 (let x = 1 in x) + (let x = 2 in x)
775 \item In our normal form (and the resulting \small{VHDL}), all binders
776 (signals) within the same function (entity) will end up in the same
777 scope. To allow this, all binders within the same function should be
780 \item When we know that all binders in an expression are unique, moving around
781 or removing a subexpression will never cause any binder conflicts. If we have
782 some way to generate fresh binders, introducing new subexpressions will not
783 cause any problems either. The only way to cause conflicts is thus to
784 duplicate an existing subexpression.
787 Given the above, our prototype maintains a unique binder invariant. This
788 means that in any given moment during normalization, all binders \emph{within
789 a single function} must be unique. To achieve this, we apply the following
792 \todo{Define fresh binders and unique supplies}
795 \item Before starting normalization, all binders in the function are made
796 unique. This is done by generating a fresh binder for every binder used. This
797 also replaces binders that did not cause any conflict, but it does ensure that
798 all binders within the function are generated by the same unique supply.
799 \refdef{fresh binder}
800 \item Whenever a new binder must be generated, we generate a fresh binder that
801 is guaranteed to be different from \emph{all binders generated so far}. This
802 can thus never introduce duplication and will maintain the invariant.
803 \item Whenever (a part of) an expression is duplicated (for example when
804 inlining), all binders in the expression are replaced with fresh binders
805 (using the same method as at the start of normalization). These fresh binders
806 can never introduce duplication, so this will maintain the invariant.
807 \item Whenever we move part of an expression around within the function, there
808 is no need to do anything special. There is obviously no way to introduce
809 duplication by moving expressions around. Since we know that each of the
810 binders is already unique, there is no way to introduce (incorrect) shadowing
814 \section{Transform passes}
815 In this section we describe the actual transforms.
817 Each transformation will be described informally first, explaining
818 the need for and goal of the transformation. Then, we will formally define
819 the transformation using the syntax introduced in
820 \in{section}[sec:normalization:transformation].
822 \subsection{General cleanup}
823 These transformations are general cleanup transformations, that aim to
824 make expressions simpler. These transformations usually clean up the
825 mess left behind by other transformations or clean up expressions to
826 expose new transformation opportunities for other transformations.
828 Most of these transformations are standard optimizations in other
829 compilers as well. However, in our compiler, most of these are not just
830 optimizations, but they are required to get our program into intended
834 \defref{substitution notation}
835 \startframedtext[width=8cm,background=box,frame=no]
836 \startalignment[center]
837 {\tfa Substitution notation}
841 In some of the transformations in this chapter, we need to perform
842 substitution on an expression. Substitution means replacing every
843 occurence of some expression (usually a variable reference) with
846 There have been a lot of different notations used in literature for
847 specifying substitution. The notation that will be used in this report
854 This means expression \lam{E} with all occurences of \lam{A} replaced
859 \subsubsection[sec:normalization:beta]{β-reduction}
860 β-reduction is a well known transformation from lambda calculus, where it is
861 the main reduction step. It reduces applications of lambda abstractions,
862 removing both the lambda abstraction and the application.
864 In our transformation system, this step helps to remove unwanted lambda
865 abstractions (basically all but the ones at the top level). Other
866 transformations (application propagation, non-representable inlining) make
867 sure that most lambda abstractions will eventually be reducable by
870 Note that β-reduction also works on type lambda abstractions and type
871 applications as well. This means the substitution below also works on
872 type variables, in the case that the binder is a type variable and teh
873 expression applied to is a type.
890 \transexample{beta}{β-reduction}{from}{to}
900 \transexample{beta-type}{β-reduction for type abstractions}{from}{to}
902 \subsubsection{Unused let binding removal}
903 This transformation removes let bindings that are never used.
904 Occasionally, \GHC's desugarer introduces some unused let bindings.
906 This normalization pass should really be not be necessary to get
907 into intended normal form (since the intended normal form
908 definition \refdef{intended normal form definition} does not
909 require that every binding is used), but in practice the
910 desugarer or simplifier emits some bindings that cannot be
911 normalized (e.g., calls to a
912 \hs{Control.Exception.Base.patError}) but are not used anywhere
913 either. To prevent the \VHDL\ generation from breaking on these
914 artifacts, this transformation removes them.
916 \todo{Do not use old-style numerals in transformations}
925 M \lam{ai} does not occur free in \lam{M}
926 ---------------------------- \lam{\forall j, 0 ≤ j ≤ n, j ≠ i} (\lam{ai} does not occur free in \lam{Ej})
952 \transexample{unusedlet}{Unused let binding removal}{from}{to}
954 \subsubsection{Empty let removal}
955 This transformation is simple: it removes recursive lets that have no bindings
956 (which usually occurs when unused let binding removal removes the last
959 Note that there is no need to define this transformation for
960 non-recursive lets, since they always contain exactly one binding.
979 \transexample{emptylet}{Empty let removal}{from}{to}
981 \subsubsection[sec:normalization:simplelet]{Simple let binding removal}
982 This transformation inlines simple let bindings, that bind some
983 binder to some other binder instead of a more complex expression (\ie\
986 This transformation is not needed to get an expression into intended
987 normal form (since these bindings are part of the intended normal
988 form), but makes the resulting \small{VHDL} a lot shorter.
990 \refdef{substitution notation}
1000 ----------------------------- \lam{b} is a variable reference
1001 letrec \lam{ai} ≠ \lam{b}
1014 \subsubsection{Cast propagation / simplification}
1015 This transform pushes casts down into the expression as far as
1016 possible. This transformation has been added to make a few
1017 specific corner cases work, but it is not clear yet if this
1018 transformation handles cast expressions completely or in the
1019 right way. See \in{section}[sec:normalization:castproblems].
1022 (let binds in E) ▶ T
1023 -------------------------
1024 let binds in (E ▶ T)
1033 -------------------------
1040 \subsubsection{Top level binding inlining}
1041 \refdef{top level binding}
1042 This transform takes simple top level bindings generated by the
1043 \small{GHC} compiler. \small{GHC} sometimes generates very simple
1044 \quote{wrapper} bindings, which are bound to just a variable
1045 reference, or contain just a (partial) function appliation with
1046 the type and dictionary arguments filled in (such as the
1047 \lam{(+)} in the example below).
1049 Note that this transformation is completely optional. It is not
1050 required to get any function into intended normal form, but it does help making
1051 the resulting VHDL output easier to read (since it removes components
1052 that do not add any real structure, but do hide away operations and
1053 cause extra clutter).
1055 This transform takes any top level binding generated by \GHC,
1056 whose normalized form contains only a single let binding.
1059 x = λa0 ... λan.let y = E in y
1062 -------------------------------------- \lam{x} is generated by the compiler
1063 λa0 ... λan.let y = E in y
1067 (+) :: Word -> Word -> Word
1068 (+) = GHC.Num.(+) @Word \$dNum
1073 GHC.Num.(+) @ Alu.Word \$dNum a b
1076 \transexample{toplevelinline}{Top level binding inlining}{from}{to}
1078 \in{Example}[ex:trans:toplevelinline] shows a typical application of
1079 the addition operator generated by \GHC. The type and dictionary
1080 arguments used here are described in
1081 \in{Section}[section:prototype:polymorphism].
1083 Without this transformation, there would be a \lam{(+)} entity
1084 in the \VHDL\ which would just add its inputs. This generates a
1085 lot of overhead in the \VHDL, which is particularly annoying
1086 when browsing the generated RTL schematic (especially since most
1087 non-alphanumerics, like all characters in \lam{(+)}, are not
1088 allowed in \VHDL\ architecture names\footnote{Technically, it is
1089 allowed to use non-alphanumerics when using extended
1090 identifiers, but it seems that none of the tooling likes
1091 extended identifiers in filenames, so it effectively does not
1092 work.}, so the entity would be called \quote{w7aA7f} or
1093 something similarly meaningless and autogenerated).
1095 \subsection{Program structure}
1096 These transformations are aimed at normalizing the overall structure
1097 into the intended form. This means ensuring there is a lambda abstraction
1098 at the top for every argument (input port or current state), putting all
1099 of the other value definitions in let bindings and making the final
1100 return value a simple variable reference.
1102 \subsubsection[sec:normalization:eta]{η-abstraction}
1103 This transformation makes sure that all arguments of a function-typed
1104 expression are named, by introducing lambda expressions. When combined with
1105 β-reduction and non-representable binding inlining, all function-typed
1106 expressions should be lambda abstractions or global identifiers.
1110 -------------- \lam{E} does not occur on a function position in an application
1111 λx.E x \lam{E} is not a lambda abstraction.
1121 foo = λa.λx.(case a of
1126 \transexample{eta}{η-abstraction}{from}{to}
1128 \subsubsection[sec:normalization:appprop]{Application propagation}
1129 This transformation is meant to propagate application expressions downwards
1130 into expressions as far as possible. This allows partial applications inside
1131 expressions to become fully applied and exposes new transformation
1132 opportunities for other transformations (like β-reduction and
1135 Since all binders in our expression are unique (see
1136 \in{section}[sec:normalization:uniq]), there is no risk that we will
1137 introduce unintended shadowing by moving an expression into a lower
1138 scope. Also, since only move expression into smaller scopes (down into
1139 our expression), there is no risk of moving a variable reference out
1140 of the scope in which it is defined.
1143 (letrec binds in E) M
1144 ------------------------
1164 \transexample{appproplet}{Application propagation for a let expression}{from}{to}
1192 \transexample{apppropcase}{Application propagation for a case expression}{from}{to}
1194 \subsubsection[sec:normalization:letrecurse]{Let recursification}
1195 This transformation makes all non-recursive lets recursive. In the
1196 end, we want a single recursive let in our normalized program, so all
1197 non-recursive lets can be converted. This also makes other
1198 transformations simpler: they only need to be specified for recursive
1199 let expressions (and simply will not apply to non-recursive let
1200 expressions until this transformation has been applied).
1207 ------------------------------------------
1214 \subsubsection{Let flattening}
1215 This transformation puts nested lets in the same scope, by lifting the
1216 binding(s) of the inner let into the outer let. Eventually, this will
1217 cause all let bindings to appear in the same scope.
1219 This transformation only applies to recursive lets, since all
1220 non-recursive lets will be made recursive (see
1221 \in{section}[sec:normalization:letrecurse]).
1223 Since we are joining two scopes together, there is no risk of moving a
1224 variable reference out of the scope where it is defined.
1230 ai = (letrec bindings in M)
1235 ------------------------------------------
1270 \transexample{letflat}{Let flattening}{from}{to}
1272 \subsubsection{Return value simplification}
1273 This transformation ensures that the return value of a function is always a
1274 simple local variable reference.
1276 This transformation only applies to the entire body of a
1277 function instead of any subexpression in a function. This is
1278 achieved by the contexts, like \lam{x = E}, though this is
1279 strictly not correct (you could read this as "if there is any
1280 function \lam{x} that binds \lam{E}, any \lam{E} can be
1281 transformed, while we only mean the \lam{E} that is bound by
1284 Note that the return value is not simplified if its not
1285 representable. Otherwise, this would cause a direct loop with
1286 the inlining of unrepresentable bindings. If the return value is
1287 not representable because it has a function type, η-abstraction
1288 should make sure that this transformation will eventually apply.
1289 If the value is not representable for other reasons, the
1290 function result itself is not representable, meaning this
1291 function is not translatable anyway.
1294 x = E \lam{E} is representable
1295 ~ \lam{E} is not a lambda abstraction
1296 E \lam{E} is not a let expression
1297 --------------------------- \lam{E} is not a local variable reference
1302 x = λv0 ... λvn.E \lam{E} is representable
1303 ~ \lam{E} is not a lambda abstraction
1304 E \lam{E} is not a let expression
1305 --------------------------- \lam{E} is not a local variable reference
1310 x = λv0 ... λvn.let ... in E
1311 ~ \lam{E} is representable
1312 E \lam{E} is not a local variable reference
1313 -----------------------------
1322 x = letrec x = add 1 2 in x
1325 \transexample{retvalsimpl}{Return value simplification}{from}{to}
1327 \todo{More examples}
1329 \subsection[sec:normalization:argsimpl]{Representable arguments simplification}
1330 This section contains just a single transformation that deals with
1331 representable arguments in applications. Non-representable arguments are
1332 handled by the transformations in
1333 \in{section}[sec:normalization:nonrep].
1335 This transformation ensures that all representable arguments will become
1336 references to local variables. This ensures they will become references
1337 to local signals in the resulting \small{VHDL}, which is required due to
1338 limitations in the component instantiation code in \VHDL\ (one can only
1339 assign a signal or constant to an input port). By ensuring that all
1340 arguments are always simple variable references, we always have a signal
1341 available to map to the input ports.
1343 To reduce a complex expression to a simple variable reference, we create
1344 a new let expression around the application, which binds the complex
1345 expression to a new variable. The original function is then applied to
1348 \refdef{global variable}
1349 Note that references to \emph{global variables} (like a top level
1350 function without arguments, but also an argumentless dataconstructors
1351 like \lam{True}) are also simplified. Only local variables generate
1352 signals in the resulting architecture. Even though argumentless
1353 dataconstructors generate constants in generated \VHDL\ code and could be
1354 mapped to an input port directly, they are still simplified to make the
1355 normal form more regular.
1357 \refdef{representable}
1360 -------------------- \lam{N} is representable
1361 letrec x = N in M x \lam{N} is not a local variable reference
1363 \refdef{local variable}
1370 letrec x = add a 1 in add x 1
1373 \transexample{argsimpl}{Argument simplification}{from}{to}
1375 \subsection[sec:normalization:built-ins]{Built-in functions}
1376 This section deals with (arguments to) built-in functions. In the
1377 intended normal form definition\refdef{intended normal form definition}
1378 we can see that there are three sorts of arguments a built-in function
1382 \item A representable local variable reference. This is the most
1383 common argument to any function. The argument simplification
1384 transformation described in \in{section}[sec:normalization:argsimpl]
1385 makes sure that \emph{any} representable argument to \emph{any}
1386 function (including built-in functions) is turned into a local variable
1388 \item (A partial application of) a top level function (either built-in on
1389 user-defined). The function extraction transformation described in
1390 this section takes care of turning every functiontyped argument into
1391 (a partial application of) a top level function.
1392 \item Any expression that is not representable and does not have a
1393 function type. Since these can be any expression, there is no
1394 transformation needed. Note that this category is exactly all
1395 expressions that are not transformed by the transformations for the
1396 previous two categories. This means that \emph{any} core expression
1397 that is used as an argument to a built-in function will be either
1398 transformed into one of the above categories, or end up in this
1399 categorie. In any case, the result is in normal form.
1402 As noted, the argument simplification will handle any representable
1403 arguments to a built-in function. The following transformation is needed
1404 to handle non-representable arguments with a function type, all other
1405 non-representable arguments do not need any special handling.
1407 \subsubsection[sec:normalization:funextract]{Function extraction}
1408 This transform deals with function-typed arguments to built-in
1410 Since built-in functions cannot be specialized (see
1411 \in{section}[sec:normalization:specialize]) to remove the arguments,
1412 these arguments are extracted into a new global function instead. In
1413 other words, we create a new top level function that has exactly the
1414 extracted argument as its body. This greatly simplifies the
1415 translation rules needed for built-in functions, since they only need
1416 to handle (partial applications of) top level functions.
1418 Any free variables occuring in the extracted arguments will become
1419 parameters to the new global function. The original argument is replaced
1420 with a reference to the new function, applied to any free variables from
1421 the original argument.
1423 This transformation is useful when applying higher-order built-in functions
1424 like \hs{map} to a lambda abstraction, for example. In this case, the code
1425 that generates \small{VHDL} for \hs{map} only needs to handle top level functions and
1426 partial applications, not any other expression (such as lambda abstractions or
1427 even more complicated expressions).
1430 M N \lam{M} is (a partial aplication of) a built-in function.
1431 --------------------- \lam{f0 ... fn} are all free local variables of \lam{N}
1432 M (x f0 ... fn) \lam{N :: a -> b}
1433 ~ \lam{N} is not a (partial application of) a top level function
1438 addList = λb.λxs.map (λa . add a b) xs
1442 addList = λb.λxs.map (f b) xs
1447 \transexample{funextract}{Function extraction}{from}{to}
1449 Note that the function \lam{f} will still need normalization after
1452 \subsection{Case normalisation}
1453 The transformations in this section ensure that case statements end up
1456 \subsubsection{Scrutinee simplification}
1457 This transform ensures that the scrutinee of a case expression is always
1458 a simple variable reference.
1463 ----------------- \lam{E} is not a local variable reference
1482 \transexample{letflat}{Case normalisation}{from}{to}
1486 \defref{wild binders}
1487 \startframedtext[width=7cm,background=box,frame=no]
1488 \startalignment[center]
1492 In a functional expression, a \emph{wild binder} refers to any
1493 binder that is never referenced. This means that even though it
1494 will be bound to a particular value, that value is never used.
1496 The Haskell syntax offers the underscore as a wild binder that
1497 cannot even be referenced (It can be seen as introducing a new,
1498 anonymous, binder everytime it is used).
1500 In these transformations, the term wild binder will sometimes be
1501 used to indicate that a binder must not be referenced.
1505 \subsubsection{Scrutinee binder removal}
1506 This transformation removes (or rather, makes wild) the binder to
1507 which the scrutinee is bound after evaluation. This is done by
1508 replacing the bndr with the scrutinee in all alternatives. To prevent
1509 duplication of work, this transformation is only applied when the
1510 scrutinee is already a simple variable reference (but the previous
1511 transformation ensures this will eventually be the case). The
1512 scrutinee binder itself is replaced by a wild binder (which is no
1515 Note that one could argue that this transformation can change the
1516 meaning of the Core expression. In the regular Core semantics, a case
1517 expression forces the evaluation of its scrutinee and can be used to
1518 implement strict evaluation. However, in the generated \VHDL,
1519 evaluation is always strict. So the semantics we assign to the Core
1520 expression (which differ only at this particular point), this
1521 transformation is completely valid.
1526 ----------------- \lam{x} is a local variable reference
1543 \transexample{scrutbndrremove}{Scrutinee binder removal}{from}{to}
1545 \subsubsection{Case normalization}
1546 This transformation ensures that all case expressions get a form
1547 that is allowed by the intended normal form. This means they
1551 \item An extractor case with a single alternative that picks a field
1552 from a datatype, \eg\ \lam{case x of (a, b) -> a}.
1553 \item A selector case with multiple alternatives and only wild binders, that
1554 makes a choice between expressions based on the constructor of another
1555 expression, \eg\ \lam{case x of Low -> a; High -> b}.
1558 For an arbitrary case, that has \lam{n} alternatives, with
1559 \lam{m} binders in each alternatives, this will result in \lam{m
1560 * n} extractor case expression to get at each variable, \lam{n}
1561 let bindings for each of the alternatives' value and a single
1562 selector case to select the right value out of these.
1564 Technically, the defintion of this transformation would require
1565 that the constructor for every alternative has exactly the same
1566 amount (\lam{m}) of arguments, but of course this transformation
1567 also applies when this is not the case.
1571 C0 v0,0 ... v0,m -> E0
1573 Cn vn,0 ... vn,m -> En
1574 --------------------------------------------------- \lam{\forall i \forall j, 0 ≤ i ≤ n, 0 ≤ i < m} (\lam{wi,j} is a wild (unused) binder)
1575 letrec The case expression is not an extractor case
1576 v0,0 = case E of C0 x0,0 .. x0,m -> x0,0 The case expression is not a selector case
1578 v0,m = case E of C0 x0,0 .. x0,m -> x0,m
1580 vn,m = case E of Cn xn,0 .. xn,m -> xn,m
1586 C0 w0,0 ... w0,m -> y0
1588 Cn wn,0 ... wn,m -> yn
1591 Note that this transformation applies to case expressions with any
1592 scrutinee. If the scrutinee is a complex expression, this might
1593 result in duplication of work (hardware). An extra condition to
1594 only apply this transformation when the scrutinee is already
1595 simple (effectively causing this transformation to be only
1596 applied after the scrutinee simplification transformation) might
1615 \transexample{selcasesimpl}{Selector case simplification}{from}{to}
1623 b = case a of (,) b c -> b
1624 c = case a of (,) b c -> c
1631 \transexample{excasesimpl}{Extractor case simplification}{from}{to}
1633 \refdef{selector case}
1634 In \in{example}[ex:trans:excasesimpl] the case expression is expanded
1635 into multiple case expressions, including a pretty useless expression
1636 (that is neither a selector or extractor case). This case can be
1637 removed by the Case removal transformation in
1638 \in{section}[sec:transformation:caseremoval].
1640 \subsubsection[sec:transformation:caseremoval]{Case removal}
1641 This transform removes any case expression with a single alternative and
1642 only wild binders.\refdef{wild binders}
1644 These "useless" case expressions are usually leftovers from case simplification
1645 on extractor case (see the previous example).
1650 ---------------------- \lam{\forall i, 0 ≤ i ≤ m} (\lam{vi} does not occur free in E)
1663 \transexample{caserem}{Case removal}{from}{to}
1665 \subsection[sec:normalization:nonrep]{Removing unrepresentable values}
1666 The transformations in this section are aimed at making all the
1667 values used in our expression representable. There are two main
1668 transformations that are applied to \emph{all} unrepresentable let
1669 bindings and function arguments. These are meant to address three
1670 different kinds of unrepresentable values: polymorphic values,
1671 higher-order values and literals. The transformation are described
1672 generically: they apply to all non-representable values. However,
1673 non-representable values that do not fall into one of these three
1674 categories will be moved around by these transformations but are
1675 unlikely to completely disappear. They usually mean the program was not
1676 valid in the first place, because unsupported types were used (for
1677 example, a program using strings).
1679 Each of these three categories will be detailed below, followed by the
1680 actual transformations.
1682 \subsubsection{Removing Polymorphism}
1683 As noted in \in{section}[sec:prototype:polymporphism],
1684 polymorphism is made explicit in Core through type and
1685 dictionary arguments. To remove the polymorphism from a
1686 function, we can simply specialize the polymorphic function for
1687 the particular type applied to it. The same goes for dictionary
1688 arguments. To remove polymorphism from let bound values, we
1689 simply inline the let bindings that have a polymorphic type,
1690 which should (eventually) make sure that the polymorphic
1691 expression is applied to a type and/or dictionary, which can
1692 then be removed by β-reduction (\in{section}[sec:normalization:beta]).
1694 Since both type and dictionary arguments are not representable,
1695 \refdef{representable}
1696 the non-representable argument specialization and
1697 non-representable let binding inlining transformations below
1698 take care of exactly this.
1700 There is one case where polymorphism cannot be completely
1701 removed: built-in functions are still allowed to be polymorphic
1702 (Since we have no function body that we could properly
1703 specialize). However, the code that generates \VHDL\ for built-in
1704 functions knows how to handle this, so this is not a problem.
1706 \subsubsection[sec:normalization:defunctionalization]{Defunctionalization}
1707 These transformations remove higher-order expressions from our
1708 program, making all values first-order. The approach used for
1709 defunctionalization uses a combination of specialization, inlining and
1710 some cleanup transformations, was also proposed in parallel research
1711 by Neil Mitchell \cite[mitchell09].
1713 Higher order values are always introduced by lambda abstractions, none
1714 of the other Core expression elements can introduce a function type.
1715 However, other expressions can \emph{have} a function type, when they
1716 have a lambda expression in their body.
1718 For example, the following expression is a higher-order expression
1719 that is not a lambda expression itself:
1721 \refdef{id function}
1728 The reference to the \lam{id} function shows that we can introduce a
1729 higher-order expression in our program without using a lambda
1730 expression directly. However, inside the definition of the \lam{id}
1731 function, we can be sure that a lambda expression is present.
1733 Looking closely at the definition of our normal form in
1734 \in{section}[sec:normalization:intendednormalform], we can see that
1735 there are three possibilities for higher-order values to appear in our
1736 intended normal form:
1739 \item[item:toplambda] Lambda abstractions can appear at the highest level of a
1740 top level function. These lambda abstractions introduce the
1741 arguments (input ports / current state) of the function.
1742 \item[item:built-inarg] (Partial applications of) top level functions can appear as an
1743 argument to a built-in function.
1744 \item[item:completeapp] (Partial applications of) top level functions can appear in
1745 function position of an application. Since a partial application
1746 cannot appear anywhere else (except as built-in function arguments),
1747 all partial applications are applied, meaning that all applications
1748 will become complete applications. However, since application of
1749 arguments happens one by one, in the expression:
1753 the subexpression \lam{f 1} has a function type. But this is
1754 allowed, since it is inside a complete application.
1757 We will take a typical function with some higher-order values as an
1758 example. The following function takes two arguments: a \lam{Bit} and a
1759 list of numbers. Depending on the first argument, each number in the
1760 list is doubled, or the list is returned unmodified. For the sake of
1761 the example, no polymorphism is shown. In reality, at least map would
1765 λy.let double = λx. x + x in
1771 This example shows a number of higher-order values that we cannot
1772 translate to \VHDL\ directly. The \lam{double} binder bound in the let
1773 expression has a function type, as well as both of the alternatives of
1774 the case expression. The first alternative is a partial application of
1775 the \lam{map} built-in function, whereas the second alternative is a
1778 To reduce all higher-order values to one of the above items, a number
1779 of transformations we have already seen are used. The η-abstraction
1780 transformation from \in{section}[sec:normalization:eta] ensures all
1781 function arguments are introduced by lambda abstraction on the highest
1782 level of a function. These lambda arguments are allowed because of
1783 \in{item}[item:toplambda] above. After η-abstraction, our example
1784 becomes a bit bigger:
1787 λy.λq.(let double = λx. x + x in
1794 η-abstraction also introduces extra applications (the application of
1795 the let expression to \lam{q} in the above example). These
1796 applications can then propagated down by the application propagation
1797 transformation (\in{section}[sec:normalization:appprop]). In our
1798 example, the \lam{q} and \lam{r} variable will be propagated into the
1799 let expression and then into the case expression:
1802 λy.λq.let double = λx. x + x in
1808 This propagation makes higher-order values become applied (in
1809 particular both of the alternatives of the case now have a
1810 representable type). Completely applied top level functions (like the
1811 first alternative) are now no longer invalid (they fall under
1812 \in{item}[item:completeapp] above). (Completely) applied lambda
1813 abstractions can be removed by β-abstraction. For our example,
1814 applying β-abstraction results in the following:
1817 λy.λq.let double = λx. x + x in
1823 As you can see in our example, all of this moves applications towards
1824 the higher-order values, but misses higher-order functions bound by
1825 let expressions. The applications cannot be moved towards these values
1826 (since they can be used in multiple places), so the values will have
1827 to be moved towards the applications. This is achieved by inlining all
1828 higher-order values bound by let applications, by the
1829 non-representable binding inlining transformation below. When applying
1830 it to our example, we get the following:
1834 Low -> map (λx. x + x) q
1838 We have nearly eliminated all unsupported higher-order values from this
1839 expressions. The one that is remaining is the first argument to the
1840 \lam{map} function. Having higher-order arguments to a built-in
1841 function like \lam{map} is allowed in the intended normal form, but
1842 only if the argument is a (partial application) of a top level
1843 function. This is easily done by introducing a new top level function
1844 and put the lambda abstraction inside. This is done by the function
1845 extraction transformation from
1846 \in{section}[sec:normalization:funextract].
1854 This also introduces a new function, that we have called \lam{func}:
1860 Note that this does not actually remove the lambda, but now it is a
1861 lambda at the highest level of a function, which is allowed in the
1862 intended normal form.
1864 There is one case that has not been discussed yet. What if the
1865 \lam{map} function in the example above was not a built-in function
1866 but a user-defined function? Then extracting the lambda expression
1867 into a new function would not be enough, since user-defined functions
1868 can never have higher-order arguments. For example, the following
1869 expression shows an example:
1872 twice :: (Word -> Word) -> Word -> Word
1873 twice = λf.λa.f (f a)
1875 main = λa.app (λx. x + x) a
1878 This example shows a function \lam{twice} that takes a function as a
1879 first argument and applies that function twice to the second argument.
1880 Again, we have made the function monomorphic for clarity, even though
1881 this function would be a lot more useful if it was polymorphic. The
1882 function \lam{main} uses \lam{twice} to apply a lambda epression twice.
1884 When faced with a user defined function, a body is available for that
1885 function. This means we could create a specialized version of the
1886 function that only works for this particular higher-order argument
1887 (\ie, we can just remove the argument and call the specialized
1888 function without the argument). This transformation is detailed below.
1889 Applying this transformation to the example gives:
1892 twice' :: Word -> Word
1893 twice' = λb.(λf.λa.f (f a)) (λx. x + x) b
1898 The \lam{main} function is now in normal form, since the only
1899 higher-order value there is the top level lambda expression. The new
1900 \lam{twice'} function is a bit complex, but the entire original body
1901 of the original \lam{twice} function is wrapped in a lambda
1902 abstraction and applied to the argument we have specialized for
1903 (\lam{λx. x + x}) and the other arguments. This complex expression can
1904 fortunately be effectively reduced by repeatedly applying β-reduction:
1907 twice' :: Word -> Word
1908 twice' = λb.(b + b) + (b + b)
1911 This example also shows that the resulting normal form might not be as
1912 efficient as we might hope it to be (it is calculating \lam{(b + b)}
1913 twice). This is discussed in more detail in
1914 \in{section}[sec:normalization:duplicatework].
1916 \subsubsection{Literals}
1917 There are a limited number of literals available in Haskell and Core.
1918 \refdef{enumerated types} When using (enumerating) algebraic
1919 datatypes, a literal is just a reference to the corresponding data
1920 constructor, which has a representable type (the algebraic datatype)
1921 and can be translated directly. This also holds for literals of the
1922 \hs{Bool} Haskell type, which is just an enumerated type.
1924 There is, however, a second type of literal that does not have a
1925 representable type: integer literals. Cλash supports using integer
1926 literals for all three integer types supported (\hs{SizedWord},
1927 \hs{SizedInt} and \hs{RangedWord}). This is implemented using
1928 Haskell's \hs{Num} type class, which offers a \hs{fromInteger} method
1929 that converts any \hs{Integer} to the Cλash datatypes.
1931 When \GHC\ sees integer literals, it will automatically insert calls to
1932 the \hs{fromInteger} method in the resulting Core expression. For
1933 example, the following expression in Haskell creates a 32 bit unsigned
1934 word with the value 1. The explicit type signature is needed, since
1935 there is no context for \GHC\ to determine the type from otherwise.
1941 This Haskell code results in the following Core expression:
1944 fromInteger @(SizedWord D32) \$dNum (smallInteger 10)
1947 The literal 10 will have the type \lam{GHC.Prim.Int\#}, which is
1948 converted into an \lam{Integer} by \lam{smallInteger}. Finally, the
1949 \lam{fromInteger} function will finally convert this into a
1950 \lam{SizedWord D32}.
1952 Both the \lam{GHC.Prim.Int\#} and \lam{Integer} types are not
1953 representable, and cannot be translated directly. Fortunately, there
1954 is no need to translate them, since \lam{fromInteger} is a built-in
1955 function that knows how to handle these values. However, this does
1956 require that the \lam{fromInteger} function is directly applied to
1957 these non-representable literal values, otherwise errors will occur.
1958 For example, the following expression is not in the intended normal
1959 form, since one of the let bindings has an unrepresentable type
1963 let l = smallInteger 10 in fromInteger @(SizedWord D32) \$dNum l
1966 By inlining these let-bindings, we can ensure that unrepresentable
1967 literals bound by a let binding end up in an application of the
1968 appropriate built-in function, where they are allowed. Since it is
1969 possible that the application of that function is in a different
1970 function than the definition of the literal value, we will always need
1971 to specialize away any unrepresentable literals that are used as
1972 function arguments. The following two transformations do exactly this.
1974 \subsubsection{Non-representable binding inlining}
1975 This transform inlines let bindings that are bound to a
1976 non-representable value. Since we can never generate a signal
1977 assignment for these bindings (we cannot declare a signal assignment
1978 with a non-representable type, for obvious reasons), we have no choice
1979 but to inline the binding to remove it.
1981 As we have seen in the previous sections, inlining these bindings
1982 solves (part of) the polymorphism, higher-order values and
1983 unrepresentable literals in an expression.
1985 \refdef{substitution notation}
1995 -------------------------- \lam{Ei} has a non-representable type.
1997 a0 = E0 [ai=>Ei] \vdots
1998 ai-1 = Ei-1 [ai=>Ei]
1999 ai+1 = Ei+1 [ai=>Ei]
2018 x = fromInteger (smallInteger 10)
2020 (λb -> add b 1) (add 1 x)
2023 \transexample{nonrepinline}{Nonrepresentable binding inlining}{from}{to}
2025 \subsubsection[sec:normalization:specialize]{Function specialization}
2026 This transform removes arguments to user-defined functions that are
2027 not representable at runtime. This is done by creating a
2028 \emph{specialized} version of the function that only works for one
2029 particular value of that argument (in other words, the argument can be
2032 Specialization means to create a specialized version of the called
2033 function, with one argument already filled in. As a simple example, in
2034 the following program (this is not actual Core, since it directly uses
2035 a literal with the unrepresentable type \lam{GHC.Prim.Int\#}).
2042 We could specialize the function \lam{f} against the literal argument
2043 1, with the following result:
2050 In some way, this transformation is similar to β-reduction, but it
2051 operates across function boundaries. It is also similar to
2052 non-representable let binding inlining above, since it sort of
2053 \quote{inlines} an expression into a called function.
2055 Special care must be taken when the argument has any free variables.
2056 If this is the case, the original argument should not be removed
2057 completely, but replaced by all the free variables of the expression.
2058 In this way, the original expression can still be evaluated inside the
2061 To prevent us from propagating the same argument over and over, a
2062 simple local variable reference is not propagated (since is has
2063 exactly one free variable, itself, we would only replace that argument
2066 This shows that any free local variables that are not runtime
2067 representable cannot be brought into normal form by this transform. We
2068 rely on an inlining or β-reduction transformation to replace such a
2069 variable with an expression we can propagate again.
2074 x Y0 ... Yi ... Yn \lam{Yi} is not representable
2075 --------------------------------------------- \lam{Yi} is not a local variable reference
2076 x' Y0 ... Yi-1 f0 ... fm Yi+1 ... Yn \lam{f0 ... fm} are all free local vars of \lam{Yi}
2077 ~ \lam{T0 ... Tn} are the types of \lam{Y0 ... Yn}
2078 x' = λ(y0 :: T0) ... λ(yi-1 :: Ty-1).
2080 λ(yi+1 :: Ty+1) ... λ(yn :: Tn).
2081 E y0 ... yi-1 Yi yi+1 ... yn
2084 This is a bit of a complex transformation. It transforms an
2085 application of the function \lam{x}, where one of the arguments
2086 (\lam{Y_i}) is not representable. A new
2087 function \lam{x'} is created that wraps the body of the old function.
2088 The body of the new function becomes a number of nested lambda
2089 abstractions, one for each of the original arguments that are left
2092 The ith argument is replaced with the free variables of
2093 \lam{Y_i}. Note that we reuse the same binders as those used in
2094 \lam{Y_i}, since we can then just use \lam{Y_i} inside the new
2095 function body and have all of the variables it uses be in scope.
2097 The argument that we are specializing for, \lam{Y_i}, is put inside
2098 the new function body. The old function body is applied to it. Since
2099 we use this new function only in place of an application with that
2100 particular argument \lam{Y_i}, behaviour should not change.
2102 Note that the types of the arguments of our new function are taken
2103 from the types of the \emph{actual} arguments (\lam{T0 ... Tn}). This
2104 means that any polymorphism in the arguments is removed, even when the
2105 corresponding explicit type lambda is not removed
2108 \todo{Examples. Perhaps reference the previous sections}
2110 \section{Unsolved problems}
2111 The above system of transformations has been implemented in the prototype
2112 and seems to work well to compile simple and more complex examples of
2113 hardware descriptions. \todo{Ref christiaan?} However, this normalization
2114 system has not seen enough review and work to be complete and work for
2115 every Core expression that is supplied to it. A number of problems
2116 have already been identified and are discussed in this section.
2118 \subsection[sec:normalization:duplicatework]{Work duplication}
2119 A possible problem of β-reduction is that it could duplicate work.
2120 When the expression applied is not a simple variable reference, but
2121 requires calculation and the binder the lambda abstraction binds to
2122 is used more than once, more hardware might be generated than strictly
2125 As an example, consider the expression:
2131 When applying β-reduction to this expression, we get:
2137 which of course calculates \lam{(a * b)} twice.
2139 A possible solution to this would be to use the following alternative
2140 transformation, which is of course no longer normal β-reduction. The
2141 followin transformation has not been tested in the prototype, but is
2142 given here for future reference:
2150 This does not seem like much of an improvement, but it does get rid of
2151 the lambda expression (and the associated higher-order value), while
2152 at the same time introducing a new let binding. Since the result of
2153 every application or case expression must be bound by a let expression
2154 in the intended normal form anyway, this is probably not a problem. If
2155 the argument happens to be a variable reference, then simple let
2156 binding removal (\in{section}[sec:normalization:simplelet]) will
2157 remove it, making the result identical to that of the original
2158 β-reduction transformation.
2160 When also applying argument simplification to the above example, we
2161 get the following expression:
2169 Looking at this, we could imagine an alternative approach: create a
2170 transformation that removes let bindings that bind identical values.
2171 In the above expression, the \lam{y} and \lam{z} variables could be
2172 merged together, resulting in the more efficient expression:
2175 let y = (a * b) in y + y
2178 \subsection[sec:normalization:non-determinism]{Non-determinism}
2179 As an example, again consider the following expression:
2185 We can apply both β-reduction (\in{section}[sec:normalization:beta])
2186 as well as argument simplification
2187 (\in{section}[sec:normalization:argsimpl]) to this expression.
2189 When applying argument simplification first and then β-reduction, we
2190 get the following expression:
2193 let y = (a * b) in y + y
2196 When applying β-reduction first and then argument simplification, we
2197 get the following expression:
2205 As you can see, this is a different expression. This means that the
2206 order of expressions, does in fact change the resulting normal form,
2207 which is something that we would like to avoid. In this particular
2208 case one of the alternatives is even clearly more efficient, so we
2209 would of course like the more efficient form to be the normal form.
2211 For this particular problem, the solutions for duplication of work
2212 seem from the previous section seem to fix the determinism of our
2213 transformation system as well. However, it is likely that there are
2214 other occurences of this problem.
2216 \subsection[sec:normalization:castproblems]{Casts}
2217 We do not fully understand the use of cast expressions in Core, so
2218 there are probably expressions involving cast expressions that cannot
2219 be brought into intended normal form by this transformation system.
2221 The uses of casts in the core system should be investigated more and
2222 transformations will probably need updating to handle them in all
2225 \subsection[sec:normalization:stateproblems]{Normalization of stateful descriptions}
2226 Currently, the intended normal form definition\refdef{intended
2227 normal form definition} offers enough freedom to describe all
2228 valid stateful descriptions, but is not limiting enough. It is
2229 possible to write descriptions which are in intended normal
2230 form, but cannot be translated into \VHDL\ in a meaningful way
2231 (\eg, a function that swaps two substates in its result, or a
2232 function that changes a substate itself instead of passing it to
2235 It is now up to the programmer to not do anything funny with
2236 these state values, whereas the normalization just tries not to
2237 mess up the flow of state values. In practice, there are
2238 situations where a Core program that \emph{could} be a valid
2239 stateful description is not translateable by the prototype. This
2240 most often happens when statefulness is mixed with pattern
2241 matching, causing a state input to be unpacked multiple times or
2242 be unpacked and repacked only in some of the code paths.
2244 Without going into detail about the exact problems (of which
2245 there are probably more than have shown up so far), it seems
2246 unlikely that these problems can be solved entirely by just
2247 improving the \VHDL\ state generation in the final stage. The
2248 normalization stage seems the best place to apply the rewriting
2249 needed to support more complex stateful descriptions. This does
2250 of course mean that the intended normal form definition must be
2251 extended as well to be more specific about how state handling
2252 should look like in normal form.
2253 \in{Section}[sec:prototype:statelimits] already contains a
2254 tight description of the limitations on the use of state
2255 variables, which could be adapted into the intended normal form.
2257 \section[sec:normalization:properties]{Provable properties}
2258 When looking at the system of transformations outlined above, there are a
2259 number of questions that we can ask ourselves. The main question is of course:
2260 \quote{Does our system work as intended?}. We can split this question into a
2261 number of subquestions:
2264 \item[q:termination] Does our system \emph{terminate}? Since our system will
2265 keep running as long as transformations apply, there is an obvious risk that
2266 it will keep running indefinitely. This typically happens when one
2267 transformation produces a result that is transformed back to the original
2268 by another transformation, or when one or more transformations keep
2269 expanding some expression.
2270 \item[q:soundness] Is our system \emph{sound}? Since our transformations
2271 continuously modify the expression, there is an obvious risk that the final
2272 normal form will not be equivalent to the original program: its meaning could
2274 \item[q:completeness] Is our system \emph{complete}? Since we have a complex
2275 system of transformations, there is an obvious risk that some expressions will
2276 not end up in our intended normal form, because we forgot some transformation.
2277 In other words: does our transformation system result in our intended normal
2278 form for all possible inputs?
2279 \item[q:determinism] Is our system \emph{deterministic}? Since we have defined
2280 no particular order in which the transformation should be applied, there is an
2281 obvious risk that different transformation orderings will result in
2282 \emph{different} normal forms. They might still both be intended normal forms
2283 (if our system is \emph{complete}) and describe correct hardware (if our
2284 system is \emph{sound}), so this property is less important than the previous
2285 three: the translator would still function properly without it.
2288 Unfortunately, the final transformation system has only been
2289 developed in the final part of the research, leaving no more time
2290 for verifying these properties. In fact, it is likely that the
2291 current transformation system still violates some of these
2292 properties in some cases (see
2293 \in{section}[sec:normalization:non-determinism] and
2294 \in{section}[sec:normalization:stateproblems]) and should be improved (or
2295 extra conditions on the input hardware descriptions should be formulated).
2297 This is most likely the case with the completeness and determinism
2298 properties, perhaps also the termination property. The soundness
2299 property probably holds, since it is easier to manually verify (each
2300 transformation can be reviewed separately).
2302 Even though no complete proofs have been made, some ideas for
2303 possible proof strategies are shown below.
2305 \subsection{Graph representation}
2306 Before looking into how to prove these properties, we will look at
2307 transformation systems from a graph perspective. We will first define
2308 the graph view and then illustrate it using a simple example from lambda
2309 calculus (which is a different system than the Cλash normalization
2310 system). The nodes of the graph are all possible Core expressions. The
2311 (directed) edges of the graph are transformations. When a transformation
2312 α applies to an expression \lam{A} to produce an expression \lam{B}, we
2313 add an edge from the node for \lam{A} to the node for \lam{B}, labeled
2316 \startuseMPgraphic{TransformGraph}
2320 newCircle.a(btex \lam{(λx.λy. (+) x y) 1} etex);
2321 newCircle.b(btex \lam{λy. (+) 1 y} etex);
2322 newCircle.c(btex \lam{(λx.(+) x) 1} etex);
2323 newCircle.d(btex \lam{(+) 1} etex);
2326 c.c = b.c + (4cm, 0cm);
2327 a.c = midpoint(b.c, c.c) + (0cm, 4cm);
2328 d.c = midpoint(b.c, c.c) - (0cm, 3cm);
2330 % β-conversion between a and b
2331 ncarc.a(a)(b) "name(bred)";
2332 ObjLabel.a(btex $\xrightarrow[normal]{}{β}$ etex) "labpathname(bred)", "labdir(rt)";
2333 ncarc.b(b)(a) "name(bexp)", "linestyle(dashed withdots)";
2334 ObjLabel.b(btex $\xleftarrow[normal]{}{β}$ etex) "labpathname(bexp)", "labdir(lft)";
2336 % η-conversion between a and c
2337 ncarc.a(a)(c) "name(ered)";
2338 ObjLabel.a(btex $\xrightarrow[normal]{}{η}$ etex) "labpathname(ered)", "labdir(rt)";
2339 ncarc.c(c)(a) "name(eexp)", "linestyle(dashed withdots)";
2340 ObjLabel.c(btex $\xleftarrow[normal]{}{η}$ etex) "labpathname(eexp)", "labdir(lft)";
2342 % η-conversion between b and d
2343 ncarc.b(b)(d) "name(ered)";
2344 ObjLabel.b(btex $\xrightarrow[normal]{}{η}$ etex) "labpathname(ered)", "labdir(rt)";
2345 ncarc.d(d)(b) "name(eexp)", "linestyle(dashed withdots)";
2346 ObjLabel.d(btex $\xleftarrow[normal]{}{η}$ etex) "labpathname(eexp)", "labdir(lft)";
2348 % β-conversion between c and d
2349 ncarc.c(c)(d) "name(bred)";
2350 ObjLabel.c(btex $\xrightarrow[normal]{}{β}$ etex) "labpathname(bred)", "labdir(rt)";
2351 ncarc.d(d)(c) "name(bexp)", "linestyle(dashed withdots)";
2352 ObjLabel.d(btex $\xleftarrow[normal]{}{β}$ etex) "labpathname(bexp)", "labdir(lft)";
2354 % Draw objects and lines
2355 drawObj(a, b, c, d);
2358 \placeexample[right][ex:TransformGraph]{Partial graph of a lambda calculus
2359 system with β and η reduction (solid lines) and expansion (dotted lines).}
2360 \boxedgraphic{TransformGraph}
2362 Of course the graph for Cλash is unbounded, since we can construct an
2363 infinite amount of Core expressions. Also, there might potentially be
2364 multiple edges between two given nodes (with different labels), though
2365 this seems unlikely to actually happen in our system.
2367 See \in{example}[ex:TransformGraph] for the graph representation of a very
2368 simple lambda calculus that contains just the expressions \lam{(λx.λy. (+) x
2369 y) 1}, \lam{λy. (+) 1 y}, \lam{(λx.(+) x) 1} and \lam{(+) 1}. The
2370 transformation system consists of β-reduction and η-reduction (solid edges) or
2371 β-expansion and η-expansion (dotted edges).
2373 \todo{Define β-reduction and η-reduction?}
2375 In such a graph a node (expression) is in normal form if it has no
2376 outgoing edges (meaning no transformation applies to it). The set of
2377 nodes without outgoing edges is called the \emph{normal set}. Similarly,
2378 the set of nodes containing expressions in intended normal form
2379 \refdef{intended normal form} is called the \emph{intended normal set}.
2381 From such a graph, we can derive some properties easily:
2383 \item A system will \emph{terminate} if there is no walk (sequence of
2384 edges, or transformations) of infinite length in the graph (this
2385 includes cycles, but can also happen without cycles).
2386 \item Soundness is not easily represented in the graph.
2387 \item A system is \emph{complete} if all of the nodes in the normal set have
2388 the intended normal form. The inverse (that all of the nodes outside of
2389 the normal set are \emph{not} in the intended normal form) is not
2390 strictly required. In other words, our normal set must be a
2391 subset of the intended normal form, but they do not need to be
2394 \item A system is deterministic if all paths starting at a particular
2395 node, which end in a node in the normal set, end at the same node.
2398 When looking at the \in{example}[ex:TransformGraph], we see that the system
2399 terminates for both the reduction and expansion systems (but note that, for
2400 expansion, this is only true because we have limited the possible
2401 expressions. In complete lambda calculus, there would be a path from
2402 \lam{(λx.λy. (+) x y) 1} to \lam{(λx.λy.(λz.(+) z) x y) 1} to
2403 \lam{(λx.λy.(λz.(λq.(+) q) z) x y) 1} etc.)
2405 If we would consider the system with both expansion and reduction, there
2406 would no longer be termination either, since there would be cycles all
2409 The reduction and expansion systems have a normal set of containing just
2410 \lam{(+) 1} or \lam{(λx.λy. (+) x y) 1} respectively. Since all paths in
2411 either system end up in these normal forms, both systems are \emph{complete}.
2412 Also, since there is only one node in the normal set, it must obviously be
2413 \emph{deterministic} as well.
2415 \subsection{Termination}
2416 In general, proving termination of an arbitrary program is a very
2417 hard problem. \todo{Ref about arbitrary termination} Fortunately,
2418 we only have to prove termination for our specific transformation
2421 A common approach for these kinds of proofs is to associate a
2422 measure with each possible expression in our system. If we can
2423 show that each transformation strictly decreases this measure
2424 (\ie, the expression transformed to has a lower measure than the
2425 expression transformed from). \todo{ref about measure-based
2426 termination proofs / analysis}
2428 A good measure for a system consisting of just β-reduction would
2429 be the number of lambda expressions in the expression. Since every
2430 application of β-reduction removes a lambda abstraction (and there
2431 is always a bounded number of lambda abstractions in every
2432 expression) we can easily see that a transformation system with
2433 just β-reduction will always terminate.
2435 For our complete system, this measure would be fairly complex
2436 (probably the sum of a lot of things). Since the (conditions on)
2437 our transformations are pretty complex, we would need to include
2438 both simple things like the number of let expressions as well as
2439 more complex things like the number of case expressions that are
2440 not yet in normal form.
2442 No real attempt has been made at finding a suitable measure for
2445 \subsection{Soundness}
2446 Soundness is a property that can be proven for each transformation
2447 separately. Since our system only runs separate transformations
2448 sequentially, if each of our transformations leaves the
2449 \emph{meaning} of the expression unchanged, then the entire system
2450 will of course leave the meaning unchanged and is thus
2453 The current prototype has only been verified in an ad-hoc fashion
2454 by inspecting (the code for) each transformation. A more formal
2455 verification would be more appropriate.
2457 To be able to formally show that each transformation properly
2458 preserves the meaning of every expression, we require an exact
2459 definition of the \emph{meaning} of every expression, so we can
2460 compare them. A definition of the operational semantics of \GHC's Core
2461 language is available \cite[sulzmann07], but this does not seem
2462 sufficient for our goals (but it is a good start).
2464 It should be possible to have a single formal definition of
2465 meaning for Core for both normal Core compilation by \GHC\ and for
2466 our compilation to \VHDL. The main difference seems to be that in
2467 hardware every expression is always evaluated, while in software
2468 it is only evaluated if needed, but it should be possible to
2469 assign a meaning to core expressions that assumes neither.
2471 Since each of the transformations can be applied to any
2472 subexpression as well, there is a constraint on our meaning
2473 definition: the meaning of an expression should depend only on the
2474 meaning of subexpressions, not on the expressions themselves. For
2475 example, the meaning of the application in \lam{f (let x = 4 in
2476 x)} should be the same as the meaning of the application in \lam{f
2477 4}, since the argument subexpression has the same meaning (though
2478 the actual expression is different).
2480 \subsection{Completeness}
2481 Proving completeness is probably not hard, but it could be a lot
2482 of work. We have seen above that to prove completeness, we must
2483 show that the normal set of our graph representation is a subset
2484 of the intended normal set.
2486 However, it is hard to systematically generate or reason about the
2487 normal set, since it is defined as any nodes to which no
2488 transformation applies. To determine this set, each transformation
2489 must be considered and when a transformation is added, the entire
2490 set should be re-evaluated. This means it is hard to show that
2491 each node in the normal set is also in the intended normal set.
2492 Reasoning about our intended normal set is easier, since we know
2493 how to generate it from its definition. \refdef{intended normal
2496 Fortunately, we can also prove the complement (which is
2497 equivalent, since $A \subseteq B \Leftrightarrow \overline{B}
2498 \subseteq \overline{A}$): show that the set of nodes not in
2499 intended normal form is a subset of the set of nodes not in normal
2500 form. In other words, show that for every expression that is not
2501 in intended normal form, that there is at least one transformation
2502 that applies to it (since that means it is not in normal form
2503 either and since $A \subseteq C \Leftrightarrow \forall x (x \in A
2504 \rightarrow x \in C)$).
2506 By systematically reviewing the entire Core language definition
2507 along with the intended normal form definition (both of which have
2508 a similar structure), it should be possible to identify all
2509 possible (sets of) core expressions that are not in intended
2510 normal form and identify a transformation that applies to it.
2512 This approach is especially useful for proving completeness of our
2513 system, since if expressions exist to which none of the
2514 transformations apply (\ie\ if the system is not yet complete), it
2515 is immediately clear which expressions these are and adding
2516 (or modifying) transformations to fix this should be relatively
2519 As observed above, applying this approach is a lot of work, since
2520 we need to check every (set of) transformation(s) separately.
2522 \todo{Perhaps do a few steps of the proofs as proof-of-concept}
2524 % vim: set sw=2 sts=2 expandtab: