1 \chapter[chap:normalization]{Normalization}
2 % A helper to print a single example in the half the page width. The example
3 % text should be in a buffer whose name is given in an argument.
5 % The align=right option really does left-alignment, but without the program
6 % will end up on a single line. The strut=no option prevents a bunch of empty
7 % space at the start of the frame.
9 \framed[offset=1mm,align=right,strut=no,background=box,frame=off]{
10 \setuptyping[option=LAM,style=sans,before=,after=,strip=auto]
12 \setuptyping[option=none,style=\tttf,strip=auto]
16 \define[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 \VHDL, and describe how the
32 \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} = letrec [\italic{binding}...] in var (representable(varvar))
321 \italic{binding} = var = \italic{rhs} (representable(rhs))
322 -- State packing and unpacking by coercion
323 | var0 = var1 :: State ty (lvar(var1))
324 | var0 = var1 :: ty (var0 :: State ty) (lvar(var1))
325 \italic{rhs} = userapp
328 | case var of C a0 ... an -> ai (lvar(var))
330 | case var of (lvar(var))
331 DEFAULT -> var0 (lvar(var0))
332 C w0 ... wn -> resvar (\forall{}i, wi \neq resvar, lvar(resvar))
333 \italic{userapp} = \italic{userfunc}
334 | \italic{userapp} {userarg}
335 \italic{userfunc} = var (gvar(var))
336 \italic{userarg} = var (lvar(var))
337 \italic{builtinapp} = \italic{builtinfunc}
338 | \italic{builtinapp} \italic{builtinarg}
339 \italic{builtinfunc} = var (bvar(var))
340 \italic{builtinarg} = \italic{coreexpr}
343 \todo{Limit builtinarg further}
345 \todo{There can still be other casts around (which the code can handle,
346 e.g., ignore), which still need to be documented here}
348 \todo{Note about the selector case. It just supports Bit and Bool
349 currently, perhaps it should be generalized in the normal form? This is
352 When looking at such a program from a hardware perspective, the top level
353 lambda's define the input ports. The value produced by the let expression is
354 the output port. Most function applications bound by the let expression
355 define a component instantiation, where the input and output ports are mapped
356 to local signals or arguments. Some of the others use a builtin
357 construction (\eg the \lam{case} statement) or call a builtin function
358 (\eg \lam{add} or \lam{sub}). For these, a hardcoded \small{VHDL} translation is
361 \section{Transformation notation}
362 To be able to concisely present transformations, we use a specific format to
363 them. It is a simple format, similar to one used in logic reasoning.
365 Such a transformation description looks like the following.
370 <original expression>
371 -------------------------- <expression conditions>
372 <transformed expresssion>
377 This format desribes a transformation that applies to \lam{original
378 expresssion} and transforms it into \lam{transformed expression}, assuming
379 that all conditions apply. In this format, there are a number of placeholders
380 in pointy brackets, most of which should be rather obvious in their meaning.
381 Nevertheless, we will more precisely specify their meaning below:
383 \startdesc{<original expression>} The expression pattern that will be matched
384 against (subexpressions of) the expression to be transformed. We call this a
385 pattern, because it can contain \emph{placeholders} (variables), which match
386 any expression or binder. Any such placeholder is said to be \emph{bound} to
387 the expression it matches. It is convention to use an uppercase latter (\eg
388 \lam{M} or \lam{E} to refer to any expression (including a simple variable
389 reference) and lowercase letters (\eg \lam{v} or \lam{b}) to refer to
390 (references to) binders.
392 For example, the pattern \lam{a + B} will match the expression
393 \lam{v + (2 * w)} (and bind \lam{a} to \lam{v} and \lam{B} to
394 \lam{(2 * 2)}), but not \lam{v + (2 * w)}.
397 \startdesc{<expression conditions>}
398 These are extra conditions on the expression that is matched. These
399 conditions can be used to further limit the cases in which the
400 transformation applies, in particular to prevent a transformation from
401 causing a loop with itself or another transformation.
403 Only if these if these conditions are \emph{all} true, this transformation
407 \startdesc{<context conditions>}
408 These are a number of extra conditions on the context of the function. In
409 particular, these conditions can require some other top level function to be
410 present, whose value matches the pattern given here. The format of each of
411 these conditions is: \lam{binder = <pattern>}.
413 Typically, the binder is some placeholder bound in the \lam{<original
414 expression>}, while the pattern contains some placeholders that are used in
415 the \lam{transformed expression}.
417 Only if a top level binder exists that matches each binder and pattern, this
418 transformation applies.
421 \startdesc{<transformed expression>}
422 This is the expression template that is the result of the transformation. If, looking
423 at the above three items, the transformation applies, the \lam{original
424 expression} is completely replaced with the \lam{<transformed expression>}.
425 We call this a template, because it can contain placeholders, referring to
426 any placeholder bound by the \lam{<original expression>} or the
427 \lam{<context conditions>}. The resulting expression will have those
428 placeholders replaced by the values bound to them.
430 Any binder (lowercase) placeholder that has no value bound to it yet will be
431 bound to (and replaced with) a fresh binder.
434 \startdesc{<context additions>}
435 These are templates for new functions to add to the context. This is a way
436 to have a transformation create new top level functiosn.
438 Each addition has the form \lam{binder = template}. As above, any
439 placeholder in the addition is replaced with the value bound to it, and any
440 binder placeholder that has no value bound to it yet will be bound to (and
441 replaced with) a fresh binder.
444 As an example, we'll look at η-abstraction:
448 -------------- \lam{E} does not occur on a function position in an application
449 λx.E x \lam{E} is not a lambda abstraction.
452 Consider the following function, which is a fairly obvious way to specify a
453 simple ALU (Note \at{example}[ex:AddSubAlu] is the normal form of this
457 alu :: Bit -> Word -> Word -> Word
458 alu = λopcode. case opcode of
463 There are a few subexpressions in this function to which we could possibly
464 apply the transformation. Since the pattern of the transformation is only
465 the placeholder \lam{E}, any expression will match that. Whether the
466 transformation applies to an expression is thus solely decided by the
467 conditions to the right of the transformation.
469 We will look at each expression in the function in a top down manner. The
470 first expression is the entire expression the function is bound to.
473 λopcode. case opcode of
478 As said, the expression pattern matches this. The type of this expression is
479 \lam{Bit -> Word -> Word -> Word}, which matches \lam{a -> b} (Note that in
480 this case \lam{a = Bit} and \lam{b = Word -> Word -> Word}).
482 Since this expression is at top level, it does not occur at a function
483 position of an application. However, The expression is a lambda abstraction,
484 so this transformation does not apply.
486 The next expression we could apply this transformation to, is the body of
487 the lambda abstraction:
495 The type of this expression is \lam{Word -> Word -> Word}, which again
496 matches \lam{a -> b}. The expression is the body of a lambda expression, so
497 it does not occur at a function position of an application. Finally, the
498 expression is not a lambda abstraction but a case expression, so all the
499 conditions match. There are no context conditions to match, so the
500 transformation applies.
502 By now, the placeholder \lam{E} is bound to the entire expression. The
503 placeholder \lam{x}, which occurs in the replacement template, is not bound
504 yet, so we need to generate a fresh binder for that. Let's use the binder
505 \lam{a}. This results in the following replacement expression:
513 Continuing with this expression, we see that the transformation does not
514 apply again (it is a lambda expression). Next we look at the body of this
523 Here, the transformation does apply, binding \lam{E} to the entire
524 expression and \lam{x} to the fresh binder \lam{b}, resulting in the
533 Again, the transformation does not apply to this lambda abstraction, so we
534 look at its body. For brevity, we'll put the case statement on one line from
538 (case opcode of Low -> (+); High -> (-)) a b
541 The type of this expression is \lam{Word}, so it does not match \lam{a -> b}
542 and the transformation does not apply. Next, we have two options for the
543 next expression to look at: The function position and argument position of
544 the application. The expression in the argument position is \lam{b}, which
545 has type \lam{Word}, so the transformation does not apply. The expression in
546 the function position is:
549 (case opcode of Low -> (+); High -> (-)) a
552 Obviously, the transformation does not apply here, since it occurs in
553 function position. In the same way the transformation does not apply to both
554 components of this expression (\lam{case opcode of Low -> (+); High -> (-)}
555 and \lam{a}), so we'll skip to the components of the case expression: The
556 scrutinee and both alternatives. Since the opcode is not a function, it does
557 not apply here, and we'll leave both alternatives as an exercise to the
558 reader. The final function, after all these transformations becomes:
561 alu :: Bit -> Word -> Word -> Word
562 alu = λopcode.λa.b. (case opcode of
563 Low -> λa1.λb1 (+) a1 b1
564 High -> λa2.λb2 (-) a2 b2) a b
567 In this case, the transformation does not apply anymore, though this might
568 not always be the case (e.g., the application of a transformation on a
569 subexpression might open up possibilities to apply the transformation
570 further up in the expression).
572 \subsection{Transformation application}
573 In this chapter we define a number of transformations, but how will we apply
574 these? As stated before, our normal form is reached as soon as no
575 transformation applies anymore. This means our application strategy is to
576 simply apply any transformation that applies, and continuing to do that with
577 the result of each transformation.
579 In particular, we define no particular order of transformations. Since
580 transformation order should not influence the resulting normal form,
581 \todo{This is not really true, but would like it to be...} this leaves
582 the implementation free to choose any application order that results in
583 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 \todo{Define substitution (notation)}
595 \subsubsection{Other concepts}
596 A \emph{global variable} is any variable that is bound at the
597 top level of a program, or an external module. A \emph{local variable} is any
598 other variable (\eg, variables local to a function, which can be bound by
599 lambda abstractions, let expressions and pattern matches of case
600 alternatives). Note that this is a slightly different notion of global versus
601 local than what \small{GHC} uses internally.
602 \defref{global variable} \defref{local variable}
604 A \emph{hardware representable} (or just \emph{representable}) type or value
605 is (a value of) a type that we can generate a signal for in hardware. For
606 example, a bit, a vector of bits, a 32 bit unsigned word, etc. Types that are
607 not runtime representable notably include (but are not limited to): Types,
608 dictionaries, functions.
609 \defref{representable}
611 A \emph{builtin function} is a function supplied by the Cλash framework, whose
612 implementation is not valid Cλash. The implementation is of course valid
613 Haskell, for simulation, but it is not expressable in Cλash.
614 \defref{builtin function} \defref{user-defined function}
616 For these functions, Cλash has a \emph{builtin hardware translation}, so calls
617 to these functions can still be translated. These are functions like
618 \lam{map}, \lam{hwor} and \lam{length}.
620 A \emph{user-defined} function is a function for which we do have a Cλash
621 implementation available.
623 \subsubsection{Functions}
624 Here, we define a number of functions that can be used below to concisely
627 \refdef{global variable}\emph{gvar(expr)} is true when \emph{expr} is a variable that references a
628 global variable. It is false when it references a local variable.
630 \refdef{local variable}\emph{lvar(expr)} is the complement of \emph{gvar}; it is true when \emph{expr}
631 references a local variable, false when it references a global variable.
633 \refdef{representable}\emph{representable(expr)} or \emph{representable(var)} is true when
634 \emph{expr} or \emph{var} is \emph{representable}.
636 \subsection{Binder uniqueness}
637 A common problem in transformation systems, is binder uniqueness. When not
638 considering this problem, it is easy to create transformations that mix up
639 bindings and cause name collisions. Take for example, the following core
643 (λa.λb.λc. a * b * c) x c
646 By applying β-reduction (see below) once, we can simplify this expression to:
652 Now, we have replaced the \lam{a} binder with a reference to the \lam{x}
653 binder. No harm done here. But note that we see multiple occurences of the
654 \lam{c} binder. The first is a binding occurence, to which the second refers.
655 The last, however refers to \emph{another} instance of \lam{c}, which is
656 bound somewhere outside of this expression. Now, if we would apply beta
657 reduction without taking heed of binder uniqueness, we would get:
663 This is obviously not what was supposed to happen! The root of this problem is
664 the reuse of binders: Identical binders can be bound in different scopes, such
665 that only the inner one is \quote{visible} in the inner expression. In the example
666 above, the \lam{c} binder was bound outside of the expression and in the inner
667 lambda expression. Inside that lambda expression, only the inner \lam{c} is
670 There are a number of ways to solve this. \small{GHC} has isolated this
671 problem to their binder substitution code, which performs \emph{deshadowing}
672 during its expression traversal. This means that any binding that shadows
673 another binding on a higher level is replaced by a new binder that does not
674 shadow any other binding. This non-shadowing invariant is enough to prevent
675 binder uniqueness problems in \small{GHC}.
677 In our transformation system, maintaining this non-shadowing invariant is
678 a bit harder to do (mostly due to implementation issues, the prototype doesn't
679 use \small{GHC}'s subsitution code). Also, we can observe the following
683 \item Deshadowing does not guarantee overall uniqueness. For example, the
684 following (slightly contrived) expression shows the identifier \lam{x} bound in
685 two seperate places (and to different values), even though no shadowing
689 (let x = 1 in x) + (let x = 2 in x)
692 \item In our normal form (and the resulting \small{VHDL}), all binders
693 (signals) will end up in the same scope. To allow this, all binders within the
694 same function should be unique.
696 \item When we know that all binders in an expression are unique, moving around
697 or removing a subexpression will never cause any binder conflicts. If we have
698 some way to generate fresh binders, introducing new subexpressions will not
699 cause any problems either. The only way to cause conflicts is thus to
700 duplicate an existing subexpression.
703 Given the above, our prototype maintains a unique binder invariant. This
704 meanst that in any given moment during normalization, all binders \emph{within
705 a single function} must be unique. To achieve this, we apply the following
708 \todo{Define fresh binders and unique supplies}
711 \item Before starting normalization, all binders in the function are made
712 unique. This is done by generating a fresh binder for every binder used. This
713 also replaces binders that did not pose any conflict, but it does ensure that
714 all binders within the function are generated by the same unique supply. See
715 \refdef{fresh binder}
716 \item Whenever a new binder must be generated, we generate a fresh binder that
717 is guaranteed to be different from \emph{all binders generated so far}. This
718 can thus never introduce duplication and will maintain the invariant.
719 \item Whenever (part of) an expression is duplicated (for example when
720 inlining), all binders in the expression are replaced with fresh binders
721 (using the same method as at the start of normalization). These fresh binders
722 can never introduce duplication, so this will maintain the invariant.
723 \item Whenever we move part of an expression around within the function, there
724 is no need to do anything special. There is obviously no way to introduce
725 duplication by moving expressions around. Since we know that each of the
726 binders is already unique, there is no way to introduce (incorrect) shadowing
730 \section{Transform passes}
731 In this section we describe the actual transforms. Here we're using
732 the core language in a notation that resembles lambda calculus.
734 Each of these transforms is meant to be applied to every (sub)expression
735 in a program, for as long as it applies. Only when none of the
736 transformations can be applied anymore, the program is in normal form (by
737 definition). We hope to be able to prove that this form will obey all of the
738 constraints defined above, but this has yet to happen (though it seems likely
741 Each of the transforms will be described informally first, explaining
742 the need for and goal of the transform. Then, a formal definition is
743 given, using a familiar syntax from the world of logic. Each transform
744 is specified as a number of conditions (above the horizontal line) and a
745 number of conclusions (below the horizontal line). The details of using
746 this notation are still a bit fuzzy, so comments are welcom.
748 \subsection{General cleanup}
749 These transformations are general cleanup transformations, that aim to
750 make expressions simpler. These transformations usually clean up the
751 mess left behind by other transformations or clean up expressions to
752 expose new transformation opportunities for other transformations.
754 Most of these transformations are standard optimizations in other
755 compilers as well. However, in our compiler, most of these are not just
756 optimizations, but they are required to get our program into normal
759 \subsubsection{β-reduction}
760 β-reduction is a well known transformation from lambda calculus, where it is
761 the main reduction step. It reduces applications of labmda abstractions,
762 removing both the lambda abstraction and the application.
764 In our transformation system, this step helps to remove unwanted lambda
765 abstractions (basically all but the ones at the top level). Other
766 transformations (application propagation, non-representable inlining) make
767 sure that most lambda abstractions will eventually be reducable by
785 \transexample{β-reduction}{from}{to}
787 \subsubsection{Empty let removal}
788 This transformation is simple: It removes recursive lets that have no bindings
789 (which usually occurs when unused let binding removal removes the last
800 \subsubsection{Simple let binding removal}
801 This transformation inlines simple let bindings (\eg a = b).
803 This transformation is not needed to get into normal form, but makes the
804 resulting \small{VHDL} a lot shorter.
815 ----------------------------- \lam{b} is a variable reference
829 \subsubsection{Unused let binding removal}
830 This transformation removes let bindings that are never used. Usually,
831 the desugarer introduces some unused let bindings.
833 This normalization pass should really be unneeded to get into normal form
834 (since unused bindings are not forbidden by the normal form), but in practice
835 the desugarer or simplifier emits some unused bindings that cannot be
836 normalized (e.g., calls to a \type{PatError} (\todo{Check this name}). Also,
837 this transformation makes the resulting \small{VHDL} a lot shorter.
847 M \lam{a} does not occur free in \lam{M}
848 ---------------------------- \forall j, 0 <= j <= n, j ≠ i (\lam{a} does not occur free in \lam{Ej})
862 \subsubsection{Cast propagation / simplification}
863 This transform pushes casts down into the expression as far as possible.
864 Since its exact role and need is not clear yet, this transformation is
867 \todo{Cast propagation}
869 \subsubsection{Top level binding inlining}
870 This transform takes simple top level bindings generated by the
871 \small{GHC} compiler. \small{GHC} sometimes generates very simple
872 \quote{wrapper} bindings, which are bound to just a variable
873 reference, or a partial application to constants or other variable
876 Note that this transformation is completely optional. It is not
877 required to get any function into normal form, but it does help making
878 the resulting VHDL output easier to read (since it removes a bunch of
879 components that are really boring).
881 This transform takes any top level binding generated by the compiler,
882 whose normalized form contains only a single let binding.
885 x = λa0 ... λan.let y = E in y
888 -------------------------------------- \lam{x} is generated by the compiler
889 λa0 ... λan.let y = E in y
893 (+) :: Word -> Word -> Word
894 (+) = GHC.Num.(+) @Word $dNum
899 GHC.Num.(+) @ Alu.Word $dNum a b
902 \transexample{Top level binding inlining}{from}{to}
904 Without this transformation, the (+) function would generate an
905 architecture which would just add its inputs. This generates a lot of
906 overhead in the VHDL, which is particularly annoying when browsing the
907 generated RTL schematic (especially since + is not allowed in VHDL
908 architecture names\footnote{Technically, it is allowed to use
909 non-alphanumerics when using extended identifiers, but it seems that
910 none of the tooling likes extended identifiers in filenames, so it
911 effectively doesn't work}, so the entity would be called
912 \quote{w7aA7f} or something similarly unreadable and autogenerated).
914 \subsection{Program structure}
915 These transformations are aimed at normalizing the overall structure
916 into the intended form. This means ensuring there is a lambda abstraction
917 at the top for every argument (input port), putting all of the other
918 value definitions in let bindings and making the final return value a
919 simple variable reference.
921 \subsubsection{η-abstraction}
922 This transformation makes sure that all arguments of a function-typed
923 expression are named, by introducing lambda expressions. When combined with
924 β-reduction and non-representable binding inlining, all function-typed
925 expressions should be lambda abstractions or global identifiers.
929 -------------- \lam{E} is not the first argument of an application.
930 λx.E x \lam{E} is not a lambda abstraction.
931 \lam{x} is a variable that does not occur free in \lam{E}.
941 foo = λa.λx.(case a of
946 \transexample{η-abstraction}{from}{to}
948 \subsubsection{Application propagation}
949 This transformation is meant to propagate application expressions downwards
950 into expressions as far as possible. This allows partial applications inside
951 expressions to become fully applied and exposes new transformation
952 opportunities for other transformations (like β-reduction and
956 (letrec binds in E) M
957 ------------------------
977 \transexample{Application propagation for a let expression}{from}{to}
1005 \transexample{Application propagation for a case expression}{from}{to}
1007 \subsubsection{Let recursification}
1008 This transformation makes all non-recursive lets recursive. In the
1009 end, we want a single recursive let in our normalized program, so all
1010 non-recursive lets can be converted. This also makes other
1011 transformations simpler: They can simply assume all lets are
1019 ------------------------------------------
1026 \subsubsection{Let flattening}
1027 This transformation puts nested lets in the same scope, by lifting the
1028 binding(s) of the inner let into a new let around the outer let. Eventually,
1029 this will cause all let bindings to appear in the same scope (they will all be
1030 in scope for the function return value).
1035 x = (letrec bindings in M)
1039 ------------------------------------------
1068 \transexample{Let flattening}{from}{to}
1070 \subsubsection{Return value simplification}
1071 This transformation ensures that the return value of a function is always a
1072 simple local variable reference.
1074 Currently implemented using lambda simplification, let simplification, and
1075 top simplification. Should change into something like the following, which
1076 works only on the result of a function instead of any subexpression. This is
1077 achieved by the contexts, like \lam{x = E}, though this is strictly not
1078 correct (you could read this as "if there is any function \lam{x} that binds
1079 \lam{E}, any \lam{E} can be transformed, while we only mean the \lam{E} that
1080 is bound by \lam{x}. This might need some extra notes or something).
1082 Note that the return value is not simplified if its not representable.
1083 Otherwise, this would cause a direct loop with the inlining of
1084 unrepresentable bindings, of course. If the return value is not
1085 representable because it has a function type, η-abstraction should
1086 make sure that this transformation will eventually apply. If the value
1087 is not representable for other reasons, the function result itself is
1088 not representable, meaning this function is not representable anyway!
1091 x = E \lam{E} is representable
1092 ~ \lam{E} is not a lambda abstraction
1093 E \lam{E} is not a let expression
1094 --------------------------- \lam{E} is not a local variable reference
1100 ~ \lam{E} is representable
1101 E \lam{E} is not a let expression
1102 --------------------------- \lam{E} is not a local variable reference
1107 x = λv0 ... λvn.let ... in E
1108 ~ \lam{E} is representable
1109 E \lam{E} is not a local variable reference
1110 ---------------------------
1119 x = letrec x = add 1 2 in x
1122 \transexample{Return value simplification}{from}{to}
1124 \subsection{Argument simplification}
1125 The transforms in this section deal with simplifying application
1126 arguments into normal form. The goal here is to:
1129 \item Make all arguments of user-defined functions (\eg, of which
1130 we have a function body) simple variable references of a runtime
1131 representable type. This is needed, since these applications will be turned
1132 into component instantiations.
1133 \item Make all arguments of builtin functions one of:
1135 \item A type argument.
1136 \item A dictionary argument.
1137 \item A type level expression.
1138 \item A variable reference of a runtime representable type.
1139 \item A variable reference or partial application of a function type.
1143 When looking at the arguments of a user-defined function, we can
1144 divide them into two categories:
1146 \item Arguments of a runtime representable type (\eg bits or vectors).
1148 These arguments can be preserved in the program, since they can
1149 be translated to input ports later on. However, since we can
1150 only connect signals to input ports, these arguments must be
1151 reduced to simple variables (for which signals will be
1152 produced). This is taken care of by the argument extraction
1154 \item Non-runtime representable typed arguments.
1156 These arguments cannot be preserved in the program, since we
1157 cannot represent them as input or output ports in the resulting
1158 \small{VHDL}. To remove them, we create a specialized version of the
1159 called function with these arguments filled in. This is done by
1160 the argument propagation transform.
1162 Typically, these arguments are type and dictionary arguments that are
1163 used to make functions polymorphic. By propagating these arguments, we
1164 are essentially doing the same which GHC does when it specializes
1165 functions: Creating multiple variants of the same function, one for
1166 each type for which it is used. Other common non-representable
1167 arguments are functions, e.g. when calling a higher order function
1168 with another function or a lambda abstraction as an argument.
1170 The reason for doing this is similar to the reasoning provided for
1171 the inlining of non-representable let bindings above. In fact, this
1172 argument propagation could be viewed as a form of cross-function
1176 \todo{Check the following itemization.}
1178 When looking at the arguments of a builtin function, we can divide them
1182 \item Arguments of a runtime representable type.
1184 As we have seen with user-defined functions, these arguments can
1185 always be reduced to a simple variable reference, by the
1186 argument extraction transform. Performing this transform for
1187 builtin functions as well, means that the translation of builtin
1188 functions can be limited to signal references, instead of
1189 needing to support all possible expressions.
1191 \item Arguments of a function type.
1193 These arguments are functions passed to higher order builtins,
1194 like \lam{map} and \lam{foldl}. Since implementing these
1195 functions for arbitrary function-typed expressions (\eg, lambda
1196 expressions) is rather comlex, we reduce these arguments to
1197 (partial applications of) global functions.
1199 We can still support arbitrary expressions from the user code,
1200 by creating a new global function containing that expression.
1201 This way, we can simply replace the argument with a reference to
1202 that new function. However, since the expression can contain any
1203 number of free variables we also have to include partial
1204 applications in our normal form.
1206 This category of arguments is handled by the function extraction
1208 \item Other unrepresentable arguments.
1210 These arguments can take a few different forms:
1211 \startdesc{Type arguments}
1212 In the core language, type arguments can only take a single
1213 form: A type wrapped in the Type constructor. Also, there is
1214 nothing that can be done with type expressions, except for
1215 applying functions to them, so we can simply leave type
1216 arguments as they are.
1218 \startdesc{Dictionary arguments}
1219 In the core language, dictionary arguments are used to find
1220 operations operating on one of the type arguments (mostly for
1221 finding class methods). Since we will not actually evaluatie
1222 the function body for builtin functions and can generate
1223 code for builtin functions by just looking at the type
1224 arguments, these arguments can be ignored and left as they
1227 \startdesc{Type level arguments}
1228 Sometimes, we want to pass a value to a builtin function, but
1229 we need to know the value at compile time. Additionally, the
1230 value has an impact on the type of the function. This is
1231 encoded using type-level values, where the actual value of the
1232 argument is not important, but the type encodes some integer,
1233 for example. Since the value is not important, the actual form
1234 of the expression does not matter either and we can leave
1235 these arguments as they are.
1237 \startdesc{Other arguments}
1238 Technically, there is still a wide array of arguments that can
1239 be passed, but does not fall into any of the above categories.
1240 However, none of the supported builtin functions requires such
1241 an argument. This leaves use with passing unsupported types to
1242 a function, such as calling \lam{head} on a list of functions.
1244 In these cases, it would be impossible to generate hardware
1245 for such a function call anyway, so we can ignore these
1248 The only way to generate hardware for builtin functions with
1249 arguments like these, is to expand the function call into an
1250 equivalent core expression (\eg, expand map into a series of
1251 function applications). But for now, we choose to simply not
1252 support expressions like these.
1255 From the above, we can conclude that we can simply ignore these
1256 other unrepresentable arguments and focus on the first two
1260 \subsubsection{Argument simplification}
1261 This transform deals with arguments to functions that
1262 are of a runtime representable type. It ensures that they will all become
1263 references to global variables, or local signals in the resulting \small{VHDL}.
1265 \todo{It seems we can map an expression to a port, not only a signal.}
1266 Perhaps this makes this transformation not needed?
1267 \todo{Say something about dataconstructors (without arguments, like True
1268 or False), which are variable references of a runtime representable
1269 type, but do not result in a signal.}
1271 To reduce a complex expression to a simple variable reference, we create
1272 a new let expression around the application, which binds the complex
1273 expression to a new variable. The original function is then applied to
1278 -------------------- \lam{N} is of a representable type
1279 letrec x = N in M x \lam{N} is not a local variable reference
1287 letrec x = add a 1 in add x 1
1290 \transexample{Argument extraction}{from}{to}
1292 \subsubsection{Function extraction}
1293 This transform deals with function-typed arguments to builtin functions.
1294 Since these arguments cannot be propagated, we choose to extract them
1295 into a new global function instead.
1297 Any free variables occuring in the extracted arguments will become
1298 parameters to the new global function. The original argument is replaced
1299 with a reference to the new function, applied to any free variables from
1300 the original argument.
1302 This transformation is useful when applying higher order builtin functions
1303 like \hs{map} to a lambda abstraction, for example. In this case, the code
1304 that generates \small{VHDL} for \hs{map} only needs to handle top level functions and
1305 partial applications, not any other expression (such as lambda abstractions or
1306 even more complicated expressions).
1309 M N \lam{M} is a (partial aplication of a) builtin function.
1310 --------------------- \lam{f0 ... fn} = free local variables of \lam{N}
1311 M (x f0 ... fn) \lam{N :: a -> b}
1312 ~ \lam{N} is not a (partial application of) a top level function
1317 map (λa . add a b) xs
1331 \transexample{Function extraction}{from}{to}
1333 Note that \lam{x0} and {x1} will still need normalization after this.
1335 \subsubsection{Argument propagation}
1336 \fxnote{This section should be generalized and describe
1337 specialization, so other transformations can refer to this (since
1338 specialization is really used in multiple categories).}
1340 This transform deals with arguments to user-defined functions that are
1341 not representable at runtime. This means these arguments cannot be
1342 preserved in the final form and most be {\em propagated}.
1344 Propagation means to create a specialized version of the called
1345 function, with the propagated argument already filled in. As a simple
1346 example, in the following program:
1353 We could {\em propagate} the constant argument 1, with the following
1361 Special care must be taken when the to-be-propagated expression has any
1362 free variables. If this is the case, the original argument should not be
1363 removed alltogether, but replaced by all the free variables of the
1364 expression. In this way, the original expression can still be evaluated
1365 inside the new function. Also, this brings us closer to our goal: All
1366 these free variables will be simple variable references.
1368 To prevent us from propagating the same argument over and over, a simple
1369 local variable reference is not propagated (since is has exactly one
1370 free variable, itself, we would only replace that argument with itself).
1372 This shows that any free local variables that are not runtime representable
1373 cannot be brought into normal form by this transform. We rely on an
1374 inlining transformation to replace such a variable with an expression we
1375 can propagate again.
1380 x Y0 ... Yi ... Yn \lam{Yi} is not of a runtime representable type
1381 --------------------------------------------- \lam{Yi} is not a local variable reference
1382 x' y0 ... yi-1 f0 ... fm Yi+1 ... Yn \lam{f0 ... fm} = free local vars of \lam{Yi}
1384 x' = λy0 ... yi-1 f0 ... fm yi+1 ... yn .
1385 E y0 ... yi-1 Yi yi+1 ... yn
1391 \subsection{Case simplification}
1392 \subsubsection{Scrutinee simplification}
1393 This transform ensures that the scrutinee of a case expression is always
1394 a simple variable reference.
1399 ----------------- \lam{E} is not a local variable reference
1418 \transexample{Let flattening}{from}{to}
1421 \subsubsection{Case simplification}
1422 This transformation ensures that all case expressions become normal form. This
1423 means they will become one of:
1425 \item An extractor case with a single alternative that picks a single field
1426 from a datatype, \eg \lam{case x of (a, b) -> a}.
1427 \item A selector case with multiple alternatives and only wild binders, that
1428 makes a choice between expressions based on the constructor of another
1429 expression, \eg \lam{case x of Low -> a; High -> b}.
1434 C0 v0,0 ... v0,m -> E0
1436 Cn vn,0 ... vn,m -> En
1437 --------------------------------------------------- \forall i \forall j, 0 <= i <= n, 0 <= i < m (\lam{wi,j} is a wild (unused) binder)
1439 v0,0 = case x of C0 v0,0 .. v0,m -> v0,0
1441 v0,m = case x of C0 v0,0 .. v0,m -> v0,m
1444 vn,m = case x of Cn vn,0 .. vn,m -> vn,m
1448 C0 w0,0 ... w0,m -> x0
1450 Cn wn,0 ... wn,m -> xn
1453 \fxnote{This transformation specified like this is complicated and misses
1454 conditions to prevent looping with itself. Perhaps it should be split here for
1473 \transexample{Selector case simplification}{from}{to}
1481 b = case a of (,) b c -> b
1482 c = case a of (,) b c -> c
1489 \transexample{Extractor case simplification}{from}{to}
1491 \subsubsection{Case removal}
1492 This transform removes any case statements with a single alternative and
1495 These "useless" case statements are usually leftovers from case simplification
1496 on extractor case (see the previous example).
1501 ---------------------- \lam{\forall i, 0 <= i <= m} (\lam{vi} does not occur free in E)
1514 \transexample{Case removal}{from}{to}
1516 \subsection{Removing polymorphism}
1517 Reference type-specialization (== argument propagation)
1519 Reference polymporphic binding inlining (== non-representable binding
1522 \subsection{Defunctionalization}
1523 These transformations remove most higher order expressions from our
1524 program, making it completely first-order (the only exception here is for
1525 arguments to builtin functions, since we can't specialize builtin
1526 function. \todo{Talk more about this somewhere}
1528 Reference higher-order-specialization (== argument propagation)
1530 \subsubsection{Non-representable binding inlining}
1531 This transform inlines let bindings that have a non-representable type. Since
1532 we can never generate a signal assignment for these bindings (we cannot
1533 declare a signal assignment with a non-representable type, for obvious
1534 reasons), we have no choice but to inline the binding to remove it.
1536 If the binding is non-representable because it is a lambda abstraction, it is
1537 likely that it will inlined into an application and β-reduction will remove
1538 the lambda abstraction and turn it into a representable expression at the
1539 inline site. The same holds for partial applications, which can be turned into
1540 full applications by inlining.
1542 Other cases of non-representable bindings we see in practice are primitive
1543 Haskell types. In most cases, these will not result in a valid normalized
1544 output, but then the input would have been invalid to start with. There is one
1545 exception to this: When a builtin function is applied to a non-representable
1546 expression, things might work out in some cases. For example, when you write a
1547 literal \hs{SizedInt} in Haskell, like \hs{1 :: SizedInt D8}, this results in
1548 the following core: \lam{fromInteger (smallInteger 10)}, where for example
1549 \lam{10 :: GHC.Prim.Int\#} and \lam{smallInteger 10 :: Integer} have
1550 non-representable types. \todo{This/these paragraph(s) should probably become a
1551 separate discussion somewhere else}
1563 -------------------------- \lam{Ei} has a non-representable type.
1587 x = fromInteger (smallInteger 10)
1589 (λb -> add b 1) (add 1 x)
1592 \transexample{None representable binding inlining}{from}{to}
1595 \section{Provable properties}
1596 When looking at the system of transformations outlined above, there are a
1597 number of questions that we can ask ourselves. The main question is of course:
1598 \quote{Does our system work as intended?}. We can split this question into a
1599 number of subquestions:
1602 \item[q:termination] Does our system \emph{terminate}? Since our system will
1603 keep running as long as transformations apply, there is an obvious risk that
1604 it will keep running indefinitely. One transformation produces a result that
1605 is transformed back to the original by another transformation, for example.
1606 \item[q:soundness] Is our system \emph{sound}? Since our transformations
1607 continuously modify the expression, there is an obvious risk that the final
1608 normal form will not be equivalent to the original program: Its meaning could
1610 \item[q:completeness] Is our system \emph{complete}? Since we have a complex
1611 system of transformations, there is an obvious risk that some expressions will
1612 not end up in our intended normal form, because we forgot some transformation.
1613 In other words: Does our transformation system result in our intended normal
1614 form for all possible inputs?
1615 \item[q:determinism] Is our system \emph{deterministic}? Since we have defined
1616 no particular order in which the transformation should be applied, there is an
1617 obvious risk that different transformation orderings will result in
1618 \emph{different} normal forms. They might still both be intended normal forms
1619 (if our system is \emph{complete}) and describe correct hardware (if our
1620 system is \emph{sound}), so this property is less important than the previous
1621 three: The translator would still function properly without it.
1624 \subsection{Graph representation}
1625 Before looking into how to prove these properties, we'll look at our
1626 transformation system from a graph perspective. The nodes of the graph are
1627 all possible Core expressions. The (directed) edges of the graph are
1628 transformations. When a transformation α applies to an expression \lam{A} to
1629 produce an expression \lam{B}, we add an edge from the node for \lam{A} to the
1630 node for \lam{B}, labeled α.
1632 \startuseMPgraphic{TransformGraph}
1636 newCircle.a(btex \lam{(λx.λy. (+) x y) 1} etex);
1637 newCircle.b(btex \lam{λy. (+) 1 y} etex);
1638 newCircle.c(btex \lam{(λx.(+) x) 1} etex);
1639 newCircle.d(btex \lam{(+) 1} etex);
1642 c.c = b.c + (4cm, 0cm);
1643 a.c = midpoint(b.c, c.c) + (0cm, 4cm);
1644 d.c = midpoint(b.c, c.c) - (0cm, 3cm);
1646 % β-conversion between a and b
1647 ncarc.a(a)(b) "name(bred)";
1648 ObjLabel.a(btex $\xrightarrow[normal]{}{β}$ etex) "labpathname(bred)", "labdir(rt)";
1649 ncarc.b(b)(a) "name(bexp)", "linestyle(dashed withdots)";
1650 ObjLabel.b(btex $\xleftarrow[normal]{}{β}$ etex) "labpathname(bexp)", "labdir(lft)";
1652 % η-conversion between a and c
1653 ncarc.a(a)(c) "name(ered)";
1654 ObjLabel.a(btex $\xrightarrow[normal]{}{η}$ etex) "labpathname(ered)", "labdir(rt)";
1655 ncarc.c(c)(a) "name(eexp)", "linestyle(dashed withdots)";
1656 ObjLabel.c(btex $\xleftarrow[normal]{}{η}$ etex) "labpathname(eexp)", "labdir(lft)";
1658 % η-conversion between b and d
1659 ncarc.b(b)(d) "name(ered)";
1660 ObjLabel.b(btex $\xrightarrow[normal]{}{η}$ etex) "labpathname(ered)", "labdir(rt)";
1661 ncarc.d(d)(b) "name(eexp)", "linestyle(dashed withdots)";
1662 ObjLabel.d(btex $\xleftarrow[normal]{}{η}$ etex) "labpathname(eexp)", "labdir(lft)";
1664 % β-conversion between c and d
1665 ncarc.c(c)(d) "name(bred)";
1666 ObjLabel.c(btex $\xrightarrow[normal]{}{β}$ etex) "labpathname(bred)", "labdir(rt)";
1667 ncarc.d(d)(c) "name(bexp)", "linestyle(dashed withdots)";
1668 ObjLabel.d(btex $\xleftarrow[normal]{}{β}$ etex) "labpathname(bexp)", "labdir(lft)";
1670 % Draw objects and lines
1671 drawObj(a, b, c, d);
1674 \placeexample[right][ex:TransformGraph]{Partial graph of a labmda calculus
1675 system with β and η reduction (solid lines) and expansion (dotted lines).}
1676 \boxedgraphic{TransformGraph}
1678 Of course our graph is unbounded, since we can construct an infinite amount of
1679 Core expressions. Also, there might potentially be multiple edges between two
1680 given nodes (with different labels), though seems unlikely to actually happen
1683 See \in{example}[ex:TransformGraph] for the graph representation of a very
1684 simple lambda calculus that contains just the expressions \lam{(λx.λy. (+) x
1685 y) 1}, \lam{λy. (+) 1 y}, \lam{(λx.(+) x) 1} and \lam{(+) 1}. The
1686 transformation system consists of β-reduction and η-reduction (solid edges) or
1687 β-reduction and η-reduction (dotted edges).
1689 \todo{Define β-reduction and η-reduction?}
1691 Note that the normal form of such a system consists of the set of nodes
1692 (expressions) without outgoing edges, since those are the expression to which
1693 no transformation applies anymore. We call this set of nodes the \emph{normal
1696 From such a graph, we can derive some properties easily:
1698 \item A system will \emph{terminate} if there is no path of infinite length
1699 in the graph (this includes cycles).
1700 \item Soundness is not easily represented in the graph.
1701 \item A system is \emph{complete} if all of the nodes in the normal set have
1702 the intended normal form. The inverse (that all of the nodes outside of
1703 the normal set are \emph{not} in the intended normal form) is not
1705 \item A system is deterministic if all paths from a node, which end in a node
1706 in the normal set, end at the same node.
1709 When looking at the \in{example}[ex:TransformGraph], we see that the system
1710 terminates for both the reduction and expansion systems (but note that, for
1711 expansion, this is only true because we've limited the possible expressions!
1712 In comlete lambda calculus, there would be a path from \lam{(λx.λy. (+) x y)
1713 1} to \lam{(λx.λy.(λz.(+) z) x y) 1} to \lam{(λx.λy.(λz.(λq.(+) q) z) x y) 1}
1716 If we would consider the system with both expansion and reduction, there would
1717 no longer be termination, since there would be cycles all over the place.
1719 The reduction and expansion systems have a normal set of containing just
1720 \lam{(+) 1} or \lam{(λx.λy. (+) x y) 1} respectively. Since all paths in
1721 either system end up in these normal forms, both systems are \emph{complete}.
1722 Also, since there is only one normal form, it must obviously be
1723 \emph{deterministic} as well.
1725 \subsection{Termination}
1730 \subsection{Soundness}
1731 Needs formal definition of semantics.
1732 Prove for each transformation seperately, implies soundness of the system.
1734 \subsection{Completeness}
1735 Show that any transformation applies to every Core expression that is not
1736 in normal form. To prove: no transformation applies => in intended form.
1737 Show the reverse: Not in intended form => transformation applies.
1739 \subsection{Determinism}