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 initial lambda
228 -- (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[][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
323 EBNF-like description in \in{definition}[def:IntendedNormal] captures
324 most of the intended structure (and generates a subset of \GHC's Core
327 There are two things missing from this definition: cast expressions are
328 sometimes allowed by the prototype, but not specified here and the below
329 definition allows uses of state that cannot be translated to \VHDL\
330 properly. These two problems are discussed in
331 \in{section}[sec:normalization:castproblems] and
332 \in{section}[sec:normalization:stateproblems] respectively.
334 Some clauses have an expression listed behind them in parentheses.
335 These are conditions that need to apply to the clause. The
336 predicates used there (\lam{lvar()}, \lam{representable()},
337 \lam{gvar()}) will be defined in
338 \in{section}[sec:normalization:predicates].
340 An expression is in normal form if it matches the first
341 definition, \emph{normal}.
343 \todo{Fix indentation}
344 \startbuffer[IntendedNormal]
345 \italic{normal} := \italic{lambda}
346 \italic{lambda} := λvar.\italic{lambda} (representable(var))
348 \italic{toplet} := letrec [\italic{binding}...] in var (representable(var))
349 \italic{binding} := var = \italic{rhs} (representable(rhs))
350 -- State packing and unpacking by coercion
351 | var0 = var1 ▶ State ty (lvar(var1))
352 | var0 = var1 ▶ ty (var1 :: State ty ∧ lvar(var1))
353 \italic{rhs} := \italic{userapp}
354 | \italic{builtinapp}
356 | case var of C a0 ... an -> ai (lvar(var))
358 | case var of (lvar(var))
359 [ DEFAULT -> var ] (lvar(var))
360 C0 w0,0 ... w0,n -> var0
362 Cm wm,0 ... wm,n -> varm (\forall{}i \forall{}j, wi,j \neq vari, lvar(vari))
363 \italic{userapp} := \italic{userfunc}
364 | \italic{userapp} {userarg}
365 \italic{userfunc} := var (gvar(var))
366 \italic{userarg} := var (lvar(var))
367 \italic{builtinapp} := \italic{builtinfunc}
368 | \italic{builtinapp} \italic{builtinarg}
369 \italic{built-infunc} := var (bvar(var))
370 \italic{built-inarg} := var (representable(var) ∧ lvar(var))
371 | \italic{partapp} (partapp :: a -> b)
372 | \italic{coreexpr} (¬representable(coreexpr) ∧ ¬(coreexpr :: a -> b))
373 \italic{partapp} := \italic{userapp}
374 | \italic{builtinapp}
377 \placedefinition[][def:IntendedNormal]{Definition of the intended normal form using an \small{EBNF}-like syntax.}
378 {\defref{intended normal form definition}
379 \typebufferlam{IntendedNormal}}
381 When looking at such a program from a hardware perspective, the top
382 level lambda abstractions (\italic{lambda}) define the input ports.
383 Lambda abstractions cannot appear anywhere else. The variable reference
384 in the body of the recursive let expression (\italic{toplet}) is the
385 output port. Most binders bound by the let expression define a
386 component instantiation (\italic{userapp}), where the input and output
387 ports are mapped to local signals (\italic{userarg}). Some of the others
388 use a built-in construction (\eg\ the \lam{case} expression) or call a
389 built-in function (\italic{builtinapp}) such as \lam{+} or \lam{map}.
390 For these, a hard-coded \small{VHDL} translation is available.
392 \section[sec:normalization:transformation]{Transformation notation}
393 To be able to concisely present transformations, we use a specific format
394 for them. It is a simple format, similar to one used in logic reasoning.
396 Such a transformation description looks like the following.
401 <original expression>
402 -------------------------- <expression conditions>
403 <transformed expression>
408 This format describes a transformation that applies to \lam{<original
409 expression>} and transforms it into \lam{<transformed expression>}, assuming
410 that all conditions are satisfied. In this format, there are a number of placeholders
411 in pointy brackets, most of which should be rather obvious in their meaning.
412 Nevertheless, we will more precisely specify their meaning below:
414 \startdesc{<original expression>} The expression pattern that will be matched
415 against (sub-expressions of) the expression to be transformed. We call this a
416 pattern, because it can contain \emph{placeholders} (variables), which match
417 any expression or binder. Any such placeholder is said to be \emph{bound} to
418 the expression it matches. It is convention to use an uppercase letter (\eg\
419 \lam{M} or \lam{E}) to refer to any expression (including a simple variable
420 reference) and lowercase letters (\eg\ \lam{v} or \lam{b}) to refer to
421 (references to) binders.
423 For example, the pattern \lam{a + B} will match the expression
424 \lam{v + (2 * w)} (binding \lam{a} to \lam{v} and \lam{B} to
425 \lam{(2 * w)}), but not \lam{(2 * w) + v}.
428 \startdesc{<expression conditions>}
429 These are extra conditions on the expression that is matched. These
430 conditions can be used to further limit the cases in which the
431 transformation applies, commonly to prevent a transformation from
432 causing a loop with itself or another transformation.
434 Only if these conditions are \emph{all} satisfied, the transformation
438 \startdesc{<context conditions>}
439 These are a number of extra conditions on the context of the function. In
440 particular, these conditions can require some (other) top level function to be
441 present, whose value matches the pattern given here. The format of each of
442 these conditions is: \lam{binder = <pattern>}.
444 Typically, the binder is some placeholder bound in the \lam{<original
445 expression>}, while the pattern contains some placeholders that are used in
446 the \lam{transformed expression}.
448 Only if a top level binder exists that matches each binder and pattern,
449 the transformation applies.
452 \startdesc{<transformed expression>}
453 This is the expression template that is the result of the transformation. If, looking
454 at the above three items, the transformation applies, the \lam{<original
455 expression>} is completely replaced by the \lam{<transformed expression>}.
456 We call this a template, because it can contain placeholders, referring to
457 any placeholder bound by the \lam{<original expression>} or the
458 \lam{<context conditions>}. The resulting expression will have those
459 placeholders replaced by the values bound to them.
461 Any binder (lowercase) placeholder that has no value bound to it yet will be
462 bound to (and replaced with) a fresh binder.
465 \startdesc{<context additions>}
466 These are templates for new functions to be added to the context.
467 This is a way to let a transformation create new top level
470 Each addition has the form \lam{binder = template}. As above, any
471 placeholder in the addition is replaced with the value bound to it, and any
472 binder placeholder that has no value bound to it yet will be bound to (and
473 replaced with) a fresh binder.
476 To understand this notation better, the step by step application of
477 the η-expansion transformation to a simple \small{ALU} will be
478 shown. Consider η-expansion, which is a common transformation from
479 lambda calculus, described using above notation as follows:
483 -------------- \lam{E} does not occur on a function position in an application
484 λx.E x \lam{E} is not a lambda abstraction.
487 η-expansion is a well known transformation from lambda calculus. What
488 this transformation does, is take any expression that has a function type
489 and turn it into a lambda expression (giving an explicit name to the
490 argument). There are some extra conditions that ensure that this
491 transformation does not apply infinitely (which are not necessarily part
492 of the conventional definition of η-expansion).
494 Consider the following function, in Core notation, which is a fairly obvious way to specify a
495 simple \small{ALU} (Note that it is not yet in normal form, but
496 \in{example}[ex:AddSubAlu] shows the normal form of this function).
497 The parentheses around the \lam{+} and \lam{-} operators are
498 commonly used in Haskell to show that the operators are used as
499 normal functions, instead of \emph{infix} operators (\eg, the
500 operators appear before their arguments, instead of in between).
503 alu :: Bit -> Word -> Word -> Word
504 alu = λopcode. case opcode of
509 There are a few sub-expressions in this function to which we could possibly
510 apply the transformation. Since the pattern of the transformation is only
511 the placeholder \lam{E}, any expression will match that. Whether the
512 transformation applies to an expression is thus solely decided by the
513 conditions to the right of the transformation.
515 We will look at each expression in the function in a top down manner. The
516 first expression is the entire expression the function is bound to.
519 λopcode. case opcode of
524 As said, the expression pattern matches this. The type of this expression is
525 \lam{Bit -> Word -> Word -> Word}, which matches \lam{a -> b} (Note that in
526 this case \lam{a = Bit} and \lam{b = Word -> Word -> Word}).
528 Since this expression is at top level, it does not occur at a function
529 position of an application. However, The expression is a lambda abstraction,
530 so this transformation does not apply.
532 The next expression we could apply this transformation to, is the body of
533 the lambda abstraction:
541 The type of this expression is \lam{Word -> Word -> Word}, which again
542 matches \lam{a -> b}. The expression is the body of a lambda expression, so
543 it does not occur at a function position of an application. Finally, the
544 expression is not a lambda abstraction but a case expression, so all the
545 conditions match. There are no context conditions to match, so the
546 transformation applies.
548 By now, the placeholder \lam{E} is bound to the entire expression. The
549 placeholder \lam{x}, which occurs in the replacement template, is not bound
550 yet, so we need to generate a fresh binder for that. Let us use the binder
551 \lam{a}. This results in the following replacement expression:
559 Continuing with this expression, we see that the transformation does not
560 apply again (it is a lambda expression). Next we look at the body of this
569 Here, the transformation does apply, binding \lam{E} to the entire
570 expression (which has type \lam{Word -> Word}) and binding \lam{x}
571 to the fresh binder \lam{b}, resulting in the replacement:
579 The transformation does not apply to this lambda abstraction, so we
580 look at its body. For brevity, we will put the case expression on one line from
584 (case opcode of Low -> (+); High -> (-)) a b
587 The type of this expression is \lam{Word}, so it does not match \lam{a -> b}
588 and the transformation does not apply. Next, we have two options for the
589 next expression to look at: the function position and argument position of
590 the application. The expression in the argument position is \lam{b}, which
591 has type \lam{Word}, so the transformation does not apply. The expression in
592 the function position is:
595 (case opcode of Low -> (+); High -> (-)) a
598 Obviously, the transformation does not apply here, since it occurs in
599 function position (which makes the second condition false). In the same
600 way the transformation does not apply to both components of this
601 expression (\lam{case opcode of Low -> (+); High -> (-)} and \lam{a}), so
602 we will skip to the components of the case expression: the scrutinee and
603 both alternatives. Since the opcode is not a function, it does not apply
606 The first alternative is \lam{(+)}. This expression has a function type
607 (the operator still needs two arguments). It does not occur in function
608 position of an application and it is not a lambda expression, so the
609 transformation applies.
611 We look at the \lam{<original expression>} pattern, which is \lam{E}.
612 This means we bind \lam{E} to \lam{(+)}. We then replace the expression
613 with the \lam{<transformed expression>}, replacing all occurrences of
614 \lam{E} with \lam{(+)}. In the \lam{<transformed expression>}, the This gives us the replacement expression:
615 \lam{λx.(+) x} (A lambda expression binding \lam{x}, with a body that
616 applies the addition operator to \lam{x}).
618 The complete function then becomes:
620 (case opcode of Low -> λa1.(+) a1; High -> (-)) a
623 Now the transformation no longer applies to the complete first alternative
624 (since it is a lambda expression). It does not apply to the addition
625 operator again, since it is now in function position in an application. It
626 does, however, apply to the application of the addition operator, since
627 that is neither a lambda expression nor does it occur in function
628 position. This means after one more application of the transformation, the
632 (case opcode of Low -> λa1.λb1.(+) a1 b1; High -> (-)) a
635 The other alternative is left as an exercise to the reader. The final
636 function, after applying η-expansion until it does no longer apply is:
639 alu :: Bit -> Word -> Word -> Word
640 alu = λopcode.λa.b. (case opcode of
641 Low -> λa1.λb1 (+) a1 b1
642 High -> λa2.λb2 (-) a2 b2) a b
645 \subsection{Transformation application}
646 In this chapter we define a number of transformations, but how will we apply
647 these? As stated before, our normal form is reached as soon as no
648 transformation applies anymore. This means our application strategy is to
649 simply apply any transformation that applies, and continuing to do that with
650 the result of each transformation.
652 In particular, we define no particular order of transformations. Since
653 transformation order should not influence the resulting normal form,
654 this leaves the implementation free to choose any application order that
655 results in an efficient implementation. Unfortunately this is not
656 entirely true for the current set of transformations. See
657 \in{section}[sec:normalization:non-determinism] for a discussion of this
660 When applying a single transformation, we try to apply it to every (sub)expression
661 in a function, not just the top level function body. This allows us to
662 keep the transformation descriptions concise and powerful.
664 \subsection{Definitions}
665 A \emph{global variable} is any variable (binder) that is bound at the
666 top level of a program, or an external module. A \emph{local variable} is any
667 other variable (\eg, variables local to a function, which can be bound by
668 lambda abstractions, let expressions and pattern matches of case
669 alternatives). This is a slightly different notion of global versus
670 local than what \small{GHC} uses internally, but for our purposes
671 the distinction \GHC\ makes is not useful.
672 \defref{global variable} \defref{local variable}
674 A \emph{hardware representable} (or just \emph{representable}) type or value
675 is (a value of) a type that we can generate a signal for in hardware. For
676 example, a bit, a vector of bits, a 32 bit unsigned word, etc. Values that are
677 not run-time representable notably include (but are not limited to): types,
678 dictionaries, functions.
679 \defref{representable}
681 A \emph{built-in function} is a function supplied by the Cλash
682 framework, whose implementation is not used to generate \VHDL. This is
683 either because it is no valid Cλash (like most list functions that need
684 recursion) or because a Cλash implementation would be unwanted (for the
685 addition operator, for example, we would rather use the \VHDL addition
686 operator to let the synthesis tool decide what kind of adder to use
687 instead of explicitly describing one in Cλash). \defref{built-in
690 These are functions like \lam{map}, \lam{hwor}, \lam{+} and \lam{length}.
692 For these functions, Cλash has a \emph{built-in hardware translation},
693 so calls to these functions can still be translated. Built-in functions
694 must have a valid Haskell implementation, of course, to allow
697 A \emph{user-defined} function is a function for which no built-in
698 translation is available and whose definition will thus need to be
699 translated to Cλash. \defref{user-defined function}
701 \subsubsection[sec:normalization:predicates]{Predicates}
702 Here, we define a number of predicates that can be used below to concisely
705 \emph{gvar(expr)} is true when \emph{expr} is a variable that references a
706 global variable. It is false when it references a local variable.
708 \emph{lvar(expr)} is the complement of \emph{gvar}; it is true when \emph{expr}
709 references a local variable, false when it references a global variable.
711 \emph{representable(expr)} is true when \emph{expr} is \emph{representable}.
713 \subsection[sec:normalization:uniq]{Binder uniqueness}
714 A common problem in transformation systems, is binder uniqueness. When not
715 considering this problem, it is easy to create transformations that mix up
716 bindings and cause name collisions. Take for example, the following Core
720 (λa.λb.λc. a * b * c) x c
723 By applying β-reduction (see \in{section}[sec:normalization:beta]) once,
724 we can simplify this expression to:
730 Now, we have replaced the \lam{a} binder with a reference to the \lam{x}
731 binder. No harm done here. But note that we see multiple occurrences of the
732 \lam{c} binder. The first is a binding occurrence, to which the second refers.
733 The last, however refers to \emph{another} instance of \lam{c}, which is
734 bound somewhere outside of this expression. Now, if we would apply beta
735 reduction without taking heed of binder uniqueness, we would get:
741 This is obviously not what was supposed to happen! The root of this problem is
742 the reuse of binders: identical binders can be bound in different,
743 but overlapping scopes. Any variable reference in those
744 overlapping scopes then refers to the variable bound in the inner
745 (smallest) scope. There is not way to refer to the variable in the
746 outer scope. This effect is usually referred to as
747 \emph{shadowing}: when a binder is bound in a scope where the
748 binder already had a value, the inner binding is said to
749 \emph{shadow} the outer binding. In the example above, the \lam{c}
750 binder was bound outside of the expression and in the inner lambda
751 expression. Inside that lambda expression, only the inner \lam{c}
754 There are a number of ways to solve this. \small{GHC} has isolated this
755 problem to their binder substitution code, which performs \emph{de-shadowing}
756 during its expression traversal. This means that any binding that shadows
757 another binding on a higher level is replaced by a new binder that does not
758 shadow any other binding. This non-shadowing invariant is enough to prevent
759 binder uniqueness problems in \small{GHC}.
761 In our transformation system, maintaining this non-shadowing invariant is
762 a bit harder to do (mostly due to implementation issues, the prototype
763 does not use \small{GHC}'s substitution code). Also, the following points
767 \item De-shadowing does not guarantee overall uniqueness. For example, the
768 following (slightly contrived) expression shows the identifier \lam{x} bound in
769 two separate places (and to different values), even though no shadowing
773 (let x = 1 in x) + (let x = 2 in x)
776 \item In our normal form (and the resulting \small{VHDL}), all binders
777 (signals) within the same function (entity) will end up in the same
778 scope. To allow this, all binders within the same function should be
781 \item When we know that all binders in an expression are unique, moving around
782 or removing a sub-expression will never cause any binder conflicts. If we have
783 some way to generate fresh binders, introducing new sub-expressions will not
784 cause any problems either. The only way to cause conflicts is thus to
785 duplicate an existing sub-expression.
788 Given the above, our prototype maintains a unique binder invariant. This
789 means that in any given moment during normalization, all binders \emph{within
790 a single function} must be unique. To achieve this, we apply the following
793 \todo{Define fresh binders and unique supplies}
796 \item Before starting normalization, all binders in the function are made
797 unique. This is done by generating a fresh binder for every binder used. This
798 also replaces binders that did not cause any conflict, but it does ensure that
799 all binders within the function are generated by the same unique supply.
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}
824 \defref{substitution notation}
825 \startframedtext[width=8cm,background=box,frame=no]
826 \startalignment[center]
827 {\tfa Substitution notation}
831 In some of the transformations in this chapter, we need to perform
832 substitution on an expression. Substitution means replacing every
833 occurrence of some expression (usually a variable reference) with
836 There have been a lot of different notations used in literature for
837 specifying substitution. The notation that will be used in this report
844 This means expression \lam{E} with all occurrences of \lam{A} replaced
849 These transformations are general cleanup transformations, that aim to
850 make expressions simpler. These transformations usually clean up the
851 mess left behind by other transformations or clean up expressions to
852 expose new transformation opportunities for other transformations.
854 Most of these transformations are standard optimizations in other
855 compilers as well. However, in our compiler, most of these are not just
856 optimizations, but they are required to get our program into intended
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 reducible 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 the
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 application 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}[sec:prototype:coretypes].
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 file names, so it effectively does not
1092 work.}, so the entity would be called \quote{w7aA7f} or
1093 something similarly meaningless and auto-generated).
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-expansion}
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-expansion}{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 The basic idea of this transformation is to take the body of a
1277 function and bind it with a let expression (so the body of that let
1278 expression becomes a variable reference that can be used as the output
1279 port). If the body of the function happens to have lambda abstractions
1280 at the top level (which is allowed by the intended normal
1281 form\refdef{intended normal form definition}), we take the body of the
1282 inner lambda instead. If that happens to be a let expression already
1283 (which is allowed by the intended normal form), we take the body of
1284 that let (which is not allowed to be anything but a variable reference
1285 according the the intended normal form).
1287 This transformation uses the context conditions in a special way.
1288 These contexts, like \lam{x = λv1 ... λvn.E}, are above the dotted
1289 line and provide a condition on the environment (\ie\ they require a
1290 certain top level binding to be present). These ensure that
1291 expressions are only transformed when they are in the functions
1292 \quote{return value} directly. This means the context conditions have
1293 to interpreted in the right way: not \quote{if there is any function
1294 \lam{x} that binds \lam{E}, any \lam{E} can be transformed}, but we
1295 mean only the \lam{E} that is bound by \lam{x}).
1297 Be careful when reading the transformations: Not the entire function
1298 from the context is transformed, just a part of it.
1300 Note that the return value is not simplified if it is not representable.
1301 Otherwise, this would cause a loop with the inlining of
1302 unrepresentable bindings in
1303 \in{section}[sec:normalization:nonrepinline]. If the return value is
1304 not representable because it has a function type, η-expansion should
1305 make sure that this transformation will eventually apply. If the
1306 value is not representable for other reasons, the function result
1307 itself is not representable, meaning this function is not translatable
1311 x = λv1 ... λvn.E \lam{n} can be zero
1312 ~ \lam{E} is representable
1313 E \lam{E} is not a lambda abstraction
1314 --------------------------- \lam{E} is not a let expression
1315 letrec y = E in y \lam{E} is not a local variable reference
1319 x = λv1 ... λvn.letrec binds in E \lam{n} can be zero
1320 ~ \lam{E} is representable
1321 letrec binds in E \lam{E} is not a local variable reference
1322 ------------------------------------
1323 letrec binds; y = E in y
1331 x = letrec y = add 1 2 in y
1334 \transexample{retvalsimpl}{Return value simplification}{from}{to}
1347 \transexample{retvalsimpllam}{Return value simplification with a lambda abstraction}{from}{to}
1364 \transexample{retvalsimpllet}{Return value simplification with a let expression}{from}{to}
1366 \subsection[sec:normalization:argsimpl]{Representable arguments simplification}
1367 This section contains just a single transformation that deals with
1368 representable arguments in applications. Non-representable arguments are
1369 handled by the transformations in
1370 \in{section}[sec:normalization:nonrep].
1372 This transformation ensures that all representable arguments will become
1373 references to local variables. This ensures they will become references
1374 to local signals in the resulting \small{VHDL}, which is required due to
1375 limitations in the component instantiation code in \VHDL\ (one can only
1376 assign a signal or constant to an input port). By ensuring that all
1377 arguments are always simple variable references, we always have a signal
1378 available to map to the input ports.
1380 To reduce a complex expression to a simple variable reference, we create
1381 a new let expression around the application, which binds the complex
1382 expression to a new variable. The original function is then applied to
1385 \refdef{global variable}
1386 Note that references to \emph{global variables} (like a top level
1387 function without arguments, but also an argumentless data-constructors
1388 like \lam{True}) are also simplified. Only local variables generate
1389 signals in the resulting architecture. Even though argumentless
1390 data-constructors generate constants in generated \VHDL\ code and could be
1391 mapped to an input port directly, they are still simplified to make the
1392 normal form more regular.
1394 \refdef{representable}
1397 -------------------- \lam{N} is representable
1398 letrec x = N in M x \lam{N} is not a local variable reference
1400 \refdef{local variable}
1407 letrec x = add a 1 in add x 1
1410 \transexample{argsimpl}{Argument simplification}{from}{to}
1412 \subsection[sec:normalization:built-ins]{Built-in functions}
1413 This section deals with (arguments to) built-in functions. In the
1414 intended normal form definition\refdef{intended normal form definition}
1415 we can see that there are three sorts of arguments a built-in function
1419 \item A representable local variable reference. This is the most
1420 common argument to any function. The argument simplification
1421 transformation described in \in{section}[sec:normalization:argsimpl]
1422 makes sure that \emph{any} representable argument to \emph{any}
1423 function (including built-in functions) is turned into a local variable
1425 \item (A partial application of) a top level function (either built-in on
1426 user-defined). The function extraction transformation described in
1427 this section takes care of turning every function-typed argument into
1428 (a partial application of) a top level function.
1429 \item Any expression that is not representable and does not have a
1430 function type. Since these can be any expression, there is no
1431 transformation needed. Note that this category is exactly all
1432 expressions that are not transformed by the transformations for the
1433 previous two categories. This means that \emph{any} Core expression
1434 that is used as an argument to a built-in function will be either
1435 transformed into one of the above categories, or end up in this
1436 category. In any case, the result is in normal form.
1439 As noted, the argument simplification will handle any representable
1440 arguments to a built-in function. The following transformation is needed
1441 to handle non-representable arguments with a function type, all other
1442 non-representable arguments do not need any special handling.
1444 \subsubsection[sec:normalization:funextract]{Function extraction}
1445 This transform deals with function-typed arguments to built-in
1447 Since built-in functions cannot be specialized (see
1448 \in{section}[sec:normalization:specialize]) to remove the arguments,
1449 these arguments are extracted into a new global function instead. In
1450 other words, we create a new top level function that has exactly the
1451 extracted argument as its body. This greatly simplifies the
1452 translation rules needed for built-in functions, since they only need
1453 to handle (partial applications of) top level functions.
1455 Any free variables occurring in the extracted arguments will become
1456 parameters to the new global function. The original argument is replaced
1457 with a reference to the new function, applied to any free variables from
1458 the original argument.
1460 This transformation is useful when applying higher-order built-in functions
1461 like \hs{map} to a lambda abstraction, for example. In this case, the code
1462 that generates \small{VHDL} for \hs{map} only needs to handle top level functions and
1463 partial applications, not any other expression (such as lambda abstractions or
1464 even more complicated expressions).
1467 M N \lam{M} is (a partial application of) a built-in function.
1468 --------------------- \lam{f0 ... fn} are all free local variables of \lam{N}
1469 M (x f0 ... fn) \lam{N :: a -> b}
1470 ~ \lam{N} is not a (partial application of) a top level function
1475 addList = λb.λxs.map (λa . add a b) xs
1479 addList = λb.λxs.map (f b) xs
1484 \transexample{funextract}{Function extraction}{from}{to}
1486 Note that the function \lam{f} will still need normalization after
1489 \subsection{Case normalization}
1490 The transformations in this section ensure that case statements end up
1493 \subsubsection{Scrutinee simplification}
1494 This transform ensures that the scrutinee of a case expression is always
1495 a simple variable reference.
1500 ----------------- \lam{E} is not a local variable reference
1519 \transexample{letflat}{Case normalization}{from}{to}
1523 \defref{wild binders}
1524 \startframedtext[width=7cm,background=box,frame=no]
1525 \startalignment[center]
1529 In a functional expression, a \emph{wild binder} refers to any
1530 binder that is never referenced. This means that even though it
1531 will be bound to a particular value, that value is never used.
1533 The Haskell syntax offers the underscore as a wild binder that
1534 cannot even be referenced (It can be seen as introducing a new,
1535 anonymous, binder every time it is used).
1537 In these transformations, the term wild binder will sometimes be
1538 used to indicate that a binder must not be referenced.
1542 \subsubsection{Scrutinee binder removal}
1543 This transformation removes (or rather, makes wild) the binder to
1544 which the scrutinee is bound after evaluation. This is done by
1545 replacing the bndr with the scrutinee in all alternatives. To prevent
1546 duplication of work, this transformation is only applied when the
1547 scrutinee is already a simple variable reference (but the previous
1548 transformation ensures this will eventually be the case). The
1549 scrutinee binder itself is replaced by a wild binder (which is no
1552 Note that one could argue that this transformation can change the
1553 meaning of the Core expression. In the regular Core semantics, a case
1554 expression forces the evaluation of its scrutinee and can be used to
1555 implement strict evaluation. However, in the generated \VHDL,
1556 evaluation is always strict. So the semantics we assign to the Core
1557 expression (which differ only at this particular point), this
1558 transformation is completely valid.
1563 ----------------- \lam{x} is a local variable reference
1580 \transexample{scrutbndrremove}{Scrutinee binder removal}{from}{to}
1582 \subsubsection{Case normalization}
1583 This transformation ensures that all case expressions get a form
1584 that is allowed by the intended normal form. This means they
1588 \item An extractor case with a single alternative that picks a field
1589 from a datatype, \eg\ \lam{case x of (a, b) ->
1590 a}.\defref{extractor case}
1591 \item A selector case with multiple alternatives and only wild binders, that
1592 makes a choice between expressions based on the constructor of another
1593 expression, \eg\ \lam{case x of Low -> a; High ->
1594 b}.\defref{selector case}
1597 For an arbitrary case, that has \lam{n} alternatives, with
1598 \lam{m} binders in each alternatives, this will result in \lam{m
1599 * n} extractor case expression to get at each variable, \lam{n}
1600 let bindings for each of the alternatives' value and a single
1601 selector case to select the right value out of these.
1603 Technically, the definition of this transformation would require
1604 that the constructor for every alternative has exactly the same
1605 amount (\lam{m}) of arguments, but of course this transformation
1606 also applies when this is not the case.
1610 C0 v0,0 ... v0,m -> E0
1612 Cn vn,0 ... vn,m -> En
1613 --------------------------------------------------- \lam{\forall i \forall j, 0 ≤ i ≤ n, 0 ≤ i < m} (\lam{wi,j} is a wild (unused) binder)
1614 letrec The case expression is not an extractor case
1615 v0,0 = case E of C0 x0,0 .. x0,m -> x0,0 The case expression is not a selector case
1617 v0,m = case E of C0 x0,0 .. x0,m -> x0,m
1619 vn,m = case E of Cn xn,0 .. xn,m -> xn,m
1625 C0 w0,0 ... w0,m -> y0
1627 Cn wn,0 ... wn,m -> yn
1630 Note that this transformation applies to case expressions with any
1631 scrutinee. If the scrutinee is a complex expression, this might
1632 result in duplication of work (hardware). An extra condition to
1633 only apply this transformation when the scrutinee is already
1634 simple (effectively causing this transformation to be only
1635 applied after the scrutinee simplification transformation) might
1654 \transexample{selcasesimpl}{Selector case simplification}{from}{to}
1662 b = case a of (,) b c -> b
1663 c = case a of (,) b c -> c
1670 \transexample{excasesimpl}{Extractor case simplification}{from}{to}
1672 \refdef{selector case}
1673 In \in{example}[ex:trans:excasesimpl] the case expression is expanded
1674 into multiple case expressions, including a pretty useless expression
1675 (that is neither a selector or extractor case). This case can be
1676 removed by the Case removal transformation in
1677 \in{section}[sec:transformation:caseremoval].
1679 \subsubsection[sec:transformation:caseremoval]{Case removal}
1680 This transform removes any case expression with a single alternative and
1681 only wild binders.\refdef{wild binders}
1683 These "useless" case expressions are usually leftovers from case simplification
1684 on extractor case (see the previous example).
1689 ---------------------- \lam{\forall i, 0 ≤ i ≤ m} (\lam{vi} does not occur free in E)
1702 \transexample{caserem}{Case removal}{from}{to}
1704 \subsection[sec:normalization:nonrep]{Removing unrepresentable values}
1705 The transformations in this section are aimed at making all the
1706 values used in our expression representable. There are two main
1707 transformations that are applied to \emph{all} unrepresentable let
1708 bindings and function arguments. These are meant to address three
1709 different kinds of unrepresentable values: polymorphic values,
1710 higher-order values and literals. The transformation are described
1711 generically: they apply to all non-representable values. However,
1712 non-representable values that do not fall into one of these three
1713 categories will be moved around by these transformations but are
1714 unlikely to completely disappear. They usually mean the program was not
1715 valid in the first place, because unsupported types were used (for
1716 example, a program using strings).
1718 Each of these three categories will be detailed below, followed by the
1719 actual transformations.
1721 \subsubsection{Removing Polymorphism}
1722 As noted in \in{section}[sec:prototype:coretypes],
1723 polymorphism is made explicit in Core through type and
1724 dictionary arguments. To remove the polymorphism from a
1725 function, we can simply specialize the polymorphic function for
1726 the particular type applied to it. The same goes for dictionary
1727 arguments. To remove polymorphism from let bound values, we
1728 simply inline the let bindings that have a polymorphic type,
1729 which should (eventually) make sure that the polymorphic
1730 expression is applied to a type and/or dictionary, which can
1731 then be removed by β-reduction (\in{section}[sec:normalization:beta]).
1733 Since both type and dictionary arguments are not representable,
1734 \refdef{representable}
1735 the non-representable argument specialization and
1736 non-representable let binding inlining transformations below
1737 take care of exactly this.
1739 There is one case where polymorphism cannot be completely
1740 removed: built-in functions are still allowed to be polymorphic
1741 (Since we have no function body that we could properly
1742 specialize). However, the code that generates \VHDL\ for built-in
1743 functions knows how to handle this, so this is not a problem.
1745 \subsubsection[sec:normalization:defunctionalization]{Defunctionalization}
1746 These transformations remove higher-order expressions from our
1747 program, making all values first-order. The approach used for
1748 defunctionalization uses a combination of specialization, inlining and
1749 some cleanup transformations, was also proposed in parallel research
1750 by Neil Mitchell \cite[mitchell09].
1752 Higher order values are always introduced by lambda abstractions, none
1753 of the other Core expression elements can introduce a function type.
1754 However, other expressions can \emph{have} a function type, when they
1755 have a lambda expression in their body.
1757 For example, the following expression is a higher-order expression
1758 that is not a lambda expression itself:
1760 \refdef{id function}
1767 The reference to the \lam{id} function shows that we can introduce a
1768 higher-order expression in our program without using a lambda
1769 expression directly. However, inside the definition of the \lam{id}
1770 function, we can be sure that a lambda expression is present.
1772 Looking closely at the definition of our normal form in
1773 \in{section}[sec:normalization:intendednormalform], we can see that
1774 there are three possibilities for higher-order values to appear in our
1775 intended normal form:
1778 \item[item:toplambda] Lambda abstractions can appear at the highest level of a
1779 top level function. These lambda abstractions introduce the
1780 arguments (input ports / current state) of the function.
1781 \item[item:built-inarg] (Partial applications of) top level functions can appear as an
1782 argument to a built-in function.
1783 \item[item:completeapp] (Partial applications of) top level functions can appear in
1784 function position of an application. Since a partial application
1785 cannot appear anywhere else (except as built-in function arguments),
1786 all partial applications are applied, meaning that all applications
1787 will become complete applications. However, since application of
1788 arguments happens one by one, in the expression:
1792 the sub-expression \lam{f 1} has a function type. But this is
1793 allowed, since it is inside a complete application.
1796 We will take a typical function with some higher-order values as an
1797 example. The following function takes two arguments: a \lam{Bit} and a
1798 list of numbers. Depending on the first argument, each number in the
1799 list is doubled, or the list is returned unmodified. For the sake of
1800 the example, no polymorphism is shown. In reality, at least map would
1804 λy.let double = λx. x + x in
1810 This example shows a number of higher-order values that we cannot
1811 translate to \VHDL\ directly. The \lam{double} binder bound in the let
1812 expression has a function type, as well as both of the alternatives of
1813 the case expression. The first alternative is a partial application of
1814 the \lam{map} built-in function, whereas the second alternative is a
1817 To reduce all higher-order values to one of the above items, a number
1818 of transformations we have already seen are used. The η-expansion
1819 transformation from \in{section}[sec:normalization:eta] ensures all
1820 function arguments are introduced by lambda abstraction on the highest
1821 level of a function. These lambda arguments are allowed because of
1822 \in{item}[item:toplambda] above. After η-expansion, our example
1823 becomes a bit bigger:
1826 λy.λq.(let double = λx. x + x in
1833 η-expansion also introduces extra applications (the application of
1834 the let expression to \lam{q} in the above example). These
1835 applications can then propagated down by the application propagation
1836 transformation (\in{section}[sec:normalization:appprop]). In our
1837 example, the \lam{q} and \lam{r} variable will be propagated into the
1838 let expression and then into the case expression:
1841 λy.λq.let double = λx. x + x in
1847 This propagation makes higher-order values become applied (in
1848 particular both of the alternatives of the case now have a
1849 representable type). Completely applied top level functions (like the
1850 first alternative) are now no longer invalid (they fall under
1851 \in{item}[item:completeapp] above). (Completely) applied lambda
1852 abstractions can be removed by β-expansion. For our example,
1853 applying β-expansion results in the following:
1856 λy.λq.let double = λx. x + x in
1862 As you can see in our example, all of this moves applications towards
1863 the higher-order values, but misses higher-order functions bound by
1864 let expressions. The applications cannot be moved towards these values
1865 (since they can be used in multiple places), so the values will have
1866 to be moved towards the applications. This is achieved by inlining all
1867 higher-order values bound by let applications, by the
1868 non-representable binding inlining transformation below. When applying
1869 it to our example, we get the following:
1873 Low -> map (λx. x + x) q
1877 We have nearly eliminated all unsupported higher-order values from this
1878 expressions. The one that is remaining is the first argument to the
1879 \lam{map} function. Having higher-order arguments to a built-in
1880 function like \lam{map} is allowed in the intended normal form, but
1881 only if the argument is a (partial application) of a top level
1882 function. This is easily done by introducing a new top level function
1883 and put the lambda abstraction inside. This is done by the function
1884 extraction transformation from
1885 \in{section}[sec:normalization:funextract].
1893 This also introduces a new function, that we have called \lam{func}:
1899 Note that this does not actually remove the lambda, but now it is a
1900 lambda at the highest level of a function, which is allowed in the
1901 intended normal form.
1903 There is one case that has not been discussed yet. What if the
1904 \lam{map} function in the example above was not a built-in function
1905 but a user-defined function? Then extracting the lambda expression
1906 into a new function would not be enough, since user-defined functions
1907 can never have higher-order arguments. For example, the following
1908 expression shows an example:
1911 twice :: (Word -> Word) -> Word -> Word
1912 twice = λf.λa.f (f a)
1914 main = λa.app (λx. x + x) a
1917 This example shows a function \lam{twice} that takes a function as a
1918 first argument and applies that function twice to the second argument.
1919 Again, we have made the function monomorphic for clarity, even though
1920 this function would be a lot more useful if it was polymorphic. The
1921 function \lam{main} uses \lam{twice} to apply a lambda expression twice.
1923 When faced with a user defined function, a body is available for that
1924 function. This means we could create a specialized version of the
1925 function that only works for this particular higher-order argument
1926 (\ie, we can just remove the argument and call the specialized
1927 function without the argument). This transformation is detailed below.
1928 Applying this transformation to the example gives:
1931 twice' :: Word -> Word
1932 twice' = λb.(λf.λa.f (f a)) (λx. x + x) b
1937 The \lam{main} function is now in normal form, since the only
1938 higher-order value there is the top level lambda expression. The new
1939 \lam{twice'} function is a bit complex, but the entire original body
1940 of the original \lam{twice} function is wrapped in a lambda
1941 abstraction and applied to the argument we have specialized for
1942 (\lam{λx. x + x}) and the other arguments. This complex expression can
1943 fortunately be effectively reduced by repeatedly applying β-reduction:
1946 twice' :: Word -> Word
1947 twice' = λb.(b + b) + (b + b)
1950 This example also shows that the resulting normal form might not be as
1951 efficient as we might hope it to be (it is calculating \lam{(b + b)}
1952 twice). This is discussed in more detail in
1953 \in{section}[sec:normalization:duplicatework].
1955 \subsubsection[sec:normalization:literals]{Literals}
1956 There are a limited number of literals available in Haskell and Core.
1957 \refdef{enumerated types} When using (enumerating) algebraic
1958 data-types, a literal is just a reference to the corresponding data
1959 constructor, which has a representable type (the algebraic datatype)
1960 and can be translated directly. This also holds for literals of the
1961 \hs{Bool} Haskell type, which is just an enumerated type.
1963 There is, however, a second type of literal that does not have a
1964 representable type: integer literals. Cλash supports using integer
1965 literals for all three integer types supported (\hs{SizedWord},
1966 \hs{SizedInt} and \hs{RangedWord}). This is implemented using
1967 Haskell's \hs{Num} type class, which offers a \hs{fromInteger} method
1968 that converts any \hs{Integer} to the Cλash data-types.
1970 When \GHC\ sees integer literals, it will automatically insert calls to
1971 the \hs{fromInteger} method in the resulting Core expression. For
1972 example, the following expression in Haskell creates a 32 bit unsigned
1973 word with the value 1. The explicit type signature is needed, since
1974 there is no context for \GHC\ to determine the type from otherwise.
1980 This Haskell code results in the following Core expression:
1983 fromInteger @(SizedWord D32) \$dNum (smallInteger 10)
1986 The literal 10 will have the type \lam{GHC.Prim.Int\#}, which is
1987 converted into an \lam{Integer} by \lam{smallInteger}. Finally, the
1988 \lam{fromInteger} function will finally convert this into a
1989 \lam{SizedWord D32}.
1991 Both the \lam{GHC.Prim.Int\#} and \lam{Integer} types are not
1992 representable, and cannot be translated directly. Fortunately, there
1993 is no need to translate them, since \lam{fromInteger} is a built-in
1994 function that knows how to handle these values. However, this does
1995 require that the \lam{fromInteger} function is directly applied to
1996 these non-representable literal values, otherwise errors will occur.
1997 For example, the following expression is not in the intended normal
1998 form, since one of the let bindings has an unrepresentable type
2002 let l = smallInteger 10 in fromInteger @(SizedWord D32) \$dNum l
2005 By inlining these let-bindings, we can ensure that unrepresentable
2006 literals bound by a let binding end up in an application of the
2007 appropriate built-in function, where they are allowed. Since it is
2008 possible that the application of that function is in a different
2009 function than the definition of the literal value, we will always need
2010 to specialize away any unrepresentable literals that are used as
2011 function arguments. The following two transformations do exactly this.
2013 \subsubsection[sec:normalization:nonrepinline]{Non-representable binding inlining}
2014 This transform inlines let bindings that are bound to a
2015 non-representable value. Since we can never generate a signal
2016 assignment for these bindings (we cannot declare a signal assignment
2017 with a non-representable type, for obvious reasons), we have no choice
2018 but to inline the binding to remove it.
2020 As we have seen in the previous sections, inlining these bindings
2021 solves (part of) the polymorphism, higher-order values and
2022 unrepresentable literals in an expression.
2024 \refdef{substitution notation}
2034 -------------------------- \lam{Ei} has a non-representable type.
2036 a0 = E0 [ai=>Ei] \vdots
2037 ai-1 = Ei-1 [ai=>Ei]
2038 ai+1 = Ei+1 [ai=>Ei]
2057 x = fromInteger (smallInteger 10)
2059 (λb -> add b 1) (add 1 x)
2062 \transexample{nonrepinline}{Non-representable binding inlining}{from}{to}
2064 \subsubsection[sec:normalization:specialize]{Function specialization}
2065 This transform removes arguments to user-defined functions that are
2066 not representable at run-time. This is done by creating a
2067 \emph{specialized} version of the function that only works for one
2068 particular value of that argument (in other words, the argument can be
2071 Specialization means to create a specialized version of the called
2072 function, with one argument already filled in. As a simple example, in
2073 the following program (this is not actual Core, since it directly uses
2074 a literal with the unrepresentable type \lam{GHC.Prim.Int\#}).
2081 We could specialize the function \lam{f} against the literal argument
2082 1, with the following result:
2089 In some way, this transformation is similar to β-reduction, but it
2090 operates across function boundaries. It is also similar to
2091 non-representable let binding inlining above, since it sort of
2092 \quote{inlines} an expression into a called function.
2094 Special care must be taken when the argument has any free variables.
2095 If this is the case, the original argument should not be removed
2096 completely, but replaced by all the free variables of the expression.
2097 In this way, the original expression can still be evaluated inside the
2100 To prevent us from propagating the same argument over and over, a
2101 simple local variable reference is not propagated (since is has
2102 exactly one free variable, itself, we would only replace that argument
2105 This shows that any free local variables that are not run-time
2106 representable cannot be brought into normal form by this transform. We
2107 rely on an inlining or β-reduction transformation to replace such a
2108 variable with an expression we can propagate again.
2113 x Y0 ... Yi ... Yn \lam{Yi} is not representable
2114 --------------------------------------------- \lam{Yi} is not a local variable reference
2115 x' Y0 ... Yi-1 f0 ... fm Yi+1 ... Yn \lam{f0 ... fm} are all free local vars of \lam{Yi}
2116 ~ \lam{T0 ... Tn} are the types of \lam{Y0 ... Yn}
2117 x' = λ(y0 :: T0) ... λ(yi-1 :: Ty-1).
2119 λ(yi+1 :: Ty+1) ... λ(yn :: Tn).
2120 E y0 ... yi-1 Yi yi+1 ... yn
2123 This is a bit of a complex transformation. It transforms an
2124 application of the function \lam{x}, where one of the arguments
2125 (\lam{Y_i}) is not representable. A new
2126 function \lam{x'} is created that wraps the body of the old function.
2127 The body of the new function becomes a number of nested lambda
2128 abstractions, one for each of the original arguments that are left
2131 The ith argument is replaced with the free variables of
2132 \lam{Y_i}. Note that we reuse the same binders as those used in
2133 \lam{Y_i}, since we can then just use \lam{Y_i} inside the new
2134 function body and have all of the variables it uses be in scope.
2136 The argument that we are specializing for, \lam{Y_i}, is put inside
2137 the new function body. The old function body is applied to it. Since
2138 we use this new function only in place of an application with that
2139 particular argument \lam{Y_i}, behavior should not change.
2141 Note that the types of the arguments of our new function are taken
2142 from the types of the \emph{actual} arguments (\lam{T0 ... Tn}). This
2143 means that any polymorphism in the arguments is removed, even when the
2144 corresponding explicit type lambda is not removed
2147 \todo{Examples. Perhaps reference the previous sections}
2149 \section{Unsolved problems}
2150 The above system of transformations has been implemented in the prototype
2151 and seems to work well to compile simple and more complex examples of
2152 hardware descriptions. \todo{Ref christiaan?} However, this normalization
2153 system has not seen enough review and work to be complete and work for
2154 every Core expression that is supplied to it. A number of problems
2155 have already been identified and are discussed in this section.
2157 \subsection[sec:normalization:duplicatework]{Work duplication}
2158 A possible problem of β-reduction is that it could duplicate work.
2159 When the expression applied is not a simple variable reference, but
2160 requires calculation and the binder the lambda abstraction binds to
2161 is used more than once, more hardware might be generated than strictly
2164 As an example, consider the expression:
2170 When applying β-reduction to this expression, we get:
2176 which of course calculates \lam{(a * b)} twice.
2178 A possible solution to this would be to use the following alternative
2179 transformation, which is of course no longer normal β-reduction. The
2180 following transformation has not been tested in the prototype, but is
2181 given here for future reference:
2189 This does not seem like much of an improvement, but it does get rid of
2190 the lambda expression (and the associated higher-order value), while
2191 at the same time introducing a new let binding. Since the result of
2192 every application or case expression must be bound by a let expression
2193 in the intended normal form anyway, this is probably not a problem. If
2194 the argument happens to be a variable reference, then simple let
2195 binding removal (\in{section}[sec:normalization:simplelet]) will
2196 remove it, making the result identical to that of the original
2197 β-reduction transformation.
2199 When also applying argument simplification to the above example, we
2200 get the following expression:
2208 Looking at this, we could imagine an alternative approach: create a
2209 transformation that removes let bindings that bind identical values.
2210 In the above expression, the \lam{y} and \lam{z} variables could be
2211 merged together, resulting in the more efficient expression:
2214 let y = (a * b) in y + y
2217 \subsection[sec:normalization:non-determinism]{Non-determinism}
2218 As an example, again consider the following expression:
2224 We can apply both β-reduction (\in{section}[sec:normalization:beta])
2225 as well as argument simplification
2226 (\in{section}[sec:normalization:argsimpl]) to this expression.
2228 When applying argument simplification first and then β-reduction, we
2229 get the following expression:
2232 let y = (a * b) in y + y
2235 When applying β-reduction first and then argument simplification, we
2236 get the following expression:
2244 As you can see, this is a different expression. This means that the
2245 order of expressions, does in fact change the resulting normal form,
2246 which is something that we would like to avoid. In this particular
2247 case one of the alternatives is even clearly more efficient, so we
2248 would of course like the more efficient form to be the normal form.
2250 For this particular problem, the solutions for duplication of work
2251 seem from the previous section seem to fix the determinism of our
2252 transformation system as well. However, it is likely that there are
2253 other occurrences of this problem.
2255 \subsection[sec:normalization:castproblems]{Casts}
2256 We do not fully understand the use of cast expressions in Core, so
2257 there are probably expressions involving cast expressions that cannot
2258 be brought into intended normal form by this transformation system.
2260 The uses of casts in the Core system should be investigated more and
2261 transformations will probably need updating to handle them in all
2264 \subsection[sec:normalization:stateproblems]{Normalization of stateful descriptions}
2265 Currently, the intended normal form definition\refdef{intended
2266 normal form definition} offers enough freedom to describe all
2267 valid stateful descriptions, but is not limiting enough. It is
2268 possible to write descriptions which are in intended normal
2269 form, but cannot be translated into \VHDL\ in a meaningful way
2270 (\eg, a function that swaps two substates in its result, or a
2271 function that changes a sub-state itself instead of passing it to
2274 It is now up to the programmer to not do anything funny with
2275 these state values, whereas the normalization just tries not to
2276 mess up the flow of state values. In practice, there are
2277 situations where a Core program that \emph{could} be a valid
2278 stateful description is not translatable by the prototype. This
2279 most often happens when statefulness is mixed with pattern
2280 matching, causing a state input to be unpacked multiple times or
2281 be unpacked and repacked only in some of the code paths.
2283 Without going into detail about the exact problems (of which
2284 there are probably more than have shown up so far), it seems
2285 unlikely that these problems can be solved entirely by just
2286 improving the \VHDL\ state generation in the final stage. The
2287 normalization stage seems the best place to apply the rewriting
2288 needed to support more complex stateful descriptions. This does
2289 of course mean that the intended normal form definition must be
2290 extended as well to be more specific about how state handling
2291 should look like in normal form.
2292 \in{Section}[sec:prototype:statelimits] already contains a
2293 tight description of the limitations on the use of state
2294 variables, which could be adapted into the intended normal form.
2296 \section[sec:normalization:properties]{Provable properties}
2297 When looking at the system of transformations outlined above, there are a
2298 number of questions that we can ask ourselves. The main question is of course:
2299 \quote{Does our system work as intended?}. We can split this question into a
2300 number of sub-questions:
2303 \item[q:termination] Does our system \emph{terminate}? Since our system will
2304 keep running as long as transformations apply, there is an obvious risk that
2305 it will keep running indefinitely. This typically happens when one
2306 transformation produces a result that is transformed back to the original
2307 by another transformation, or when one or more transformations keep
2308 expanding some expression.
2309 \item[q:soundness] Is our system \emph{sound}? Since our transformations
2310 continuously modify the expression, there is an obvious risk that the final
2311 normal form will not be equivalent to the original program: its meaning could
2313 \item[q:completeness] Is our system \emph{complete}? Since we have a complex
2314 system of transformations, there is an obvious risk that some expressions will
2315 not end up in our intended normal form, because we forgot some transformation.
2316 In other words: does our transformation system result in our intended normal
2317 form for all possible inputs?
2318 \item[q:determinism] Is our system \emph{deterministic}? Since we have defined
2319 no particular order in which the transformation should be applied, there is an
2320 obvious risk that different transformation orderings will result in
2321 \emph{different} normal forms. They might still both be intended normal forms
2322 (if our system is \emph{complete}) and describe correct hardware (if our
2323 system is \emph{sound}), so this property is less important than the previous
2324 three: the translator would still function properly without it.
2327 Unfortunately, the final transformation system has only been
2328 developed in the final part of the research, leaving no more time
2329 for verifying these properties. In fact, it is likely that the
2330 current transformation system still violates some of these
2331 properties in some cases (see
2332 \in{section}[sec:normalization:non-determinism] and
2333 \in{section}[sec:normalization:stateproblems]) and should be improved (or
2334 extra conditions on the input hardware descriptions should be formulated).
2336 This is most likely the case with the completeness and determinism
2337 properties, perhaps also the termination property. The soundness
2338 property probably holds, since it is easier to manually verify (each
2339 transformation can be reviewed separately).
2341 Even though no complete proofs have been made, some ideas for
2342 possible proof strategies are shown below.
2344 \subsection{Graph representation}
2345 Before looking into how to prove these properties, we will look at
2346 transformation systems from a graph perspective. We will first define
2347 the graph view and then illustrate it using a simple example from lambda
2348 calculus (which is a different system than the Cλash normalization
2349 system). The nodes of the graph are all possible Core expressions. The
2350 (directed) edges of the graph are transformations. When a transformation
2351 α applies to an expression \lam{A} to produce an expression \lam{B}, we
2352 add an edge from the node for \lam{A} to the node for \lam{B}, labeled
2355 \startuseMPgraphic{TransformGraph}
2359 newCircle.a(btex \lam{(λx.λy. (+) x y) 1} etex);
2360 newCircle.b(btex \lam{λy. (+) 1 y} etex);
2361 newCircle.c(btex \lam{(λx.(+) x) 1} etex);
2362 newCircle.d(btex \lam{(+) 1} etex);
2365 c.c = b.c + (4cm, 0cm);
2366 a.c = midpoint(b.c, c.c) + (0cm, 4cm);
2367 d.c = midpoint(b.c, c.c) - (0cm, 3cm);
2369 % β-conversion between a and b
2370 ncarc.a(a)(b) "name(bred)";
2371 ObjLabel.a(btex $\xrightarrow[normal]{}{β}$ etex) "labpathname(bred)", "labdir(rt)";
2372 ncarc.b(b)(a) "name(bexp)", "linestyle(dashed withdots)";
2373 ObjLabel.b(btex $\xleftarrow[normal]{}{β}$ etex) "labpathname(bexp)", "labdir(lft)";
2375 % η-conversion between a and c
2376 ncarc.a(a)(c) "name(ered)";
2377 ObjLabel.a(btex $\xrightarrow[normal]{}{η}$ etex) "labpathname(ered)", "labdir(rt)";
2378 ncarc.c(c)(a) "name(eexp)", "linestyle(dashed withdots)";
2379 ObjLabel.c(btex $\xleftarrow[normal]{}{η}$ etex) "labpathname(eexp)", "labdir(lft)";
2381 % η-conversion between b and d
2382 ncarc.b(b)(d) "name(ered)";
2383 ObjLabel.b(btex $\xrightarrow[normal]{}{η}$ etex) "labpathname(ered)", "labdir(rt)";
2384 ncarc.d(d)(b) "name(eexp)", "linestyle(dashed withdots)";
2385 ObjLabel.d(btex $\xleftarrow[normal]{}{η}$ etex) "labpathname(eexp)", "labdir(lft)";
2387 % β-conversion between c and d
2388 ncarc.c(c)(d) "name(bred)";
2389 ObjLabel.c(btex $\xrightarrow[normal]{}{β}$ etex) "labpathname(bred)", "labdir(rt)";
2390 ncarc.d(d)(c) "name(bexp)", "linestyle(dashed withdots)";
2391 ObjLabel.d(btex $\xleftarrow[normal]{}{β}$ etex) "labpathname(bexp)", "labdir(lft)";
2393 % Draw objects and lines
2394 drawObj(a, b, c, d);
2397 \placeexample[right][ex:TransformGraph]{Partial graph of a lambda calculus
2398 system with β and η reduction (solid lines) and expansion (dotted lines).}
2399 \boxedgraphic{TransformGraph}
2401 Of course the graph for Cλash is unbounded, since we can construct an
2402 infinite amount of Core expressions. Also, there might potentially be
2403 multiple edges between two given nodes (with different labels), though
2404 this seems unlikely to actually happen in our system.
2406 See \in{example}[ex:TransformGraph] for the graph representation of a very
2407 simple lambda calculus that contains just the expressions \lam{(λx.λy. (+) x
2408 y) 1}, \lam{λy. (+) 1 y}, \lam{(λx.(+) x) 1} and \lam{(+) 1}. The
2409 transformation system consists of β-reduction and η-reduction (solid edges) or
2410 β-expansion and η-expansion (dotted edges).
2412 \todo{Define β-reduction and η-reduction?}
2414 In such a graph a node (expression) is in normal form if it has no
2415 outgoing edges (meaning no transformation applies to it). The set of
2416 nodes without outgoing edges is called the \emph{normal set}. Similarly,
2417 the set of nodes containing expressions in intended normal form
2418 \refdef{intended normal form definition} is called the \emph{intended normal set}.
2420 From such a graph, we can derive some properties easily:
2422 \item A system will \emph{terminate} if there is no walk (sequence of
2423 edges, or transformations) of infinite length in the graph (this
2424 includes cycles, but can also happen without cycles).
2425 \item Soundness is not easily represented in the graph.
2426 \item A system is \emph{complete} if all of the nodes in the normal set have
2427 the intended normal form. The inverse (that all of the nodes outside of
2428 the normal set are \emph{not} in the intended normal form) is not
2429 strictly required. In other words, our normal set must be a
2430 subset of the intended normal form, but they do not need to be
2433 \item A system is deterministic if all paths starting at a particular
2434 node, which end in a node in the normal set, end at the same node.
2437 When looking at the \in{example}[ex:TransformGraph], we see that the system
2438 terminates for both the reduction and expansion systems (but note that, for
2439 expansion, this is only true because we have limited the possible
2440 expressions. In complete lambda calculus, there would be a path from
2441 \lam{(λx.λy. (+) x y) 1} to \lam{(λx.λy.(λz.(+) z) x y) 1} to
2442 \lam{(λx.λy.(λz.(λq.(+) q) z) x y) 1} etc.)
2444 If we would consider the system with both expansion and reduction, there
2445 would no longer be termination either, since there would be cycles all
2448 The reduction and expansion systems have a normal set of containing just
2449 \lam{(+) 1} or \lam{(λx.λy. (+) x y) 1} respectively. Since all paths in
2450 either system end up in these normal forms, both systems are \emph{complete}.
2451 Also, since there is only one node in the normal set, it must obviously be
2452 \emph{deterministic} as well.
2454 \subsection{Termination}
2455 In general, proving termination of an arbitrary program is a very
2456 hard problem. \todo{Ref about arbitrary termination} Fortunately,
2457 we only have to prove termination for our specific transformation
2460 A common approach for these kinds of proofs is to associate a
2461 measure with each possible expression in our system. If we can
2462 show that each transformation strictly decreases this measure
2463 (\ie, the expression transformed to has a lower measure than the
2464 expression transformed from). \todo{ref about measure-based
2465 termination proofs / analysis}
2467 A good measure for a system consisting of just β-reduction would
2468 be the number of lambda expressions in the expression. Since every
2469 application of β-reduction removes a lambda abstraction (and there
2470 is always a bounded number of lambda abstractions in every
2471 expression) we can easily see that a transformation system with
2472 just β-reduction will always terminate.
2474 For our complete system, this measure would be fairly complex
2475 (probably the sum of a lot of things). Since the (conditions on)
2476 our transformations are pretty complex, we would need to include
2477 both simple things like the number of let expressions as well as
2478 more complex things like the number of case expressions that are
2479 not yet in normal form.
2481 No real attempt has been made at finding a suitable measure for
2484 \subsection{Soundness}
2485 Soundness is a property that can be proven for each transformation
2486 separately. Since our system only runs separate transformations
2487 sequentially, if each of our transformations leaves the
2488 \emph{meaning} of the expression unchanged, then the entire system
2489 will of course leave the meaning unchanged and is thus
2492 The current prototype has only been verified in an ad hoc fashion
2493 by inspecting (the code for) each transformation. A more formal
2494 verification would be more appropriate.
2496 To be able to formally show that each transformation properly
2497 preserves the meaning of every expression, we require an exact
2498 definition of the \emph{meaning} of every expression, so we can
2499 compare them. A definition of the operational semantics of \GHC's Core
2500 language is available \cite[sulzmann07], but this does not seem
2501 sufficient for our goals (but it is a good start).
2503 It should be possible to have a single formal definition of
2504 meaning for Core for both normal Core compilation by \GHC\ and for
2505 our compilation to \VHDL. The main difference seems to be that in
2506 hardware every expression is always evaluated, while in software
2507 it is only evaluated if needed, but it should be possible to
2508 assign a meaning to Core expressions that assumes neither.
2510 Since each of the transformations can be applied to any
2511 sub-expression as well, there is a constraint on our meaning
2512 definition: the meaning of an expression should depend only on the
2513 meaning of sub-expressions, not on the expressions themselves. For
2514 example, the meaning of the application in \lam{f (let x = 4 in
2515 x)} should be the same as the meaning of the application in \lam{f
2516 4}, since the argument sub-expression has the same meaning (though
2517 the actual expression is different).
2519 \subsection{Completeness}
2520 Proving completeness is probably not hard, but it could be a lot
2521 of work. We have seen above that to prove completeness, we must
2522 show that the normal set of our graph representation is a subset
2523 of the intended normal set.
2525 However, it is hard to systematically generate or reason about the
2526 normal set, since it is defined as any nodes to which no
2527 transformation applies. To determine this set, each transformation
2528 must be considered and when a transformation is added, the entire
2529 set should be re-evaluated. This means it is hard to show that
2530 each node in the normal set is also in the intended normal set.
2531 Reasoning about our intended normal set is easier, since we know
2532 how to generate it from its definition. \refdef{intended normal
2535 Fortunately, we can also prove the complement (which is
2536 equivalent, since $A \subseteq B \Leftrightarrow \overline{B}
2537 \subseteq \overline{A}$): show that the set of nodes not in
2538 intended normal form is a subset of the set of nodes not in normal
2539 form. In other words, show that for every expression that is not
2540 in intended normal form, that there is at least one transformation
2541 that applies to it (since that means it is not in normal form
2542 either and since $A \subseteq C \Leftrightarrow \forall x (x \in A
2543 \rightarrow x \in C)$).
2545 By systematically reviewing the entire Core language definition
2546 along with the intended normal form definition (both of which have
2547 a similar structure), it should be possible to identify all
2548 possible (sets of) Core expressions that are not in intended
2549 normal form and identify a transformation that applies to it.
2551 This approach is especially useful for proving completeness of our
2552 system, since if expressions exist to which none of the
2553 transformations apply (\ie\ if the system is not yet complete), it
2554 is immediately clear which expressions these are and adding
2555 (or modifying) transformations to fix this should be relatively
2558 As observed above, applying this approach is a lot of work, since
2559 we need to check every (set of) transformation(s) separately.
2561 \todo{Perhaps do a few steps of the proofs as proof-of-concept}
2562 \subsection{Determinism}
2563 A well-known technique for proving determinism in lambda calculus
2564 and other reduction systems, is using the Church-Rosser property
2565 \cite[church36]. A reduction system has the CR property if and only if:
2567 \placedefinition[here]{Church-Rosser theorem}
2568 {\lam{\forall A, B, C \exists D (A ->> B ∧ A ->> C => B ->> D ∧ C ->> D)}}
2570 Here, \lam{A ->> B} means \lam{A} \emph{reduces to} \lam{B}. In
2571 other words, there is a set of transformations that can transform
2572 \lam{A} to \lam{B}. \lam{=>} is used to mean \emph{implies}.
2574 For a transformation system holding the Church-Rosser property, it
2575 is easy to show that it is in fact deterministic. Showing that this
2576 property actually holds is a harder problem, but has been
2577 done for some reduction systems in the lambda calculus
2578 \cite[klop80]\ \cite[barendregt84]. Doing the same for our
2579 transformation system is probably more complicated, but not
2582 % vim: set sw=2 sts=2 expandtab: