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
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12 \setuptyping[option=none,style=\tttf,strip=auto]
16 \define[4]\transexample{
17 \placeexample[here][ex:trans:#1]{#2}
18 \startcombination[2*1]
19 {\example{#3}}{Original program}
20 {\example{#4}}{Transformed program}
24 The first step in the core to \small{VHDL} translation process, is normalization. We
25 aim to bring the core description into a simpler form, which we can
26 subsequently translate into \small{VHDL} easily. This normal form is needed because
27 the full core language is more expressive than \small{VHDL} in some
28 areas (higher-order expressions, limited polymorphism using type
29 classes, etc.) and because core can describe expressions that do not
30 have a direct hardware interpretation.
33 The transformations described here have a well-defined goal: to bring the
34 program in a well-defined form that is directly translatable to
35 \VHDL, while fully preserving the semantics of the program. We refer
36 to this form as the \emph{normal form} of the program. The formal
37 definition of this normal form is quite simple:
39 \placedefinition{}{\startboxed A program is in \emph{normal form} if none of the
40 transformations from this chapter apply.\stopboxed}
42 Of course, this is an \quote{easy} definition of the normal form, since our
43 program will end up in normal form automatically. The more interesting part is
44 to see if this normal form actually has the properties we would like it to
47 But, before getting into more definitions and details about this normal
48 form, let us try to get a feeling for it first. The easiest way to do this
49 is by describing the things that are unwanted in the intended normal form.
52 \item Any \emph{polymorphism} must be removed. When laying down hardware, we
53 cannot generate any signals that can have multiple types. All types must be
54 completely known to generate hardware.
56 \item All \emph{higher-order} constructions must be removed. We cannot
57 generate a hardware signal that contains a function, so all values,
58 arguments and return values used must be first order.
60 \item All complex \emph{nested scopes} must be removed. In the \small{VHDL}
61 description, every signal is in a single scope. Also, full expressions are
62 not supported everywhere (in particular port maps can only map signal
63 names and constants, not complete expressions). To make the \small{VHDL}
64 generation easy, a separate binder must be bound to ever application or
68 \todo{Intermezzo: functions vs plain values}
70 A very simple example of a program in normal form is given in
71 \in{example}[ex:MulSum]. As you can see, all arguments to the function (which
72 will become input ports in the generated \VHDL) are at the outer level.
73 This means that the body of the inner lambda abstraction is never a
74 function, but always a plain value.
76 As the body of the inner lambda abstraction, we see a single (recursive)
77 let expression, that binds two variables (\lam{mul} and \lam{sum}). These
78 variables will be signals in the generated \VHDL, bound to the output port
79 of the \lam{*} and \lam{+} components.
81 The final line (the \quote{return value} of the function) selects the
82 \lam{sum} signal to be the output port of the function. This \quote{return
83 value} can always only be a variable reference, never a more complex
86 \todo{Add generated VHDL}
89 alu :: Bit -> Word -> Word -> Word
98 \startuseMPgraphic{MulSum}
99 save a, b, c, mul, add, sum;
102 newCircle.a(btex $a$ etex) "framed(false)";
103 newCircle.b(btex $b$ etex) "framed(false)";
104 newCircle.c(btex $c$ etex) "framed(false)";
105 newCircle.sum(btex $sum$ etex) "framed(false)";
108 newCircle.mul(btex * etex);
109 newCircle.add(btex + etex);
111 a.c - b.c = (0cm, 2cm);
112 b.c - c.c = (0cm, 2cm);
113 add.c = c.c + (2cm, 0cm);
114 mul.c = midpoint(a.c, b.c) + (2cm, 0cm);
115 sum.c = add.c + (2cm, 0cm);
118 % Draw objects and lines
119 drawObj(a, b, c, mul, add, sum);
121 ncarc(a)(mul) "arcangle(15)";
122 ncarc(b)(mul) "arcangle(-15)";
128 \placeexample[here][ex:MulSum]{Simple architecture consisting of a
129 multiplier and a subtractor.}
130 \startcombination[2*1]
131 {\typebufferlam{MulSum}}{Core description in normal form.}
132 {\boxedgraphic{MulSum}}{The architecture described by the normal form.}
135 \in{Example}[ex:MulSum] showed a function that just applied two
136 other functions (multiplication and addition), resulting in a simple
137 architecture with two components and some connections. There is of
138 course also some mechanism for choice in the normal form. In a
139 normal Core program, the \emph{case} expression can be used in a few
140 different ways to describe choice. In normal form, this is limited
141 to a very specific form.
143 \in{Example}[ex:AddSubAlu] shows an example describing a
144 simple \small{ALU}, which chooses between two operations based on an opcode
145 bit. The main structure is similar to \in{example}[ex:MulSum], but this
146 time the \lam{res} variable is bound to a case expression. This case
147 expression scrutinizes the variable \lam{opcode} (and scrutinizing more
148 complex expressions is not supported). The case expression can select a
149 different variable based on the constructor of \lam{opcode}.
150 \refdef{case expression}
152 \startbuffer[AddSubAlu]
153 alu :: Bit -> Word -> Word -> Word
165 \startuseMPgraphic{AddSubAlu}
166 save opcode, a, b, add, sub, mux, res;
169 newCircle.opcode(btex $opcode$ etex) "framed(false)";
170 newCircle.a(btex $a$ etex) "framed(false)";
171 newCircle.b(btex $b$ etex) "framed(false)";
172 newCircle.res(btex $res$ etex) "framed(false)";
174 newCircle.add(btex + etex);
175 newCircle.sub(btex - etex);
178 opcode.c - a.c = (0cm, 2cm);
179 add.c - a.c = (4cm, 0cm);
180 sub.c - b.c = (4cm, 0cm);
181 a.c - b.c = (0cm, 3cm);
182 mux.c = midpoint(add.c, sub.c) + (1.5cm, 0cm);
183 res.c - mux.c = (1.5cm, 0cm);
186 % Draw objects and lines
187 drawObj(opcode, a, b, res, add, sub, mux);
189 ncline(a)(add) "posA(e)";
190 ncline(b)(sub) "posA(e)";
191 nccurve(a)(sub) "posA(e)", "angleA(0)";
192 nccurve(b)(add) "posA(e)", "angleA(0)";
193 nccurve(add)(mux) "posB(inpa)", "angleB(0)";
194 nccurve(sub)(mux) "posB(inpb)", "angleB(0)";
195 nccurve(opcode)(mux) "posB(n)", "angleA(0)", "angleB(-90)";
196 ncline(mux)(res) "posA(out)";
199 \placeexample[here][ex:AddSubAlu]{Simple \small{ALU} supporting two operations.}
200 \startcombination[2*1]
201 {\typebufferlam{AddSubAlu}}{Core description in normal form.}
202 {\boxedgraphic{AddSubAlu}}{The architecture described by the normal form.}
205 As a more complete example, consider
206 \in{example}[ex:NormalComplete]. This example shows everything that
207 is allowed in normal form, except for built-in higher-order functions
208 (like \lam{map}). The graphical version of the architecture contains
209 a slightly simplified version, since the state tuple packing and
210 unpacking have been left out. Instead, two separate registers are
211 drawn. Most synthesis tools will further optimize this architecture by
212 removing the multiplexers at the register input and instead use the write
213 enable port of the register (when it is available), but we want to show
214 the architecture as close to the description as possible.
216 As you can see from the previous examples, the generation of the final
217 architecture from the normal form is straightforward. In each of the
218 examples, there is a direct match between the normal form structure,
219 the generated VHDL and the architecture shown in the images.
221 \startbuffer[NormalComplete]
224 -> State (Word, Word)
225 -> (State (Word, Word), Word)
227 -- All arguments are an inital lambda (address, data, packed state)
229 -- There are nested let expressions at top level
231 -- Unpack the state by coercion (\eg, cast from
232 -- State (Word, Word) to (Word, Word))
233 s = sp ▶ (Word, Word)
234 -- Extract both registers from the state
235 r1 = case s of (a, b) -> a
236 r2 = case s of (a, b) -> b
237 -- Calling some other user-defined function.
239 -- Conditional connections
251 -- pack the state by coercion (\eg, cast from
252 -- (Word, Word) to State (Word, Word))
253 sp' = s' ▶ State (Word, Word)
254 -- Pack our return value
261 \startuseMPgraphic{NormalComplete}
262 save a, d, r, foo, muxr, muxout, out;
265 newCircle.a(btex \lam{a} etex) "framed(false)";
266 newCircle.d(btex \lam{d} etex) "framed(false)";
267 newCircle.out(btex \lam{out} etex) "framed(false)";
269 %newCircle.add(btex + etex);
270 newBox.foo(btex \lam{foo} etex);
271 newReg.r1(btex $\lam{r1}$ etex) "dx(4mm)", "dy(6mm)";
272 newReg.r2(btex $\lam{r2}$ etex) "dx(4mm)", "dy(6mm)", "reflect(true)";
274 % Reflect over the vertical axis
275 reflectObj(muxr1)((0,0), (0,1));
278 rotateObj(muxout)(-90);
280 d.c = foo.c + (0cm, 1.5cm);
281 a.c = (xpart r2.c + 2cm, ypart d.c - 0.5cm);
282 foo.c = midpoint(muxr1.c, muxr2.c) + (0cm, 2cm);
283 muxr1.c = r1.c + (0cm, 2cm);
284 muxr2.c = r2.c + (0cm, 2cm);
285 r2.c = r1.c + (4cm, 0cm);
287 muxout.c = midpoint(r1.c, r2.c) - (0cm, 2cm);
288 out.c = muxout.c - (0cm, 1.5cm);
290 % % Draw objects and lines
291 drawObj(a, d, foo, r1, r2, muxr1, muxr2, muxout, out);
294 nccurve(foo)(muxr1) "angleA(-90)", "posB(inpa)", "angleB(180)";
295 nccurve(foo)(muxr2) "angleA(-90)", "posB(inpb)", "angleB(0)";
296 nccurve(muxr1)(r1) "posA(out)", "angleA(180)", "posB(d)", "angleB(0)";
297 nccurve(r1)(muxr1) "posA(out)", "angleA(0)", "posB(inpb)", "angleB(180)";
298 nccurve(muxr2)(r2) "posA(out)", "angleA(0)", "posB(d)", "angleB(180)";
299 nccurve(r2)(muxr2) "posA(out)", "angleA(180)", "posB(inpa)", "angleB(0)";
300 nccurve(r1)(muxout) "posA(out)", "angleA(0)", "posB(inpb)", "angleB(-90)";
301 nccurve(r2)(muxout) "posA(out)", "angleA(180)", "posB(inpa)", "angleB(-90)";
303 nccurve(a)(muxout) "angleA(-90)", "angleB(180)", "posB(sel)";
304 nccurve(a)(muxr1) "angleA(180)", "angleB(-90)", "posB(sel)";
305 nccurve(a)(muxr2) "angleA(180)", "angleB(-90)", "posB(sel)";
306 ncline(muxout)(out) "posA(out)";
309 \todo{Don't split registers in this image?}
310 \placeexample[here][ex:NormalComplete]{Simple architecture consisting of an adder and a
312 \startcombination[2*1]
313 {\typebufferlam{NormalComplete}}{Core description in normal form.}
314 {\boxedgraphic{NormalComplete}}{The architecture described by the normal form.}
319 \subsection[sec:normalization:intendednormalform]{Intended normal form definition}
320 Now we have some intuition for the normal form, we can describe how we want
321 the normal form to look like in a slightly more formal manner. The following
322 EBNF-like description captures most of the intended structure (and
323 generates a subset of \GHC's core format).
325 There are two things missing: cast expressions are sometimes
326 allowed by the prototype, but not specified here and the below
327 definition allows uses of state that cannot be translated to \VHDL\
328 properly. These two problems are discussed in
329 \in{section}[sec:normalization:castproblems] and
330 \in{section}[sec:normalization:stateproblems] respectively.
332 Some clauses have an expression listed behind them in parentheses.
333 These are conditions that need to apply to the clause. The
334 predicates used there (\lam{lvar()}, \lam{representable()},
335 \lam{gvar()}) will be defined in
336 \in{section}[sec:normalization:predicates].
338 An expression is in normal form if it matches the first
339 definition, \emph{normal}.
341 \todo{Fix indentation}
342 \startbuffer[IntendedNormal]
343 \italic{normal} := \italic{lambda}
344 \italic{lambda} := λvar.\italic{lambda} (representable(var))
346 \italic{toplet} := letrec [\italic{binding}...] in var (representable(var))
347 \italic{binding} := var = \italic{rhs} (representable(rhs))
348 -- State packing and unpacking by coercion
349 | var0 = var1 ▶ State ty (lvar(var1))
350 | var0 = var1 ▶ ty (var1 :: State ty ∧ lvar(var1))
351 \italic{rhs} := \italic{userapp}
352 | \italic{builtinapp}
354 | case var of C a0 ... an -> ai (lvar(var))
356 | case var of (lvar(var))
357 [ DEFAULT -> var ] (lvar(var))
358 C0 w0,0 ... w0,n -> var0
360 Cm wm,0 ... wm,n -> varm (\forall{}i \forall{}j, wi,j \neq vari, lvar(vari))
361 \italic{userapp} := \italic{userfunc}
362 | \italic{userapp} {userarg}
363 \italic{userfunc} := var (gvar(var))
364 \italic{userarg} := var (lvar(var))
365 \italic{builtinapp} := \italic{builtinfunc}
366 | \italic{builtinapp} \italic{builtinarg}
367 \italic{built-infunc} := var (bvar(var))
368 \italic{built-inarg} := var (representable(var) ∧ lvar(var))
369 | \italic{partapp} (partapp :: a -> b)
370 | \italic{coreexpr} (¬representable(coreexpr) ∧ ¬(coreexpr :: a -> b))
371 \italic{partapp} := \italic{userapp}
372 | \italic{builtinapp}
375 \placedefinition[][def:IntendedNormal]{Definition of the intended nnormal form using an \small{EBNF}-like syntax.}
376 {\defref{intended normal form definition}
377 \typebufferlam{IntendedNormal}}
379 When looking at such a program from a hardware perspective, the top
380 level lambda abstractions (\italic{lambda}) define the input ports.
381 Lambda abstractions cannot appear anywhere else. The variable reference
382 in the body of the recursive let expression (\italic{toplet}) is the
383 output port. Most binders bound by the let expression define a
384 component instantiation (\italic{userapp}), where the input and output
385 ports are mapped to local signals (\italic{userarg}). Some of the others
386 use a built-in construction (\eg\ the \lam{case} expression) or call a
387 built-in function (\italic{builtinapp}) such as \lam{+} or \lam{map}.
388 For these, a hardcoded \small{VHDL} translation is available.
390 \section[sec:normalization:transformation]{Transformation notation}
391 To be able to concisely present transformations, we use a specific format
392 for them. It is a simple format, similar to one used in logic reasoning.
394 Such a transformation description looks like the following.
399 <original expression>
400 -------------------------- <expression conditions>
401 <transformed expression>
406 This format describes a transformation that applies to \lam{<original
407 expression>} and transforms it into \lam{<transformed expression>}, assuming
408 that all conditions are satisfied. In this format, there are a number of placeholders
409 in pointy brackets, most of which should be rather obvious in their meaning.
410 Nevertheless, we will more precisely specify their meaning below:
412 \startdesc{<original expression>} The expression pattern that will be matched
413 against (subexpressions of) the expression to be transformed. We call this a
414 pattern, because it can contain \emph{placeholders} (variables), which match
415 any expression or binder. Any such placeholder is said to be \emph{bound} to
416 the expression it matches. It is convention to use an uppercase letter (\eg\
417 \lam{M} or \lam{E}) to refer to any expression (including a simple variable
418 reference) and lowercase letters (\eg\ \lam{v} or \lam{b}) to refer to
419 (references to) binders.
421 For example, the pattern \lam{a + B} will match the expression
422 \lam{v + (2 * w)} (binding \lam{a} to \lam{v} and \lam{B} to
423 \lam{(2 * w)}), but not \lam{(2 * w) + v}.
426 \startdesc{<expression conditions>}
427 These are extra conditions on the expression that is matched. These
428 conditions can be used to further limit the cases in which the
429 transformation applies, commonly to prevent a transformation from
430 causing a loop with itself or another transformation.
432 Only if these conditions are \emph{all} satisfied, the transformation
436 \startdesc{<context conditions>}
437 These are a number of extra conditions on the context of the function. In
438 particular, these conditions can require some (other) top level function to be
439 present, whose value matches the pattern given here. The format of each of
440 these conditions is: \lam{binder = <pattern>}.
442 Typically, the binder is some placeholder bound in the \lam{<original
443 expression>}, while the pattern contains some placeholders that are used in
444 the \lam{transformed expression}.
446 Only if a top level binder exists that matches each binder and pattern,
447 the transformation applies.
450 \startdesc{<transformed expression>}
451 This is the expression template that is the result of the transformation. If, looking
452 at the above three items, the transformation applies, the \lam{<original
453 expression>} is completely replaced by the \lam{<transformed expression>}.
454 We call this a template, because it can contain placeholders, referring to
455 any placeholder bound by the \lam{<original expression>} or the
456 \lam{<context conditions>}. The resulting expression will have those
457 placeholders replaced by the values bound to them.
459 Any binder (lowercase) placeholder that has no value bound to it yet will be
460 bound to (and replaced with) a fresh binder.
463 \startdesc{<context additions>}
464 These are templates for new functions to be added to the context.
465 This is a way to let a transformation create new top level
468 Each addition has the form \lam{binder = template}. As above, any
469 placeholder in the addition is replaced with the value bound to it, and any
470 binder placeholder that has no value bound to it yet will be bound to (and
471 replaced with) a fresh binder.
474 To understand this notation better, the step by step application of
475 the η-abstraction transformation to a simple \small{ALU} will be
476 shown. Consider η-abstraction, which is a common transformation from
477 labmda calculus, described using above notation as follows:
481 -------------- \lam{E} does not occur on a function position in an application
482 λx.E x \lam{E} is not a lambda abstraction.
485 η-abstraction is a well known transformation from lambda calculus. What
486 this transformation does, is take any expression that has a function type
487 and turn it into a lambda expression (giving an explicit name to the
488 argument). There are some extra conditions that ensure that this
489 transformation does not apply infinitely (which are not necessarily part
490 of the conventional definition of η-abstraction).
492 Consider the following function, in Core notation, which is a fairly obvious way to specify a
493 simple \small{ALU} (Note that it is not yet in normal form, but
494 \in{example}[ex:AddSubAlu] shows the normal form of this function).
495 The parentheses around the \lam{+} and \lam{-} operators are
496 commonly used in Haskell to show that the operators are used as
497 normal functions, instead of \emph{infix} operators (\eg, the
498 operators appear before their arguments, instead of in between).
501 alu :: Bit -> Word -> Word -> Word
502 alu = λopcode. case opcode of
507 There are a few subexpressions in this function to which we could possibly
508 apply the transformation. Since the pattern of the transformation is only
509 the placeholder \lam{E}, any expression will match that. Whether the
510 transformation applies to an expression is thus solely decided by the
511 conditions to the right of the transformation.
513 We will look at each expression in the function in a top down manner. The
514 first expression is the entire expression the function is bound to.
517 λopcode. case opcode of
522 As said, the expression pattern matches this. The type of this expression is
523 \lam{Bit -> Word -> Word -> Word}, which matches \lam{a -> b} (Note that in
524 this case \lam{a = Bit} and \lam{b = Word -> Word -> Word}).
526 Since this expression is at top level, it does not occur at a function
527 position of an application. However, The expression is a lambda abstraction,
528 so this transformation does not apply.
530 The next expression we could apply this transformation to, is the body of
531 the lambda abstraction:
539 The type of this expression is \lam{Word -> Word -> Word}, which again
540 matches \lam{a -> b}. The expression is the body of a lambda expression, so
541 it does not occur at a function position of an application. Finally, the
542 expression is not a lambda abstraction but a case expression, so all the
543 conditions match. There are no context conditions to match, so the
544 transformation applies.
546 By now, the placeholder \lam{E} is bound to the entire expression. The
547 placeholder \lam{x}, which occurs in the replacement template, is not bound
548 yet, so we need to generate a fresh binder for that. Let us use the binder
549 \lam{a}. This results in the following replacement expression:
557 Continuing with this expression, we see that the transformation does not
558 apply again (it is a lambda expression). Next we look at the body of this
567 Here, the transformation does apply, binding \lam{E} to the entire
568 expression (which has type \lam{Word -> Word}) and binding \lam{x}
569 to the fresh binder \lam{b}, resulting in the replacement:
577 The transformation does not apply to this lambda abstraction, so we
578 look at its body. For brevity, we will put the case expression on one line from
582 (case opcode of Low -> (+); High -> (-)) a b
585 The type of this expression is \lam{Word}, so it does not match \lam{a -> b}
586 and the transformation does not apply. Next, we have two options for the
587 next expression to look at: the function position and argument position of
588 the application. The expression in the argument position is \lam{b}, which
589 has type \lam{Word}, so the transformation does not apply. The expression in
590 the function position is:
593 (case opcode of Low -> (+); High -> (-)) a
596 Obviously, the transformation does not apply here, since it occurs in
597 function position (which makes the second condition false). In the same
598 way the transformation does not apply to both components of this
599 expression (\lam{case opcode of Low -> (+); High -> (-)} and \lam{a}), so
600 we will skip to the components of the case expression: the scrutinee and
601 both alternatives. Since the opcode is not a function, it does not apply
604 The first alternative is \lam{(+)}. This expression has a function type
605 (the operator still needs two arguments). It does not occur in function
606 position of an application and it is not a lambda expression, so the
607 transformation applies.
609 We look at the \lam{<original expression>} pattern, which is \lam{E}.
610 This means we bind \lam{E} to \lam{(+)}. We then replace the expression
611 with the \lam{<transformed expression>}, replacing all occurences of
612 \lam{E} with \lam{(+)}. In the \lam{<transformed expression>}, the This gives us the replacement expression:
613 \lam{λx.(+) x} (A lambda expression binding \lam{x}, with a body that
614 applies the addition operator to \lam{x}).
616 The complete function then becomes:
618 (case opcode of Low -> λa1.(+) a1; High -> (-)) a
621 Now the transformation no longer applies to the complete first alternative
622 (since it is a lambda expression). It does not apply to the addition
623 operator again, since it is now in function position in an application. It
624 does, however, apply to the application of the addition operator, since
625 that is neither a lambda expression nor does it occur in function
626 position. This means after one more application of the transformation, the
630 (case opcode of Low -> λa1.λb1.(+) a1 b1; High -> (-)) a
633 The other alternative is left as an exercise to the reader. The final
634 function, after applying η-abstraction until it does no longer apply is:
637 alu :: Bit -> Word -> Word -> Word
638 alu = λopcode.λa.b. (case opcode of
639 Low -> λa1.λb1 (+) a1 b1
640 High -> λa2.λb2 (-) a2 b2) a b
643 \subsection{Transformation application}
644 In this chapter we define a number of transformations, but how will we apply
645 these? As stated before, our normal form is reached as soon as no
646 transformation applies anymore. This means our application strategy is to
647 simply apply any transformation that applies, and continuing to do that with
648 the result of each transformation.
650 In particular, we define no particular order of transformations. Since
651 transformation order should not influence the resulting normal form,
652 this leaves the implementation free to choose any application order that
653 results in an efficient implementation. Unfortunately this is not
654 entirely true for the current set of transformations. See
655 \in{section}[sec:normalization:non-determinism] for a discussion of this
658 When applying a single transformation, we try to apply it to every (sub)expression
659 in a function, not just the top level function body. This allows us to
660 keep the transformation descriptions concise and powerful.
662 \subsection{Definitions}
663 A \emph{global variable} is any variable (binder) that is bound at the
664 top level of a program, or an external module. A \emph{local variable} is any
665 other variable (\eg, variables local to a function, which can be bound by
666 lambda abstractions, let expressions and pattern matches of case
667 alternatives). This is a slightly different notion of global versus
668 local than what \small{GHC} uses internally, but for our purposes
669 the distinction \GHC\ makes is not useful.
670 \defref{global variable} \defref{local variable}
672 A \emph{hardware representable} (or just \emph{representable}) type or value
673 is (a value of) a type that we can generate a signal for in hardware. For
674 example, a bit, a vector of bits, a 32 bit unsigned word, etc. Values that are
675 not runtime representable notably include (but are not limited to): types,
676 dictionaries, functions.
677 \defref{representable}
679 A \emph{built-in function} is a function supplied by the Cλash
680 framework, whose implementation is not used to generate \VHDL. This is
681 either because it is no valid Cλash (like most list functions that need
682 recursion) or because a Cλash implementation would be unwanted (for the
683 addition operator, for example, we would rather use the \VHDL addition
684 operator to let the synthesis tool decide what kind of adder to use
685 instead of explicitly describing one in Cλash). \defref{built-in
688 These are functions like \lam{map}, \lam{hwor}, \lam{+} and \lam{length}.
690 For these functions, Cλash has a \emph{built-in hardware translation},
691 so calls to these functions can still be translated. Built-in functions
692 must have a valid Haskell implementation, of course, to allow
695 A \emph{user-defined} function is a function for which no built-in
696 translation is available and whose definition will thus need to be
697 translated to Cλash. \defref{user-defined function}
699 \subsubsection[sec:normalization:predicates]{Predicates}
700 Here, we define a number of predicates that can be used below to concisely
703 \emph{gvar(expr)} is true when \emph{expr} is a variable that references a
704 global variable. It is false when it references a local variable.
706 \emph{lvar(expr)} is the complement of \emph{gvar}; it is true when \emph{expr}
707 references a local variable, false when it references a global variable.
709 \emph{representable(expr)} is true when \emph{expr} is \emph{representable}.
711 \subsection[sec:normalization:uniq]{Binder uniqueness}
712 A common problem in transformation systems, is binder uniqueness. When not
713 considering this problem, it is easy to create transformations that mix up
714 bindings and cause name collisions. Take for example, the following core
718 (λa.λb.λc. a * b * c) x c
721 By applying β-reduction (see \in{section}[sec:normalization:beta]) once,
722 we can simplify this expression to:
728 Now, we have replaced the \lam{a} binder with a reference to the \lam{x}
729 binder. No harm done here. But note that we see multiple occurences of the
730 \lam{c} binder. The first is a binding occurence, to which the second refers.
731 The last, however refers to \emph{another} instance of \lam{c}, which is
732 bound somewhere outside of this expression. Now, if we would apply beta
733 reduction without taking heed of binder uniqueness, we would get:
739 This is obviously not what was supposed to happen! The root of this problem is
740 the reuse of binders: identical binders can be bound in different,
741 but overlapping scopes. Any variable reference in those
742 overlapping scopes then refers to the variable bound in the inner
743 (smallest) scope. There is not way to refer to the variable in the
744 outer scope. This effect is usually referred to as
745 \emph{shadowing}: when a binder is bound in a scope where the
746 binder already had a value, the inner binding is said to
747 \emph{shadow} the outer binding. In the example above, the \lam{c}
748 binder was bound outside of the expression and in the inner lambda
749 expression. Inside that lambda expression, only the inner \lam{c}
752 There are a number of ways to solve this. \small{GHC} has isolated this
753 problem to their binder substitution code, which performs \emph{deshadowing}
754 during its expression traversal. This means that any binding that shadows
755 another binding on a higher level is replaced by a new binder that does not
756 shadow any other binding. This non-shadowing invariant is enough to prevent
757 binder uniqueness problems in \small{GHC}.
759 In our transformation system, maintaining this non-shadowing invariant is
760 a bit harder to do (mostly due to implementation issues, the prototype
761 does not use \small{GHC}'s subsitution code). Also, the following points
765 \item Deshadowing does not guarantee overall uniqueness. For example, the
766 following (slightly contrived) expression shows the identifier \lam{x} bound in
767 two seperate places (and to different values), even though no shadowing
771 (let x = 1 in x) + (let x = 2 in x)
774 \item In our normal form (and the resulting \small{VHDL}), all binders
775 (signals) within the same function (entity) will end up in the same
776 scope. To allow this, all binders within the same function should be
779 \item When we know that all binders in an expression are unique, moving around
780 or removing a subexpression will never cause any binder conflicts. If we have
781 some way to generate fresh binders, introducing new subexpressions will not
782 cause any problems either. The only way to cause conflicts is thus to
783 duplicate an existing subexpression.
786 Given the above, our prototype maintains a unique binder invariant. This
787 means that in any given moment during normalization, all binders \emph{within
788 a single function} must be unique. To achieve this, we apply the following
791 \todo{Define fresh binders and unique supplies}
794 \item Before starting normalization, all binders in the function are made
795 unique. This is done by generating a fresh binder for every binder used. This
796 also replaces binders that did not cause any conflict, but it does ensure that
797 all binders within the function are generated by the same unique supply.
798 \refdef{fresh binder}
799 \item Whenever a new binder must be generated, we generate a fresh binder that
800 is guaranteed to be different from \emph{all binders generated so far}. This
801 can thus never introduce duplication and will maintain the invariant.
802 \item Whenever (a part of) an expression is duplicated (for example when
803 inlining), all binders in the expression are replaced with fresh binders
804 (using the same method as at the start of normalization). These fresh binders
805 can never introduce duplication, so this will maintain the invariant.
806 \item Whenever we move part of an expression around within the function, there
807 is no need to do anything special. There is obviously no way to introduce
808 duplication by moving expressions around. Since we know that each of the
809 binders is already unique, there is no way to introduce (incorrect) shadowing
813 \section{Transform passes}
814 In this section we describe the actual transforms.
816 Each transformation will be described informally first, explaining
817 the need for and goal of the transformation. Then, we will formally define
818 the transformation using the syntax introduced in
819 \in{section}[sec:normalization:transformation].
821 \subsection{General cleanup}
822 These transformations are general cleanup transformations, that aim to
823 make expressions simpler. These transformations usually clean up the
824 mess left behind by other transformations or clean up expressions to
825 expose new transformation opportunities for other transformations.
827 Most of these transformations are standard optimizations in other
828 compilers as well. However, in our compiler, most of these are not just
829 optimizations, but they are required to get our program into intended
833 \defref{substitution notation}
834 \startframedtext[width=8cm,background=box,frame=no]
835 \startalignment[center]
836 {\tfa Substitution notation}
840 In some of the transformations in this chapter, we need to perform
841 substitution on an expression. Substitution means replacing every
842 occurence of some expression (usually a variable reference) with
845 There have been a lot of different notations used in literature for
846 specifying substitution. The notation that will be used in this report
853 This means expression \lam{E} with all occurences of \lam{A} replaced
858 \subsubsection[sec:normalization:beta]{β-reduction}
859 β-reduction is a well known transformation from lambda calculus, where it is
860 the main reduction step. It reduces applications of lambda abstractions,
861 removing both the lambda abstraction and the application.
863 In our transformation system, this step helps to remove unwanted lambda
864 abstractions (basically all but the ones at the top level). Other
865 transformations (application propagation, non-representable inlining) make
866 sure that most lambda abstractions will eventually be reducable by
869 Note that β-reduction also works on type lambda abstractions and type
870 applications as well. This means the substitution below also works on
871 type variables, in the case that the binder is a type variable and teh
872 expression applied to is a type.
889 \transexample{beta}{β-reduction}{from}{to}
899 \transexample{beta-type}{β-reduction for type abstractions}{from}{to}
901 \subsubsection{Unused let binding removal}
902 This transformation removes let bindings that are never used.
903 Occasionally, \GHC's desugarer introduces some unused let bindings.
905 This normalization pass should really be not be necessary to get
906 into intended normal form (since the intended normal form
907 definition \refdef{intended normal form definition} does not
908 require that every binding is used), but in practice the
909 desugarer or simplifier emits some bindings that cannot be
910 normalized (e.g., calls to a
911 \hs{Control.Exception.Base.patError}) but are not used anywhere
912 either. To prevent the \VHDL\ generation from breaking on these
913 artifacts, this transformation removes them.
915 \todo{Do not use old-style numerals in transformations}
924 M \lam{ai} does not occur free in \lam{M}
925 ---------------------------- \lam{\forall j, 0 ≤ j ≤ n, j ≠ i} (\lam{ai} does not occur free in \lam{Ej})
951 \transexample{unusedlet}{Unused let binding removal}{from}{to}
953 \subsubsection{Empty let removal}
954 This transformation is simple: it removes recursive lets that have no bindings
955 (which usually occurs when unused let binding removal removes the last
958 Note that there is no need to define this transformation for
959 non-recursive lets, since they always contain exactly one binding.
978 \transexample{emptylet}{Empty let removal}{from}{to}
980 \subsubsection[sec:normalization:simplelet]{Simple let binding removal}
981 This transformation inlines simple let bindings, that bind some
982 binder to some other binder instead of a more complex expression (\ie\
985 This transformation is not needed to get an expression into intended
986 normal form (since these bindings are part of the intended normal
987 form), but makes the resulting \small{VHDL} a lot shorter.
989 \refdef{substitution notation}
999 ----------------------------- \lam{b} is a variable reference
1000 letrec \lam{ai} ≠ \lam{b}
1013 \subsubsection{Cast propagation / simplification}
1014 This transform pushes casts down into the expression as far as
1015 possible. This transformation has been added to make a few
1016 specific corner cases work, but it is not clear yet if this
1017 transformation handles cast expressions completely or in the
1018 right way. See \in{section}[sec:normalization:castproblems].
1021 (let binds in E) ▶ T
1022 -------------------------
1023 let binds in (E ▶ T)
1032 -------------------------
1039 \subsubsection{Top level binding inlining}
1040 \refdef{top level binding}
1041 This transform takes simple top level bindings generated by the
1042 \small{GHC} compiler. \small{GHC} sometimes generates very simple
1043 \quote{wrapper} bindings, which are bound to just a variable
1044 reference, or contain just a (partial) function appliation with
1045 the type and dictionary arguments filled in (such as the
1046 \lam{(+)} in the example below).
1048 Note that this transformation is completely optional. It is not
1049 required to get any function into intended normal form, but it does help making
1050 the resulting VHDL output easier to read (since it removes components
1051 that do not add any real structure, but do hide away operations and
1052 cause extra clutter).
1054 This transform takes any top level binding generated by \GHC,
1055 whose normalized form contains only a single let binding.
1058 x = λa0 ... λan.let y = E in y
1061 -------------------------------------- \lam{x} is generated by the compiler
1062 λa0 ... λan.let y = E in y
1066 (+) :: Word -> Word -> Word
1067 (+) = GHC.Num.(+) @Word \$dNum
1072 GHC.Num.(+) @ Alu.Word \$dNum a b
1075 \transexample{toplevelinline}{Top level binding inlining}{from}{to}
1077 \in{Example}[ex:trans:toplevelinline] shows a typical application of
1078 the addition operator generated by \GHC. The type and dictionary
1079 arguments used here are described in
1080 \in{Section}[section:prototype:polymorphism].
1082 Without this transformation, there would be a \lam{(+)} entity
1083 in the \VHDL\ which would just add its inputs. This generates a
1084 lot of overhead in the \VHDL, which is particularly annoying
1085 when browsing the generated RTL schematic (especially since most
1086 non-alphanumerics, like all characters in \lam{(+)}, are not
1087 allowed in \VHDL\ architecture names\footnote{Technically, it is
1088 allowed to use non-alphanumerics when using extended
1089 identifiers, but it seems that none of the tooling likes
1090 extended identifiers in filenames, so it effectively does not
1091 work.}, so the entity would be called \quote{w7aA7f} or
1092 something similarly meaningless and autogenerated).
1094 \subsection{Program structure}
1095 These transformations are aimed at normalizing the overall structure
1096 into the intended form. This means ensuring there is a lambda abstraction
1097 at the top for every argument (input port or current state), putting all
1098 of the other value definitions in let bindings and making the final
1099 return value a simple variable reference.
1101 \subsubsection[sec:normalization:eta]{η-abstraction}
1102 This transformation makes sure that all arguments of a function-typed
1103 expression are named, by introducing lambda expressions. When combined with
1104 β-reduction and non-representable binding inlining, all function-typed
1105 expressions should be lambda abstractions or global identifiers.
1109 -------------- \lam{E} does not occur on a function position in an application
1110 λx.E x \lam{E} is not a lambda abstraction.
1120 foo = λa.λx.(case a of
1125 \transexample{eta}{η-abstraction}{from}{to}
1127 \subsubsection[sec:normalization:appprop]{Application propagation}
1128 This transformation is meant to propagate application expressions downwards
1129 into expressions as far as possible. This allows partial applications inside
1130 expressions to become fully applied and exposes new transformation
1131 opportunities for other transformations (like β-reduction and
1134 Since all binders in our expression are unique (see
1135 \in{section}[sec:normalization:uniq]), there is no risk that we will
1136 introduce unintended shadowing by moving an expression into a lower
1137 scope. Also, since only move expression into smaller scopes (down into
1138 our expression), there is no risk of moving a variable reference out
1139 of the scope in which it is defined.
1142 (letrec binds in E) M
1143 ------------------------
1163 \transexample{appproplet}{Application propagation for a let expression}{from}{to}
1191 \transexample{apppropcase}{Application propagation for a case expression}{from}{to}
1193 \subsubsection[sec:normalization:letrecurse]{Let recursification}
1194 This transformation makes all non-recursive lets recursive. In the
1195 end, we want a single recursive let in our normalized program, so all
1196 non-recursive lets can be converted. This also makes other
1197 transformations simpler: they only need to be specified for recursive
1198 let expressions (and simply will not apply to non-recursive let
1199 expressions until this transformation has been applied).
1206 ------------------------------------------
1213 \subsubsection{Let flattening}
1214 This transformation puts nested lets in the same scope, by lifting the
1215 binding(s) of the inner let into the outer let. Eventually, this will
1216 cause all let bindings to appear in the same scope.
1218 This transformation only applies to recursive lets, since all
1219 non-recursive lets will be made recursive (see
1220 \in{section}[sec:normalization:letrecurse]).
1222 Since we are joining two scopes together, there is no risk of moving a
1223 variable reference out of the scope where it is defined.
1229 ai = (letrec bindings in M)
1234 ------------------------------------------
1269 \transexample{letflat}{Let flattening}{from}{to}
1271 \subsubsection{Return value simplification}
1272 This transformation ensures that the return value of a function is always a
1273 simple local variable reference.
1275 This transformation only applies to the entire body of a
1276 function instead of any subexpression in a function. This is
1277 achieved by the contexts, like \lam{x = E}, though this is
1278 strictly not correct (you could read this as "if there is any
1279 function \lam{x} that binds \lam{E}, any \lam{E} can be
1280 transformed, while we only mean the \lam{E} that is bound by
1283 Note that the return value is not simplified if its not
1284 representable. Otherwise, this would cause a direct loop with
1285 the inlining of unrepresentable bindings. If the return value is
1286 not representable because it has a function type, η-abstraction
1287 should make sure that this transformation will eventually apply.
1288 If the value is not representable for other reasons, the
1289 function result itself is not representable, meaning this
1290 function is not translatable anyway.
1293 x = E \lam{E} is representable
1294 ~ \lam{E} is not a lambda abstraction
1295 E \lam{E} is not a let expression
1296 --------------------------- \lam{E} is not a local variable reference
1302 ~ \lam{E} is representable
1303 E \lam{E} is not a let expression
1304 --------------------------- \lam{E} is not a local variable reference
1309 x = λv0 ... λvn.let ... in E
1310 ~ \lam{E} is representable
1311 E \lam{E} is not a local variable reference
1312 -----------------------------
1321 x = letrec x = add 1 2 in x
1324 \transexample{retvalsimpl}{Return value simplification}{from}{to}
1326 \todo{More examples}
1328 \subsection[sec:normalization:argsimpl]{Representable arguments simplification}
1329 This section contains just a single transformation that deals with
1330 representable arguments in applications. Non-representable arguments are
1331 handled by the transformations in
1332 \in{section}[sec:normalization:nonrep].
1334 This transformation ensures that all representable arguments will become
1335 references to local variables. This ensures they will become references
1336 to local signals in the resulting \small{VHDL}, which is required due to
1337 limitations in the component instantiation code in \VHDL\ (one can only
1338 assign a signal or constant to an input port). By ensuring that all
1339 arguments are always simple variable references, we always have a signal
1340 available to map to the input ports.
1342 To reduce a complex expression to a simple variable reference, we create
1343 a new let expression around the application, which binds the complex
1344 expression to a new variable. The original function is then applied to
1347 \refdef{global variable}
1348 Note that references to \emph{global variables} (like a top level
1349 function without arguments, but also an argumentless dataconstructors
1350 like \lam{True}) are also simplified. Only local variables generate
1351 signals in the resulting architecture. Even though argumentless
1352 dataconstructors generate constants in generated \VHDL\ code and could be
1353 mapped to an input port directly, they are still simplified to make the
1354 normal form more regular.
1356 \refdef{representable}
1359 -------------------- \lam{N} is representable
1360 letrec x = N in M x \lam{N} is not a local variable reference
1362 \refdef{local variable}
1369 letrec x = add a 1 in add x 1
1372 \transexample{argsimpl}{Argument simplification}{from}{to}
1374 \subsection[sec:normalization:built-ins]{Built-in functions}
1375 This section deals with (arguments to) built-in functions. In the
1376 intended normal form definition\refdef{intended normal form definition}
1377 we can see that there are three sorts of arguments a built-in function
1381 \item A representable local variable reference. This is the most
1382 common argument to any function. The argument simplification
1383 transformation described in \in{section}[sec:normalization:argsimpl]
1384 makes sure that \emph{any} representable argument to \emph{any}
1385 function (including built-in functions) is turned into a local variable
1387 \item (A partial application of) a top level function (either built-in on
1388 user-defined). The function extraction transformation described in
1389 this section takes care of turning every functiontyped argument into
1390 (a partial application of) a top level function.
1391 \item Any expression that is not representable and does not have a
1392 function type. Since these can be any expression, there is no
1393 transformation needed. Note that this category is exactly all
1394 expressions that are not transformed by the transformations for the
1395 previous two categories. This means that \emph{any} core expression
1396 that is used as an argument to a built-in function will be either
1397 transformed into one of the above categories, or end up in this
1398 categorie. In any case, the result is in normal form.
1401 As noted, the argument simplification will handle any representable
1402 arguments to a built-in function. The following transformation is needed
1403 to handle non-representable arguments with a function type, all other
1404 non-representable arguments do not need any special handling.
1406 \subsubsection[sec:normalization:funextract]{Function extraction}
1407 This transform deals with function-typed arguments to built-in
1409 Since built-in functions cannot be specialized (see
1410 \in{section}[sec:normalization:specialize]) to remove the arguments,
1411 these arguments are extracted into a new global function instead. In
1412 other words, we create a new top level function that has exactly the
1413 extracted argument as its body. This greatly simplifies the
1414 translation rules needed for built-in functions, since they only need
1415 to handle (partial applications of) top level functions.
1417 Any free variables occuring in the extracted arguments will become
1418 parameters to the new global function. The original argument is replaced
1419 with a reference to the new function, applied to any free variables from
1420 the original argument.
1422 This transformation is useful when applying higher-order built-in functions
1423 like \hs{map} to a lambda abstraction, for example. In this case, the code
1424 that generates \small{VHDL} for \hs{map} only needs to handle top level functions and
1425 partial applications, not any other expression (such as lambda abstractions or
1426 even more complicated expressions).
1429 M N \lam{M} is (a partial aplication of) a built-in function.
1430 --------------------- \lam{f0 ... fn} are all free local variables of \lam{N}
1431 M (x f0 ... fn) \lam{N :: a -> b}
1432 ~ \lam{N} is not a (partial application of) a top level function
1437 addList = λb.λxs.map (λa . add a b) xs
1441 addList = λb.λxs.map (f b) xs
1446 \transexample{funextract}{Function extraction}{from}{to}
1448 Note that the function \lam{f} will still need normalization after
1451 \subsection{Case normalisation}
1452 The transformations in this section ensure that case statements end up
1455 \subsubsection{Scrutinee simplification}
1456 This transform ensures that the scrutinee of a case expression is always
1457 a simple variable reference.
1462 ----------------- \lam{E} is not a local variable reference
1481 \transexample{letflat}{Case normalisation}{from}{to}
1485 \defref{wild binders}
1486 \startframedtext[width=7cm,background=box,frame=no]
1487 \startalignment[center]
1491 In a functional expression, a \emph{wild binder} refers to any
1492 binder that is never referenced. This means that even though it
1493 will be bound to a particular value, that value is never used.
1495 The Haskell syntax offers the underscore as a wild binder that
1496 cannot even be referenced (It can be seen as introducing a new,
1497 anonymous, binder everytime it is used).
1499 In these transformations, the term wild binder will sometimes be
1500 used to indicate that a binder must not be referenced.
1504 \subsubsection{Case normalization}
1505 This transformation ensures that all case expressions get a form
1506 that is allowed by the intended normal form. This means they
1510 \item An extractor case with a single alternative that picks a field
1511 from a datatype, \eg\ \lam{case x of (a, b) -> a}.
1512 \item A selector case with multiple alternatives and only wild binders, that
1513 makes a choice between expressions based on the constructor of another
1514 expression, \eg\ \lam{case x of Low -> a; High -> b}.
1517 For an arbitrary case, that has \lam{n} alternatives, with
1518 \lam{m} binders in each alternatives, this will result in \lam{m
1519 * n} extractor case expression to get at each variable, \lam{n}
1520 let bindings for each of the alternatives' value and a single
1521 selector case to select the right value out of these.
1523 Technically, the defintion of this transformation would require
1524 that the constructor for every alternative has exactly the same
1525 amount (\lam{m}) of arguments, but of course this transformation
1526 also applies when this is not the case.
1530 C0 v0,0 ... v0,m -> E0
1532 Cn vn,0 ... vn,m -> En
1533 --------------------------------------------------- \lam{\forall i \forall j, 0 ≤ i ≤ n, 0 ≤ i < m} (\lam{wi,j} is a wild (unused) binder)
1534 letrec The case expression is not an extractor case
1535 v0,0 = case E of C0 x0,0 .. x0,m -> x0,0 The case expression is not a selector case
1537 v0,m = case E of C0 x0,0 .. x0,m -> x0,m
1539 vn,m = case E of Cn xn,0 .. xn,m -> xn,m
1545 C0 w0,0 ... w0,m -> y0
1547 Cn wn,0 ... wn,m -> yn
1550 Note that this transformation applies to case expressions with any
1551 scrutinee. If the scrutinee is a complex expression, this might
1552 result in duplication of work (hardware). An extra condition to
1553 only apply this transformation when the scrutinee is already
1554 simple (effectively causing this transformation to be only
1555 applied after the scrutinee simplification transformation) might
1574 \transexample{selcasesimpl}{Selector case simplification}{from}{to}
1582 b = case a of (,) b c -> b
1583 c = case a of (,) b c -> c
1590 \transexample{excasesimpl}{Extractor case simplification}{from}{to}
1592 \refdef{selector case}
1593 In \in{example}[ex:trans:excasesimpl] the case expression is expanded
1594 into multiple case expressions, including a pretty useless expression
1595 (that is neither a selector or extractor case). This case can be
1596 removed by the Case removal transformation in
1597 \in{section}[sec:transformation:caseremoval].
1599 \subsubsection[sec:transformation:caseremoval]{Case removal}
1600 This transform removes any case expression with a single alternative and
1601 only wild binders.\refdef{wild binders}
1603 These "useless" case expressions are usually leftovers from case simplification
1604 on extractor case (see the previous example).
1609 ---------------------- \lam{\forall i, 0 ≤ i ≤ m} (\lam{vi} does not occur free in E)
1622 \transexample{caserem}{Case removal}{from}{to}
1624 \subsection[sec:normalization:nonrep]{Removing unrepresentable values}
1625 The transformations in this section are aimed at making all the
1626 values used in our expression representable. There are two main
1627 transformations that are applied to \emph{all} unrepresentable let
1628 bindings and function arguments. These are meant to address three
1629 different kinds of unrepresentable values: polymorphic values,
1630 higher-order values and literals. The transformation are described
1631 generically: they apply to all non-representable values. However,
1632 non-representable values that do not fall into one of these three
1633 categories will be moved around by these transformations but are
1634 unlikely to completely disappear. They usually mean the program was not
1635 valid in the first place, because unsupported types were used (for
1636 example, a program using strings).
1638 Each of these three categories will be detailed below, followed by the
1639 actual transformations.
1641 \subsubsection{Removing Polymorphism}
1642 As noted in \in{section}[sec:prototype:polymporphism],
1643 polymorphism is made explicit in Core through type and
1644 dictionary arguments. To remove the polymorphism from a
1645 function, we can simply specialize the polymorphic function for
1646 the particular type applied to it. The same goes for dictionary
1647 arguments. To remove polymorphism from let bound values, we
1648 simply inline the let bindings that have a polymorphic type,
1649 which should (eventually) make sure that the polymorphic
1650 expression is applied to a type and/or dictionary, which can
1651 then be removed by β-reduction (\in{section}[sec:normalization:beta]).
1653 Since both type and dictionary arguments are not representable,
1654 \refdef{representable}
1655 the non-representable argument specialization and
1656 non-representable let binding inlining transformations below
1657 take care of exactly this.
1659 There is one case where polymorphism cannot be completely
1660 removed: built-in functions are still allowed to be polymorphic
1661 (Since we have no function body that we could properly
1662 specialize). However, the code that generates \VHDL\ for built-in
1663 functions knows how to handle this, so this is not a problem.
1665 \subsubsection[sec:normalization:defunctionalization]{Defunctionalization}
1666 These transformations remove higher-order expressions from our
1667 program, making all values first-order. The approach used for
1668 defunctionalization uses a combination of specialization, inlining and
1669 some cleanup transformations, was also proposed in parallel research
1670 by Neil Mitchell \cite[mitchell09].
1672 Higher order values are always introduced by lambda abstractions, none
1673 of the other Core expression elements can introduce a function type.
1674 However, other expressions can \emph{have} a function type, when they
1675 have a lambda expression in their body.
1677 For example, the following expression is a higher-order expression
1678 that is not a lambda expression itself:
1680 \refdef{id function}
1687 The reference to the \lam{id} function shows that we can introduce a
1688 higher-order expression in our program without using a lambda
1689 expression directly. However, inside the definition of the \lam{id}
1690 function, we can be sure that a lambda expression is present.
1692 Looking closely at the definition of our normal form in
1693 \in{section}[sec:normalization:intendednormalform], we can see that
1694 there are three possibilities for higher-order values to appear in our
1695 intended normal form:
1698 \item[item:toplambda] Lambda abstractions can appear at the highest level of a
1699 top level function. These lambda abstractions introduce the
1700 arguments (input ports / current state) of the function.
1701 \item[item:built-inarg] (Partial applications of) top level functions can appear as an
1702 argument to a built-in function.
1703 \item[item:completeapp] (Partial applications of) top level functions can appear in
1704 function position of an application. Since a partial application
1705 cannot appear anywhere else (except as built-in function arguments),
1706 all partial applications are applied, meaning that all applications
1707 will become complete applications. However, since application of
1708 arguments happens one by one, in the expression:
1712 the subexpression \lam{f 1} has a function type. But this is
1713 allowed, since it is inside a complete application.
1716 We will take a typical function with some higher-order values as an
1717 example. The following function takes two arguments: a \lam{Bit} and a
1718 list of numbers. Depending on the first argument, each number in the
1719 list is doubled, or the list is returned unmodified. For the sake of
1720 the example, no polymorphism is shown. In reality, at least map would
1724 λy.let double = λx. x + x in
1730 This example shows a number of higher-order values that we cannot
1731 translate to \VHDL\ directly. The \lam{double} binder bound in the let
1732 expression has a function type, as well as both of the alternatives of
1733 the case expression. The first alternative is a partial application of
1734 the \lam{map} built-in function, whereas the second alternative is a
1737 To reduce all higher-order values to one of the above items, a number
1738 of transformations we have already seen are used. The η-abstraction
1739 transformation from \in{section}[sec:normalization:eta] ensures all
1740 function arguments are introduced by lambda abstraction on the highest
1741 level of a function. These lambda arguments are allowed because of
1742 \in{item}[item:toplambda] above. After η-abstraction, our example
1743 becomes a bit bigger:
1746 λy.λq.(let double = λx. x + x in
1753 η-abstraction also introduces extra applications (the application of
1754 the let expression to \lam{q} in the above example). These
1755 applications can then propagated down by the application propagation
1756 transformation (\in{section}[sec:normalization:appprop]). In our
1757 example, the \lam{q} and \lam{r} variable will be propagated into the
1758 let expression and then into the case expression:
1761 λy.λq.let double = λx. x + x in
1767 This propagation makes higher-order values become applied (in
1768 particular both of the alternatives of the case now have a
1769 representable type). Completely applied top level functions (like the
1770 first alternative) are now no longer invalid (they fall under
1771 \in{item}[item:completeapp] above). (Completely) applied lambda
1772 abstractions can be removed by β-abstraction. For our example,
1773 applying β-abstraction results in the following:
1776 λy.λq.let double = λx. x + x in
1782 As you can see in our example, all of this moves applications towards
1783 the higher-order values, but misses higher-order functions bound by
1784 let expressions. The applications cannot be moved towards these values
1785 (since they can be used in multiple places), so the values will have
1786 to be moved towards the applications. This is achieved by inlining all
1787 higher-order values bound by let applications, by the
1788 non-representable binding inlining transformation below. When applying
1789 it to our example, we get the following:
1793 Low -> map (λx. x + x) q
1797 We have nearly eliminated all unsupported higher-order values from this
1798 expressions. The one that is remaining is the first argument to the
1799 \lam{map} function. Having higher-order arguments to a built-in
1800 function like \lam{map} is allowed in the intended normal form, but
1801 only if the argument is a (partial application) of a top level
1802 function. This is easily done by introducing a new top level function
1803 and put the lambda abstraction inside. This is done by the function
1804 extraction transformation from
1805 \in{section}[sec:normalization:funextract].
1813 This also introduces a new function, that we have called \lam{func}:
1819 Note that this does not actually remove the lambda, but now it is a
1820 lambda at the highest level of a function, which is allowed in the
1821 intended normal form.
1823 There is one case that has not been discussed yet. What if the
1824 \lam{map} function in the example above was not a built-in function
1825 but a user-defined function? Then extracting the lambda expression
1826 into a new function would not be enough, since user-defined functions
1827 can never have higher-order arguments. For example, the following
1828 expression shows an example:
1831 twice :: (Word -> Word) -> Word -> Word
1832 twice = λf.λa.f (f a)
1834 main = λa.app (λx. x + x) a
1837 This example shows a function \lam{twice} that takes a function as a
1838 first argument and applies that function twice to the second argument.
1839 Again, we have made the function monomorphic for clarity, even though
1840 this function would be a lot more useful if it was polymorphic. The
1841 function \lam{main} uses \lam{twice} to apply a lambda epression twice.
1843 When faced with a user defined function, a body is available for that
1844 function. This means we could create a specialized version of the
1845 function that only works for this particular higher-order argument
1846 (\ie, we can just remove the argument and call the specialized
1847 function without the argument). This transformation is detailed below.
1848 Applying this transformation to the example gives:
1851 twice' :: Word -> Word
1852 twice' = λb.(λf.λa.f (f a)) (λx. x + x) b
1857 The \lam{main} function is now in normal form, since the only
1858 higher-order value there is the top level lambda expression. The new
1859 \lam{twice'} function is a bit complex, but the entire original body
1860 of the original \lam{twice} function is wrapped in a lambda
1861 abstraction and applied to the argument we have specialized for
1862 (\lam{λx. x + x}) and the other arguments. This complex expression can
1863 fortunately be effectively reduced by repeatedly applying β-reduction:
1866 twice' :: Word -> Word
1867 twice' = λb.(b + b) + (b + b)
1870 This example also shows that the resulting normal form might not be as
1871 efficient as we might hope it to be (it is calculating \lam{(b + b)}
1872 twice). This is discussed in more detail in
1873 \in{section}[sec:normalization:duplicatework].
1875 \subsubsection{Literals}
1876 There are a limited number of literals available in Haskell and Core.
1877 \refdef{enumerated types} When using (enumerating) algebraic
1878 datatypes, a literal is just a reference to the corresponding data
1879 constructor, which has a representable type (the algebraic datatype)
1880 and can be translated directly. This also holds for literals of the
1881 \hs{Bool} Haskell type, which is just an enumerated type.
1883 There is, however, a second type of literal that does not have a
1884 representable type: integer literals. Cλash supports using integer
1885 literals for all three integer types supported (\hs{SizedWord},
1886 \hs{SizedInt} and \hs{RangedWord}). This is implemented using
1887 Haskell's \hs{Num} type class, which offers a \hs{fromInteger} method
1888 that converts any \hs{Integer} to the Cλash datatypes.
1890 When \GHC\ sees integer literals, it will automatically insert calls to
1891 the \hs{fromInteger} method in the resulting Core expression. For
1892 example, the following expression in Haskell creates a 32 bit unsigned
1893 word with the value 1. The explicit type signature is needed, since
1894 there is no context for \GHC\ to determine the type from otherwise.
1900 This Haskell code results in the following Core expression:
1903 fromInteger @(SizedWord D32) \$dNum (smallInteger 10)
1906 The literal 10 will have the type \lam{GHC.Prim.Int\#}, which is
1907 converted into an \lam{Integer} by \lam{smallInteger}. Finally, the
1908 \lam{fromInteger} function will finally convert this into a
1909 \lam{SizedWord D32}.
1911 Both the \lam{GHC.Prim.Int\#} and \lam{Integer} types are not
1912 representable, and cannot be translated directly. Fortunately, there
1913 is no need to translate them, since \lam{fromInteger} is a built-in
1914 function that knows how to handle these values. However, this does
1915 require that the \lam{fromInteger} function is directly applied to
1916 these non-representable literal values, otherwise errors will occur.
1917 For example, the following expression is not in the intended normal
1918 form, since one of the let bindings has an unrepresentable type
1922 let l = smallInteger 10 in fromInteger @(SizedWord D32) \$dNum l
1925 By inlining these let-bindings, we can ensure that unrepresentable
1926 literals bound by a let binding end up in an application of the
1927 appropriate built-in function, where they are allowed. Since it is
1928 possible that the application of that function is in a different
1929 function than the definition of the literal value, we will always need
1930 to specialize away any unrepresentable literals that are used as
1931 function arguments. The following two transformations do exactly this.
1933 \subsubsection{Non-representable binding inlining}
1934 This transform inlines let bindings that are bound to a
1935 non-representable value. Since we can never generate a signal
1936 assignment for these bindings (we cannot declare a signal assignment
1937 with a non-representable type, for obvious reasons), we have no choice
1938 but to inline the binding to remove it.
1940 As we have seen in the previous sections, inlining these bindings
1941 solves (part of) the polymorphism, higher-order values and
1942 unrepresentable literals in an expression.
1944 \refdef{substitution notation}
1954 -------------------------- \lam{Ei} has a non-representable type.
1956 a0 = E0 [ai=>Ei] \vdots
1957 ai-1 = Ei-1 [ai=>Ei]
1958 ai+1 = Ei+1 [ai=>Ei]
1977 x = fromInteger (smallInteger 10)
1979 (λb -> add b 1) (add 1 x)
1982 \transexample{nonrepinline}{Nonrepresentable binding inlining}{from}{to}
1984 \subsubsection[sec:normalization:specialize]{Function specialization}
1985 This transform removes arguments to user-defined functions that are
1986 not representable at runtime. This is done by creating a
1987 \emph{specialized} version of the function that only works for one
1988 particular value of that argument (in other words, the argument can be
1991 Specialization means to create a specialized version of the called
1992 function, with one argument already filled in. As a simple example, in
1993 the following program (this is not actual Core, since it directly uses
1994 a literal with the unrepresentable type \lam{GHC.Prim.Int\#}).
2001 We could specialize the function \lam{f} against the literal argument
2002 1, with the following result:
2009 In some way, this transformation is similar to β-reduction, but it
2010 operates across function boundaries. It is also similar to
2011 non-representable let binding inlining above, since it sort of
2012 \quote{inlines} an expression into a called function.
2014 Special care must be taken when the argument has any free variables.
2015 If this is the case, the original argument should not be removed
2016 completely, but replaced by all the free variables of the expression.
2017 In this way, the original expression can still be evaluated inside the
2020 To prevent us from propagating the same argument over and over, a
2021 simple local variable reference is not propagated (since is has
2022 exactly one free variable, itself, we would only replace that argument
2025 This shows that any free local variables that are not runtime
2026 representable cannot be brought into normal form by this transform. We
2027 rely on an inlining or β-reduction transformation to replace such a
2028 variable with an expression we can propagate again.
2033 x Y0 ... Yi ... Yn \lam{Yi} is not representable
2034 --------------------------------------------- \lam{Yi} is not a local variable reference
2035 x' Y0 ... Yi-1 f0 ... fm Yi+1 ... Yn \lam{f0 ... fm} are all free local vars of \lam{Yi}
2036 ~ \lam{T0 ... Tn} are the types of \lam{Y0 ... Yn}
2037 x' = λ(y0 :: T0) ... λ(yi-1 :: Ty-1).
2039 λ(yi+1 :: Ty+1) ... λ(yn :: Tn).
2040 E y0 ... yi-1 Yi yi+1 ... yn
2043 This is a bit of a complex transformation. It transforms an
2044 application of the function \lam{x}, where one of the arguments
2045 (\lam{Y_i}) is not representable. A new
2046 function \lam{x'} is created that wraps the body of the old function.
2047 The body of the new function becomes a number of nested lambda
2048 abstractions, one for each of the original arguments that are left
2051 The ith argument is replaced with the free variables of
2052 \lam{Y_i}. Note that we reuse the same binders as those used in
2053 \lam{Y_i}, since we can then just use \lam{Y_i} inside the new
2054 function body and have all of the variables it uses be in scope.
2056 The argument that we are specializing for, \lam{Y_i}, is put inside
2057 the new function body. The old function body is applied to it. Since
2058 we use this new function only in place of an application with that
2059 particular argument \lam{Y_i}, behaviour should not change.
2061 Note that the types of the arguments of our new function are taken
2062 from the types of the \emph{actual} arguments (\lam{T0 ... Tn}). This
2063 means that any polymorphism in the arguments is removed, even when the
2064 corresponding explicit type lambda is not removed
2067 \todo{Examples. Perhaps reference the previous sections}
2069 \section{Unsolved problems}
2070 The above system of transformations has been implemented in the prototype
2071 and seems to work well to compile simple and more complex examples of
2072 hardware descriptions. \todo{Ref christiaan?} However, this normalization
2073 system has not seen enough review and work to be complete and work for
2074 every Core expression that is supplied to it. A number of problems
2075 have already been identified and are discussed in this section.
2077 \subsection[sec:normalization:duplicatework]{Work duplication}
2078 A possible problem of β-reduction is that it could duplicate work.
2079 When the expression applied is not a simple variable reference, but
2080 requires calculation and the binder the lambda abstraction binds to
2081 is used more than once, more hardware might be generated than strictly
2084 As an example, consider the expression:
2090 When applying β-reduction to this expression, we get:
2096 which of course calculates \lam{(a * b)} twice.
2098 A possible solution to this would be to use the following alternative
2099 transformation, which is of course no longer normal β-reduction. The
2100 followin transformation has not been tested in the prototype, but is
2101 given here for future reference:
2109 This does not seem like much of an improvement, but it does get rid of
2110 the lambda expression (and the associated higher-order value), while
2111 at the same time introducing a new let binding. Since the result of
2112 every application or case expression must be bound by a let expression
2113 in the intended normal form anyway, this is probably not a problem. If
2114 the argument happens to be a variable reference, then simple let
2115 binding removal (\in{section}[sec:normalization:simplelet]) will
2116 remove it, making the result identical to that of the original
2117 β-reduction transformation.
2119 When also applying argument simplification to the above example, we
2120 get the following expression:
2128 Looking at this, we could imagine an alternative approach: create a
2129 transformation that removes let bindings that bind identical values.
2130 In the above expression, the \lam{y} and \lam{z} variables could be
2131 merged together, resulting in the more efficient expression:
2134 let y = (a * b) in y + y
2137 \subsection[sec:normalization:non-determinism]{Non-determinism}
2138 As an example, again consider the following expression:
2144 We can apply both β-reduction (\in{section}[sec:normalization:beta])
2145 as well as argument simplification
2146 (\in{section}[sec:normalization:argsimpl]) to this expression.
2148 When applying argument simplification first and then β-reduction, we
2149 get the following expression:
2152 let y = (a * b) in y + y
2155 When applying β-reduction first and then argument simplification, we
2156 get the following expression:
2164 As you can see, this is a different expression. This means that the
2165 order of expressions, does in fact change the resulting normal form,
2166 which is something that we would like to avoid. In this particular
2167 case one of the alternatives is even clearly more efficient, so we
2168 would of course like the more efficient form to be the normal form.
2170 For this particular problem, the solutions for duplication of work
2171 seem from the previous section seem to fix the determinism of our
2172 transformation system as well. However, it is likely that there are
2173 other occurences of this problem.
2175 \subsection[sec:normalization:castproblems]{Casts}
2176 We do not fully understand the use of cast expressions in Core, so
2177 there are probably expressions involving cast expressions that cannot
2178 be brought into intended normal form by this transformation system.
2180 The uses of casts in the core system should be investigated more and
2181 transformations will probably need updating to handle them in all
2184 \subsection[sec:normalization:stateproblems]{Normalization of stateful descriptions}
2185 Currently, the intended normal form definition\refdef{intended
2186 normal form definition} offers enough freedom to describe all
2187 valid stateful descriptions, but is not limiting enough. It is
2188 possible to write descriptions which are in intended normal
2189 form, but cannot be translated into \VHDL\ in a meaningful way
2190 (\eg, a function that swaps two substates in its result, or a
2191 function that changes a substate itself instead of passing it to
2194 It is now up to the programmer to not do anything funny with
2195 these state values, whereas the normalization just tries not to
2196 mess up the flow of state values. In practice, there are
2197 situations where a Core program that \emph{could} be a valid
2198 stateful description is not translateable by the prototype. This
2199 most often happens when statefulness is mixed with pattern
2200 matching, causing a state input to be unpacked multiple times or
2201 be unpacked and repacked only in some of the code paths.
2203 Without going into detail about the exact problems (of which
2204 there are probably more than have shown up so far), it seems
2205 unlikely that these problems can be solved entirely by just
2206 improving the \VHDL\ state generation in the final stage. The
2207 normalization stage seems the best place to apply the rewriting
2208 needed to support more complex stateful descriptions. This does
2209 of course mean that the intended normal form definition must be
2210 extended as well to be more specific about how state handling
2211 should look like in normal form.
2212 \in{Section}[sec:prototype:statelimits] already contains a
2213 tight description of the limitations on the use of state
2214 variables, which could be adapted into the intended normal form.
2216 \section[sec:normalization:properties]{Provable properties}
2217 When looking at the system of transformations outlined above, there are a
2218 number of questions that we can ask ourselves. The main question is of course:
2219 \quote{Does our system work as intended?}. We can split this question into a
2220 number of subquestions:
2223 \item[q:termination] Does our system \emph{terminate}? Since our system will
2224 keep running as long as transformations apply, there is an obvious risk that
2225 it will keep running indefinitely. This typically happens when one
2226 transformation produces a result that is transformed back to the original
2227 by another transformation, or when one or more transformations keep
2228 expanding some expression.
2229 \item[q:soundness] Is our system \emph{sound}? Since our transformations
2230 continuously modify the expression, there is an obvious risk that the final
2231 normal form will not be equivalent to the original program: its meaning could
2233 \item[q:completeness] Is our system \emph{complete}? Since we have a complex
2234 system of transformations, there is an obvious risk that some expressions will
2235 not end up in our intended normal form, because we forgot some transformation.
2236 In other words: does our transformation system result in our intended normal
2237 form for all possible inputs?
2238 \item[q:determinism] Is our system \emph{deterministic}? Since we have defined
2239 no particular order in which the transformation should be applied, there is an
2240 obvious risk that different transformation orderings will result in
2241 \emph{different} normal forms. They might still both be intended normal forms
2242 (if our system is \emph{complete}) and describe correct hardware (if our
2243 system is \emph{sound}), so this property is less important than the previous
2244 three: the translator would still function properly without it.
2247 Unfortunately, the final transformation system has only been
2248 developed in the final part of the research, leaving no more time
2249 for verifying these properties. In fact, it is likely that the
2250 current transformation system still violates some of these
2251 properties in some cases (see
2252 \in{section}[sec:normalization:non-determinism] and
2253 \in{section}[sec:normalization:stateproblems]) and should be improved (or
2254 extra conditions on the input hardware descriptions should be formulated).
2256 This is most likely the case with the completeness and determinism
2257 properties, perhaps also the termination property. The soundness
2258 property probably holds, since it is easier to manually verify (each
2259 transformation can be reviewed separately).
2261 Even though no complete proofs have been made, some ideas for
2262 possible proof strategies are shown below.
2264 \subsection{Graph representation}
2265 Before looking into how to prove these properties, we will look at
2266 transformation systems from a graph perspective. We will first define
2267 the graph view and then illustrate it using a simple example from lambda
2268 calculus (which is a different system than the Cλash normalization
2269 system). The nodes of the graph are all possible Core expressions. The
2270 (directed) edges of the graph are transformations. When a transformation
2271 α applies to an expression \lam{A} to produce an expression \lam{B}, we
2272 add an edge from the node for \lam{A} to the node for \lam{B}, labeled
2275 \startuseMPgraphic{TransformGraph}
2279 newCircle.a(btex \lam{(λx.λy. (+) x y) 1} etex);
2280 newCircle.b(btex \lam{λy. (+) 1 y} etex);
2281 newCircle.c(btex \lam{(λx.(+) x) 1} etex);
2282 newCircle.d(btex \lam{(+) 1} etex);
2285 c.c = b.c + (4cm, 0cm);
2286 a.c = midpoint(b.c, c.c) + (0cm, 4cm);
2287 d.c = midpoint(b.c, c.c) - (0cm, 3cm);
2289 % β-conversion between a and b
2290 ncarc.a(a)(b) "name(bred)";
2291 ObjLabel.a(btex $\xrightarrow[normal]{}{β}$ etex) "labpathname(bred)", "labdir(rt)";
2292 ncarc.b(b)(a) "name(bexp)", "linestyle(dashed withdots)";
2293 ObjLabel.b(btex $\xleftarrow[normal]{}{β}$ etex) "labpathname(bexp)", "labdir(lft)";
2295 % η-conversion between a and c
2296 ncarc.a(a)(c) "name(ered)";
2297 ObjLabel.a(btex $\xrightarrow[normal]{}{η}$ etex) "labpathname(ered)", "labdir(rt)";
2298 ncarc.c(c)(a) "name(eexp)", "linestyle(dashed withdots)";
2299 ObjLabel.c(btex $\xleftarrow[normal]{}{η}$ etex) "labpathname(eexp)", "labdir(lft)";
2301 % η-conversion between b and d
2302 ncarc.b(b)(d) "name(ered)";
2303 ObjLabel.b(btex $\xrightarrow[normal]{}{η}$ etex) "labpathname(ered)", "labdir(rt)";
2304 ncarc.d(d)(b) "name(eexp)", "linestyle(dashed withdots)";
2305 ObjLabel.d(btex $\xleftarrow[normal]{}{η}$ etex) "labpathname(eexp)", "labdir(lft)";
2307 % β-conversion between c and d
2308 ncarc.c(c)(d) "name(bred)";
2309 ObjLabel.c(btex $\xrightarrow[normal]{}{β}$ etex) "labpathname(bred)", "labdir(rt)";
2310 ncarc.d(d)(c) "name(bexp)", "linestyle(dashed withdots)";
2311 ObjLabel.d(btex $\xleftarrow[normal]{}{β}$ etex) "labpathname(bexp)", "labdir(lft)";
2313 % Draw objects and lines
2314 drawObj(a, b, c, d);
2317 \placeexample[right][ex:TransformGraph]{Partial graph of a lambda calculus
2318 system with β and η reduction (solid lines) and expansion (dotted lines).}
2319 \boxedgraphic{TransformGraph}
2321 Of course the graph for Cλash is unbounded, since we can construct an
2322 infinite amount of Core expressions. Also, there might potentially be
2323 multiple edges between two given nodes (with different labels), though
2324 this seems unlikely to actually happen in our system.
2326 See \in{example}[ex:TransformGraph] for the graph representation of a very
2327 simple lambda calculus that contains just the expressions \lam{(λx.λy. (+) x
2328 y) 1}, \lam{λy. (+) 1 y}, \lam{(λx.(+) x) 1} and \lam{(+) 1}. The
2329 transformation system consists of β-reduction and η-reduction (solid edges) or
2330 β-expansion and η-expansion (dotted edges).
2332 \todo{Define β-reduction and η-reduction?}
2334 In such a graph a node (expression) is in normal form if it has no
2335 outgoing edges (meaning no transformation applies to it). The set of
2336 nodes without outgoing edges is called the \emph{normal set}. Similarly,
2337 the set of nodes containing expressions in intended normal form
2338 \refdef{intended normal form} is called the \emph{intended normal set}.
2340 From such a graph, we can derive some properties easily:
2342 \item A system will \emph{terminate} if there is no walk (sequence of
2343 edges, or transformations) of infinite length in the graph (this
2344 includes cycles, but can also happen without cycles).
2345 \item Soundness is not easily represented in the graph.
2346 \item A system is \emph{complete} if all of the nodes in the normal set have
2347 the intended normal form. The inverse (that all of the nodes outside of
2348 the normal set are \emph{not} in the intended normal form) is not
2349 strictly required. In other words, our normal set must be a
2350 subset of the intended normal form, but they do not need to be
2353 \item A system is deterministic if all paths starting at a particular
2354 node, which end in a node in the normal set, end at the same node.
2357 When looking at the \in{example}[ex:TransformGraph], we see that the system
2358 terminates for both the reduction and expansion systems (but note that, for
2359 expansion, this is only true because we have limited the possible
2360 expressions. In complete lambda calculus, there would be a path from
2361 \lam{(λx.λy. (+) x y) 1} to \lam{(λx.λy.(λz.(+) z) x y) 1} to
2362 \lam{(λx.λy.(λz.(λq.(+) q) z) x y) 1} etc.)
2364 If we would consider the system with both expansion and reduction, there
2365 would no longer be termination either, since there would be cycles all
2368 The reduction and expansion systems have a normal set of containing just
2369 \lam{(+) 1} or \lam{(λx.λy. (+) x y) 1} respectively. Since all paths in
2370 either system end up in these normal forms, both systems are \emph{complete}.
2371 Also, since there is only one node in the normal set, it must obviously be
2372 \emph{deterministic} as well.
2374 \subsection{Termination}
2375 In general, proving termination of an arbitrary program is a very
2376 hard problem. \todo{Ref about arbitrary termination} Fortunately,
2377 we only have to prove termination for our specific transformation
2380 A common approach for these kinds of proofs is to associate a
2381 measure with each possible expression in our system. If we can
2382 show that each transformation strictly decreases this measure
2383 (\ie, the expression transformed to has a lower measure than the
2384 expression transformed from). \todo{ref about measure-based
2385 termination proofs / analysis}
2387 A good measure for a system consisting of just β-reduction would
2388 be the number of lambda expressions in the expression. Since every
2389 application of β-reduction removes a lambda abstraction (and there
2390 is always a bounded number of lambda abstractions in every
2391 expression) we can easily see that a transformation system with
2392 just β-reduction will always terminate.
2394 For our complete system, this measure would be fairly complex
2395 (probably the sum of a lot of things). Since the (conditions on)
2396 our transformations are pretty complex, we would need to include
2397 both simple things like the number of let expressions as well as
2398 more complex things like the number of case expressions that are
2399 not yet in normal form.
2401 No real attempt has been made at finding a suitable measure for
2404 \subsection{Soundness}
2405 Soundness is a property that can be proven for each transformation
2406 separately. Since our system only runs separate transformations
2407 sequentially, if each of our transformations leaves the
2408 \emph{meaning} of the expression unchanged, then the entire system
2409 will of course leave the meaning unchanged and is thus
2412 The current prototype has only been verified in an ad-hoc fashion
2413 by inspecting (the code for) each transformation. A more formal
2414 verification would be more appropriate.
2416 To be able to formally show that each transformation properly
2417 preserves the meaning of every expression, we require an exact
2418 definition of the \emph{meaning} of every expression, so we can
2419 compare them. A definition of the operational semantics of \GHC's Core
2420 language is available \cite[sulzmann07], but this does not seem
2421 sufficient for our goals (but it is a good start).
2423 It should be possible to have a single formal definition of
2424 meaning for Core for both normal Core compilation by \GHC\ and for
2425 our compilation to \VHDL. The main difference seems to be that in
2426 hardware every expression is always evaluated, while in software
2427 it is only evaluated if needed, but it should be possible to
2428 assign a meaning to core expressions that assumes neither.
2430 Since each of the transformations can be applied to any
2431 subexpression as well, there is a constraint on our meaning
2432 definition: the meaning of an expression should depend only on the
2433 meaning of subexpressions, not on the expressions themselves. For
2434 example, the meaning of the application in \lam{f (let x = 4 in
2435 x)} should be the same as the meaning of the application in \lam{f
2436 4}, since the argument subexpression has the same meaning (though
2437 the actual expression is different).
2439 \subsection{Completeness}
2440 Proving completeness is probably not hard, but it could be a lot
2441 of work. We have seen above that to prove completeness, we must
2442 show that the normal set of our graph representation is a subset
2443 of the intended normal set.
2445 However, it is hard to systematically generate or reason about the
2446 normal set, since it is defined as any nodes to which no
2447 transformation applies. To determine this set, each transformation
2448 must be considered and when a transformation is added, the entire
2449 set should be re-evaluated. This means it is hard to show that
2450 each node in the normal set is also in the intended normal set.
2451 Reasoning about our intended normal set is easier, since we know
2452 how to generate it from its definition. \refdef{intended normal
2455 Fortunately, we can also prove the complement (which is
2456 equivalent, since $A \subseteq B \Leftrightarrow \overline{B}
2457 \subseteq \overline{A}$): show that the set of nodes not in
2458 intended normal form is a subset of the set of nodes not in normal
2459 form. In other words, show that for every expression that is not
2460 in intended normal form, that there is at least one transformation
2461 that applies to it (since that means it is not in normal form
2462 either and since $A \subseteq C \Leftrightarrow \forall x (x \in A
2463 \rightarrow x \in C)$).
2465 By systematically reviewing the entire Core language definition
2466 along with the intended normal form definition (both of which have
2467 a similar structure), it should be possible to identify all
2468 possible (sets of) core expressions that are not in intended
2469 normal form and identify a transformation that applies to it.
2471 This approach is especially useful for proving completeness of our
2472 system, since if expressions exist to which none of the
2473 transformations apply (\ie\ if the system is not yet complete), it
2474 is immediately clear which expressions these are and adding
2475 (or modifying) transformations to fix this should be relatively
2478 As observed above, applying this approach is a lot of work, since
2479 we need to check every (set of) transformation(s) separately.
2481 \todo{Perhaps do a few steps of the proofs as proof-of-concept}
2483 % vim: set sw=2 sts=2 expandtab: