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} = 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?
351 When looking at such a program from a hardware perspective, the top level
352 lambda's define the input ports. The value produced by the let expression is
353 the output port. Most function applications bound by the let expression
354 define a component instantiation, where the input and output ports are mapped
355 to local signals or arguments. Some of the others use a builtin
356 construction (\eg the \lam{case} statement) or call a builtin function
357 (\eg \lam{add} or \lam{sub}). For these, a hardcoded \small{VHDL} translation is
360 \section{Transformation notation}
361 To be able to concisely present transformations, we use a specific format to
362 them. It is a simple format, similar to one used in logic reasoning.
364 Such a transformation description looks like the following.
369 <original expression>
370 -------------------------- <expression conditions>
371 <transformed expresssion>
376 This format desribes a transformation that applies to \lam{original
377 expresssion} and transforms it into \lam{transformed expression}, assuming
378 that all conditions apply. In this format, there are a number of placeholders
379 in pointy brackets, most of which should be rather obvious in their meaning.
380 Nevertheless, we will more precisely specify their meaning below:
382 \startdesc{<original expression>} The expression pattern that will be matched
383 against (subexpressions of) the expression to be transformed. We call this a
384 pattern, because it can contain \emph{placeholders} (variables), which match
385 any expression or binder. Any such placeholder is said to be \emph{bound} to
386 the expression it matches. It is convention to use an uppercase latter (\eg
387 \lam{M} or \lam{E} to refer to any expression (including a simple variable
388 reference) and lowercase letters (\eg \lam{v} or \lam{b}) to refer to
389 (references to) binders.
391 For example, the pattern \lam{a + B} will match the expression
392 \lam{v + (2 * w)} (and bind \lam{a} to \lam{v} and \lam{B} to
393 \lam{(2 * 2)}), but not \lam{v + (2 * w)}.
396 \startdesc{<expression conditions>}
397 These are extra conditions on the expression that is matched. These
398 conditions can be used to further limit the cases in which the
399 transformation applies, in particular to prevent a transformation from
400 causing a loop with itself or another transformation.
402 Only if these if these conditions are \emph{all} true, this transformation
406 \startdesc{<context conditions>}
407 These are a number of extra conditions on the context of the function. In
408 particular, these conditions can require some other top level function to be
409 present, whose value matches the pattern given here. The format of each of
410 these conditions is: \lam{binder = <pattern>}.
412 Typically, the binder is some placeholder bound in the \lam{<original
413 expression>}, while the pattern contains some placeholders that are used in
414 the \lam{transformed expression}.
416 Only if a top level binder exists that matches each binder and pattern, this
417 transformation applies.
420 \startdesc{<transformed expression>}
421 This is the expression template that is the result of the transformation. If, looking
422 at the above three items, the transformation applies, the \lam{original
423 expression} is completely replaced with the \lam{<transformed expression>}.
424 We call this a template, because it can contain placeholders, referring to
425 any placeholder bound by the \lam{<original expression>} or the
426 \lam{<context conditions>}. The resulting expression will have those
427 placeholders replaced by the values bound to them.
429 Any binder (lowercase) placeholder that has no value bound to it yet will be
430 bound to (and replaced with) a fresh binder.
433 \startdesc{<context additions>}
434 These are templates for new functions to add to the context. This is a way
435 to have a transformation create new top level functiosn.
437 Each addition has the form \lam{binder = template}. As above, any
438 placeholder in the addition is replaced with the value bound to it, and any
439 binder placeholder that has no value bound to it yet will be bound to (and
440 replaced with) a fresh binder.
443 As an example, we'll look at η-abstraction:
447 -------------- \lam{E} does not occur on a function position in an application
448 λx.E x \lam{E} is not a lambda abstraction.
451 Consider the following function, which is a fairly obvious way to specify a
452 simple ALU (Note \at{example}[ex:AddSubAlu] is the normal form of this
456 alu :: Bit -> Word -> Word -> Word
457 alu = λopcode. case opcode of
462 There are a few subexpressions in this function to which we could possibly
463 apply the transformation. Since the pattern of the transformation is only
464 the placeholder \lam{E}, any expression will match that. Whether the
465 transformation applies to an expression is thus solely decided by the
466 conditions to the right of the transformation.
468 We will look at each expression in the function in a top down manner. The
469 first expression is the entire expression the function is bound to.
472 λopcode. case opcode of
477 As said, the expression pattern matches this. The type of this expression is
478 \lam{Bit -> Word -> Word -> Word}, which matches \lam{a -> b} (Note that in
479 this case \lam{a = Bit} and \lam{b = Word -> Word -> Word}).
481 Since this expression is at top level, it does not occur at a function
482 position of an application. However, The expression is a lambda abstraction,
483 so this transformation does not apply.
485 The next expression we could apply this transformation to, is the body of
486 the lambda abstraction:
494 The type of this expression is \lam{Word -> Word -> Word}, which again
495 matches \lam{a -> b}. The expression is the body of a lambda expression, so
496 it does not occur at a function position of an application. Finally, the
497 expression is not a lambda abstraction but a case expression, so all the
498 conditions match. There are no context conditions to match, so the
499 transformation applies.
501 By now, the placeholder \lam{E} is bound to the entire expression. The
502 placeholder \lam{x}, which occurs in the replacement template, is not bound
503 yet, so we need to generate a fresh binder for that. Let's use the binder
504 \lam{a}. This results in the following replacement expression:
512 Continuing with this expression, we see that the transformation does not
513 apply again (it is a lambda expression). Next we look at the body of this
522 Here, the transformation does apply, binding \lam{E} to the entire
523 expression and \lam{x} to the fresh binder \lam{b}, resulting in the
532 Again, the transformation does not apply to this lambda abstraction, so we
533 look at its body. For brevity, we'll put the case statement on one line from
537 (case opcode of Low -> (+); High -> (-)) a b
540 The type of this expression is \lam{Word}, so it does not match \lam{a -> b}
541 and the transformation does not apply. Next, we have two options for the
542 next expression to look at: The function position and argument position of
543 the application. The expression in the argument position is \lam{b}, which
544 has type \lam{Word}, so the transformation does not apply. The expression in
545 the function position is:
548 (case opcode of Low -> (+); High -> (-)) a
551 Obviously, the transformation does not apply here, since it occurs in
552 function position. In the same way the transformation does not apply to both
553 components of this expression (\lam{case opcode of Low -> (+); High -> (-)}
554 and \lam{a}), so we'll skip to the components of the case expression: The
555 scrutinee and both alternatives. Since the opcode is not a function, it does
556 not apply here, and we'll leave both alternatives as an exercise to the
557 reader. The final function, after all these transformations becomes:
560 alu :: Bit -> Word -> Word -> Word
561 alu = λopcode.λa.b. (case opcode of
562 Low -> λa1.λb1 (+) a1 b1
563 High -> λa2.λb2 (-) a2 b2) a b
566 In this case, the transformation does not apply anymore, though this might
567 not always be the case (e.g., the application of a transformation on a
568 subexpression might open up possibilities to apply the transformation
569 further up in the expression).
571 \subsection{Transformation application}
572 In this chapter we define a number of transformations, but how will we apply
573 these? As stated before, our normal form is reached as soon as no
574 transformation applies anymore. This means our application strategy is to
575 simply apply any transformation that applies, and continuing to do that with
576 the result of each transformation.
578 In particular, we define no particular order of transformations. Since
579 transformation order should not influence the resulting normal form (see TODO
580 ref), this leaves the implementation free to choose any application order that
581 results in an efficient implementation.
583 When applying a single transformation, we try to apply it to every (sub)expression
584 in a function, not just the top level function. This allows us to keep the
585 transformation descriptions concise and powerful.
587 \subsection{Definitions}
588 In the following sections, we will be using a number of functions and
589 notations, which we will define here.
591 TODO: Define substitution
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 unused let binding removal removes the last
798 \subsubsection{Simple let binding removal}
799 This transformation inlines simple let bindings (\eg a = b).
801 This transformation is not needed to get into normal form, but makes the
802 resulting \small{VHDL} a lot shorter.
813 ----------------------------- \lam{b} is a variable reference
827 \subsubsection{Unused let binding removal}
828 This transformation removes let bindings that are never used. Usually,
829 the desugarer introduces some unused let bindings.
831 This normalization pass should really be unneeded to get into normal form
832 (since unused bindings are not forbidden by the normal form), but in practice
833 the desugarer or simplifier emits some unused bindings that cannot be
834 normalized (e.g., calls to a \type{PatError} (TODO: Check this name)). Also,
835 this transformation makes the resulting \small{VHDL} a lot shorter.
845 M \lam{a} does not occur free in \lam{M}
846 ---------------------------- \forall j, 0 <= j <= n, j ≠ i (\lam{a} does not occur free in \lam{Ej})
860 \subsubsection{Cast propagation / simplification}
861 This transform pushes casts down into the expression as far as possible.
862 Since its exact role and need is not clear yet, this transformation is
865 \subsubsection{Top level binding inlining}
866 This transform takes simple top level bindings generated by the
867 \small{GHC} compiler. \small{GHC} sometimes generates very simple
868 \quote{wrapper} bindings, which are bound to just a variable
869 reference, or a partial application to constants or other variable
872 Note that this transformation is completely optional. It is not
873 required to get any function into normal form, but it does help making
874 the resulting VHDL output easier to read (since it removes a bunch of
875 components that are really boring).
877 This transform takes any top level binding generated by the compiler,
878 whose normalized form contains only a single let binding.
881 x = λa0 ... λan.let y = E in y
884 -------------------------------------- \lam{x} is generated by the compiler
885 λa0 ... λan.let y = E in y
889 (+) :: Word -> Word -> Word
890 (+) = GHC.Num.(+) @Word $dNum
895 GHC.Num.(+) @ Alu.Word $dNum a b
898 \transexample{Top level binding inlining}{from}{to}
900 Without this transformation, the (+) function would generate an
901 architecture which would just add its inputs. This generates a lot of
902 overhead in the VHDL, which is particularly annoying when browsing the
903 generated RTL schematic (especially since + is not allowed in VHDL
904 architecture names\footnote{Technically, it is allowed when using
905 extended identifiers, but it seems that none of the tooling likes
906 extended identifiers in filenames, so it effectively doesn't work}, so
907 the entity would be called \quote{w7aA7f} or something similarly
908 unreadable and autogenerated).
910 \subsection{Program structure}
911 These transformations are aimed at normalizing the overall structure
912 into the intended form. This means ensuring there is a lambda abstraction
913 at the top for every argument (input port), putting all of the other
914 value definitions in let bindings and making the final return value a
915 simple variable reference.
917 \subsubsection{η-abstraction}
918 This transformation makes sure that all arguments of a function-typed
919 expression are named, by introducing lambda expressions. When combined with
920 β-reduction and non-representable binding inlining, all function-typed
921 expressions should be lambda abstractions or global identifiers.
925 -------------- \lam{E} is not the first argument of an application.
926 λx.E x \lam{E} is not a lambda abstraction.
927 \lam{x} is a variable that does not occur free in \lam{E}.
937 foo = λa.λx.(case a of
942 \transexample{η-abstraction}{from}{to}
944 \subsubsection{Application propagation}
945 This transformation is meant to propagate application expressions downwards
946 into expressions as far as possible. This allows partial applications inside
947 expressions to become fully applied and exposes new transformation
948 opportunities for other transformations (like β-reduction and
952 (letrec binds in E) M
973 \transexample{Application propagation for a let expression}{from}{to}
1001 \transexample{Application propagation for a case expression}{from}{to}
1003 \subsubsection{Let recursification}
1004 This transformation makes all non-recursive lets recursive. In the
1005 end, we want a single recursive let in our normalized program, so all
1006 non-recursive lets can be converted. This also makes other
1007 transformations simpler: They can simply assume all lets are
1015 ------------------------------------------
1022 \subsubsection{Let flattening}
1023 This transformation puts nested lets in the same scope, by lifting the
1024 binding(s) of the inner let into a new let around the outer let. Eventually,
1025 this will cause all let bindings to appear in the same scope (they will all be
1026 in scope for the function return value).
1031 x = (letrec bindings in M)
1035 ------------------------------------------
1064 \transexample{Let flattening}{from}{to}
1066 \subsubsection{Return value simplification}
1067 This transformation ensures that the return value of a function is always a
1068 simple local variable reference.
1070 Currently implemented using lambda simplification, let simplification, and
1071 top simplification. Should change into something like the following, which
1072 works only on the result of a function instead of any subexpression. This is
1073 achieved by the contexts, like \lam{x = E}, though this is strictly not
1074 correct (you could read this as "if there is any function \lam{x} that binds
1075 \lam{E}, any \lam{E} can be transformed, while we only mean the \lam{E} that
1076 is bound by \lam{x}. This might need some extra notes or something).
1078 Note that the return value is not simplified if its not representable.
1079 Otherwise, this would cause a direct loop with the inlining of
1080 unrepresentable bindings, of course. If the return value is not
1081 representable because it has a function type, η-abstraction should
1082 make sure that this transformation will eventually apply. If the value
1083 is not representable for other reasons, the function result itself is
1084 not representable, meaning this function is not representable anyway!
1087 x = E \lam{E} is representable
1088 ~ \lam{E} is not a lambda abstraction
1089 E \lam{E} is not a let expression
1090 --------------------------- \lam{E} is not a local variable reference
1096 ~ \lam{E} is representable
1097 E \lam{E} is not a let expression
1098 --------------------------- \lam{E} is not a local variable reference
1103 x = λv0 ... λvn.let ... in E
1104 ~ \lam{E} is representable
1105 E \lam{E} is not a local variable reference
1106 ---------------------------
1115 x = letrec x = add 1 2 in x
1118 \transexample{Return value simplification}{from}{to}
1120 \subsection{Argument simplification}
1121 The transforms in this section deal with simplifying application
1122 arguments into normal form. The goal here is to:
1125 \item Make all arguments of user-defined functions (\eg, of which
1126 we have a function body) simple variable references of a runtime
1127 representable type. This is needed, since these applications will be turned
1128 into component instantiations.
1129 \item Make all arguments of builtin functions one of:
1131 \item A type argument.
1132 \item A dictionary argument.
1133 \item A type level expression.
1134 \item A variable reference of a runtime representable type.
1135 \item A variable reference or partial application of a function type.
1139 When looking at the arguments of a user-defined function, we can
1140 divide them into two categories:
1142 \item Arguments of a runtime representable type (\eg bits or vectors).
1144 These arguments can be preserved in the program, since they can
1145 be translated to input ports later on. However, since we can
1146 only connect signals to input ports, these arguments must be
1147 reduced to simple variables (for which signals will be
1148 produced). This is taken care of by the argument extraction
1150 \item Non-runtime representable typed arguments.
1152 These arguments cannot be preserved in the program, since we
1153 cannot represent them as input or output ports in the resulting
1154 \small{VHDL}. To remove them, we create a specialized version of the
1155 called function with these arguments filled in. This is done by
1156 the argument propagation transform.
1158 Typically, these arguments are type and dictionary arguments that are
1159 used to make functions polymorphic. By propagating these arguments, we
1160 are essentially doing the same which GHC does when it specializes
1161 functions: Creating multiple variants of the same function, one for
1162 each type for which it is used. Other common non-representable
1163 arguments are functions, e.g. when calling a higher order function
1164 with another function or a lambda abstraction as an argument.
1166 The reason for doing this is similar to the reasoning provided for
1167 the inlining of non-representable let bindings above. In fact, this
1168 argument propagation could be viewed as a form of cross-function
1172 TODO: Check the following itemization.
1174 When looking at the arguments of a builtin function, we can divide them
1178 \item Arguments of a runtime representable type.
1180 As we have seen with user-defined functions, these arguments can
1181 always be reduced to a simple variable reference, by the
1182 argument extraction transform. Performing this transform for
1183 builtin functions as well, means that the translation of builtin
1184 functions can be limited to signal references, instead of
1185 needing to support all possible expressions.
1187 \item Arguments of a function type.
1189 These arguments are functions passed to higher order builtins,
1190 like \lam{map} and \lam{foldl}. Since implementing these
1191 functions for arbitrary function-typed expressions (\eg, lambda
1192 expressions) is rather comlex, we reduce these arguments to
1193 (partial applications of) global functions.
1195 We can still support arbitrary expressions from the user code,
1196 by creating a new global function containing that expression.
1197 This way, we can simply replace the argument with a reference to
1198 that new function. However, since the expression can contain any
1199 number of free variables we also have to include partial
1200 applications in our normal form.
1202 This category of arguments is handled by the function extraction
1204 \item Other unrepresentable arguments.
1206 These arguments can take a few different forms:
1207 \startdesc{Type arguments}
1208 In the core language, type arguments can only take a single
1209 form: A type wrapped in the Type constructor. Also, there is
1210 nothing that can be done with type expressions, except for
1211 applying functions to them, so we can simply leave type
1212 arguments as they are.
1214 \startdesc{Dictionary arguments}
1215 In the core language, dictionary arguments are used to find
1216 operations operating on one of the type arguments (mostly for
1217 finding class methods). Since we will not actually evaluatie
1218 the function body for builtin functions and can generate
1219 code for builtin functions by just looking at the type
1220 arguments, these arguments can be ignored and left as they
1223 \startdesc{Type level arguments}
1224 Sometimes, we want to pass a value to a builtin function, but
1225 we need to know the value at compile time. Additionally, the
1226 value has an impact on the type of the function. This is
1227 encoded using type-level values, where the actual value of the
1228 argument is not important, but the type encodes some integer,
1229 for example. Since the value is not important, the actual form
1230 of the expression does not matter either and we can leave
1231 these arguments as they are.
1233 \startdesc{Other arguments}
1234 Technically, there is still a wide array of arguments that can
1235 be passed, but does not fall into any of the above categories.
1236 However, none of the supported builtin functions requires such
1237 an argument. This leaves use with passing unsupported types to
1238 a function, such as calling \lam{head} on a list of functions.
1240 In these cases, it would be impossible to generate hardware
1241 for such a function call anyway, so we can ignore these
1244 The only way to generate hardware for builtin functions with
1245 arguments like these, is to expand the function call into an
1246 equivalent core expression (\eg, expand map into a series of
1247 function applications). But for now, we choose to simply not
1248 support expressions like these.
1251 From the above, we can conclude that we can simply ignore these
1252 other unrepresentable arguments and focus on the first two
1256 \subsubsection{Argument simplification}
1257 This transform deals with arguments to functions that
1258 are of a runtime representable type. It ensures that they will all become
1259 references to global variables, or local signals in the resulting \small{VHDL}.
1261 TODO: It seems we can map an expression to a port, not only a signal.
1262 Perhaps this makes this transformation not needed?
1263 TODO: Say something about dataconstructors (without arguments, like True
1264 or False), which are variable references of a runtime representable
1265 type, but do not result in a signal.
1267 To reduce a complex expression to a simple variable reference, we create
1268 a new let expression around the application, which binds the complex
1269 expression to a new variable. The original function is then applied to
1274 -------------------- \lam{N} is of a representable type
1275 letrec x = N in M x \lam{N} is not a local variable reference
1283 letrec x = add a 1 in add x 1
1286 \transexample{Argument extraction}{from}{to}
1288 \subsubsection{Function extraction}
1289 This transform deals with function-typed arguments to builtin functions.
1290 Since these arguments cannot be propagated, we choose to extract them
1291 into a new global function instead.
1293 Any free variables occuring in the extracted arguments will become
1294 parameters to the new global function. The original argument is replaced
1295 with a reference to the new function, applied to any free variables from
1296 the original argument.
1298 This transformation is useful when applying higher order builtin functions
1299 like \hs{map} to a lambda abstraction, for example. In this case, the code
1300 that generates \small{VHDL} for \hs{map} only needs to handle top level functions and
1301 partial applications, not any other expression (such as lambda abstractions or
1302 even more complicated expressions).
1305 M N \lam{M} is a (partial aplication of a) builtin function.
1306 --------------------- \lam{f0 ... fn} = free local variables of \lam{N}
1307 M (x f0 ... fn) \lam{N :: a -> b}
1308 ~ \lam{N} is not a (partial application of) a top level function
1313 map (λa . add a b) xs
1327 \transexample{Function extraction}{from}{to}
1329 Note that \lam{x0} and {x1} will still need normalization after this.
1331 \subsubsection{Argument propagation}
1332 This transform deals with arguments to user-defined functions that are
1333 not representable at runtime. This means these arguments cannot be
1334 preserved in the final form and most be {\em propagated}.
1336 Propagation means to create a specialized version of the called
1337 function, with the propagated argument already filled in. As a simple
1338 example, in the following program:
1345 We could {\em propagate} the constant argument 1, with the following
1353 Special care must be taken when the to-be-propagated expression has any
1354 free variables. If this is the case, the original argument should not be
1355 removed alltogether, but replaced by all the free variables of the
1356 expression. In this way, the original expression can still be evaluated
1357 inside the new function. Also, this brings us closer to our goal: All
1358 these free variables will be simple variable references.
1360 To prevent us from propagating the same argument over and over, a simple
1361 local variable reference is not propagated (since is has exactly one
1362 free variable, itself, we would only replace that argument with itself).
1364 This shows that any free local variables that are not runtime representable
1365 cannot be brought into normal form by this transform. We rely on an
1366 inlining transformation to replace such a variable with an expression we
1367 can propagate again.
1372 x Y0 ... Yi ... Yn \lam{Yi} is not of a runtime representable type
1373 --------------------------------------------- \lam{Yi} is not a local variable reference
1374 x' y0 ... yi-1 f0 ... fm Yi+1 ... Yn \lam{f0 ... fm} = free local vars of \lam{Yi}
1376 x' = λy0 ... yi-1 f0 ... fm yi+1 ... yn .
1377 E y0 ... yi-1 Yi yi+1 ... yn
1383 \subsection{Case simplification}
1384 \subsubsection{Scrutinee simplification}
1385 This transform ensures that the scrutinee of a case expression is always
1386 a simple variable reference.
1391 ----------------- \lam{E} is not a local variable reference
1410 \transexample{Let flattening}{from}{to}
1413 \subsubsection{Case simplification}
1414 This transformation ensures that all case expressions become normal form. This
1415 means they will become one of:
1417 \item An extractor case with a single alternative that picks a single field
1418 from a datatype, \eg \lam{case x of (a, b) -> a}.
1419 \item A selector case with multiple alternatives and only wild binders, that
1420 makes a choice between expressions based on the constructor of another
1421 expression, \eg \lam{case x of Low -> a; High -> b}.
1426 C0 v0,0 ... v0,m -> E0
1428 Cn vn,0 ... vn,m -> En
1429 --------------------------------------------------- \forall i \forall j, 0 <= i <= n, 0 <= i < m (\lam{wi,j} is a wild (unused) binder)
1431 v0,0 = case x of C0 v0,0 .. v0,m -> v0,0
1433 v0,m = case x of C0 v0,0 .. v0,m -> v0,m
1436 vn,m = case x of Cn vn,0 .. vn,m -> vn,m
1440 C0 w0,0 ... w0,m -> x0
1442 Cn wn,0 ... wn,m -> xn
1445 TODO: This transformation specified like this is complicated and misses
1446 conditions to prevent looping with itself. Perhaps we should split it here for
1465 \transexample{Selector case simplification}{from}{to}
1473 b = case a of (,) b c -> b
1474 c = case a of (,) b c -> c
1481 \transexample{Extractor case simplification}{from}{to}
1483 \subsubsection{Case removal}
1484 This transform removes any case statements with a single alternative and
1487 These "useless" case statements are usually leftovers from case simplification
1488 on extractor case (see the previous example).
1493 ---------------------- \lam{\forall i, 0 <= i <= m} (\lam{vi} does not occur free in E)
1506 \transexample{Case removal}{from}{to}
1508 \subsection{Removing polymorphism}
1509 Reference type-specialization (== argument propagation)
1511 Reference polymporphic binding inlining (== non-representable binding
1514 \subsection{Defunctionalization}
1515 These transformations remove most higher order expressions from our
1516 program, making it completely first-order (the only exception here is for
1517 arguments to builtin functions, since we can't specialize builtin
1518 function. TODO: Talk more about this somewhere).
1520 Reference higher-order-specialization (== argument propagation)
1522 \subsubsection{Non-representable binding inlining}
1523 This transform inlines let bindings that have a non-representable type. Since
1524 we can never generate a signal assignment for these bindings (we cannot
1525 declare a signal assignment with a non-representable type, for obvious
1526 reasons), we have no choice but to inline the binding to remove it.
1528 If the binding is non-representable because it is a lambda abstraction, it is
1529 likely that it will inlined into an application and β-reduction will remove
1530 the lambda abstraction and turn it into a representable expression at the
1531 inline site. The same holds for partial applications, which can be turned into
1532 full applications by inlining.
1534 Other cases of non-representable bindings we see in practice are primitive
1535 Haskell types. In most cases, these will not result in a valid normalized
1536 output, but then the input would have been invalid to start with. There is one
1537 exception to this: When a builtin function is applied to a non-representable
1538 expression, things might work out in some cases. For example, when you write a
1539 literal \hs{SizedInt} in Haskell, like \hs{1 :: SizedInt D8}, this results in
1540 the following core: \lam{fromInteger (smallInteger 10)}, where for example
1541 \lam{10 :: GHC.Prim.Int\#} and \lam{smallInteger 10 :: Integer} have
1542 non-representable types. TODO: This/these paragraph(s) should probably become a
1543 separate discussion somewhere else.
1555 -------------------------- \lam{Ei} has a non-representable type.
1579 x = fromInteger (smallInteger 10)
1581 (λb -> add b 1) (add 1 x)
1584 \transexample{None representable binding inlining}{from}{to}
1587 \section{Provable properties}
1588 When looking at the system of transformations outlined above, there are a
1589 number of questions that we can ask ourselves. The main question is of course:
1590 \quote{Does our system work as intended?}. We can split this question into a
1591 number of subquestions:
1594 \item[q:termination] Does our system \emph{terminate}? Since our system will
1595 keep running as long as transformations apply, there is an obvious risk that
1596 it will keep running indefinitely. One transformation produces a result that
1597 is transformed back to the original by another transformation, for example.
1598 \item[q:soundness] Is our system \emph{sound}? Since our transformations
1599 continuously modify the expression, there is an obvious risk that the final
1600 normal form will not be equivalent to the original program: Its meaning could
1602 \item[q:completeness] Is our system \emph{complete}? Since we have a complex
1603 system of transformations, there is an obvious risk that some expressions will
1604 not end up in our intended normal form, because we forgot some transformation.
1605 In other words: Does our transformation system result in our intended normal
1606 form for all possible inputs?
1607 \item[q:determinism] Is our system \emph{deterministic}? Since we have defined
1608 no particular order in which the transformation should be applied, there is an
1609 obvious risk that different transformation orderings will result in
1610 \emph{different} normal forms. They might still both be intended normal forms
1611 (if our system is \emph{complete}) and describe correct hardware (if our
1612 system is \emph{sound}), so this property is less important than the previous
1613 three: The translator would still function properly without it.
1616 \subsection{Graph representation}
1617 Before looking into how to prove these properties, we'll look at our
1618 transformation system from a graph perspective. The nodes of the graph are
1619 all possible Core expressions. The (directed) edges of the graph are
1620 transformations. When a transformation α applies to an expression \lam{A} to
1621 produce an expression \lam{B}, we add an edge from the node for \lam{A} to the
1622 node for \lam{B}, labeled α.
1624 \startuseMPgraphic{TransformGraph}
1628 newCircle.a(btex \lam{(λx.λy. (+) x y) 1} etex);
1629 newCircle.b(btex \lam{λy. (+) 1 y} etex);
1630 newCircle.c(btex \lam{(λx.(+) x) 1} etex);
1631 newCircle.d(btex \lam{(+) 1} etex);
1634 c.c = b.c + (4cm, 0cm);
1635 a.c = midpoint(b.c, c.c) + (0cm, 4cm);
1636 d.c = midpoint(b.c, c.c) - (0cm, 3cm);
1638 % β-conversion between a and b
1639 ncarc.a(a)(b) "name(bred)";
1640 ObjLabel.a(btex $\xrightarrow[normal]{}{β}$ etex) "labpathname(bred)", "labdir(rt)";
1641 ncarc.b(b)(a) "name(bexp)", "linestyle(dashed withdots)";
1642 ObjLabel.b(btex $\xleftarrow[normal]{}{β}$ etex) "labpathname(bexp)", "labdir(lft)";
1644 % η-conversion between a and c
1645 ncarc.a(a)(c) "name(ered)";
1646 ObjLabel.a(btex $\xrightarrow[normal]{}{η}$ etex) "labpathname(ered)", "labdir(rt)";
1647 ncarc.c(c)(a) "name(eexp)", "linestyle(dashed withdots)";
1648 ObjLabel.c(btex $\xleftarrow[normal]{}{η}$ etex) "labpathname(eexp)", "labdir(lft)";
1650 % η-conversion between b and d
1651 ncarc.b(b)(d) "name(ered)";
1652 ObjLabel.b(btex $\xrightarrow[normal]{}{η}$ etex) "labpathname(ered)", "labdir(rt)";
1653 ncarc.d(d)(b) "name(eexp)", "linestyle(dashed withdots)";
1654 ObjLabel.d(btex $\xleftarrow[normal]{}{η}$ etex) "labpathname(eexp)", "labdir(lft)";
1656 % β-conversion between c and d
1657 ncarc.c(c)(d) "name(bred)";
1658 ObjLabel.c(btex $\xrightarrow[normal]{}{β}$ etex) "labpathname(bred)", "labdir(rt)";
1659 ncarc.d(d)(c) "name(bexp)", "linestyle(dashed withdots)";
1660 ObjLabel.d(btex $\xleftarrow[normal]{}{β}$ etex) "labpathname(bexp)", "labdir(lft)";
1662 % Draw objects and lines
1663 drawObj(a, b, c, d);
1666 \placeexample[right][ex:TransformGraph]{Partial graph of a labmda calculus
1667 system with β and η reduction (solid lines) and expansion (dotted lines).}
1668 \boxedgraphic{TransformGraph}
1670 Of course our graph is unbounded, since we can construct an infinite amount of
1671 Core expressions. Also, there might potentially be multiple edges between two
1672 given nodes (with different labels), though seems unlikely to actually happen
1675 See \in{example}[ex:TransformGraph] for the graph representation of a very
1676 simple lambda calculus that contains just the expressions \lam{(λx.λy. (+) x
1677 y) 1}, \lam{λy. (+) 1 y}, \lam{(λx.(+) x) 1} and \lam{(+) 1}. The
1678 transformation system consists of β-reduction and η-reduction (solid edges) or
1679 β-reduction and η-reduction (dotted edges).
1681 TODO: Define β-reduction and η-reduction?
1683 Note that the normal form of such a system consists of the set of nodes
1684 (expressions) without outgoing edges, since those are the expression to which
1685 no transformation applies anymore. We call this set of nodes the \emph{normal
1688 From such a graph, we can derive some properties easily:
1690 \item A system will \emph{terminate} if there is no path of infinite length
1691 in the graph (this includes cycles).
1692 \item Soundness is not easily represented in the graph.
1693 \item A system is \emph{complete} if all of the nodes in the normal set have
1694 the intended normal form. The inverse (that all of the nodes outside of
1695 the normal set are \emph{not} in the intended normal form) is not
1697 \item A system is deterministic if all paths from a node, which end in a node
1698 in the normal set, end at the same node.
1701 When looking at the \in{example}[ex:TransformGraph], we see that the system
1702 terminates for both the reduction and expansion systems (but note that, for
1703 expansion, this is only true because we've limited the possible expressions!
1704 In comlete lambda calculus, there would be a path from \lam{(λx.λy. (+) x y)
1705 1} to \lam{(λx.λy.(λz.(+) z) x y) 1} to \lam{(λx.λy.(λz.(λq.(+) q) z) x y) 1}
1708 If we would consider the system with both expansion and reduction, there would
1709 no longer be termination, since there would be cycles all over the place.
1711 The reduction and expansion systems have a normal set of containing just
1712 \lam{(+) 1} or \lam{(λx.λy. (+) x y) 1} respectively. Since all paths in
1713 either system end up in these normal forms, both systems are \emph{complete}.
1714 Also, since there is only one normal form, it must obviously be
1715 \emph{deterministic} as well.
1717 \subsection{Termination}
1722 \subsection{Soundness}
1723 Needs formal definition of semantics.
1724 Prove for each transformation seperately, implies soundness of the system.
1726 \subsection{Completeness}
1727 Show that any transformation applies to every Core expression that is not
1728 in normal form. To prove: no transformation applies => in intended form.
1729 Show the reverse: Not in intended form => transformation applies.
1731 \subsection{Determinism}