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).
256 %\transform{Extended β-reduction}
259 %\trans{(λx.E) M}{E[M/x]}
263 %\trans{(let binds in E) M}{let binds in E M}
275 b = (let y = 3 in add y) 2
284 b = let y = 3 in add y 2
289 \transexample{Extended β-reduction}{from}{to}
291 \subsection{Argument simplification}
292 The transforms in this section deal with simplifying application
293 arguments into normal form. The goal here is to:
296 \item Make all arguments of user-defined functions (\eg, of which
297 we have a function body) simple variable references of a runtime
299 \item Make all arguments of builtin functions either:
301 \item A type argument.
302 \item A dictionary argument.
303 \item A type level expression.
304 \item A variable reference of a runtime representable type.
305 \item A variable reference or partial application of a function type.
309 When looking at the arguments of a user-defined function, we can
310 divide them into two categories:
312 \item Arguments with a runtime representable type (\eg bits or vectors).
314 These arguments can be preserved in the program, since they can
315 be translated to input ports later on. However, since we can
316 only connect signals to input ports, these arguments must be
317 reduced to simple variables (for which signals will be
318 produced). This is taken care of by the argument extraction
320 \item Non-runtime representable typed arguments.
322 These arguments cannot be preserved in the program, since we
323 cannot represent them as input or output ports in the resulting
324 VHDL. To remove them, we create a specialized version of the
325 called function with these arguments filled in. This is done by
326 the argument propagation transform.
329 When looking at the arguments of a builtin function, we can divide them
333 \item Arguments with a runtime representable type.
335 As we have seen with user-defined functions, these arguments can
336 always be reduced to a simple variable reference, by the
337 argument extraction transform. Performing this transform for
338 builtin functions as well, means that the translation of builtin
339 functions can be limited to signal references, instead of
340 needing to support all possible expressions.
342 \item Arguments with a function type.
344 These arguments are functions passed to higher order builtins,
345 like \lam{map} and \lam{foldl}. Since implementing these
346 functions for arbitrary function-typed expressions (\eg, lambda
347 expressions) is rather comlex, we reduce these arguments to
348 (partial applications of) global functions.
350 We can still support arbitrary expressions from the user code,
351 by creating a new global function containing that expression.
352 This way, we can simply replace the argument with a reference to
353 that new function. However, since the expression can contain any
354 number of free variables we also have to include partial
355 applications in our normal form.
357 This category of arguments is handled by the function extraction
359 \item Other unrepresentable arguments.
361 These arguments can take a few different forms:
362 \startdesc{Type arguments}
363 In the core language, type arguments can only take a single
364 form: A type wrapped in the Type constructor. Also, there is
365 nothing that can be done with type expressions, except for
366 applying functions to them, so we can simply leave type
367 arguments as they are.
369 \startdesc{Dictionary arguments}
370 In the core language, dictionary arguments are used to find
371 operations operating on one of the type arguments (mostly for
372 finding class methods). Since we will not actually evaluatie
373 the function body for builtin functions and can generate
374 code for builtin functions by just looking at the type
375 arguments, these arguments can be ignored and left as they
378 \startdesc{Type level arguments}
379 Sometimes, we want to pass a value to a builtin function, but
380 we need to know the value at compile time. Additionally, the
381 value has an impact on the type of the function. This is
382 encoded using type-level values, where the actual value of the
383 argument is not important, but the type encodes some integer,
384 for example. Since the value is not important, the actual form
385 of the expression does not matter either and we can leave
386 these arguments as they are.
388 \startdesc{Other arguments}
389 Technically, there is still a wide array of arguments that can
390 be passed, but does not fall into any of the above categories.
391 However, none of the supported builtin functions requires such
392 an argument. This leaves use with passing unsupported types to
393 a function, such as calling \lam{head} on a list of functions.
395 In these cases, it would be impossible to generate hardware
396 for such a function call anyway, so we can ignore these
399 The only way to generate hardware for builtin functions with
400 arguments like these, is to expand the function call into an
401 equivalent core expression (\eg, expand map into a series of
402 function applications). But for now, we choose to simply not
403 support expressions like these.
406 From the above, we can conclude that we can simply ignore these
407 other unrepresentable arguments and focus on the first two
411 \subsubsection{Argument extraction}
412 This transform deals with arguments to functions that
413 are of a runtime representable type.
415 TODO: It seems we can map an expression to a port, not only a signal.
416 Perhaps this makes this transformation not needed?
417 TODO: Say something about dataconstructors (without arguments, like True
418 or False), which are variable references of a runtime representable
419 type, but do not result in a signal.
421 To reduce a complex expression to a simple variable reference, we create
422 a new let expression around the application, which binds the complex
423 expression to a new variable. The original function is then applied to
426 %\transform{Argument extract}
428 %\lam{Y} is of a hardware representable type
430 %\lam{Y} is not a variable referene
434 %\trans{X Y}{let z = Y in X z}
437 \subsubsection{Function extraction}
438 This transform deals with function-typed arguments to builtin functions.
439 Since these arguments cannot be propagated, we choose to extract them
440 into a new global function instead.
442 Any free variables occuring in the extracted arguments will become
443 parameters to the new global function. The original argument is replaced
444 with a reference to the new function, applied to any free variables from
445 the original argument.
447 %\transform{Function extraction}
449 %\lam{X} is a (partial application of) a builtin function
451 %\lam{Y} is not an application
453 %\lam{Y} is not a variable reference
457 %\lam{f0 ... fm} = free local vars of \lam{Y}
459 %\lam{y} is a new global variable
461 %\lam{y = λf0 ... fn.Y}
463 %\trans{X Y}{X (y f0 ... fn)}
466 \subsubsection{Argument propagation}
467 This transform deals with arguments to user-defined functions that are
468 not representable at runtime. This means these arguments cannot be
469 preserved in the final form and most be {\em propagated}.
471 Propagation means to create a specialized version of the called
472 function, with the propagated argument already filled in. As a simple
473 example, in the following program:
480 we could {\em propagate} the constant argument 1, with the following
488 Special care must be taken when the to-be-propagated expression has any
489 free variables. If this is the case, the original argument should not be
490 removed alltogether, but replaced by all the free variables of the
491 expression. In this way, the original expression can still be evaluated
492 inside the new function. Also, this brings us closer to our goal: All
493 these free variables will be simple variable references.
495 To prevent us from propagating the same argument over and over, a simple
496 local variable reference is not propagated (since is has exactly one
497 free variable, itself, we would only replace that argument with itself).
499 This shows that any free local variables that are not runtime representable
500 cannot be brought into normal form by this transform. We rely on an
501 inlining transformation to replace such a variable with an expression we
504 TODO: Move these definitions somewhere sensible.
506 Definition: A global variable is any variable that is bound at the
507 top level of a program. A local variable is any other variable.
509 Definition: A hardware representable type is a type that we can generate
510 a signal for in hardware. For example, a bit, a vector of bits, a 32 bit
511 unsigned word, etc. Types that are not runtime representable notably
512 include (but are not limited to): Types, dictionaries, functions.
514 Definition: A builtin function is a function for which a builtin
515 hardware translation is available, because its actual definition is not
516 translatable. A user-defined function is any other function.
521 x Y0 ... Yi ... Yn \lam{Yi} is not of a runtime representable type
522 --------------------------------------------- \lam{Yi} is not a local variable reference
523 x' = λy0 ... yi-1 f0 ... fm yi+1 ... yn . \lam{f0 ... fm} = free local vars of \lam{Y_i}
524 E y0 ... yi-1 Yi yi+1 ... yn
526 x' y0 ... yi-1 f0 ... fm Yi+1 ... Yn
529 %\transform{Argument propagation}
531 %\lam{x} is a global variable, bound to a user-defined function
535 %\lam{Y_i} is not of a runtime representable type
537 %\lam{Y_i} is not a local variable reference
541 %\lam{f0 ... fm} = free local vars of \lam{Y_i}
543 %\lam{x'} is a new global variable
545 %\lam{x' = λy0 ... yi-1 f0 ... fm yi+1 ... yn . E y0 ... yi-1 Yi yi+1 ... yn}
547 %\trans{x Y0 ... Yi ... Yn}{x' y0 ... yi-1 f0 ... fm Yi+1 ... Yn}
550 %TODO: The above definition looks too complicated... Can we find
551 %something more concise?
553 \subsection{Cast propagation}
554 This transform pushes casts down into the expression as far as possible.
555 \subsection{Let recursification}
556 This transform makes all lets recursive.
557 \subsection{Let simplification}
558 This transform makes the result value of all let expressions a simple
560 \subsection{Let flattening}
561 This transform turns two nested lets (\lam{let x = (let ... in ...) in
562 ...}) into a single let.
563 \subsection{Simple let binding removal}
564 This transforms inlines simple let bindings (\eg a = b).
565 \subsection{Function inlining}
566 This transform inlines let bindings of a funtion type. TODO: This should
567 be generelized to anything that is non representable at runtime, or
569 \subsection{Scrutinee simplification}
570 This transform ensures that the scrutinee of a case expression is always
571 a simple variable reference.
572 \subsection{Case binder wildening}
573 This transform replaces all binders of a each case alternative with a
574 wild binder (\ie, one that is never referred to). This will possibly
575 introduce a number of new "selector" case statements, that only select
576 one element from an algebraic datatype and bind it to a variable.
577 \subsection{Case value simplification}
578 This transform simplifies the result value of each case alternative by
579 binding the value in a let expression and replacing the value by a
580 simple variable reference.
581 \subsection{Case removal}
582 This transform removes any case statements with a single alternative and
585 \subsection{Example sequence}
587 This section lists an example expression, with a sequence of transforms
588 applied to it. The exact transforms given here probably don't exactly
589 match the transforms given above anymore, but perhaps this can clarify
590 the big picture a bit.
592 TODO: Update or remove this section.
610 After top-level η-abstraction:
629 After (extended) β-reduction:
647 After return value extraction:
666 Scrutinee simplification does not apply.
668 After case binder wildening:
673 a = case s of (a, _) -> a
674 b = case s of (_, b) -> b
675 r = case s of (_, _) ->
678 Low -> let op' = case b of
687 After case value simplification
692 a = case s of (a, _) -> a
693 b = case s of (_, b) -> b
694 r = case s of (_, _) -> r'
696 rl = let rll = λc.λd.c
709 After let flattening:
714 a = case s of (a, _) -> a
715 b = case s of (_, b) -> b
716 r = case s of (_, _) -> r'
730 After function inlining:
735 a = case s of (a, _) -> a
736 b = case s of (_, b) -> b
737 r = case s of (_, _) -> r'
749 After (extended) β-reduction again:
754 a = case s of (a, _) -> a
755 b = case s of (_, b) -> b
756 r = case s of (_, _) -> r'
768 After case value simplification again:
773 a = case s of (a, _) -> a
774 b = case s of (_, b) -> b
775 r = case s of (_, _) -> r'
793 a = case s of (a, _) -> a
794 b = case s of (_, b) -> b
808 After let bind removal:
813 a = case s of (a, _) -> a
814 b = case s of (_, b) -> b
827 Application simplification is not applicable.