1 \chapter{Normalization}
3 % A helper to print a single example in the half the page width. The example
4 % text should be in a buffer whose name is given in an argument.
6 % The align=right option really does left-alignment, but without the program
7 % will end up on a single line. The strut=no option prevents a bunch of empty
8 % space at the start of the frame.
10 \framed[offset=1mm,align=right,strut=no]{
11 \setuptyping[option=LAM,style=sans,before=,after=]
13 \setuptyping[option=none,style=\tttf]
18 % A transformation example
19 \definefloat[example][examples]
20 \setupcaption[example][location=top] % Put captions on top
22 \define[3]\transexample{
23 \placeexample[here]{#1}
24 \startcombination[2*1]
25 {\example{#2}}{Original program}
26 {\example{#3}}{Transformed program}
30 %\define[3]\transexampleh{
31 %% \placeexample[here]{#1}
32 %% \startcombination[1*2]
33 %% {\example{#2}}{Original program}
34 %% {\example{#3}}{Transformed program}
38 The first step in the core to VHDL translation process, is normalization. We
39 aim to bring the core description into a simpler form, which we can
40 subsequently translate into VHDL easily. This normal form is needed because
41 the full core language is more expressive than VHDL in some areas and because
42 core can describe expressions that do not have a direct hardware
45 TODO: Describe core properties not supported in VHDL, and describe how the
46 VHDL we want to generate should look like.
49 The transformations described here have a well-defined goal: To bring the
50 program in a well-defined form that is directly translatable to hardware,
51 while fully preserving the semantics of the program.
53 This {\em normal form} is again a Core program, but with a very specific
54 structure. A function in normal form has nested lambda's at the top, which
55 produce a let expression. This let expression binds every function application
56 in the function and produces a simple identifier. Every bound value in
57 the let expression is either a simple function application or a case
58 expression to extract a single element from a tuple returned by a
61 An example of a program in canonical form would be:
64 -- All arguments are an inital lambda
66 -- There are nested let expressions at top level
68 -- Unpack the state by coercion
69 s = sp :: (Word, Word)
70 -- Extract both registers from the state
71 r1 = case s of (fst, snd) -> fst
72 r2 = case s of (fst, snd) -> snd
73 -- Calling some other user-defined function.
75 -- Conditional connections
87 -- Packing the state by coercion
88 sp' = s' :: State (Word, Word)
89 -- Pack our return value
97 \italic{normal} = \italic{lambda}
98 \italic{lambda} = λvar.\italic{lambda} (representable(typeof(var)))
100 \italic{toplet} = let \italic{binding} in \italic{toplet}
101 | letrec [\italic{binding}] in \italic{toplet}
102 | var (representable(typeof(var)), fvar(var))
103 \italic{binding} = var = \italic{rhs} (representable(typeof(rhs)))
104 -- State packing and unpacking by coercion
105 | var0 = var1 :: State ty (fvar(var1))
106 | var0 = var1 :: ty (var0 :: State ty) (fvar(var1))
107 \italic{rhs} = userapp
110 | case var of C a0 ... an -> ai (fvar(var))
112 | case var of (fvar(var))
113 DEFAULT -> var0 (fvar(var0))
114 C w0 ... wn -> resvar (\forall{}i, wi \neq resvar, fvar(resvar))
115 \italic{userapp} = \italic{userfunc}
116 | \italic{userapp} {userarg}
117 \italic{userfunc} = var (tvar(var))
118 \italic{userarg} = var (fvar(var))
119 \italic{builtinapp} = \italic{builtinfunc}
120 | \italic{builtinapp} \italic{builtinarg}
121 \italic{builtinfunc} = var (bvar(var))
122 \italic{builtinarg} = \italic{coreexpr}
125 -- TODO: Define tvar, fvar, typeof, representable
126 -- TODO: Limit builtinarg further
128 -- TODO: There can still be other casts around (which the code can handle,
129 e.g., ignore), which still need to be documented here.
131 -- TODO: Note about the selector case. It just supports Bit and Bool
132 currently, perhaps it should be generalized in the normal form?
134 When looking at such a program from a hardware perspective, the top level
135 lambda's define the input ports. The value produced by the let expression is
136 the output port. Most function applications bound by the let expression
137 define a component instantiation, where the input and output ports are mapped
138 to local signals or arguments. Some of the others use a builtin
139 construction (\eg the \lam{case} statement) or call a builtin function
140 (\eg \lam{add} or \lam{sub}). For these, a hardcoded VHDL translation is
143 \subsection{Normal definition}
144 Formally, the normal form is a core program obeying the following
145 constraints. TODO: Update this section, this is probably not completely
146 accurate or relevant anymore.
148 \startitemize[R,inmargin]
149 %\item All top level binds must have the form $\expr{\bind{fun}{lamexpr}}$.
150 %$fun$ is an identifier that will be bound as a global identifier.
151 %\item A $lamexpr$ has the form $\expr{\lam{arg}{lamexpr}}$ or
152 %$\expr{letexpr}$. $arg$ is an identifier which will be bound as an $argument$.
153 %\item[letexpr] A $letexpr$ has the form $\expr{\letexpr{letbinds}{retexpr}}$.
154 %\item $letbinds$ is a list with elements of the form
155 %$\expr{\bind{res}{appexpr}}$ or $\expr{\bind{res}{builtinexpr}}$, where $res$ is
156 %an identifier that will be bound as local identifier. The type of the bound
157 %value must be a $hardware\;type$.
158 %\item[builtinexpr] A $builtinexpr$ is an expression that can be mapped to an
159 %equivalent VHDL expression. Since there are many supported forms for this,
160 %these are defined in a separate table.
161 %\item An $appexpr$ has the form $\expr{fun}$ or $\expr{\app{appexpr}{x}}$,
162 %where $fun$ is a global identifier and $x$ is a local identifier.
163 %\item[retexpr] A $retexpr$ has the form $\expr{x}$ or $\expr{tupexpr}$, where $x$ is a local identifier that is bound as an $argument$ or $result$. A $retexpr$ must
164 %be of a $hardware\;type$.
165 %\item A $tupexpr$ has the form $\expr{con}$ or $\expr{\app{tupexpr}{x}}$,
166 %where $con$ is a tuple constructor ({\em e.g.} $(,)$ or $(,,,)$) and $x$ is
168 %\item A $hardware\;type$ is a type that can be directly translated to
169 %hardware. This includes the types $Bit$, $SizedWord$, tuples containing
170 %elements of $hardware\;type$s, and will include others. This explicitely
171 %excludes function types.
174 TODO: Say something about uniqueness of identifiers
176 \subsection{Builtin expressions}
177 A $builtinexpr$, as defined at \in[builtinexpr] can have any of the following forms.
179 \startitemize[m,inmargin]
181 %$tuple\_extract=\expr{\case{t}{\alt{\app{con}{x_0\;x_1\;..\;x_n}}{x_i}}}$,
182 %where $t$ can be any local identifier, $con$ is a tuple constructor ({\em
183 %e.g.} $(,)$ or $(,,,)$), $x_0$ to $x_n$ can be any identifier, and $x_i$ can
184 %be any of $x_0$ to $x_n$. A case expression must have a $hardware\;type$.
185 %\item TODO: Many more!
188 \section{Transform passes}
190 In this section we describe the actual transforms. Here we're using
191 the core language in a notation that resembles lambda calculus.
193 Each of these transforms is meant to be applied to every (sub)expression
194 in a program, for as long as it applies. Only when none of the
195 expressions can be applied anymore, the program is in normal form. We
196 hope to be able to prove that this form will obey all of the constraints
197 defined above, but this has yet to happen (though it seems likely that
200 Each of the transforms will be described informally first, explaining
201 the need for and goal of the transform. Then, a formal definition is
202 given, using a familiar syntax from the world of logic. Each transform
203 is specified as a number of conditions (above the horizontal line) and a
204 number of conclusions (below the horizontal line). The details of using
205 this notation are still a bit fuzzy, so comments are welcom.
207 TODO: Formally describe the "apply to every (sub)expression" in terms of
208 rules with full transformations in the conditions.
210 \subsection{η-abstraction}
211 This transformation makes sure that all arguments of a function-typed
212 expression are named, by introducing lambda expressions. When combined with
213 β-reduction and function inlining below, all function-typed expressions should
214 be lambda abstractions or global identifiers.
218 -------------- \lam{E} is not the first argument of an application.
219 λx.E x \lam{E} is not a lambda abstraction.
220 \lam{x} is a variable that does not occur free in \lam{E}.
224 foo = λa -> case a of
230 foo = λa.λx -> (case a of
235 \transexample{η-abstraction}{from}{to}
237 \subsection{Extended β-reduction}
238 This transformation is meant to propagate application expressions downwards
239 into expressions as far as possible. In lambda calculus, this reduction
240 is known as β-reduction, but it is of course only defined for
241 applications of lambda abstractions. We extend this reduction to also
242 work for the rest of core (case and let expressions).
264 For lambda expressions:
277 b = (let y = 3 in add y) 2
287 b = let y = 3 in add y 2
292 \transexample{Extended β-reduction}{from}{to}
294 \subsection{Let derecursification}
296 \subsection{Let flattening}
297 This transform turns two nested lets (\lam{let x = (let ... in ...) in
298 ...}) into a single let.
300 \subsection{Empty let removal}
302 \subsection{Simple let binding removal}
303 This transforms inlines simple let bindings (\eg a = b).
305 \subsection{Unused let binding removal}
307 \subsection{Non-representable binding inlining}
308 This transform inlines let bindings of a funtion type. TODO: This should
309 be generelized to anything that is non representable at runtime, or
312 \subsection{Scrutinee simplification}
313 This transform ensures that the scrutinee of a case expression is always
314 a simple variable reference.
316 \subsection{Case simplification}
318 \subsection{Case removal}
319 This transform removes any case statements with a single alternative and
322 \subsection{Argument simplification}
323 The transforms in this section deal with simplifying application
324 arguments into normal form. The goal here is to:
327 \item Make all arguments of user-defined functions (\eg, of which
328 we have a function body) simple variable references of a runtime
330 \item Make all arguments of builtin functions either:
332 \item A type argument.
333 \item A dictionary argument.
334 \item A type level expression.
335 \item A variable reference of a runtime representable type.
336 \item A variable reference or partial application of a function type.
340 When looking at the arguments of a user-defined function, we can
341 divide them into two categories:
343 \item Arguments with a runtime representable type (\eg bits or vectors).
345 These arguments can be preserved in the program, since they can
346 be translated to input ports later on. However, since we can
347 only connect signals to input ports, these arguments must be
348 reduced to simple variables (for which signals will be
349 produced). This is taken care of by the argument extraction
351 \item Non-runtime representable typed arguments.
353 These arguments cannot be preserved in the program, since we
354 cannot represent them as input or output ports in the resulting
355 VHDL. To remove them, we create a specialized version of the
356 called function with these arguments filled in. This is done by
357 the argument propagation transform.
360 When looking at the arguments of a builtin function, we can divide them
364 \item Arguments with a runtime representable type.
366 As we have seen with user-defined functions, these arguments can
367 always be reduced to a simple variable reference, by the
368 argument extraction transform. Performing this transform for
369 builtin functions as well, means that the translation of builtin
370 functions can be limited to signal references, instead of
371 needing to support all possible expressions.
373 \item Arguments with a function type.
375 These arguments are functions passed to higher order builtins,
376 like \lam{map} and \lam{foldl}. Since implementing these
377 functions for arbitrary function-typed expressions (\eg, lambda
378 expressions) is rather comlex, we reduce these arguments to
379 (partial applications of) global functions.
381 We can still support arbitrary expressions from the user code,
382 by creating a new global function containing that expression.
383 This way, we can simply replace the argument with a reference to
384 that new function. However, since the expression can contain any
385 number of free variables we also have to include partial
386 applications in our normal form.
388 This category of arguments is handled by the function extraction
390 \item Other unrepresentable arguments.
392 These arguments can take a few different forms:
393 \startdesc{Type arguments}
394 In the core language, type arguments can only take a single
395 form: A type wrapped in the Type constructor. Also, there is
396 nothing that can be done with type expressions, except for
397 applying functions to them, so we can simply leave type
398 arguments as they are.
400 \startdesc{Dictionary arguments}
401 In the core language, dictionary arguments are used to find
402 operations operating on one of the type arguments (mostly for
403 finding class methods). Since we will not actually evaluatie
404 the function body for builtin functions and can generate
405 code for builtin functions by just looking at the type
406 arguments, these arguments can be ignored and left as they
409 \startdesc{Type level arguments}
410 Sometimes, we want to pass a value to a builtin function, but
411 we need to know the value at compile time. Additionally, the
412 value has an impact on the type of the function. This is
413 encoded using type-level values, where the actual value of the
414 argument is not important, but the type encodes some integer,
415 for example. Since the value is not important, the actual form
416 of the expression does not matter either and we can leave
417 these arguments as they are.
419 \startdesc{Other arguments}
420 Technically, there is still a wide array of arguments that can
421 be passed, but does not fall into any of the above categories.
422 However, none of the supported builtin functions requires such
423 an argument. This leaves use with passing unsupported types to
424 a function, such as calling \lam{head} on a list of functions.
426 In these cases, it would be impossible to generate hardware
427 for such a function call anyway, so we can ignore these
430 The only way to generate hardware for builtin functions with
431 arguments like these, is to expand the function call into an
432 equivalent core expression (\eg, expand map into a series of
433 function applications). But for now, we choose to simply not
434 support expressions like these.
437 From the above, we can conclude that we can simply ignore these
438 other unrepresentable arguments and focus on the first two
442 \subsubsection{Argument extraction}
443 This transform deals with arguments to functions that
444 are of a runtime representable type.
446 TODO: It seems we can map an expression to a port, not only a signal.
447 Perhaps this makes this transformation not needed?
448 TODO: Say something about dataconstructors (without arguments, like True
449 or False), which are variable references of a runtime representable
450 type, but do not result in a signal.
452 To reduce a complex expression to a simple variable reference, we create
453 a new let expression around the application, which binds the complex
454 expression to a new variable. The original function is then applied to
457 %\transform{Argument extract}
459 %\lam{Y} is of a hardware representable type
461 %\lam{Y} is not a variable referene
465 %\trans{X Y}{let z = Y in X z}
468 \subsubsection{Function extraction}
469 This transform deals with function-typed arguments to builtin functions.
470 Since these arguments cannot be propagated, we choose to extract them
471 into a new global function instead.
473 Any free variables occuring in the extracted arguments will become
474 parameters to the new global function. The original argument is replaced
475 with a reference to the new function, applied to any free variables from
476 the original argument.
478 %\transform{Function extraction}
480 %\lam{X} is a (partial application of) a builtin function
482 %\lam{Y} is not an application
484 %\lam{Y} is not a variable reference
488 %\lam{f0 ... fm} = free local vars of \lam{Y}
490 %\lam{y} is a new global variable
492 %\lam{y = λf0 ... fn.Y}
494 %\trans{X Y}{X (y f0 ... fn)}
497 \subsubsection{Argument propagation}
498 This transform deals with arguments to user-defined functions that are
499 not representable at runtime. This means these arguments cannot be
500 preserved in the final form and most be {\em propagated}.
502 Propagation means to create a specialized version of the called
503 function, with the propagated argument already filled in. As a simple
504 example, in the following program:
511 we could {\em propagate} the constant argument 1, with the following
519 Special care must be taken when the to-be-propagated expression has any
520 free variables. If this is the case, the original argument should not be
521 removed alltogether, but replaced by all the free variables of the
522 expression. In this way, the original expression can still be evaluated
523 inside the new function. Also, this brings us closer to our goal: All
524 these free variables will be simple variable references.
526 To prevent us from propagating the same argument over and over, a simple
527 local variable reference is not propagated (since is has exactly one
528 free variable, itself, we would only replace that argument with itself).
530 This shows that any free local variables that are not runtime representable
531 cannot be brought into normal form by this transform. We rely on an
532 inlining transformation to replace such a variable with an expression we
535 TODO: Move these definitions somewhere sensible.
537 Definition: A global variable is any variable that is bound at the
538 top level of a program. A local variable is any other variable.
540 Definition: A hardware representable type is a type that we can generate
541 a signal for in hardware. For example, a bit, a vector of bits, a 32 bit
542 unsigned word, etc. Types that are not runtime representable notably
543 include (but are not limited to): Types, dictionaries, functions.
545 Definition: A builtin function is a function for which a builtin
546 hardware translation is available, because its actual definition is not
547 translatable. A user-defined function is any other function.
552 x Y0 ... Yi ... Yn \lam{Yi} is not of a runtime representable type
553 --------------------------------------------- \lam{Yi} is not a local variable reference
554 x' = λy0 ... yi-1 f0 ... fm yi+1 ... yn . \lam{f0 ... fm} = free local vars of \lam{Y_i}
555 E y0 ... yi-1 Yi yi+1 ... yn
557 x' y0 ... yi-1 f0 ... fm Yi+1 ... Yn
560 \subsection{Cast propagation / simplification}
561 This transform pushes casts down into the expression as far as possible.
563 \subsection{Return value simplification}
564 Currently implemented using lambda simplification, let simplification, and
565 top simplification. Should change.
567 \subsection{Example sequence}
569 This section lists an example expression, with a sequence of transforms
570 applied to it. The exact transforms given here probably don't exactly
571 match the transforms given above anymore, but perhaps this can clarify
572 the big picture a bit.
574 TODO: Update or remove this section.
592 After top-level η-abstraction:
611 After (extended) β-reduction:
629 After return value extraction:
648 Scrutinee simplification does not apply.
650 After case binder wildening:
655 a = case s of (a, _) -> a
656 b = case s of (_, b) -> b
657 r = case s of (_, _) ->
660 Low -> let op' = case b of
669 After case value simplification
674 a = case s of (a, _) -> a
675 b = case s of (_, b) -> b
676 r = case s of (_, _) -> r'
678 rl = let rll = λc.λd.c
691 After let flattening:
696 a = case s of (a, _) -> a
697 b = case s of (_, b) -> b
698 r = case s of (_, _) -> r'
712 After function inlining:
717 a = case s of (a, _) -> a
718 b = case s of (_, b) -> b
719 r = case s of (_, _) -> r'
731 After (extended) β-reduction again:
736 a = case s of (a, _) -> a
737 b = case s of (_, b) -> b
738 r = case s of (_, _) -> r'
750 After case value simplification again:
755 a = case s of (a, _) -> a
756 b = case s of (_, b) -> b
757 r = case s of (_, _) -> r'
775 a = case s of (a, _) -> a
776 b = case s of (_, b) -> b
790 After let bind removal:
795 a = case s of (a, _) -> a
796 b = case s of (_, b) -> b
809 Application simplification is not applicable.