1 \chapter[chap:normalization]{Normalization}
2 % A helper to print a single example in the half the page width. The example
3 % text should be in a buffer whose name is given in an argument.
5 % The align=right option really does left-alignment, but without the program
6 % will end up on a single line. The strut=no option prevents a bunch of empty
7 % space at the start of the frame.
9 \framed[offset=1mm,align=right,strut=no,background=box,frame=off]{
10 \setuptyping[option=LAM,style=sans,before=,after=]
12 \setuptyping[option=none,style=\tttf]
16 \define[3]\transexample{
17 \placeexample[here]{#1}
18 \startcombination[2*1]
19 {\example{#2}}{Original program}
20 {\example{#3}}{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 areas and because
28 core can describe expressions that do not have a direct hardware
31 TODO: Describe core properties not supported in \small{VHDL}, and describe how the
32 \small{VHDL} we want to generate should look like.
35 The transformations described here have a well-defined goal: To bring the
36 program in a well-defined form that is directly translatable to hardware,
37 while fully preserving the semantics of the program. We refer to this form as
38 the \emph{normal form} of the program. The formal definition of this normal
41 \placedefinition{}{A program is in \emph{normal form} if none of the
42 transformations from this chapter apply.}
44 Of course, this is an \quote{easy} definition of the normal form, since our
45 program will end up in normal form automatically. The more interesting part is
46 to see if this normal form actually has the properties we would like it to
49 But, before getting into more definitions and details about this normal form,
50 let's try to get a feeling for it first. The easiest way to do this is by
51 describing the things we want to not have in a normal form.
54 \item Any \emph{polymorphism} must be removed. When laying down hardware, we
55 can't generate any signals that can have multiple types. All types must be
56 completely known to generate hardware.
58 \item Any \emph{higher order} constructions must be removed. We can't
59 generate a hardware signal that contains a function, so all values,
60 arguments and returns values used must be first order.
62 \item Any complex \emph{nested scopes} must be removed. In the \small{VHDL}
63 description, every signal is in a single scope. Also, full expressions are
64 not supported everywhere (in particular port maps can only map signal names,
65 not expressions). To make the \small{VHDL} generation easy, all values must be bound
66 on the \quote{top level}.
69 TODO: Intermezzo: functions vs plain values
71 A very simple example of a program in normal form is given in
72 \in{example}[ex:MulSum]. As you can see, all arguments to the function (which
73 will become input ports in the final hardware) are at the top. This means that
74 the body of the final lambda abstraction is never a function, but always a
77 After the lambda abstractions, we see a single let expression, that binds two
78 variables (\lam{mul} and \lam{sum}). These variables will be signals in the
79 final hardware, bound to the output port of the \lam{*} and \lam{+}
82 The final line (the \quote{return value} of the function) selects the
83 \lam{sum} signal to be the output port of the function. This \quote{return
84 value} can always only be a variable reference, never a more complex
88 alu :: Bit -> Word -> Word -> Word
97 \startuseMPgraphic{MulSum}
98 save a, b, c, mul, add, sum;
101 newCircle.a(btex $a$ etex) "framed(false)";
102 newCircle.b(btex $b$ etex) "framed(false)";
103 newCircle.c(btex $c$ etex) "framed(false)";
104 newCircle.sum(btex $res$ etex) "framed(false)";
107 newCircle.mul(btex - etex);
108 newCircle.add(btex + etex);
110 a.c - b.c = (0cm, 2cm);
111 b.c - c.c = (0cm, 2cm);
112 add.c = c.c + (2cm, 0cm);
113 mul.c = midpoint(a.c, b.c) + (2cm, 0cm);
114 sum.c = add.c + (2cm, 0cm);
117 % Draw objects and lines
118 drawObj(a, b, c, mul, add, sum);
120 ncarc(a)(mul) "arcangle(15)";
121 ncarc(b)(mul) "arcangle(-15)";
127 \placeexample[here][ex:MulSum]{Simple architecture consisting of an adder and a
129 \startcombination[2*1]
130 {\typebufferlam{MulSum}}{Core description in normal form.}
131 {\boxedgraphic{MulSum}}{The architecture described by the normal form.}
134 The previous example described composing an architecture by calling other
135 functions (operators), resulting in a simple architecture with component and
136 connection. There is of course also some mechanism for choice in the normal
137 form. In a normal Core program, the \emph{case} expression can be used in a
138 few different ways to describe choice. In normal form, this is limited to a
141 \in{Example}[ex:AddSubAlu] shows an example describing a
142 simple \small{ALU}, which chooses between two operations based on an opcode
143 bit. The main structure is the same as in \in{example}[ex:MulSum], but this
144 time the \lam{res} variable is bound to a case expression. This case
145 expression scrutinizes the variable \lam{opcode} (and scrutinizing more
146 complex expressions is not supported). The case expression can select a
147 different variable based on the constructor of \lam{opcode}.
149 \startbuffer[AddSubAlu]
150 alu :: Bit -> Word -> Word -> Word
162 \startuseMPgraphic{AddSubAlu}
163 save opcode, a, b, add, sub, mux, res;
166 newCircle.opcode(btex $opcode$ etex) "framed(false)";
167 newCircle.a(btex $a$ etex) "framed(false)";
168 newCircle.b(btex $b$ etex) "framed(false)";
169 newCircle.res(btex $res$ etex) "framed(false)";
171 newCircle.add(btex + etex);
172 newCircle.sub(btex - etex);
175 opcode.c - a.c = (0cm, 2cm);
176 add.c - a.c = (4cm, 0cm);
177 sub.c - b.c = (4cm, 0cm);
178 a.c - b.c = (0cm, 3cm);
179 mux.c = midpoint(add.c, sub.c) + (1.5cm, 0cm);
180 res.c - mux.c = (1.5cm, 0cm);
183 % Draw objects and lines
184 drawObj(opcode, a, b, res, add, sub, mux);
186 ncline(a)(add) "posA(e)";
187 ncline(b)(sub) "posA(e)";
188 nccurve(a)(sub) "posA(e)", "angleA(0)";
189 nccurve(b)(add) "posA(e)", "angleA(0)";
190 nccurve(add)(mux) "posB(inpa)", "angleB(0)";
191 nccurve(sub)(mux) "posB(inpb)", "angleB(0)";
192 nccurve(opcode)(mux) "posB(n)", "angleA(0)", "angleB(-90)";
193 ncline(mux)(res) "posA(out)";
196 \placeexample[here][ex:AddSubAlu]{Simple \small{ALU} supporting two operations.}
197 \startcombination[2*1]
198 {\typebufferlam{AddSubAlu}}{Core description in normal form.}
199 {\boxedgraphic{AddSubAlu}}{The architecture described by the normal form.}
202 As a more complete example, consider \in{example}[ex:NormalComplete]. This
203 example contains everything that is supported in normal form, with the
204 exception of builtin higher order functions. The graphical version of the
205 architecture contains a slightly simplified version, since the state tuple
206 packing and unpacking have been left out. Instead, two seperate registers are
207 drawn. Also note that most synthesis tools will further optimize this
208 architecture by removing the multiplexers at the register input and replace
209 them with some logic in the clock inputs, but we want to show the architecture
210 as close to the description as possible.
212 \startbuffer[NormalComplete]
215 -> State (Word, Word)
216 -> (State (Word, Word), Word)
218 -- All arguments are an inital lambda
220 -- There are nested let expressions at top level
222 -- Unpack the state by coercion (\eg, cast from
223 -- State (Word, Word) to (Word, Word))
224 s = sp :: (Word, Word)
225 -- Extract both registers from the state
226 r1 = case s of (fst, snd) -> fst
227 r2 = case s of (fst, snd) -> snd
228 -- Calling some other user-defined function.
230 -- Conditional connections
242 -- pack the state by coercion (\eg, cast from
243 -- (Word, Word) to State (Word, Word))
244 sp' = s' :: State (Word, Word)
245 -- Pack our return value
252 \startuseMPgraphic{NormalComplete}
253 save a, d, r, foo, muxr, muxout, out;
256 newCircle.a(btex \lam{a} etex) "framed(false)";
257 newCircle.d(btex \lam{d} etex) "framed(false)";
258 newCircle.out(btex \lam{out} etex) "framed(false)";
260 %newCircle.add(btex + etex);
261 newBox.foo(btex \lam{foo} etex);
262 newReg.r1(btex $\lam{r1}$ etex) "dx(4mm)", "dy(6mm)";
263 newReg.r2(btex $\lam{r2}$ etex) "dx(4mm)", "dy(6mm)", "reflect(true)";
265 % Reflect over the vertical axis
266 reflectObj(muxr1)((0,0), (0,1));
269 rotateObj(muxout)(-90);
271 d.c = foo.c + (0cm, 1.5cm);
272 a.c = (xpart r2.c + 2cm, ypart d.c - 0.5cm);
273 foo.c = midpoint(muxr1.c, muxr2.c) + (0cm, 2cm);
274 muxr1.c = r1.c + (0cm, 2cm);
275 muxr2.c = r2.c + (0cm, 2cm);
276 r2.c = r1.c + (4cm, 0cm);
278 muxout.c = midpoint(r1.c, r2.c) - (0cm, 2cm);
279 out.c = muxout.c - (0cm, 1.5cm);
281 % % Draw objects and lines
282 drawObj(a, d, foo, r1, r2, muxr1, muxr2, muxout, out);
285 nccurve(foo)(muxr1) "angleA(-90)", "posB(inpa)", "angleB(180)";
286 nccurve(foo)(muxr2) "angleA(-90)", "posB(inpb)", "angleB(0)";
287 nccurve(muxr1)(r1) "posA(out)", "angleA(180)", "posB(d)", "angleB(0)";
288 nccurve(r1)(muxr1) "posA(out)", "angleA(0)", "posB(inpb)", "angleB(180)";
289 nccurve(muxr2)(r2) "posA(out)", "angleA(0)", "posB(d)", "angleB(180)";
290 nccurve(r2)(muxr2) "posA(out)", "angleA(180)", "posB(inpa)", "angleB(0)";
291 nccurve(r1)(muxout) "posA(out)", "angleA(0)", "posB(inpb)", "angleB(-90)";
292 nccurve(r2)(muxout) "posA(out)", "angleA(180)", "posB(inpa)", "angleB(-90)";
294 nccurve(a)(muxout) "angleA(-90)", "angleB(180)", "posB(sel)";
295 nccurve(a)(muxr1) "angleA(180)", "angleB(-90)", "posB(sel)";
296 nccurve(a)(muxr2) "angleA(180)", "angleB(-90)", "posB(sel)";
297 ncline(muxout)(out) "posA(out)";
300 \placeexample[here][ex:NormalComplete]{Simple architecture consisting of an adder and a
302 \startcombination[2*1]
303 {\typebufferlam{NormalComplete}}{Core description in normal form.}
304 {\boxedgraphic{NormalComplete}}{The architecture described by the normal form.}
307 \subsection{Intended normal form definition}
308 Now we have some intuition for the normal form, we can describe how we want
309 the normal form to look like in a slightly more formal manner. The following
310 EBNF-like description completely captures the intended structure (and
311 generates a subset of GHC's core format).
313 Some clauses have an expression listed in parentheses. These are conditions
314 that need to apply to the clause.
317 \italic{normal} = \italic{lambda}
318 \italic{lambda} = λvar.\italic{lambda} (representable(var))
320 \italic{toplet} = let \italic{binding} in \italic{toplet}
321 | letrec [\italic{binding}] in \italic{toplet}
322 | var (representable(varvar))
323 \italic{binding} = var = \italic{rhs} (representable(rhs))
324 -- State packing and unpacking by coercion
325 | var0 = var1 :: State ty (lvar(var1))
326 | var0 = var1 :: ty (var0 :: State ty) (lvar(var1))
327 \italic{rhs} = userapp
330 | case var of C a0 ... an -> ai (lvar(var))
332 | case var of (lvar(var))
333 DEFAULT -> var0 (lvar(var0))
334 C w0 ... wn -> resvar (\forall{}i, wi \neq resvar, lvar(resvar))
335 \italic{userapp} = \italic{userfunc}
336 | \italic{userapp} {userarg}
337 \italic{userfunc} = var (gvar(var))
338 \italic{userarg} = var (lvar(var))
339 \italic{builtinapp} = \italic{builtinfunc}
340 | \italic{builtinapp} \italic{builtinarg}
341 \italic{builtinfunc} = var (bvar(var))
342 \italic{builtinarg} = \italic{coreexpr}
345 -- TODO: Limit builtinarg further
347 -- TODO: There can still be other casts around (which the code can handle,
348 e.g., ignore), which still need to be documented here.
350 -- TODO: Note about the selector case. It just supports Bit and Bool
351 currently, perhaps it should be generalized in the normal form?
353 When looking at such a program from a hardware perspective, the top level
354 lambda's define the input ports. The value produced by the let expression is
355 the output port. Most function applications bound by the let expression
356 define a component instantiation, where the input and output ports are mapped
357 to local signals or arguments. Some of the others use a builtin
358 construction (\eg the \lam{case} statement) or call a builtin function
359 (\eg \lam{add} or \lam{sub}). For these, a hardcoded \small{VHDL} translation is
362 \section{Transformation notation}
363 To be able to concisely present transformations, we use a specific format to
364 them. It is a simple format, similar to one used in logic reasoning.
366 Such a transformation description looks like the following.
371 <original expression>
372 -------------------------- <expression conditions>
373 <transformed expresssion>
378 This format desribes a transformation that applies to \lam{original
379 expresssion} and transforms it into \lam{transformed expression}, assuming
380 that all conditions apply. In this format, there are a number of placeholders
381 in pointy brackets, most of which should be rather obvious in their meaning.
382 Nevertheless, we will more precisely specify their meaning below:
384 \startdesc{<original expression>} The expression pattern that will be matched
385 against (subexpressions of) the expression to be transformed. We call this a
386 pattern, because it can contain \emph{placeholders} (variables), which match
387 any expression or binder. Any such placeholder is said to be \emph{bound} to
388 the expression it matches. It is convention to use an uppercase latter (\eg
389 \lam{M} or \lam{E} to refer to any expression (including a simple variable
390 reference) and lowercase letters (\eg \lam{v} or \lam{b}) to refer to
391 (references to) binders.
393 For example, the pattern \lam{a + B} will match the expression
394 \lam{v + (2 * w)} (and bind \lam{a} to \lam{v} and \lam{B} to
395 \lam{(2 * 2)}), but not \lam{v + (2 * w)}.
398 \startdesc{<expression conditions>}
399 These are extra conditions on the expression that is matched. These
400 conditions can be used to further limit the cases in which the
401 transformation applies, in particular to prevent a transformation from
402 causing a loop with itself or another transformation.
404 Only if these if these conditions are \emph{all} true, this transformation
408 \startdesc{<context conditions>}
409 These are a number of extra conditions on the context of the function. In
410 particular, these conditions can require some other top level function to be
411 present, whose value matches the pattern given here. The format of each of
412 these conditions is: \lam{binder = <pattern>}.
414 Typically, the binder is some placeholder bound in the \lam{<original
415 expression>}, while the pattern contains some placeholders that are used in
416 the \lam{transformed expression}.
418 Only if a top level binder exists that matches each binder and pattern, this
419 transformation applies.
422 \startdesc{<transformed expression>}
423 This is the expression template that is the result of the transformation. If, looking
424 at the above three items, the transformation applies, the \lam{original
425 expression} is completely replaced with the \lam{<transformed expression>}.
426 We call this a template, because it can contain placeholders, referring to
427 any placeholder bound by the \lam{<original expression>} or the
428 \lam{<context conditions>}. The resulting expression will have those
429 placeholders replaced by the values bound to them.
431 Any binder (lowercase) placeholder that has no value bound to it yet will be
432 bound to (and replaced with) a fresh binder.
435 \startdesc{<context additions>}
436 These are templates for new functions to add to the context. This is a way
437 to have a transformation create new top level functiosn.
439 Each addition has the form \lam{binder = template}. As above, any
440 placeholder in the addition is replaced with the value bound to it, and any
441 binder placeholder that has no value bound to it yet will be bound to (and
442 replaced with) a fresh binder.
445 As an example, we'll look at η-abstraction:
449 -------------- \lam{E} does not occur on a function position in an application
450 λx.E x \lam{E} is not a lambda abstraction.
453 Consider the following function, which is a fairly obvious way to specify a
454 simple ALU (Note \at{example}[ex:AddSubAlu] is the normal form of this
458 alu :: Bit -> Word -> Word -> Word
459 alu = λopcode. case opcode of
464 There are a few subexpressions in this function to which we could possibly
465 apply the transformation. Since the pattern of the transformation is only
466 the placeholder \lam{E}, any expression will match that. Whether the
467 transformation applies to an expression is thus solely decided by the
468 conditions to the right of the transformation.
470 We will look at each expression in the function in a top down manner. The
471 first expression is the entire expression the function is bound to.
474 λopcode. case opcode of
479 As said, the expression pattern matches this. The type of this expression is
480 \lam{Bit -> Word -> Word -> Word}, which matches \lam{a -> b} (Note that in
481 this case \lam{a = Bit} and \lam{b = Word -> Word -> Word}).
483 Since this expression is at top level, it does not occur at a function
484 position of an application. However, The expression is a lambda abstraction,
485 so this transformation does not apply.
487 The next expression we could apply this transformation to, is the body of
488 the lambda abstraction:
496 The type of this expression is \lam{Word -> Word -> Word}, which again
497 matches \lam{a -> b}. The expression is the body of a lambda expression, so
498 it does not occur at a function position of an application. Finally, the
499 expression is not a lambda abstraction but a case expression, so all the
500 conditions match. There are no context conditions to match, so the
501 transformation applies.
503 By now, the placeholder \lam{E} is bound to the entire expression. The
504 placeholder \lam{x}, which occurs in the replacement template, is not bound
505 yet, so we need to generate a fresh binder for that. Let's use the binder
506 \lam{a}. This results in the following replacement expression:
514 Continuing with this expression, we see that the transformation does not
515 apply again (it is a lambda expression). Next we look at the body of this
524 Here, the transformation does apply, binding \lam{E} to the entire
525 expression and \lam{x} to the fresh binder \lam{b}, resulting in the
534 Again, the transformation does not apply to this lambda abstraction, so we
535 look at its body. For brevity, we'll put the case statement on one line from
539 (case opcode of Low -> (+); High -> (-)) a b
542 The type of this expression is \lam{Word}, so it does not match \lam{a -> b}
543 and the transformation does not apply. Next, we have two options for the
544 next expression to look at: The function position and argument position of
545 the application. The expression in the argument position is \lam{b}, which
546 has type \lam{Word}, so the transformation does not apply. The expression in
547 the function position is:
550 (case opcode of Low -> (+); High -> (-)) a
553 Obviously, the transformation does not apply here, since it occurs in
554 function position. In the same way the transformation does not apply to both
555 components of this expression (\lam{case opcode of Low -> (+); High -> (-)}
556 and \lam{a}), so we'll skip to the components of the case expression: The
557 scrutinee and both alternatives. Since the opcode is not a function, it does
558 not apply here, and we'll leave both alternatives as an exercise to the
559 reader. The final function, after all these transformations becomes:
562 alu :: Bit -> Word -> Word -> Word
563 alu = λopcode.λa.b. (case opcode of
564 Low -> λa1.λb1 (+) a1 b1
565 High -> λa2.λb2 (-) a2 b2) a b
568 In this case, the transformation does not apply anymore, though this might
569 not always be the case (e.g., the application of a transformation on a
570 subexpression might open up possibilities to apply the transformation
571 further up in the expression).
573 \subsection{Transformation application}
574 In this chapter we define a number of transformations, but how will we apply
575 these? As stated before, our normal form is reached as soon as no
576 transformation applies anymore. This means our application strategy is to
577 simply apply any transformation that applies, and continuing to do that with
578 the result of each transformation.
580 In particular, we define no particular order of transformations. Since
581 transformation order should not influence the resulting normal form (see TODO
582 ref), this leaves the implementation free to choose any application order that
583 results in an efficient implementation.
585 When applying a single transformation, we try to apply it to every (sub)expression
586 in a function, not just the top level function. This allows us to keep the
587 transformation descriptions concise and powerful.
589 \subsection{Definitions}
590 In the following sections, we will be using a number of functions and
591 notations, which we will define here.
593 \subsubsection{Other concepts}
594 A \emph{global variable} is any variable that is bound at the
595 top level of a program, or an external module. A \emph{local variable} is any
596 other variable (\eg, variables local to a function, which can be bound by
597 lambda abstractions, let expressions and pattern matches of case
598 alternatives). Note that this is a slightly different notion of global versus
599 local than what \small{GHC} uses internally.
600 \defref{global variable} \defref{local variable}
602 A \emph{hardware representable} (or just \emph{representable}) type or value
603 is (a value of) a type that we can generate a signal for in hardware. For
604 example, a bit, a vector of bits, a 32 bit unsigned word, etc. Types that are
605 not runtime representable notably include (but are not limited to): Types,
606 dictionaries, functions.
607 \defref{representable}
609 A \emph{builtin function} is a function supplied by the Cλash framework, whose
610 implementation is not valid Cλash. The implementation is of course valid
611 Haskell, for simulation, but it is not expressable in Cλash.
612 \defref{builtin function} \defref{user-defined function}
614 For these functions, Cλash has a \emph{builtin hardware translation}, so calls
615 to these functions can still be translated. These are functions like
616 \lam{map}, \lam{hwor} and \lam{length}.
618 A \emph{user-defined} function is a function for which we do have a Cλash
619 implementation available.
621 \subsubsection{Functions}
622 Here, we define a number of functions that can be used below to concisely
625 \refdef{global variable}\emph{gvar(expr)} is true when \emph{expr} is a variable that references a
626 global variable. It is false when it references a local variable.
628 \refdef{local variable}\emph{lvar(expr)} is the complement of \emph{gvar}; it is true when \emph{expr}
629 references a local variable, false when it references a global variable.
631 \refdef{representable}\emph{representable(expr)} or \emph{representable(var)} is true when
632 \emph{expr} or \emph{var} is \emph{representable}.
634 \subsection{Binder uniqueness}
635 A common problem in transformation systems, is binder uniqueness. When not
636 considering this problem, it is easy to create transformations that mix up
637 bindings and cause name collisions. Take for example, the following core
641 (λa.λb.λc. a * b * c) x c
644 By applying β-reduction (see below) once, we can simplify this expression to:
650 Now, we have replaced the \lam{a} binder with a reference to the \lam{x}
651 binder. No harm done here. But note that we see multiple occurences of the
652 \lam{c} binder. The first is a binding occurence, to which the second refers.
653 The last, however refers to \emph{another} instance of \lam{c}, which is
654 bound somewhere outside of this expression. Now, if we would apply beta
655 reduction without taking heed of binder uniqueness, we would get:
661 This is obviously not what was supposed to happen! The root of this problem is
662 the reuse of binders: Identical binders can be bound in different scopes, such
663 that only the inner one is \quote{visible} in the inner expression. In the example
664 above, the \lam{c} binder was bound outside of the expression and in the inner
665 lambda expression. Inside that lambda expression, only the inner \lam{c} is
668 There are a number of ways to solve this. \small{GHC} has isolated this
669 problem to their binder substitution code, which performs \emph{deshadowing}
670 during its expression traversal. This means that any binding that shadows
671 another binding on a higher level is replaced by a new binder that does not
672 shadow any other binding. This non-shadowing invariant is enough to prevent
673 binder uniqueness problems in \small{GHC}.
675 In our transformation system, maintaining this non-shadowing invariant is
676 a bit harder to do (mostly due to implementation issues, the prototype doesn't
677 use \small{GHC}'s subsitution code). Also, we can observe the following
681 \item Deshadowing does not guarantee overall uniqueness. For example, the
682 following (slightly contrived) expression shows the identifier \lam{x} bound in
683 two seperate places (and to different values), even though no shadowing
687 (let x = 1 in x) + (let x = 2 in x)
690 \item In our normal form (and the resulting \small{VHDL}), all binders
691 (signals) will end up in the same scope. To allow this, all binders within the
692 same function should be unique.
694 \item When we know that all binders in an expression are unique, moving around
695 or removing a subexpression will never cause any binder conflicts. If we have
696 some way to generate fresh binders, introducing new subexpressions will not
697 cause any problems either. The only way to cause conflicts is thus to
698 duplicate an existing subexpression.
701 Given the above, our prototype maintains a unique binder invariant. This
702 meanst that in any given moment during normalization, all binders \emph{within
703 a single function} must be unique. To achieve this, we apply the following
706 TODO: Define fresh binders and unique supplies
709 \item Before starting normalization, all binders in the function are made
710 unique. This is done by generating a fresh binder for every binder used. This
711 also replaces binders that did not pose any conflict, but it does ensure that
712 all binders within the function are generated by the same unique supply. See
713 (TODO: ref fresh binder).
714 \item Whenever a new binder must be generated, we generate a fresh binder that
715 is guaranteed to be different from \emph{all binders generated so far}. This
716 can thus never introduce duplication and will maintain the invariant.
717 \item Whenever (part of) an expression is duplicated (for example when
718 inlining), all binders in the expression are replaced with fresh binders
719 (using the same method as at the start of normalization). These fresh binders
720 can never introduce duplication, so this will maintain the invariant.
721 \item Whenever we move part of an expression around within the function, there
722 is no need to do anything special. There is obviously no way to introduce
723 duplication by moving expressions around. Since we know that each of the
724 binders is already unique, there is no way to introduce (incorrect) shadowing
728 \section{Transform passes}
729 In this section we describe the actual transforms. Here we're using
730 the core language in a notation that resembles lambda calculus.
732 Each of these transforms is meant to be applied to every (sub)expression
733 in a program, for as long as it applies. Only when none of the
734 transformations can be applied anymore, the program is in normal form (by
735 definition). We hope to be able to prove that this form will obey all of the
736 constraints defined above, but this has yet to happen (though it seems likely
739 Each of the transforms will be described informally first, explaining
740 the need for and goal of the transform. Then, a formal definition is
741 given, using a familiar syntax from the world of logic. Each transform
742 is specified as a number of conditions (above the horizontal line) and a
743 number of conclusions (below the horizontal line). The details of using
744 this notation are still a bit fuzzy, so comments are welcom.
746 \subsection{General cleanup}
747 These transformations are general cleanup transformations, that aim to
748 make expressions simpler. These transformations usually clean up the
749 mess left behind by other transformations or clean up expressions to
750 expose new transformation opportunities for other transformations.
752 Most of these transformations are standard optimizations in other
753 compilers as well. However, in our compiler, most of these are not just
754 optimizations, but they are required to get our program into normal
757 \subsubsection{β-reduction}
758 β-reduction is a well known transformation from lambda calculus, where it is
759 the main reduction step. It reduces applications of labmda abstractions,
760 removing both the lambda abstraction and the application.
762 In our transformation system, this step helps to remove unwanted lambda
763 abstractions (basically all but the ones at the top level). Other
764 transformations (application propagation, non-representable inlining) make
765 sure that most lambda abstractions will eventually be reducable by
783 \transexample{β-reduction}{from}{to}
785 \subsubsection{Empty let removal}
786 This transformation is simple: It removes recursive lets that have no bindings
787 (which usually occurs when let derecursification removes the last binding from
796 \subsubsection{Simple let binding removal}
797 This transformation inlines simple let bindings (\eg a = b).
799 This transformation is not needed to get into normal form, but makes the
800 resulting \small{VHDL} a lot shorter.
826 \subsubsection{Unused let binding removal}
827 This transformation removes let bindings that are never used. Usually,
828 the desugarer introduces some unused let bindings.
830 This normalization pass should really be unneeded to get into normal form
831 (since ununsed bindings are not forbidden by the normal form), but in practice
832 the desugarer or simplifier emits some unused bindings that cannot be
833 normalized (e.g., calls to a \type{PatError} (TODO: Check this name)). Also,
834 this transformation makes the resulting \small{VHDL} a lot shorter.
838 ---------------------------- \lam{a} does not occur free in \lam{M}
849 ---------------------------- \lam{a} does not occur free in \lam{M}
857 \subsubsection{Cast propagation / simplification}
858 This transform pushes casts down into the expression as far as possible.
859 Since its exact role and need is not clear yet, this transformation is
862 \subsubsection{Compiler generated top level binding inlining}
865 \section{Program structure}
866 These transformations are aimed at normalizing the overall structure
867 into the intended form. This means ensuring there is a lambda abstraction
868 at the top for every argument (input port), putting all of the other
869 value definitions in let bindings and making the final return value a
870 simple variable reference.
872 \subsubsection{η-abstraction}
873 This transformation makes sure that all arguments of a function-typed
874 expression are named, by introducing lambda expressions. When combined with
875 β-reduction and non-representable binding inlining, all function-typed
876 expressions should be lambda abstractions or global identifiers.
880 -------------- \lam{E} is not the first argument of an application.
881 λx.E x \lam{E} is not a lambda abstraction.
882 \lam{x} is a variable that does not occur free in \lam{E}.
892 foo = λa.λx.(case a of
897 \transexample{η-abstraction}{from}{to}
899 \subsubsection{Application propagation}
900 This transformation is meant to propagate application expressions downwards
901 into expressions as far as possible. This allows partial applications inside
902 expressions to become fully applied and exposes new transformation
903 opportunities for other transformations (like β-reduction and
928 \transexample{Application propagation for a let expression}{from}{to}
956 \transexample{Application propagation for a case expression}{from}{to}
958 \subsubsection{Let derecursification}
959 This transformation is meant to make lets non-recursive whenever possible.
960 This might allow other optimizations to do their work better. TODO: Why is
963 \subsubsection{Let flattening}
964 This transformation puts nested lets in the same scope, by lifting the
965 binding(s) of the inner let into a new let around the outer let. Eventually,
966 this will cause all let bindings to appear in the same scope (they will all be
967 in scope for the function return value).
969 Note that this transformation does not try to be smart when faced with
970 recursive lets, it will just leave the lets recursive (possibly joining a
971 recursive and non-recursive let into a single recursive let). The let
972 dederecursification transformation will do this instead.
975 letnonrec x = (let bindings in M) in N
976 ------------------------------------------
977 let bindings in (letnonrec x = M) in N
983 x = (let bindings in M)
987 ------------------------------------------
1006 b = let c = 3 in a + c
1027 \transexample{Let flattening}{from}{to}
1029 \subsubsection{Return value simplification}
1030 This transformation ensures that the return value of a function is always a
1031 simple local variable reference.
1033 Currently implemented using lambda simplification, let simplification, and
1034 top simplification. Should change into something like the following, which
1035 works only on the result of a function instead of any subexpression. This is
1036 achieved by the contexts, like \lam{x = E}, though this is strictly not
1037 correct (you could read this as "if there is any function \lam{x} that binds
1038 \lam{E}, any \lam{E} can be transformed, while we only mean the \lam{E} that
1039 is bound by \lam{x}. This might need some extra notes or something).
1041 Note that the return value is not simplified if its not representable.
1042 Otherwise, this would cause a direct loop with the inlining of
1043 unrepresentable bindings, of course. If the return value is not
1044 representable because it has a function type, η-abstraction should
1045 make sure that this transformation will eventually apply. If the value
1046 is not representable for other reasons, the function result itself is
1047 not representable, meaning this function is not representable anyway!
1050 x = E \lam{E} is representable
1051 ~ \lam{E} is not a lambda abstraction
1052 E \lam{E} is not a let expression
1053 --------------------------- \lam{E} is not a local variable reference
1059 ~ \lam{E} is representable
1060 E \lam{E} is not a let expression
1061 --------------------------- \lam{E} is not a local variable reference
1066 x = λv0 ... λvn.let ... in E
1067 ~ \lam{E} is representable
1068 E \lam{E} is not a local variable reference
1069 ---------------------------
1078 x = let x = add 1 2 in x
1081 \transexample{Return value simplification}{from}{to}
1083 \subsection{Argument simplification}
1084 The transforms in this section deal with simplifying application
1085 arguments into normal form. The goal here is to:
1088 \item Make all arguments of user-defined functions (\eg, of which
1089 we have a function body) simple variable references of a runtime
1090 representable type. This is needed, since these applications will be turned
1091 into component instantiations.
1092 \item Make all arguments of builtin functions one of:
1094 \item A type argument.
1095 \item A dictionary argument.
1096 \item A type level expression.
1097 \item A variable reference of a runtime representable type.
1098 \item A variable reference or partial application of a function type.
1102 When looking at the arguments of a user-defined function, we can
1103 divide them into two categories:
1105 \item Arguments of a runtime representable type (\eg bits or vectors).
1107 These arguments can be preserved in the program, since they can
1108 be translated to input ports later on. However, since we can
1109 only connect signals to input ports, these arguments must be
1110 reduced to simple variables (for which signals will be
1111 produced). This is taken care of by the argument extraction
1113 \item Non-runtime representable typed arguments.
1115 These arguments cannot be preserved in the program, since we
1116 cannot represent them as input or output ports in the resulting
1117 \small{VHDL}. To remove them, we create a specialized version of the
1118 called function with these arguments filled in. This is done by
1119 the argument propagation transform.
1121 Typically, these arguments are type and dictionary arguments that are
1122 used to make functions polymorphic. By propagating these arguments, we
1123 are essentially doing the same which GHC does when it specializes
1124 functions: Creating multiple variants of the same function, one for
1125 each type for which it is used. Other common non-representable
1126 arguments are functions, e.g. when calling a higher order function
1127 with another function or a lambda abstraction as an argument.
1129 The reason for doing this is similar to the reasoning provided for
1130 the inlining of non-representable let bindings above. In fact, this
1131 argument propagation could be viewed as a form of cross-function
1135 TODO: Check the following itemization.
1137 When looking at the arguments of a builtin function, we can divide them
1141 \item Arguments of a runtime representable type.
1143 As we have seen with user-defined functions, these arguments can
1144 always be reduced to a simple variable reference, by the
1145 argument extraction transform. Performing this transform for
1146 builtin functions as well, means that the translation of builtin
1147 functions can be limited to signal references, instead of
1148 needing to support all possible expressions.
1150 \item Arguments of a function type.
1152 These arguments are functions passed to higher order builtins,
1153 like \lam{map} and \lam{foldl}. Since implementing these
1154 functions for arbitrary function-typed expressions (\eg, lambda
1155 expressions) is rather comlex, we reduce these arguments to
1156 (partial applications of) global functions.
1158 We can still support arbitrary expressions from the user code,
1159 by creating a new global function containing that expression.
1160 This way, we can simply replace the argument with a reference to
1161 that new function. However, since the expression can contain any
1162 number of free variables we also have to include partial
1163 applications in our normal form.
1165 This category of arguments is handled by the function extraction
1167 \item Other unrepresentable arguments.
1169 These arguments can take a few different forms:
1170 \startdesc{Type arguments}
1171 In the core language, type arguments can only take a single
1172 form: A type wrapped in the Type constructor. Also, there is
1173 nothing that can be done with type expressions, except for
1174 applying functions to them, so we can simply leave type
1175 arguments as they are.
1177 \startdesc{Dictionary arguments}
1178 In the core language, dictionary arguments are used to find
1179 operations operating on one of the type arguments (mostly for
1180 finding class methods). Since we will not actually evaluatie
1181 the function body for builtin functions and can generate
1182 code for builtin functions by just looking at the type
1183 arguments, these arguments can be ignored and left as they
1186 \startdesc{Type level arguments}
1187 Sometimes, we want to pass a value to a builtin function, but
1188 we need to know the value at compile time. Additionally, the
1189 value has an impact on the type of the function. This is
1190 encoded using type-level values, where the actual value of the
1191 argument is not important, but the type encodes some integer,
1192 for example. Since the value is not important, the actual form
1193 of the expression does not matter either and we can leave
1194 these arguments as they are.
1196 \startdesc{Other arguments}
1197 Technically, there is still a wide array of arguments that can
1198 be passed, but does not fall into any of the above categories.
1199 However, none of the supported builtin functions requires such
1200 an argument. This leaves use with passing unsupported types to
1201 a function, such as calling \lam{head} on a list of functions.
1203 In these cases, it would be impossible to generate hardware
1204 for such a function call anyway, so we can ignore these
1207 The only way to generate hardware for builtin functions with
1208 arguments like these, is to expand the function call into an
1209 equivalent core expression (\eg, expand map into a series of
1210 function applications). But for now, we choose to simply not
1211 support expressions like these.
1214 From the above, we can conclude that we can simply ignore these
1215 other unrepresentable arguments and focus on the first two
1219 \subsubsection{Argument simplification}
1220 This transform deals with arguments to functions that
1221 are of a runtime representable type. It ensures that they will all become
1222 references to global variables, or local signals in the resulting \small{VHDL}.
1224 TODO: It seems we can map an expression to a port, not only a signal.
1225 Perhaps this makes this transformation not needed?
1226 TODO: Say something about dataconstructors (without arguments, like True
1227 or False), which are variable references of a runtime representable
1228 type, but do not result in a signal.
1230 To reduce a complex expression to a simple variable reference, we create
1231 a new let expression around the application, which binds the complex
1232 expression to a new variable. The original function is then applied to
1237 -------------------- \lam{N} is of a representable type
1238 let x = N in M x \lam{N} is not a local variable reference
1246 let x = add a 1 in add x 1
1249 \transexample{Argument extraction}{from}{to}
1251 \subsubsection{Function extraction}
1252 This transform deals with function-typed arguments to builtin functions.
1253 Since these arguments cannot be propagated, we choose to extract them
1254 into a new global function instead.
1256 Any free variables occuring in the extracted arguments will become
1257 parameters to the new global function. The original argument is replaced
1258 with a reference to the new function, applied to any free variables from
1259 the original argument.
1261 This transformation is useful when applying higher order builtin functions
1262 like \hs{map} to a lambda abstraction, for example. In this case, the code
1263 that generates \small{VHDL} for \hs{map} only needs to handle top level functions and
1264 partial applications, not any other expression (such as lambda abstractions or
1265 even more complicated expressions).
1268 M N \lam{M} is a (partial aplication of a) builtin function.
1269 --------------------- \lam{f0 ... fn} = free local variables of \lam{N}
1270 M x f0 ... fn \lam{N :: a -> b}
1271 ~ \lam{N} is not a (partial application of) a top level function
1276 map (λa . add a b) xs
1290 \transexample{Function extraction}{from}{to}
1292 \subsubsection{Argument propagation}
1293 This transform deals with arguments to user-defined functions that are
1294 not representable at runtime. This means these arguments cannot be
1295 preserved in the final form and most be {\em propagated}.
1297 Propagation means to create a specialized version of the called
1298 function, with the propagated argument already filled in. As a simple
1299 example, in the following program:
1306 we could {\em propagate} the constant argument 1, with the following
1314 Special care must be taken when the to-be-propagated expression has any
1315 free variables. If this is the case, the original argument should not be
1316 removed alltogether, but replaced by all the free variables of the
1317 expression. In this way, the original expression can still be evaluated
1318 inside the new function. Also, this brings us closer to our goal: All
1319 these free variables will be simple variable references.
1321 To prevent us from propagating the same argument over and over, a simple
1322 local variable reference is not propagated (since is has exactly one
1323 free variable, itself, we would only replace that argument with itself).
1325 This shows that any free local variables that are not runtime representable
1326 cannot be brought into normal form by this transform. We rely on an
1327 inlining transformation to replace such a variable with an expression we
1328 can propagate again.
1333 x Y0 ... Yi ... Yn \lam{Yi} is not of a runtime representable type
1334 --------------------------------------------- \lam{Yi} is not a local variable reference
1335 x' y0 ... yi-1 f0 ... fm Yi+1 ... Yn \lam{f0 ... fm} = free local vars of \lam{Yi}
1337 x' = λy0 ... yi-1 f0 ... fm yi+1 ... yn .
1338 E y0 ... yi-1 Yi yi+1 ... yn
1344 \subsection{Case simplification}
1345 \subsubsection{Scrutinee simplification}
1346 This transform ensures that the scrutinee of a case expression is always
1347 a simple variable reference.
1352 ----------------- \lam{E} is not a local variable reference
1371 \transexample{Let flattening}{from}{to}
1374 \subsubsection{Case simplification}
1375 This transformation ensures that all case expressions become normal form. This
1376 means they will become one of:
1378 \item An extractor case with a single alternative that picks a single field
1379 from a datatype, \eg \lam{case x of (a, b) -> a}.
1380 \item A selector case with multiple alternatives and only wild binders, that
1381 makes a choice between expressions based on the constructor of another
1382 expression, \eg \lam{case x of Low -> a; High -> b}.
1387 C0 v0,0 ... v0,m -> E0
1389 Cn vn,0 ... vn,m -> En
1390 --------------------------------------------------- \forall i \forall j, 0 <= i <= n, 0 <= i < m (\lam{wi,j} is a wild (unused) binder)
1392 v0,0 = case x of C0 v0,0 .. v0,m -> v0,0
1394 v0,m = case x of C0 v0,0 .. v0,m -> v0,m
1397 vn,m = case x of Cn vn,0 .. vn,m -> vn,m
1401 C0 w0,0 ... w0,m -> x0
1403 Cn wn,0 ... wn,m -> xn
1406 TODO: This transformation specified like this is complicated and misses
1407 conditions to prevent looping with itself. Perhaps we should split it here for
1426 \transexample{Selector case simplification}{from}{to}
1434 b = case a of (,) b c -> b
1435 c = case a of (,) b c -> c
1442 \transexample{Extractor case simplification}{from}{to}
1444 \subsubsection{Case removal}
1445 This transform removes any case statements with a single alternative and
1448 These "useless" case statements are usually leftovers from case simplification
1449 on extractor case (see the previous example).
1454 ---------------------- \lam{\forall i, 0 <= i <= m} (\lam{vi} does not occur free in E)
1467 \transexample{Case removal}{from}{to}
1469 \subsection{Removing polymorphism}
1470 Reference type-specialization (== argument propagation)
1472 Reference polymporphic binding inlining (== non-representable binding
1475 \subsection{Defunctionalization}
1476 These transformations remove most higher order expressions from our
1477 program, making it completely first-order (the only exception here is for
1478 arguments to builtin functions, since we can't specialize builtin
1479 function. TODO: Talk more about this somewhere).
1481 Reference higher-order-specialization (== argument propagation)
1483 \subsubsection{Non-representable binding inlining}
1484 This transform inlines let bindings that have a non-representable type. Since
1485 we can never generate a signal assignment for these bindings (we cannot
1486 declare a signal assignment with a non-representable type, for obvious
1487 reasons), we have no choice but to inline the binding to remove it.
1489 If the binding is non-representable because it is a lambda abstraction, it is
1490 likely that it will inlined into an application and β-reduction will remove
1491 the lambda abstraction and turn it into a representable expression at the
1492 inline site. The same holds for partial applications, which can be turned into
1493 full applications by inlining.
1495 Other cases of non-representable bindings we see in practice are primitive
1496 Haskell types. In most cases, these will not result in a valid normalized
1497 output, but then the input would have been invalid to start with. There is one
1498 exception to this: When a builtin function is applied to a non-representable
1499 expression, things might work out in some cases. For example, when you write a
1500 literal \hs{SizedInt} in Haskell, like \hs{1 :: SizedInt D8}, this results in
1501 the following core: \lam{fromInteger (smallInteger 10)}, where for example
1502 \lam{10 :: GHC.Prim.Int\#} and \lam{smallInteger 10 :: Integer} have
1503 non-representable types. TODO: This/these paragraph(s) should probably become a
1504 separate discussion somewhere else.
1507 letnonrec a = E in M
1508 -------------------------- \lam{E} has a non-representable type.
1519 -------------------------- \lam{E} has a non-representable type.
1539 x = fromInteger (smallInteger 10)
1541 (λa -> add a 1) (add 1 x)
1544 \transexample{Let flattening}{from}{to}
1547 \section{Provable properties}
1548 When looking at the system of transformations outlined above, there are a
1549 number of questions that we can ask ourselves. The main question is of course:
1550 \quote{Does our system work as intended?}. We can split this question into a
1551 number of subquestions:
1554 \item[q:termination] Does our system \emph{terminate}? Since our system will
1555 keep running as long as transformations apply, there is an obvious risk that
1556 it will keep running indefinitely. One transformation produces a result that
1557 is transformed back to the original by another transformation, for example.
1558 \item[q:soundness] Is our system \emph{sound}? Since our transformations
1559 continuously modify the expression, there is an obvious risk that the final
1560 normal form will not be equivalent to the original program: Its meaning could
1562 \item[q:completeness] Is our system \emph{complete}? Since we have a complex
1563 system of transformations, there is an obvious risk that some expressions will
1564 not end up in our intended normal form, because we forgot some transformation.
1565 In other words: Does our transformation system result in our intended normal
1566 form for all possible inputs?
1567 \item[q:determinism] Is our system \emph{deterministic}? Since we have defined
1568 no particular order in which the transformation should be applied, there is an
1569 obvious risk that different transformation orderings will result in
1570 \emph{different} normal forms. They might still both be intended normal forms
1571 (if our system is \emph{complete}) and describe correct hardware (if our
1572 system is \emph{sound}), so this property is less important than the previous
1573 three: The translator would still function properly without it.
1576 \subsection{Graph representation}
1577 Before looking into how to prove these properties, we'll look at our
1578 transformation system from a graph perspective. The nodes of the graph are
1579 all possible Core expressions. The (directed) edges of the graph are
1580 transformations. When a transformation α applies to an expression \lam{A} to
1581 produce an expression \lam{B}, we add an edge from the node for \lam{A} to the
1582 node for \lam{B}, labeled α.
1584 \startuseMPgraphic{TransformGraph}
1588 newCircle.a(btex \lam{(λx.λy. (+) x y) 1} etex);
1589 newCircle.b(btex \lam{λy. (+) 1 y} etex);
1590 newCircle.c(btex \lam{(λx.(+) x) 1} etex);
1591 newCircle.d(btex \lam{(+) 1} etex);
1594 c.c = b.c + (4cm, 0cm);
1595 a.c = midpoint(b.c, c.c) + (0cm, 4cm);
1596 d.c = midpoint(b.c, c.c) - (0cm, 3cm);
1598 % β-conversion between a and b
1599 ncarc.a(a)(b) "name(bred)";
1600 ObjLabel.a(btex $\xrightarrow[normal]{}{β}$ etex) "labpathname(bred)", "labdir(rt)";
1601 ncarc.b(b)(a) "name(bexp)", "linestyle(dashed withdots)";
1602 ObjLabel.b(btex $\xleftarrow[normal]{}{β}$ etex) "labpathname(bexp)", "labdir(lft)";
1604 % η-conversion between a and c
1605 ncarc.a(a)(c) "name(ered)";
1606 ObjLabel.a(btex $\xrightarrow[normal]{}{η}$ etex) "labpathname(ered)", "labdir(rt)";
1607 ncarc.c(c)(a) "name(eexp)", "linestyle(dashed withdots)";
1608 ObjLabel.c(btex $\xleftarrow[normal]{}{η}$ etex) "labpathname(eexp)", "labdir(lft)";
1610 % η-conversion between b and d
1611 ncarc.b(b)(d) "name(ered)";
1612 ObjLabel.b(btex $\xrightarrow[normal]{}{η}$ etex) "labpathname(ered)", "labdir(rt)";
1613 ncarc.d(d)(b) "name(eexp)", "linestyle(dashed withdots)";
1614 ObjLabel.d(btex $\xleftarrow[normal]{}{η}$ etex) "labpathname(eexp)", "labdir(lft)";
1616 % β-conversion between c and d
1617 ncarc.c(c)(d) "name(bred)";
1618 ObjLabel.c(btex $\xrightarrow[normal]{}{β}$ etex) "labpathname(bred)", "labdir(rt)";
1619 ncarc.d(d)(c) "name(bexp)", "linestyle(dashed withdots)";
1620 ObjLabel.d(btex $\xleftarrow[normal]{}{β}$ etex) "labpathname(bexp)", "labdir(lft)";
1622 % Draw objects and lines
1623 drawObj(a, b, c, d);
1626 \placeexample[right][ex:TransformGraph]{Partial graph of a labmda calculus
1627 system with β and η reduction (solid lines) and expansion (dotted lines).}
1628 \boxedgraphic{TransformGraph}
1630 Of course our graph is unbounded, since we can construct an infinite amount of
1631 Core expressions. Also, there might potentially be multiple edges between two
1632 given nodes (with different labels), though seems unlikely to actually happen
1635 See \in{example}[ex:TransformGraph] for the graph representation of a very
1636 simple lambda calculus that contains just the expressions \lam{(λx.λy. (+) x
1637 y) 1}, \lam{λy. (+) 1 y}, \lam{(λx.(+) x) 1} and \lam{(+) 1}. The
1638 transformation system consists of β-reduction and η-reduction (solid edges) or
1639 β-reduction and η-reduction (dotted edges).
1641 TODO: Define β-reduction and η-reduction?
1643 Note that the normal form of such a system consists of the set of nodes
1644 (expressions) without outgoing edges, since those are the expression to which
1645 no transformation applies anymore. We call this set of nodes the \emph{normal
1648 From such a graph, we can derive some properties easily:
1650 \item A system will \emph{terminate} if there is no path of infinite length
1651 in the graph (this includes cycles).
1652 \item Soundness is not easily represented in the graph.
1653 \item A system is \emph{complete} if all of the nodes in the normal set have
1654 the intended normal form. The inverse (that all of the nodes outside of
1655 the normal set are \emph{not} in the intended normal form) is not
1657 \item A system is deterministic if all paths from a node, which end in a node
1658 in the normal set, end at the same node.
1661 When looking at the \in{example}[ex:TransformGraph], we see that the system
1662 terminates for both the reduction and expansion systems (but note that, for
1663 expansion, this is only true because we've limited the possible expressions!
1664 In comlete lambda calculus, there would be a path from \lam{(λx.λy. (+) x y)
1665 1} to \lam{(λx.λy.(λz.(+) z) x y) 1} to \lam{(λx.λy.(λz.(λq.(+) q) z) x y) 1}
1668 If we would consider the system with both expansion and reduction, there would
1669 no longer be termination, since there would be cycles all over the place.
1671 The reduction and expansion systems have a normal set of containing just
1672 \lam{(+) 1} or \lam{(λx.λy. (+) x y) 1} respectively. Since all paths in
1673 either system end up in these normal forms, both systems are \emph{complete}.
1674 Also, since there is only one normal form, it must obviously be
1675 \emph{deterministic} as well.
1677 \subsection{Termination}
1682 \subsection{Soundness}
1683 Needs formal definition of semantics.
1684 Prove for each transformation seperately, implies soundness of the system.
1686 \subsection{Completeness}
1687 Show that any transformation applies to every Core expression that is not
1688 in normal form. To prove: no transformation applies => in intended form.
1689 Show the reverse: Not in intended form => transformation applies.
1691 \subsection{Determinism}