1 \chapter[chap:normalization]{Normalization}
3 % A helper to print a single example in the half the page width. The example
4 % text should be in a buffer whose name is given in an argument.
6 % The align=right option really does left-alignment, but without the program
7 % will end up on a single line. The strut=no option prevents a bunch of empty
8 % space at the start of the frame.
10 \framed[offset=1mm,align=right,strut=no,background=box,frame=off]{
11 \setuptyping[option=LAM,style=sans,before=,after=]
13 \setuptyping[option=none,style=\tttf]
18 \define[3]\transexample{
19 \placeexample[here]{#1}
20 \startcombination[2*1]
21 {\example{#2}}{Original program}
22 {\example{#3}}{Transformed program}
26 %\define[3]\transexampleh{
27 %% \placeexample[here]{#1}
28 %% \startcombination[1*2]
29 %% {\example{#2}}{Original program}
30 %% {\example{#3}}{Transformed program}
34 The first step in the core to \small{VHDL} translation process, is normalization. We
35 aim to bring the core description into a simpler form, which we can
36 subsequently translate into \small{VHDL} easily. This normal form is needed because
37 the full core language is more expressive than \small{VHDL} in some areas and because
38 core can describe expressions that do not have a direct hardware
41 TODO: Describe core properties not supported in \small{VHDL}, and describe how the
42 \small{VHDL} we want to generate should look like.
45 The transformations described here have a well-defined goal: To bring the
46 program in a well-defined form that is directly translatable to hardware,
47 while fully preserving the semantics of the program. We refer to this form as
48 the \emph{normal form} of the program. The formal definition of this normal
51 \placedefinition{}{A program is in \emph{normal form} if none of the
52 transformations from this chapter apply.}
54 Of course, this is an \quote{easy} definition of the normal form, since our
55 program will end up in normal form automatically. The more interesting part is
56 to see if this normal form actually has the properties we would like it to
59 But, before getting into more definitions and details about this normal form,
60 let's try to get a feeling for it first. The easiest way to do this is by
61 describing the things we want to not have in a normal form.
64 \item Any \emph{polymorphism} must be removed. When laying down hardware, we
65 can't generate any signals that can have multiple types. All types must be
66 completely known to generate hardware.
68 \item Any \emph{higher order} constructions must be removed. We can't
69 generate a hardware signal that contains a function, so all values,
70 arguments and returns values used must be first order.
72 \item Any complex \emph{nested scopes} must be removed. In the \small{VHDL}
73 description, every signal is in a single scope. Also, full expressions are
74 not supported everywhere (in particular port maps can only map signal names,
75 not expressions). To make the \small{VHDL} generation easy, all values must be bound
76 on the \quote{top level}.
79 TODO: Intermezzo: functions vs plain values
81 A very simple example of a program in normal form is given in
82 \in{example}[ex:MulSum]. As you can see, all arguments to the function (which
83 will become input ports in the final hardware) are at the top. This means that
84 the body of the final lambda abstraction is never a function, but always a
87 After the lambda abstractions, we see a single let expression, that binds two
88 variables (\lam{mul} and \lam{sum}). These variables will be signals in the
89 final hardware, bound to the output port of the \lam{*} and \lam{+}
92 The final line (the \quote{return value} of the function) selects the
93 \lam{sum} signal to be the output port of the function. This \quote{return
94 value} can always only be a variable reference, never a more complex
98 alu :: Bit -> Word -> Word -> Word
107 \startuseMPgraphic{MulSum}
108 save a, b, c, mul, add, sum;
111 newCircle.a(btex $a$ etex) "framed(false)";
112 newCircle.b(btex $b$ etex) "framed(false)";
113 newCircle.c(btex $c$ etex) "framed(false)";
114 newCircle.sum(btex $res$ etex) "framed(false)";
117 newCircle.mul(btex - etex);
118 newCircle.add(btex + etex);
120 a.c - b.c = (0cm, 2cm);
121 b.c - c.c = (0cm, 2cm);
122 add.c = c.c + (2cm, 0cm);
123 mul.c = midpoint(a.c, b.c) + (2cm, 0cm);
124 sum.c = add.c + (2cm, 0cm);
127 % Draw objects and lines
128 drawObj(a, b, c, mul, add, sum);
130 ncarc(a)(mul) "arcangle(15)";
131 ncarc(b)(mul) "arcangle(-15)";
137 \placeexample[here][ex:MulSum]{Simple architecture consisting of an adder and a
139 \startcombination[2*1]
140 {\typebufferlam{MulSum}}{Core description in normal form.}
141 {\boxedgraphic{MulSum}}{The architecture described by the normal form.}
144 The previous example described composing an architecture by calling other
145 functions (operators), resulting in a simple architecture with component and
146 connection. There is of course also some mechanism for choice in the normal
147 form. In a normal Core program, the \emph{case} expression can be used in a
148 few different ways to describe choice. In normal form, this is limited to a
151 \in{Example}[ex:AddSubAlu] shows an example describing a
152 simple \small{ALU}, which chooses between two operations based on an opcode
153 bit. The main structure is the same as in \in{example}[ex:MulSum], but this
154 time the \lam{res} variable is bound to a case expression. This case
155 expression scrutinizes the variable \lam{opcode} (and scrutinizing more
156 complex expressions is not supported). The case expression can select a
157 different variable based on the constructor of \lam{opcode}.
159 \startbuffer[AddSubAlu]
160 alu :: Bit -> Word -> Word -> Word
172 \startuseMPgraphic{AddSubAlu}
173 save opcode, a, b, add, sub, mux, res;
176 newCircle.opcode(btex $opcode$ etex) "framed(false)";
177 newCircle.a(btex $a$ etex) "framed(false)";
178 newCircle.b(btex $b$ etex) "framed(false)";
179 newCircle.res(btex $res$ etex) "framed(false)";
181 newCircle.add(btex + etex);
182 newCircle.sub(btex - etex);
185 opcode.c - a.c = (0cm, 2cm);
186 add.c - a.c = (4cm, 0cm);
187 sub.c - b.c = (4cm, 0cm);
188 a.c - b.c = (0cm, 3cm);
189 mux.c = midpoint(add.c, sub.c) + (1.5cm, 0cm);
190 res.c - mux.c = (1.5cm, 0cm);
193 % Draw objects and lines
194 drawObj(opcode, a, b, res, add, sub, mux);
196 ncline(a)(add) "posA(e)";
197 ncline(b)(sub) "posA(e)";
198 nccurve(a)(sub) "posA(e)", "angleA(0)";
199 nccurve(b)(add) "posA(e)", "angleA(0)";
200 nccurve(add)(mux) "posB(inpa)", "angleB(0)";
201 nccurve(sub)(mux) "posB(inpb)", "angleB(0)";
202 nccurve(opcode)(mux) "posB(n)", "angleA(0)", "angleB(-90)";
203 ncline(mux)(res) "posA(out)";
206 \placeexample[here][ex:AddSubAlu]{Simple \small{ALU} supporting two operations.}
207 \startcombination[2*1]
208 {\typebufferlam{AddSubAlu}}{Core description in normal form.}
209 {\boxedgraphic{AddSubAlu}}{The architecture described by the normal form.}
212 As a more complete example, consider \in{example}[ex:NormalComplete]. This
213 example contains everything that is supported in normal form, with the
214 exception of builtin higher order functions. The graphical version of the
215 architecture contains a slightly simplified version, since the state tuple
216 packing and unpacking have been left out. Instead, two seperate registers are
217 drawn. Also note that most synthesis tools will further optimize this
218 architecture by removing the multiplexers at the register input and replace
219 them with some logic in the clock inputs, but we want to show the architecture
220 as close to the description as possible.
222 \startbuffer[NormalComplete]
225 -> State (Word, Word)
226 -> (State (Word, Word), Word)
228 -- All arguments are an inital lambda
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 (fst, snd) -> fst
237 r2 = case s of (fst, snd) -> snd
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 \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.}
317 \subsection{Intended normal form definition}
318 Now we have some intuition for the normal form, we can describe how we want
319 the normal form to look like in a slightly more formal manner. The following
320 EBNF-like description completely captures the intended structure (and
321 generates a subset of GHC's core format).
323 Some clauses have an expression listed in parentheses. These are conditions
324 that need to apply to the clause.
327 \italic{normal} = \italic{lambda}
328 \italic{lambda} = λvar.\italic{lambda} (representable(var))
330 \italic{toplet} = let \italic{binding} in \italic{toplet}
331 | letrec [\italic{binding}] in \italic{toplet}
332 | var (representable(varvar))
333 \italic{binding} = var = \italic{rhs} (representable(rhs))
334 -- State packing and unpacking by coercion
335 | var0 = var1 :: State ty (lvar(var1))
336 | var0 = var1 :: ty (var0 :: State ty) (lvar(var1))
337 \italic{rhs} = userapp
340 | case var of C a0 ... an -> ai (lvar(var))
342 | case var of (lvar(var))
343 DEFAULT -> var0 (lvar(var0))
344 C w0 ... wn -> resvar (\forall{}i, wi \neq resvar, lvar(resvar))
345 \italic{userapp} = \italic{userfunc}
346 | \italic{userapp} {userarg}
347 \italic{userfunc} = var (gvar(var))
348 \italic{userarg} = var (lvar(var))
349 \italic{builtinapp} = \italic{builtinfunc}
350 | \italic{builtinapp} \italic{builtinarg}
351 \italic{builtinfunc} = var (bvar(var))
352 \italic{builtinarg} = \italic{coreexpr}
355 -- TODO: Limit builtinarg further
357 -- TODO: There can still be other casts around (which the code can handle,
358 e.g., ignore), which still need to be documented here.
360 -- TODO: Note about the selector case. It just supports Bit and Bool
361 currently, perhaps it should be generalized in the normal form?
363 When looking at such a program from a hardware perspective, the top level
364 lambda's define the input ports. The value produced by the let expression is
365 the output port. Most function applications bound by the let expression
366 define a component instantiation, where the input and output ports are mapped
367 to local signals or arguments. Some of the others use a builtin
368 construction (\eg the \lam{case} statement) or call a builtin function
369 (\eg \lam{add} or \lam{sub}). For these, a hardcoded \small{VHDL} translation is
372 \section{Transformation notation}
373 To be able to concisely present transformations, we use a specific format to
374 them. It is a simple format, similar to one used in logic reasoning.
376 Such a transformation description looks like the following.
381 <original expression>
382 -------------------------- <expression conditions>
383 <transformed expresssion>
388 This format desribes a transformation that applies to \lam{original
389 expresssion} and transforms it into \lam{transformed expression}, assuming
390 that all conditions apply. In this format, there are a number of placeholders
391 in pointy brackets, most of which should be rather obvious in their meaning.
392 Nevertheless, we will more precisely specify their meaning below:
394 \startdesc{<original expression>} The expression pattern that will be matched
395 against (subexpressions of) the expression to be transformed. We call this a
396 pattern, because it can contain \emph{placeholders} (variables), which match
397 any expression or binder. Any such placeholder is said to be \emph{bound} to
398 the expression it matches. It is convention to use an uppercase latter (\eg
399 \lam{M} or \lam{E} to refer to any expression (including a simple variable
400 reference) and lowercase letters (\eg \lam{v} or \lam{b}) to refer to
401 (references to) binders.
403 For example, the pattern \lam{a + B} will match the expression
404 \lam{v + (2 * w)} (and bind \lam{a} to \lam{v} and \lam{B} to
405 \lam{(2 * 2)}), but not \lam{v + (2 * w)}.
408 \startdesc{<expression conditions>}
409 These are extra conditions on the expression that is matched. These
410 conditions can be used to further limit the cases in which the
411 transformation applies, in particular to prevent a transformation from
412 causing a loop with itself or another transformation.
414 Only if these if these conditions are \emph{all} true, this transformation
418 \startdesc{<context conditions>}
419 These are a number of extra conditions on the context of the function. In
420 particular, these conditions can require some other top level function to be
421 present, whose value matches the pattern given here. The format of each of
422 these conditions is: \lam{binder = <pattern>}.
424 Typically, the binder is some placeholder bound in the \lam{<original
425 expression>}, while the pattern contains some placeholders that are used in
426 the \lam{transformed expression}.
428 Only if a top level binder exists that matches each binder and pattern, this
429 transformation applies.
432 \startdesc{<transformed expression>}
433 This is the expression template that is the result of the transformation. If, looking
434 at the above three items, the transformation applies, the \lam{original
435 expression} is completely replaced with the \lam{<transformed expression>}.
436 We call this a template, because it can contain placeholders, referring to
437 any placeholder bound by the \lam{<original expression>} or the
438 \lam{<context conditions>}. The resulting expression will have those
439 placeholders replaced by the values bound to them.
441 Any binder (lowercase) placeholder that has no value bound to it yet will be
442 bound to (and replaced with) a fresh binder.
445 \startdesc{<context additions>}
446 These are templates for new functions to add to the context. This is a way
447 to have a transformation create new top level functiosn.
449 Each addition has the form \lam{binder = template}. As above, any
450 placeholder in the addition is replaced with the value bound to it, and any
451 binder placeholder that has no value bound to it yet will be bound to (and
452 replaced with) a fresh binder.
455 As an example, we'll look at η-abstraction:
459 -------------- \lam{E} does not occur on a function position in an application
460 λx.E x \lam{E} is not a lambda abstraction.
463 Consider the following function, which is a fairly obvious way to specify a
464 simple ALU (Note \at{example}[ex:AddSubAlu] is the normal form of this
468 alu :: Bit -> Word -> Word -> Word
469 alu = λopcode. case opcode of
474 There are a few subexpressions in this function to which we could possibly
475 apply the transformation. Since the pattern of the transformation is only
476 the placeholder \lam{E}, any expression will match that. Whether the
477 transformation applies to an expression is thus solely decided by the
478 conditions to the right of the transformation.
480 We will look at each expression in the function in a top down manner. The
481 first expression is the entire expression the function is bound to.
484 λopcode. case opcode of
489 As said, the expression pattern matches this. The type of this expression is
490 \lam{Bit -> Word -> Word -> Word}, which matches \lam{a -> b} (Note that in
491 this case \lam{a = Bit} and \lam{b = Word -> Word -> Word}).
493 Since this expression is at top level, it does not occur at a function
494 position of an application. However, The expression is a lambda abstraction,
495 so this transformation does not apply.
497 The next expression we could apply this transformation to, is the body of
498 the lambda abstraction:
506 The type of this expression is \lam{Word -> Word -> Word}, which again
507 matches \lam{a -> b}. The expression is the body of a lambda expression, so
508 it does not occur at a function position of an application. Finally, the
509 expression is not a lambda abstraction but a case expression, so all the
510 conditions match. There are no context conditions to match, so the
511 transformation applies.
513 By now, the placeholder \lam{E} is bound to the entire expression. The
514 placeholder \lam{x}, which occurs in the replacement template, is not bound
515 yet, so we need to generate a fresh binder for that. Let's use the binder
516 \lam{a}. This results in the following replacement expression:
524 Continuing with this expression, we see that the transformation does not
525 apply again (it is a lambda expression). Next we look at the body of this
534 Here, the transformation does apply, binding \lam{E} to the entire
535 expression and \lam{x} to the fresh binder \lam{b}, resulting in the
544 Again, the transformation does not apply to this lambda abstraction, so we
545 look at its body. For brevity, we'll put the case statement on one line from
549 (case opcode of Low -> (+); High -> (-)) a b
552 The type of this expression is \lam{Word}, so it does not match \lam{a -> b}
553 and the transformation does not apply. Next, we have two options for the
554 next expression to look at: The function position and argument position of
555 the application. The expression in the argument position is \lam{b}, which
556 has type \lam{Word}, so the transformation does not apply. The expression in
557 the function position is:
560 (case opcode of Low -> (+); High -> (-)) a
563 Obviously, the transformation does not apply here, since it occurs in
564 function position. In the same way the transformation does not apply to both
565 components of this expression (\lam{case opcode of Low -> (+); High -> (-)}
566 and \lam{a}), so we'll skip to the components of the case expression: The
567 scrutinee and both alternatives. Since the opcode is not a function, it does
568 not apply here, and we'll leave both alternatives as an exercise to the
569 reader. The final function, after all these transformations becomes:
572 alu :: Bit -> Word -> Word -> Word
573 alu = λopcode.λa.b. (case opcode of
574 Low -> λa1.λb1 (+) a1 b1
575 High -> λa2.λb2 (-) a2 b2) a b
578 In this case, the transformation does not apply anymore, though this might
579 not always be the case (e.g., the application of a transformation on a
580 subexpression might open up possibilities to apply the transformation
581 further up in the expression).
583 \subsection{Transformation application}
584 In this chapter we define a number of transformations, but how will we apply
585 these? As stated before, our normal form is reached as soon as no
586 transformation applies anymore. This means our application strategy is to
587 simply apply any transformation that applies, and continuing to do that with
588 the result of each transformation.
590 In particular, we define no particular order of transformations. Since
591 transformation order should not influence the resulting normal form (see TODO
592 ref), this leaves the implementation free to choose any application order that
593 results in an efficient implementation.
595 When applying a single transformation, we try to apply it to every (sub)expression
596 in a function, not just the top level function. This allows us to keep the
597 transformation descriptions concise and powerful.
599 \subsection{Definitions}
600 In the following sections, we will be using a number of functions and
601 notations, which we will define here.
603 \subsubsection{Other concepts}
604 A \emph{global variable} is any variable that is bound at the
605 top level of a program, or an external module. A \emph{local variable} is any
606 other variable (\eg, variables local to a function, which can be bound by
607 lambda abstractions, let expressions and pattern matches of case
608 alternatives). Note that this is a slightly different notion of global versus
609 local than what \small{GHC} uses internally.
610 \defref{global variable} \defref{local variable}
612 A \emph{hardware representable} (or just \emph{representable}) type or value
613 is (a value of) a type that we can generate a signal for in hardware. For
614 example, a bit, a vector of bits, a 32 bit unsigned word, etc. Types that are
615 not runtime representable notably include (but are not limited to): Types,
616 dictionaries, functions.
617 \defref{representable}
619 A \emph{builtin function} is a function supplied by the Cλash framework, whose
620 implementation is not valid Cλash. The implementation is of course valid
621 Haskell, for simulation, but it is not expressable in Cλash.
622 \defref{builtin function} \defref{user-defined function}
624 For these functions, Cλash has a \emph{builtin hardware translation}, so calls
625 to these functions can still be translated. These are functions like
626 \lam{map}, \lam{hwor} and \lam{length}.
628 A \emph{user-defined} function is a function for which we do have a Cλash
629 implementation available.
631 \subsubsection{Functions}
632 Here, we define a number of functions that can be used below to concisely
635 \refdef{global variable}\emph{gvar(expr)} is true when \emph{expr} is a variable that references a
636 global variable. It is false when it references a local variable.
638 \refdef{local variable}\emph{lvar(expr)} is the complement of \emph{gvar}; it is true when \emph{expr}
639 references a local variable, false when it references a global variable.
641 \refdef{representable}\emph{representable(expr)} or \emph{representable(var)} is true when
642 \emph{expr} or \emph{var} is \emph{representable}.
644 \subsection{Binder uniqueness}
645 A common problem in transformation systems, is binder uniqueness. When not
646 considering this problem, it is easy to create transformations that mix up
647 bindings and cause name collisions. Take for example, the following core
651 (λa.λb.λc. a * b * c) x c
654 By applying β-reduction (see below) once, we can simplify this expression to:
660 Now, we have replaced the \lam{a} binder with a reference to the \lam{x}
661 binder. No harm done here. But note that we see multiple occurences of the
662 \lam{c} binder. The first is a binding occurence, to which the second refers.
663 The last, however refers to \emph{another} instance of \lam{c}, which is
664 bound somewhere outside of this expression. Now, if we would apply beta
665 reduction without taking heed of binder uniqueness, we would get:
671 This is obviously not what was supposed to happen! The root of this problem is
672 the reuse of binders: Identical binders can be bound in different scopes, such
673 that only the inner one is \quote{visible} in the inner expression. In the example
674 above, the \lam{c} binder was bound outside of the expression and in the inner
675 lambda expression. Inside that lambda expression, only the inner \lam{c} is
678 There are a number of ways to solve this. \small{GHC} has isolated this
679 problem to their binder substitution code, which performs \emph{deshadowing}
680 during its expression traversal. This means that any binding that shadows
681 another binding on a higher level is replaced by a new binder that does not
682 shadow any other binding. This non-shadowing invariant is enough to prevent
683 binder uniqueness problems in \small{GHC}.
685 In our transformation system, maintaining this non-shadowing invariant is
686 a bit harder to do (mostly due to implementation issues, the prototype doesn't
687 use \small{GHC}'s subsitution code). Also, we can observe the following
691 \item Deshadowing does not guarantee overall uniqueness. For example, the
692 following (slightly contrived) expression shows the identifier \lam{x} bound in
693 two seperate places (and to different values), even though no shadowing
697 (let x = 1 in x) + (let x = 2 in x)
700 \item In our normal form (and the resulting \small{VHDL}), all binders
701 (signals) will end up in the same scope. To allow this, all binders within the
702 same function should be unique.
704 \item When we know that all binders in an expression are unique, moving around
705 or removing a subexpression will never cause any binder conflicts. If we have
706 some way to generate fresh binders, introducing new subexpressions will not
707 cause any problems either. The only way to cause conflicts is thus to
708 duplicate an existing subexpression.
711 Given the above, our prototype maintains a unique binder invariant. This
712 meanst that in any given moment during normalization, all binders \emph{within
713 a single function} must be unique. To achieve this, we apply the following
716 TODO: Define fresh binders and unique supplies
719 \item Before starting normalization, all binders in the function are made
720 unique. This is done by generating a fresh binder for every binder used. This
721 also replaces binders that did not pose any conflict, but it does ensure that
722 all binders within the function are generated by the same unique supply. See
723 (TODO: ref fresh binder).
724 \item Whenever a new binder must be generated, we generate a fresh binder that
725 is guaranteed to be different from \emph{all binders generated so far}. This
726 can thus never introduce duplication and will maintain the invariant.
727 \item Whenever (part of) an expression is duplicated (for example when
728 inlining), all binders in the expression are replaced with fresh binders
729 (using the same method as at the start of normalization). These fresh binders
730 can never introduce duplication, so this will maintain the invariant.
731 \item Whenever we move part of an expression around within the function, there
732 is no need to do anything special. There is obviously no way to introduce
733 duplication by moving expressions around. Since we know that each of the
734 binders is already unique, there is no way to introduce (incorrect) shadowing
738 \section{Transform passes}
739 In this section we describe the actual transforms. Here we're using
740 the core language in a notation that resembles lambda calculus.
742 Each of these transforms is meant to be applied to every (sub)expression
743 in a program, for as long as it applies. Only when none of the
744 transformations can be applied anymore, the program is in normal form (by
745 definition). We hope to be able to prove that this form will obey all of the
746 constraints defined above, but this has yet to happen (though it seems likely
749 Each of the transforms will be described informally first, explaining
750 the need for and goal of the transform. Then, a formal definition is
751 given, using a familiar syntax from the world of logic. Each transform
752 is specified as a number of conditions (above the horizontal line) and a
753 number of conclusions (below the horizontal line). The details of using
754 this notation are still a bit fuzzy, so comments are welcom.
756 \subsection{General cleanup}
758 \subsubsection{β-reduction}
759 β-reduction is a well known transformation from lambda calculus, where it is
760 the main reduction step. It reduces applications of labmda abstractions,
761 removing both the lambda abstraction and the application.
763 In our transformation system, this step helps to remove unwanted lambda
764 abstractions (basically all but the ones at the top level). Other
765 transformations (application propagation, non-representable inlining) make
766 sure that most lambda abstractions will eventually be reducable by
769 TODO: Define substitution syntax
786 \transexample{β-reduction}{from}{to}
788 \subsubsection{Application propagation}
789 This transformation is meant to propagate application expressions downwards
790 into expressions as far as possible. This allows partial applications inside
791 expressions to become fully applied and exposes new transformation
792 possibilities for other transformations (like β-reduction).
816 \transexample{Application propagation for a let expression}{from}{to}
844 \transexample{Application propagation for a case expression}{from}{to}
846 \subsubsection{Empty let removal}
847 This transformation is simple: It removes recursive lets that have no bindings
848 (which usually occurs when let derecursification removes the last binding from
857 \subsubsection{Simple let binding removal}
858 This transformation inlines simple let bindings (\eg a = b).
860 This transformation is not needed to get into normal form, but makes the
861 resulting \small{VHDL} a lot shorter.
887 \subsubsection{Unused let binding removal}
888 This transformation removes let bindings that are never used. Usually,
889 the desugarer introduces some unused let bindings.
891 This normalization pass should really be unneeded to get into normal form
892 (since ununsed bindings are not forbidden by the normal form), but in practice
893 the desugarer or simplifier emits some unused bindings that cannot be
894 normalized (e.g., calls to a \type{PatError} (TODO: Check this name)). Also,
895 this transformation makes the resulting \small{VHDL} a lot shorter.
899 ---------------------------- \lam{a} does not occur free in \lam{M}
910 ---------------------------- \lam{a} does not occur free in \lam{M}
918 \subsubsection{Cast propagation / simplification}
919 This transform pushes casts down into the expression as far as possible.
920 Since its exact role and need is not clear yet, this transformation is
923 \subsubsection{Compiler generated top level binding inlining}
926 \section{Program structure}
928 \subsubsection{η-abstraction}
929 This transformation makes sure that all arguments of a function-typed
930 expression are named, by introducing lambda expressions. When combined with
931 β-reduction and function inlining below, all function-typed expressions should
932 be lambda abstractions or global identifiers.
936 -------------- \lam{E} is not the first argument of an application.
937 λx.E x \lam{E} is not a lambda abstraction.
938 \lam{x} is a variable that does not occur free in \lam{E}.
948 foo = λa.λx.(case a of
953 \transexample{η-abstraction}{from}{to}
955 \subsubsection{Let derecursification}
956 This transformation is meant to make lets non-recursive whenever possible.
957 This might allow other optimizations to do their work better. TODO: Why is
960 \subsubsection{Let flattening}
961 This transformation puts nested lets in the same scope, by lifting the
962 binding(s) of the inner let into a new let around the outer let. Eventually,
963 this will cause all let bindings to appear in the same scope (they will all be
964 in scope for the function return value).
966 Note that this transformation does not try to be smart when faced with
967 recursive lets, it will just leave the lets recursive (possibly joining a
968 recursive and non-recursive let into a single recursive let). The let
969 rederecursification transformation will do this instead.
972 letnonrec x = (let bindings in M) in N
973 ------------------------------------------
974 let bindings in (letnonrec x = M) in N
980 x = (let bindings in M)
984 ------------------------------------------
1003 b = let c = 3 in a + c
1024 \transexample{Let flattening}{from}{to}
1026 \subsubsection{Return value simplification}
1027 This transformation ensures that the return value of a function is always a
1028 simple local variable reference.
1030 Currently implemented using lambda simplification, let simplification, and
1031 top simplification. Should change into something like the following, which
1032 works only on the result of a function instead of any subexpression. This is
1033 achieved by the contexts, like \lam{x = E}, though this is strictly not
1034 correct (you could read this as "if there is any function \lam{x} that binds
1035 \lam{E}, any \lam{E} can be transformed, while we only mean the \lam{E} that
1036 is bound by \lam{x}. This might need some extra notes or something).
1039 x = E \lam{E} is representable
1040 ~ \lam{E} is not a lambda abstraction
1041 E \lam{E} is not a let expression
1042 --------------------------- \lam{E} is not a local variable reference
1048 ~ \lam{E} is representable
1049 E \lam{E} is not a let expression
1050 --------------------------- \lam{E} is not a local variable reference
1055 x = λv0 ... λvn.let ... in E
1056 ~ \lam{E} is representable
1057 E \lam{E} is not a local variable reference
1058 ---------------------------
1067 x = let x = add 1 2 in x
1070 \transexample{Return value simplification}{from}{to}
1072 \subsection{Argument simplification}
1073 The transforms in this section deal with simplifying application
1074 arguments into normal form. The goal here is to:
1077 \item Make all arguments of user-defined functions (\eg, of which
1078 we have a function body) simple variable references of a runtime
1079 representable type. This is needed, since these applications will be turned
1080 into component instantiations.
1081 \item Make all arguments of builtin functions one of:
1083 \item A type argument.
1084 \item A dictionary argument.
1085 \item A type level expression.
1086 \item A variable reference of a runtime representable type.
1087 \item A variable reference or partial application of a function type.
1091 When looking at the arguments of a user-defined function, we can
1092 divide them into two categories:
1094 \item Arguments of a runtime representable type (\eg bits or vectors).
1096 These arguments can be preserved in the program, since they can
1097 be translated to input ports later on. However, since we can
1098 only connect signals to input ports, these arguments must be
1099 reduced to simple variables (for which signals will be
1100 produced). This is taken care of by the argument extraction
1102 \item Non-runtime representable typed arguments.
1104 These arguments cannot be preserved in the program, since we
1105 cannot represent them as input or output ports in the resulting
1106 \small{VHDL}. To remove them, we create a specialized version of the
1107 called function with these arguments filled in. This is done by
1108 the argument propagation transform.
1110 Typically, these arguments are type and dictionary arguments that are
1111 used to make functions polymorphic. By propagating these arguments, we
1112 are essentially doing the same which GHC does when it specializes
1113 functions: Creating multiple variants of the same function, one for
1114 each type for which it is used. Other common non-representable
1115 arguments are functions, e.g. when calling a higher order function
1116 with another function or a lambda abstraction as an argument.
1118 The reason for doing this is similar to the reasoning provided for
1119 the inlining of non-representable let bindings above. In fact, this
1120 argument propagation could be viewed as a form of cross-function
1124 TODO: Check the following itemization.
1126 When looking at the arguments of a builtin function, we can divide them
1130 \item Arguments of a runtime representable type.
1132 As we have seen with user-defined functions, these arguments can
1133 always be reduced to a simple variable reference, by the
1134 argument extraction transform. Performing this transform for
1135 builtin functions as well, means that the translation of builtin
1136 functions can be limited to signal references, instead of
1137 needing to support all possible expressions.
1139 \item Arguments of a function type.
1141 These arguments are functions passed to higher order builtins,
1142 like \lam{map} and \lam{foldl}. Since implementing these
1143 functions for arbitrary function-typed expressions (\eg, lambda
1144 expressions) is rather comlex, we reduce these arguments to
1145 (partial applications of) global functions.
1147 We can still support arbitrary expressions from the user code,
1148 by creating a new global function containing that expression.
1149 This way, we can simply replace the argument with a reference to
1150 that new function. However, since the expression can contain any
1151 number of free variables we also have to include partial
1152 applications in our normal form.
1154 This category of arguments is handled by the function extraction
1156 \item Other unrepresentable arguments.
1158 These arguments can take a few different forms:
1159 \startdesc{Type arguments}
1160 In the core language, type arguments can only take a single
1161 form: A type wrapped in the Type constructor. Also, there is
1162 nothing that can be done with type expressions, except for
1163 applying functions to them, so we can simply leave type
1164 arguments as they are.
1166 \startdesc{Dictionary arguments}
1167 In the core language, dictionary arguments are used to find
1168 operations operating on one of the type arguments (mostly for
1169 finding class methods). Since we will not actually evaluatie
1170 the function body for builtin functions and can generate
1171 code for builtin functions by just looking at the type
1172 arguments, these arguments can be ignored and left as they
1175 \startdesc{Type level arguments}
1176 Sometimes, we want to pass a value to a builtin function, but
1177 we need to know the value at compile time. Additionally, the
1178 value has an impact on the type of the function. This is
1179 encoded using type-level values, where the actual value of the
1180 argument is not important, but the type encodes some integer,
1181 for example. Since the value is not important, the actual form
1182 of the expression does not matter either and we can leave
1183 these arguments as they are.
1185 \startdesc{Other arguments}
1186 Technically, there is still a wide array of arguments that can
1187 be passed, but does not fall into any of the above categories.
1188 However, none of the supported builtin functions requires such
1189 an argument. This leaves use with passing unsupported types to
1190 a function, such as calling \lam{head} on a list of functions.
1192 In these cases, it would be impossible to generate hardware
1193 for such a function call anyway, so we can ignore these
1196 The only way to generate hardware for builtin functions with
1197 arguments like these, is to expand the function call into an
1198 equivalent core expression (\eg, expand map into a series of
1199 function applications). But for now, we choose to simply not
1200 support expressions like these.
1203 From the above, we can conclude that we can simply ignore these
1204 other unrepresentable arguments and focus on the first two
1208 \subsubsection{Argument simplification}
1209 This transform deals with arguments to functions that
1210 are of a runtime representable type. It ensures that they will all become
1211 references to global variables, or local signals in the resulting \small{VHDL}.
1213 TODO: It seems we can map an expression to a port, not only a signal.
1214 Perhaps this makes this transformation not needed?
1215 TODO: Say something about dataconstructors (without arguments, like True
1216 or False), which are variable references of a runtime representable
1217 type, but do not result in a signal.
1219 To reduce a complex expression to a simple variable reference, we create
1220 a new let expression around the application, which binds the complex
1221 expression to a new variable. The original function is then applied to
1226 -------------------- \lam{N} is of a representable type
1227 let x = N in M x \lam{N} is not a local variable reference
1235 let x = add a 1 in add x 1
1238 \transexample{Argument extraction}{from}{to}
1240 \subsubsection{Function extraction}
1241 This transform deals with function-typed arguments to builtin functions.
1242 Since these arguments cannot be propagated, we choose to extract them
1243 into a new global function instead.
1245 Any free variables occuring in the extracted arguments will become
1246 parameters to the new global function. The original argument is replaced
1247 with a reference to the new function, applied to any free variables from
1248 the original argument.
1250 This transformation is useful when applying higher order builtin functions
1251 like \hs{map} to a lambda abstraction, for example. In this case, the code
1252 that generates \small{VHDL} for \hs{map} only needs to handle top level functions and
1253 partial applications, not any other expression (such as lambda abstractions or
1254 even more complicated expressions).
1257 M N \lam{M} is a (partial aplication of a) builtin function.
1258 --------------------- \lam{f0 ... fn} = free local variables of \lam{N}
1259 M x f0 ... fn \lam{N :: a -> b}
1260 ~ \lam{N} is not a (partial application of) a top level function
1265 map (λa . add a b) xs
1279 \transexample{Function extraction}{from}{to}
1281 \subsubsection{Argument propagation}
1282 This transform deals with arguments to user-defined functions that are
1283 not representable at runtime. This means these arguments cannot be
1284 preserved in the final form and most be {\em propagated}.
1286 Propagation means to create a specialized version of the called
1287 function, with the propagated argument already filled in. As a simple
1288 example, in the following program:
1295 we could {\em propagate} the constant argument 1, with the following
1303 Special care must be taken when the to-be-propagated expression has any
1304 free variables. If this is the case, the original argument should not be
1305 removed alltogether, but replaced by all the free variables of the
1306 expression. In this way, the original expression can still be evaluated
1307 inside the new function. Also, this brings us closer to our goal: All
1308 these free variables will be simple variable references.
1310 To prevent us from propagating the same argument over and over, a simple
1311 local variable reference is not propagated (since is has exactly one
1312 free variable, itself, we would only replace that argument with itself).
1314 This shows that any free local variables that are not runtime representable
1315 cannot be brought into normal form by this transform. We rely on an
1316 inlining transformation to replace such a variable with an expression we
1317 can propagate again.
1322 x Y0 ... Yi ... Yn \lam{Yi} is not of a runtime representable type
1323 --------------------------------------------- \lam{Yi} is not a local variable reference
1324 x' y0 ... yi-1 f0 ... fm Yi+1 ... Yn \lam{f0 ... fm} = free local vars of \lam{Yi}
1326 x' = λy0 ... yi-1 f0 ... fm yi+1 ... yn .
1327 E y0 ... yi-1 Yi yi+1 ... yn
1333 \subsection{Case simplification}
1334 \subsubsection{Scrutinee simplification}
1335 This transform ensures that the scrutinee of a case expression is always
1336 a simple variable reference.
1341 ----------------- \lam{E} is not a local variable reference
1360 \transexample{Let flattening}{from}{to}
1363 \subsubsection{Case simplification}
1364 This transformation ensures that all case expressions become normal form. This
1365 means they will become one of:
1367 \item An extractor case with a single alternative that picks a single field
1368 from a datatype, \eg \lam{case x of (a, b) -> a}.
1369 \item A selector case with multiple alternatives and only wild binders, that
1370 makes a choice between expressions based on the constructor of another
1371 expression, \eg \lam{case x of Low -> a; High -> b}.
1376 C0 v0,0 ... v0,m -> E0
1378 Cn vn,0 ... vn,m -> En
1379 --------------------------------------------------- \forall i \forall j, 0 <= i <= n, 0 <= i < m (\lam{wi,j} is a wild (unused) binder)
1381 v0,0 = case x of C0 v0,0 .. v0,m -> v0,0
1383 v0,m = case x of C0 v0,0 .. v0,m -> v0,m
1386 vn,m = case x of Cn vn,0 .. vn,m -> vn,m
1390 C0 w0,0 ... w0,m -> x0
1392 Cn wn,0 ... wn,m -> xn
1395 TODO: This transformation specified like this is complicated and misses
1396 conditions to prevent looping with itself. Perhaps we should split it here for
1415 \transexample{Selector case simplification}{from}{to}
1423 b = case a of (,) b c -> b
1424 c = case a of (,) b c -> c
1431 \transexample{Extractor case simplification}{from}{to}
1433 \subsubsection{Case removal}
1434 This transform removes any case statements with a single alternative and
1437 These "useless" case statements are usually leftovers from case simplification
1438 on extractor case (see the previous example).
1443 ---------------------- \lam{\forall i, 0 <= i <= m} (\lam{vi} does not occur free in E)
1456 \transexample{Case removal}{from}{to}
1458 \subsection{Monomorphisation}
1459 TODO: Better name for this section
1461 Reference type-specialization (== argument propagation)
1463 \subsubsection{Defunctionalization}
1464 Reference higher-order-specialization (== argument propagation)
1466 \subsubsection{Non-representable binding inlining}
1467 This transform inlines let bindings that have a non-representable type. Since
1468 we can never generate a signal assignment for these bindings (we cannot
1469 declare a signal assignment with a non-representable type, for obvious
1470 reasons), we have no choice but to inline the binding to remove it.
1472 If the binding is non-representable because it is a lambda abstraction, it is
1473 likely that it will inlined into an application and β-reduction will remove
1474 the lambda abstraction and turn it into a representable expression at the
1475 inline site. The same holds for partial applications, which can be turned into
1476 full applications by inlining.
1478 Other cases of non-representable bindings we see in practice are primitive
1479 Haskell types. In most cases, these will not result in a valid normalized
1480 output, but then the input would have been invalid to start with. There is one
1481 exception to this: When a builtin function is applied to a non-representable
1482 expression, things might work out in some cases. For example, when you write a
1483 literal \hs{SizedInt} in Haskell, like \hs{1 :: SizedInt D8}, this results in
1484 the following core: \lam{fromInteger (smallInteger 10)}, where for example
1485 \lam{10 :: GHC.Prim.Int\#} and \lam{smallInteger 10 :: Integer} have
1486 non-representable types. TODO: This/these paragraph(s) should probably become a
1487 separate discussion somewhere else.
1490 letnonrec a = E in M
1491 -------------------------- \lam{E} has a non-representable type.
1502 -------------------------- \lam{E} has a non-representable type.
1522 x = fromInteger (smallInteger 10)
1524 (λa -> add a 1) (add 1 x)
1527 \transexample{Let flattening}{from}{to}
1530 \section{Provable properties}
1531 When looking at the system of transformations outlined above, there are a
1532 number of questions that we can ask ourselves. The main question is of course:
1533 \quote{Does our system work as intended?}. We can split this question into a
1534 number of subquestions:
1537 \item[q:termination] Does our system \emph{terminate}? Since our system will
1538 keep running as long as transformations apply, there is an obvious risk that
1539 it will keep running indefinitely. One transformation produces a result that
1540 is transformed back to the original by another transformation, for example.
1541 \item[q:soundness] Is our system \emph{sound}? Since our transformations
1542 continuously modify the expression, there is an obvious risk that the final
1543 normal form will not be equivalent to the original program: Its meaning could
1545 \item[q:completeness] Is our system \emph{complete}? Since we have a complex
1546 system of transformations, there is an obvious risk that some expressions will
1547 not end up in our intended normal form, because we forgot some transformation.
1548 In other words: Does our transformation system result in our intended normal
1549 form for all possible inputs?
1550 \item[q:determinism] Is our system \emph{deterministic}? Since we have defined
1551 no particular order in which the transformation should be applied, there is an
1552 obvious risk that different transformation orderings will result in
1553 \emph{different} normal forms. They might still both be intended normal forms
1554 (if our system is \emph{complete}) and describe correct hardware (if our
1555 system is \emph{sound}), so this property is less important than the previous
1556 three: The translator would still function properly without it.
1559 \subsection{Graph representation}
1560 Before looking into how to prove these properties, we'll look at our
1561 transformation system from a graph perspective. The nodes of the graph are
1562 all possible Core expressions. The (directed) edges of the graph are
1563 transformations. When a transformation α applies to an expression \lam{A} to
1564 produce an expression \lam{B}, we add an edge from the node for \lam{A} to the
1565 node for \lam{B}, labeled α.
1567 \startuseMPgraphic{TransformGraph}
1571 newCircle.a(btex \lam{(λx.λy. (+) x y) 1} etex);
1572 newCircle.b(btex \lam{λy. (+) 1 y} etex);
1573 newCircle.c(btex \lam{(λx.(+) x) 1} etex);
1574 newCircle.d(btex \lam{(+) 1} etex);
1577 c.c = b.c + (4cm, 0cm);
1578 a.c = midpoint(b.c, c.c) + (0cm, 4cm);
1579 d.c = midpoint(b.c, c.c) - (0cm, 3cm);
1581 % β-conversion between a and b
1582 ncarc.a(a)(b) "name(bred)";
1583 ObjLabel.a(btex $\xrightarrow[normal]{}{β}$ etex) "labpathname(bred)", "labdir(rt)";
1584 ncarc.b(b)(a) "name(bexp)", "linestyle(dashed withdots)";
1585 ObjLabel.b(btex $\xleftarrow[normal]{}{β}$ etex) "labpathname(bexp)", "labdir(lft)";
1587 % η-conversion between a and c
1588 ncarc.a(a)(c) "name(ered)";
1589 ObjLabel.a(btex $\xrightarrow[normal]{}{η}$ etex) "labpathname(ered)", "labdir(rt)";
1590 ncarc.c(c)(a) "name(eexp)", "linestyle(dashed withdots)";
1591 ObjLabel.c(btex $\xleftarrow[normal]{}{η}$ etex) "labpathname(eexp)", "labdir(lft)";
1593 % η-conversion between b and d
1594 ncarc.b(b)(d) "name(ered)";
1595 ObjLabel.b(btex $\xrightarrow[normal]{}{η}$ etex) "labpathname(ered)", "labdir(rt)";
1596 ncarc.d(d)(b) "name(eexp)", "linestyle(dashed withdots)";
1597 ObjLabel.d(btex $\xleftarrow[normal]{}{η}$ etex) "labpathname(eexp)", "labdir(lft)";
1599 % β-conversion between c and d
1600 ncarc.c(c)(d) "name(bred)";
1601 ObjLabel.c(btex $\xrightarrow[normal]{}{β}$ etex) "labpathname(bred)", "labdir(rt)";
1602 ncarc.d(d)(c) "name(bexp)", "linestyle(dashed withdots)";
1603 ObjLabel.d(btex $\xleftarrow[normal]{}{β}$ etex) "labpathname(bexp)", "labdir(lft)";
1605 % Draw objects and lines
1606 drawObj(a, b, c, d);
1609 \placeexample[right][ex:TransformGraph]{Partial graph of a labmda calculus
1610 system with β and η reduction (solid lines) and expansion (dotted lines).}
1611 \boxedgraphic{TransformGraph}
1613 Of course our graph is unbounded, since we can construct an infinite amount of
1614 Core expressions. Also, there might potentially be multiple edges between two
1615 given nodes (with different labels), though seems unlikely to actually happen
1618 See \in{example}[ex:TransformGraph] for the graph representation of a very
1619 simple lambda calculus that contains just the expressions \lam{(λx.λy. (+) x
1620 y) 1}, \lam{λy. (+) 1 y}, \lam{(λx.(+) x) 1} and \lam{(+) 1}. The
1621 transformation system consists of β-reduction and η-reduction (solid edges) or
1622 β-reduction and η-reduction (dotted edges).
1624 TODO: Define β-reduction and η-reduction?
1626 Note that the normal form of such a system consists of the set of nodes
1627 (expressions) without outgoing edges, since those are the expression to which
1628 no transformation applies anymore. We call this set of nodes the \emph{normal
1631 From such a graph, we can derive some properties easily:
1633 \item A system will \emph{terminate} if there is no path of infinite length
1634 in the graph (this includes cycles).
1635 \item Soundness is not easily represented in the graph.
1636 \item A system is \emph{complete} if all of the nodes in the normal set have
1637 the intended normal form. The inverse (that all of the nodes outside of
1638 the normal set are \emph{not} in the intended normal form) is not
1640 \item A system is deterministic if all paths from a node, which end in a node
1641 in the normal set, end at the same node.
1644 When looking at the \in{example}[ex:TransformGraph], we see that the system
1645 terminates for both the reduction and expansion systems (but note that, for
1646 expansion, this is only true because we've limited the possible expressions!
1647 In comlete lambda calculus, there would be a path from \lam{(λx.λy. (+) x y)
1648 1} to \lam{(λx.λy.(λz.(+) z) x y) 1} to \lam{(λx.λy.(λz.(λq.(+) q) z) x y) 1}
1651 If we would consider the system with both expansion and reduction, there would
1652 no longer be termination, since there would be cycles all over the place.
1654 The reduction and expansion systems have a normal set of containing just
1655 \lam{(+) 1} or \lam{(λx.λy. (+) x y) 1} respectively. Since all paths in
1656 either system end up in these normal forms, both systems are \emph{complete}.
1657 Also, since there is only one normal form, it must obviously be
1658 \emph{deterministic} as well.
1660 \subsection{Termination}
1665 \subsection{Soundness}
1666 Needs formal definition of semantics.
1667 Prove for each transformation seperately, implies soundness of the system.
1669 \subsection{Completeness}
1670 Show that any transformation applies to every Core expression that is not
1671 in normal form. To prove: no transformation applies => in intended form.
1672 Show the reverse: Not in intended form => transformation applies.
1674 \subsection{Determinism}