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 is one let expression at the top level
68 -- Calling some other user-defined function.
70 -- Extracting result values from a tuple
71 a = case s of (a, b) -> a
72 b = case s of (a, b) -> b
73 -- Some builtin expressions
76 -- Conditional connections
88 When looking at such a program from a hardware perspective, the top level
89 lambda's define the input ports. The value produced by the let expression is
90 the output port. Most function applications bound by the let expression
91 define a component instantiation, where the input and output ports are mapped
92 to local signals or arguments. Some of the others use a builtin
93 construction (\eg the \lam{case} statement) or call a builtin function
94 (\eg \lam{add} or \lam{sub}). For these, a hardcoded VHDL translation is
97 \subsection{Normal definition}
98 Formally, the normal form is a core program obeying the following
99 constraints. TODO: Update this section, this is probably not completely
100 accurate or relevant anymore.
102 \startitemize[R,inmargin]
103 %\item All top level binds must have the form $\expr{\bind{fun}{lamexpr}}$.
104 %$fun$ is an identifier that will be bound as a global identifier.
105 %\item A $lamexpr$ has the form $\expr{\lam{arg}{lamexpr}}$ or
106 %$\expr{letexpr}$. $arg$ is an identifier which will be bound as an $argument$.
107 %\item[letexpr] A $letexpr$ has the form $\expr{\letexpr{letbinds}{retexpr}}$.
108 %\item $letbinds$ is a list with elements of the form
109 %$\expr{\bind{res}{appexpr}}$ or $\expr{\bind{res}{builtinexpr}}$, where $res$ is
110 %an identifier that will be bound as local identifier. The type of the bound
111 %value must be a $hardware\;type$.
112 %\item[builtinexpr] A $builtinexpr$ is an expression that can be mapped to an
113 %equivalent VHDL expression. Since there are many supported forms for this,
114 %these are defined in a separate table.
115 %\item An $appexpr$ has the form $\expr{fun}$ or $\expr{\app{appexpr}{x}}$,
116 %where $fun$ is a global identifier and $x$ is a local identifier.
117 %\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
118 %be of a $hardware\;type$.
119 %\item A $tupexpr$ has the form $\expr{con}$ or $\expr{\app{tupexpr}{x}}$,
120 %where $con$ is a tuple constructor ({\em e.g.} $(,)$ or $(,,,)$) and $x$ is
122 %\item A $hardware\;type$ is a type that can be directly translated to
123 %hardware. This includes the types $Bit$, $SizedWord$, tuples containing
124 %elements of $hardware\;type$s, and will include others. This explicitely
125 %excludes function types.
128 TODO: Say something about uniqueness of identifiers
130 \subsection{Builtin expressions}
131 A $builtinexpr$, as defined at \in[builtinexpr] can have any of the following forms.
133 \startitemize[m,inmargin]
135 %$tuple\_extract=\expr{\case{t}{\alt{\app{con}{x_0\;x_1\;..\;x_n}}{x_i}}}$,
136 %where $t$ can be any local identifier, $con$ is a tuple constructor ({\em
137 %e.g.} $(,)$ or $(,,,)$), $x_0$ to $x_n$ can be any identifier, and $x_i$ can
138 %be any of $x_0$ to $x_n$. A case expression must have a $hardware\;type$.
139 %\item TODO: Many more!
142 \section{Transform passes}
144 In this section we describe the actual transforms. Here we're using
145 the core language in a notation that resembles lambda calculus.
147 Each of these transforms is meant to be applied to every (sub)expression
148 in a program, for as long as it applies. Only when none of the
149 expressions can be applied anymore, the program is in normal form. We
150 hope to be able to prove that this form will obey all of the constraints
151 defined above, but this has yet to happen (though it seems likely that
154 Each of the transforms will be described informally first, explaining
155 the need for and goal of the transform. Then, a formal definition is
156 given, using a familiar syntax from the world of logic. Each transform
157 is specified as a number of conditions (above the horizontal line) and a
158 number of conclusions (below the horizontal line). The details of using
159 this notation are still a bit fuzzy, so comments are welcom.
161 TODO: Formally describe the "apply to every (sub)expression" in terms of
162 rules with full transformations in the conditions.
164 \subsection{η-abstraction}
165 This transformation makes sure that all arguments of a function-typed
166 expression are named, by introducing lambda expressions. When combined with
167 β-reduction and function inlining below, all function-typed expressions should
168 be lambda abstractions or global identifiers.
172 -------------- \lam{E} is not the first argument of an application.
173 λx.E x \lam{E} is not a lambda abstraction.
174 \lam{x} is a variable that does not occur free in \lam{E}.
178 foo = λa -> case a of
184 foo = λa.λx -> (case a of
189 \transexample{η-abstraction}{from}{to}
191 \subsection{Extended β-reduction}
192 This transformation is meant to propagate application expressions downwards
193 into expressions as far as possible. In lambda calculus, this reduction
194 is known as β-reduction, but it is of course only defined for
195 applications of lambda abstractions. We extend this reduction to also
196 work for the rest of core (case and let expressions).
210 %\transform{Extended β-reduction}
213 %\trans{(λx.E) M}{E[M/x]}
217 %\trans{(let binds in E) M}{let binds in E M}
229 b = (let y = 3 in add y) 2
238 b = let y = 3 in add y 2
243 \transexample{Extended β-reduction}{from}{to}
245 \subsection{Argument simplification}
246 The transforms in this section deal with simplifying application
247 arguments into normal form. The goal here is to:
250 \item Make all arguments of user-defined functions (\eg, of which
251 we have a function body) simple variable references of a runtime
253 \item Make all arguments of builtin functions either:
255 \item A type argument.
256 \item A dictionary argument.
257 \item A type level expression.
258 \item A variable reference of a runtime representable type.
259 \item A variable reference or partial application of a function type.
263 When looking at the arguments of a user-defined function, we can
264 divide them into two categories:
266 \item Arguments with a runtime representable type (\eg bits or vectors).
268 These arguments can be preserved in the program, since they can
269 be translated to input ports later on. However, since we can
270 only connect signals to input ports, these arguments must be
271 reduced to simple variables (for which signals will be
272 produced). This is taken care of by the argument extraction
274 \item Non-runtime representable typed arguments.
276 These arguments cannot be preserved in the program, since we
277 cannot represent them as input or output ports in the resulting
278 VHDL. To remove them, we create a specialized version of the
279 called function with these arguments filled in. This is done by
280 the argument propagation transform.
283 When looking at the arguments of a builtin function, we can divide them
287 \item Arguments with a runtime representable type.
289 As we have seen with user-defined functions, these arguments can
290 always be reduced to a simple variable reference, by the
291 argument extraction transform. Performing this transform for
292 builtin functions as well, means that the translation of builtin
293 functions can be limited to signal references, instead of
294 needing to support all possible expressions.
296 \item Arguments with a function type.
298 These arguments are functions passed to higher order builtins,
299 like \lam{map} and \lam{foldl}. Since implementing these
300 functions for arbitrary function-typed expressions (\eg, lambda
301 expressions) is rather comlex, we reduce these arguments to
302 (partial applications of) global functions.
304 We can still support arbitrary expressions from the user code,
305 by creating a new global function containing that expression.
306 This way, we can simply replace the argument with a reference to
307 that new function. However, since the expression can contain any
308 number of free variables we also have to include partial
309 applications in our normal form.
311 This category of arguments is handled by the function extraction
313 \item Other unrepresentable arguments.
315 These arguments can take a few different forms:
316 \startdesc{Type arguments}
317 In the core language, type arguments can only take a single
318 form: A type wrapped in the Type constructor. Also, there is
319 nothing that can be done with type expressions, except for
320 applying functions to them, so we can simply leave type
321 arguments as they are.
323 \startdesc{Dictionary arguments}
324 In the core language, dictionary arguments are used to find
325 operations operating on one of the type arguments (mostly for
326 finding class methods). Since we will not actually evaluatie
327 the function body for builtin functions and can generate
328 code for builtin functions by just looking at the type
329 arguments, these arguments can be ignored and left as they
332 \startdesc{Type level arguments}
333 Sometimes, we want to pass a value to a builtin function, but
334 we need to know the value at compile time. Additionally, the
335 value has an impact on the type of the function. This is
336 encoded using type-level values, where the actual value of the
337 argument is not important, but the type encodes some integer,
338 for example. Since the value is not important, the actual form
339 of the expression does not matter either and we can leave
340 these arguments as they are.
342 \startdesc{Other arguments}
343 Technically, there is still a wide array of arguments that can
344 be passed, but does not fall into any of the above categories.
345 However, none of the supported builtin functions requires such
346 an argument. This leaves use with passing unsupported types to
347 a function, such as calling \lam{head} on a list of functions.
349 In these cases, it would be impossible to generate hardware
350 for such a function call anyway, so we can ignore these
353 The only way to generate hardware for builtin functions with
354 arguments like these, is to expand the function call into an
355 equivalent core expression (\eg, expand map into a series of
356 function applications). But for now, we choose to simply not
357 support expressions like these.
360 From the above, we can conclude that we can simply ignore these
361 other unrepresentable arguments and focus on the first two
365 \subsubsection{Argument extraction}
366 This transform deals with arguments to functions that
367 are of a runtime representable type.
369 TODO: It seems we can map an expression to a port, not only a signal.
370 Perhaps this makes this transformation not needed?
371 TODO: Say something about dataconstructors (without arguments, like True
372 or False), which are variable references of a runtime representable
373 type, but do not result in a signal.
375 To reduce a complex expression to a simple variable reference, we create
376 a new let expression around the application, which binds the complex
377 expression to a new variable. The original function is then applied to
380 %\transform{Argument extract}
382 %\lam{Y} is of a hardware representable type
384 %\lam{Y} is not a variable referene
388 %\trans{X Y}{let z = Y in X z}
391 \subsubsection{Function extraction}
392 This transform deals with function-typed arguments to builtin functions.
393 Since these arguments cannot be propagated, we choose to extract them
394 into a new global function instead.
396 Any free variables occuring in the extracted arguments will become
397 parameters to the new global function. The original argument is replaced
398 with a reference to the new function, applied to any free variables from
399 the original argument.
401 %\transform{Function extraction}
403 %\lam{X} is a (partial application of) a builtin function
405 %\lam{Y} is not an application
407 %\lam{Y} is not a variable reference
411 %\lam{f0 ... fm} = free local vars of \lam{Y}
413 %\lam{y} is a new global variable
415 %\lam{y = λf0 ... fn.Y}
417 %\trans{X Y}{X (y f0 ... fn)}
420 \subsubsection{Argument propagation}
421 This transform deals with arguments to user-defined functions that are
422 not representable at runtime. This means these arguments cannot be
423 preserved in the final form and most be {\em propagated}.
425 Propagation means to create a specialized version of the called
426 function, with the propagated argument already filled in. As a simple
427 example, in the following program:
434 we could {\em propagate} the constant argument 1, with the following
442 Special care must be taken when the to-be-propagated expression has any
443 free variables. If this is the case, the original argument should not be
444 removed alltogether, but replaced by all the free variables of the
445 expression. In this way, the original expression can still be evaluated
446 inside the new function. Also, this brings us closer to our goal: All
447 these free variables will be simple variable references.
449 To prevent us from propagating the same argument over and over, a simple
450 local variable reference is not propagated (since is has exactly one
451 free variable, itself, we would only replace that argument with itself).
453 This shows that any free local variables that are not runtime representable
454 cannot be brought into normal form by this transform. We rely on an
455 inlining transformation to replace such a variable with an expression we
458 TODO: Move these definitions somewhere sensible.
460 Definition: A global variable is any variable that is bound at the
461 top level of a program. A local variable is any other variable.
463 Definition: A hardware representable type is a type that we can generate
464 a signal for in hardware. For example, a bit, a vector of bits, a 32 bit
465 unsigned word, etc. Types that are not runtime representable notably
466 include (but are not limited to): Types, dictionaries, functions.
468 Definition: A builtin function is a function for which a builtin
469 hardware translation is available, because its actual definition is not
470 translatable. A user-defined function is any other function.
475 x Y0 ... Yi ... Yn \lam{Y_i} is not of a runtime representable type
476 --------------------------------------------- \lam{Y_i} is not a local variable reference
477 x' = λy0 ... yi-1 f0 ... fm yi+1 ... yn . \lam{f0 ... fm} = free local vars of \lam{Y_i}
478 E y0 ... yi-1 Yi yi+1 ... yn
480 x' y0 ... yi-1 f0 ... fm Yi+1 ... Yn
483 %\transform{Argument propagation}
485 %\lam{x} is a global variable, bound to a user-defined function
489 %\lam{Y_i} is not of a runtime representable type
491 %\lam{Y_i} is not a local variable reference
495 %\lam{f0 ... fm} = free local vars of \lam{Y_i}
497 %\lam{x'} is a new global variable
499 %\lam{x' = λy0 ... yi-1 f0 ... fm yi+1 ... yn . E y0 ... yi-1 Yi yi+1 ... yn}
501 %\trans{x Y0 ... Yi ... Yn}{x' y0 ... yi-1 f0 ... fm Yi+1 ... Yn}
504 %TODO: The above definition looks too complicated... Can we find
505 %something more concise?
507 \subsection{Cast propagation}
508 This transform pushes casts down into the expression as far as possible.
509 \subsection{Let recursification}
510 This transform makes all lets recursive.
511 \subsection{Let simplification}
512 This transform makes the result value of all let expressions a simple
514 \subsection{Let flattening}
515 This transform turns two nested lets (\lam{let x = (let ... in ...) in
516 ...}) into a single let.
517 \subsection{Simple let binding removal}
518 This transforms inlines simple let bindings (\eg a = b).
519 \subsection{Function inlining}
520 This transform inlines let bindings of a funtion type. TODO: This should
521 be generelized to anything that is non representable at runtime, or
523 \subsection{Scrutinee simplification}
524 This transform ensures that the scrutinee of a case expression is always
525 a simple variable reference.
526 \subsection{Case binder wildening}
527 This transform replaces all binders of a each case alternative with a
528 wild binder (\ie, one that is never referred to). This will possibly
529 introduce a number of new "selector" case statements, that only select
530 one element from an algebraic datatype and bind it to a variable.
531 \subsection{Case value simplification}
532 This transform simplifies the result value of each case alternative by
533 binding the value in a let expression and replacing the value by a
534 simple variable reference.
535 \subsection{Case removal}
536 This transform removes any case statements with a single alternative and
539 \subsection{Example sequence}
541 This section lists an example expression, with a sequence of transforms
542 applied to it. The exact transforms given here probably don't exactly
543 match the transforms given above anymore, but perhaps this can clarify
544 the big picture a bit.
546 TODO: Update or remove this section.
564 After top-level η-abstraction:
583 After (extended) β-reduction:
601 After return value extraction:
620 Scrutinee simplification does not apply.
622 After case binder wildening:
627 a = case s of (a, _) -> a
628 b = case s of (_, b) -> b
629 r = case s of (_, _) ->
632 Low -> let op' = case b of
641 After case value simplification
646 a = case s of (a, _) -> a
647 b = case s of (_, b) -> b
648 r = case s of (_, _) -> r'
650 rl = let rll = λc.λd.c
663 After let flattening:
668 a = case s of (a, _) -> a
669 b = case s of (_, b) -> b
670 r = case s of (_, _) -> r'
684 After function inlining:
689 a = case s of (a, _) -> a
690 b = case s of (_, b) -> b
691 r = case s of (_, _) -> r'
703 After (extended) β-reduction again:
708 a = case s of (a, _) -> a
709 b = case s of (_, b) -> b
710 r = case s of (_, _) -> r'
722 After case value simplification again:
727 a = case s of (a, _) -> a
728 b = case s of (_, b) -> b
729 r = case s of (_, _) -> r'
747 a = case s of (a, _) -> a
748 b = case s of (_, b) -> b
762 After let bind removal:
767 a = case s of (a, _) -> a
768 b = case s of (_, b) -> b
781 Application simplification is not applicable.