}
}
- \define[3]\transexample{
- \placeexample[here]{#1}
+ \define[4]\transexample{
+ \placeexample[here][ex:trans:#1]{#2}
\startcombination[2*1]
- {\example{#2}}{Original program}
- {\example{#3}}{Transformed program}
+ {\example{#3}}{Original program}
+ {\example{#4}}{Transformed program}
\stopcombination
}
- The first step in the core to \small{VHDL} translation process, is normalization. We
- aim to bring the core description into a simpler form, which we can
+ The first step in the Core to \small{VHDL} translation process, is normalization. We
+ aim to bring the Core description into a simpler form, which we can
subsequently translate into \small{VHDL} easily. This normal form is needed because
- the full core language is more expressive than \small{VHDL} in some areas and because
- core can describe expressions that do not have a direct hardware
- interpretation.
-
- TODO: Describe core properties not supported in \small{VHDL}, and describe how the
- \small{VHDL} we want to generate should look like.
+ the full Core language is more expressive than \small{VHDL} in some
+ areas (higher-order expressions, limited polymorphism using type
+ classes, etc.) and because Core can describe expressions that do not
+ have a direct hardware interpretation.
\section{Normal form}
- The transformations described here have a well-defined goal: To bring the
- program in a well-defined form that is directly translatable to hardware,
- while fully preserving the semantics of the program. We refer to this form as
- the \emph{normal form} of the program. The formal definition of this normal
- form is quite simple:
+ The transformations described here have a well-defined goal: to bring the
+ program in a well-defined form that is directly translatable to
+ \VHDL, while fully preserving the semantics of the program. We refer
+ to this form as the \emph{normal form} of the program. The formal
+ definition of this normal form is quite simple:
- \placedefinition{}{A program is in \emph{normal form} if none of the
- transformations from this chapter apply.}
+ \placedefinition[force]{}{\startboxed A program is in \emph{normal form} if none of the
+ transformations from this chapter apply.\stopboxed}
Of course, this is an \quote{easy} definition of the normal form, since our
program will end up in normal form automatically. The more interesting part is
to see if this normal form actually has the properties we would like it to
have.
- But, before getting into more definitions and details about this normal form,
- let's try to get a feeling for it first. The easiest way to do this is by
- describing the things we want to not have in a normal form.
+ But, before getting into more definitions and details about this normal
+ form, let us try to get a feeling for it first. The easiest way to do this
+ is by describing the things that are unwanted in the intended normal form.
\startitemize
\item Any \emph{polymorphism} must be removed. When laying down hardware, we
- can't generate any signals that can have multiple types. All types must be
+ cannot generate any signals that can have multiple types. All types must be
completely known to generate hardware.
- \item Any \emph{higher order} constructions must be removed. We can't
+ \item All \emph{higher-order} constructions must be removed. We cannot
generate a hardware signal that contains a function, so all values,
- arguments and returns values used must be first order.
+ arguments and return values used must be first order.
- \item Any complex \emph{nested scopes} must be removed. In the \small{VHDL}
+ \item All complex \emph{nested scopes} must be removed. In the \small{VHDL}
description, every signal is in a single scope. Also, full expressions are
- not supported everywhere (in particular port maps can only map signal names,
- not expressions). To make the \small{VHDL} generation easy, all values must be bound
- on the \quote{top level}.
+ not supported everywhere (in particular port maps can only map signal
+ names and constants, not complete expressions). To make the \small{VHDL}
+ generation easy, a separate binder must be bound to ever application or
+ other expression.
\stopitemize
- TODO: Intermezzo: functions vs plain values
-
- A very simple example of a program in normal form is given in
- \in{example}[ex:MulSum]. As you can see, all arguments to the function (which
- will become input ports in the final hardware) are at the top. This means that
- the body of the final lambda abstraction is never a function, but always a
- plain value.
-
- After the lambda abstractions, we see a single let expression, that binds two
- variables (\lam{mul} and \lam{sum}). These variables will be signals in the
- final hardware, bound to the output port of the \lam{*} and \lam{+}
- components.
-
- The final line (the \quote{return value} of the function) selects the
- \lam{sum} signal to be the output port of the function. This \quote{return
- value} can always only be a variable reference, never a more complex
- expression.
-
\startbuffer[MulSum]
alu :: Bit -> Word -> Word -> Word
alu = λa.λb.λc.
newCircle.a(btex $a$ etex) "framed(false)";
newCircle.b(btex $b$ etex) "framed(false)";
newCircle.c(btex $c$ etex) "framed(false)";
- newCircle.sum(btex $res$ etex) "framed(false)";
+ newCircle.sum(btex $sum$ etex) "framed(false)";
% Components
- newCircle.mul(btex - etex);
+ newCircle.mul(btex * etex);
newCircle.add(btex + etex);
a.c - b.c = (0cm, 2cm);
ncline(add)(sum);
\stopuseMPgraphic
- \placeexample[here][ex:MulSum]{Simple architecture consisting of an adder and a
- subtractor.}
+ \placeexample[][ex:MulSum]{Simple architecture consisting of a
+ multiplier and a subtractor.}
\startcombination[2*1]
{\typebufferlam{MulSum}}{Core description in normal form.}
{\boxedgraphic{MulSum}}{The architecture described by the normal form.}
\stopcombination
- The previous example described composing an architecture by calling other
- functions (operators), resulting in a simple architecture with component and
- connection. There is of course also some mechanism for choice in the normal
- form. In a normal Core program, the \emph{case} expression can be used in a
- few different ways to describe choice. In normal form, this is limited to a
- very specific form.
+ \todo{Intermezzo: functions vs plain values}
+
+ A very simple example of a program in normal form is given in
+ \in{example}[ex:MulSum]. As you can see, all arguments to the function (which
+ will become input ports in the generated \VHDL) are at the outer level.
+ This means that the body of the inner lambda abstraction is never a
+ function, but always a plain value.
+
+ As the body of the inner lambda abstraction, we see a single (recursive)
+ let expression, that binds two variables (\lam{mul} and \lam{sum}). These
+ variables will be signals in the generated \VHDL, bound to the output port
+ of the \lam{*} and \lam{+} components.
+
+ The final line (the \quote{return value} of the function) selects the
+ \lam{sum} signal to be the output port of the function. This \quote{return
+ value} can always only be a variable reference, never a more complex
+ expression.
+
+ \todo{Add generated VHDL}
+
+ \in{Example}[ex:MulSum] showed a function that just applied two
+ other functions (multiplication and addition), resulting in a simple
+ architecture with two components and some connections. There is of
+ course also some mechanism for choice in the normal form. In a
+ normal Core program, the \emph{case} expression can be used in a few
+ different ways to describe choice. In normal form, this is limited
+ to a very specific form.
\in{Example}[ex:AddSubAlu] shows an example describing a
simple \small{ALU}, which chooses between two operations based on an opcode
- bit. The main structure is the same as in \in{example}[ex:MulSum], but this
+ bit. The main structure is similar to \in{example}[ex:MulSum], but this
time the \lam{res} variable is bound to a case expression. This case
expression scrutinizes the variable \lam{opcode} (and scrutinizing more
complex expressions is not supported). The case expression can select a
different variable based on the constructor of \lam{opcode}.
+ \refdef{case expression}
\startbuffer[AddSubAlu]
alu :: Bit -> Word -> Word -> Word
ncline(mux)(res) "posA(out)";
\stopuseMPgraphic
- \placeexample[here][ex:AddSubAlu]{Simple \small{ALU} supporting two operations.}
+ \placeexample[][ex:AddSubAlu]{Simple \small{ALU} supporting two operations.}
\startcombination[2*1]
{\typebufferlam{AddSubAlu}}{Core description in normal form.}
{\boxedgraphic{AddSubAlu}}{The architecture described by the normal form.}
\stopcombination
- As a more complete example, consider \in{example}[ex:NormalComplete]. This
- example contains everything that is supported in normal form, with the
- exception of builtin higher order functions. The graphical version of the
- architecture contains a slightly simplified version, since the state tuple
- packing and unpacking have been left out. Instead, two seperate registers are
- drawn. Also note that most synthesis tools will further optimize this
- architecture by removing the multiplexers at the register input and replace
- them with some logic in the clock inputs, but we want to show the architecture
- as close to the description as possible.
+ As a more complete example, consider
+ \in{example}[ex:NormalComplete]. This example shows everything that
+ is allowed in normal form, except for built-in higher-order functions
+ (like \lam{map}). The graphical version of the architecture contains
+ a slightly simplified version, since the state tuple packing and
+ unpacking have been left out. Instead, two separate registers are
+ drawn. Most synthesis tools will further optimize this architecture by
+ removing the multiplexers at the register input and instead use the write
+ enable port of the register (when it is available), but we want to show
+ the architecture as close to the description as possible.
+
+ As you can see from the previous examples, the generation of the final
+ architecture from the normal form is straightforward. In each of the
+ examples, there is a direct match between the normal form structure,
+ the generated VHDL and the architecture shown in the images.
\startbuffer[NormalComplete]
regbank :: Bit
-> State (Word, Word)
-> (State (Word, Word), Word)
- -- All arguments are an inital lambda
+ -- All arguments are an initial lambda
+ -- (address, data, packed state)
regbank = λa.λd.λsp.
-- There are nested let expressions at top level
let
-- Unpack the state by coercion (\eg, cast from
-- State (Word, Word) to (Word, Word))
- s = sp :: (Word, Word)
+ s = sp ▶ (Word, Word)
-- Extract both registers from the state
- r1 = case s of (fst, snd) -> fst
- r2 = case s of (fst, snd) -> snd
+ r1 = case s of (a, b) -> a
+ r2 = case s of (a, b) -> b
-- Calling some other user-defined function.
d' = foo d
-- Conditional connections
s' = (,) r1' r2'
-- pack the state by coercion (\eg, cast from
-- (Word, Word) to State (Word, Word))
- sp' = s' :: State (Word, Word)
+ sp' = s' ▶ State (Word, Word)
-- Pack our return value
res = (,) sp' out
in
ncline(muxout)(out) "posA(out)";
\stopuseMPgraphic
- \placeexample[here][ex:NormalComplete]{Simple architecture consisting of an adder and a
+ \todo{Don't split registers in this image?}
+ \placeexample[][ex:NormalComplete]{Simple architecture consisting of an adder and a
subtractor.}
\startcombination[2*1]
{\typebufferlam{NormalComplete}}{Core description in normal form.}
{\boxedgraphic{NormalComplete}}{The architecture described by the normal form.}
\stopcombination
+
- \subsection{Intended normal form definition}
- Now we have some intuition for the normal form, we can describe how we want
- the normal form to look like in a slightly more formal manner. The following
- EBNF-like description completely captures the intended structure (and
- generates a subset of GHC's core format).
-
- Some clauses have an expression listed in parentheses. These are conditions
- that need to apply to the clause.
- \startlambda
- \italic{normal} = \italic{lambda}
- \italic{lambda} = λvar.\italic{lambda} (representable(var))
+ \subsection[sec:normalization:intendednormalform]{Intended normal form definition}
+ Now we have some intuition for the normal form, we can describe how we want
+ the normal form to look like in a slightly more formal manner. The
+ EBNF-like description in \in{definition}[def:IntendedNormal] captures
+ most of the intended structure (and generates a subset of \GHC's Core
+ format).
+
+ There are two things missing from this definition: cast expressions are
+ sometimes allowed by the prototype, but not specified here and the below
+ definition allows uses of state that cannot be translated to \VHDL\
+ properly. These two problems are discussed in
+ \in{section}[sec:normalization:castproblems] and
+ \in{section}[sec:normalization:stateproblems] respectively.
+
+ Some clauses have an expression listed behind them in parentheses.
+ These are conditions that need to apply to the clause. The
+ predicates used there (\lam{lvar()}, \lam{representable()},
+ \lam{gvar()}) will be defined in
+ \in{section}[sec:normalization:predicates].
+
+ An expression is in normal form if it matches the first
+ definition, \emph{normal}.
+
+ \todo{Fix indentation}
+ \startbuffer[IntendedNormal]
+ \italic{normal} := \italic{lambda}
+ \italic{lambda} := λvar.\italic{lambda} (representable(var))
| \italic{toplet}
- \italic{toplet} = letrec [\italic{binding}...] in var (representable(varvar))
- \italic{binding} = var = \italic{rhs} (representable(rhs))
+ \italic{toplet} := letrec [\italic{binding}...] in var (representable(var))
+ \italic{binding} := var = \italic{rhs} (representable(rhs))
-- State packing and unpacking by coercion
- | var0 = var1 :: State ty (lvar(var1))
- | var0 = var1 :: ty (var0 :: State ty) (lvar(var1))
- \italic{rhs} = userapp
- | builtinapp
+ | var0 = var1 ▶ State ty (lvar(var1))
+ | var0 = var1 ▶ ty (var1 :: State ty ∧ lvar(var1))
+ \italic{rhs} := \italic{userapp}
+ | \italic{builtinapp}
-- Extractor case
- | case var of C a0 ... an -> ai (lvar(var))
+ | case var of C a0 ... an -> ai (lvar(var))
-- Selector case
- | case var of (lvar(var))
- DEFAULT -> var0 (lvar(var0))
- C w0 ... wn -> resvar (\forall{}i, wi \neq resvar, lvar(resvar))
- \italic{userapp} = \italic{userfunc}
+ | case var of (lvar(var))
+ [ DEFAULT -> var ] (lvar(var))
+ C0 w0,0 ... w0,n -> var0
+ \vdots
+ Cm wm,0 ... wm,n -> varm (\forall{}i \forall{}j, wi,j \neq vari, lvar(vari))
+ \italic{userapp} := \italic{userfunc}
| \italic{userapp} {userarg}
- \italic{userfunc} = var (gvar(var))
- \italic{userarg} = var (lvar(var))
- \italic{builtinapp} = \italic{builtinfunc}
+ \italic{userfunc} := var (gvar(var))
+ \italic{userarg} := var (lvar(var))
+ \italic{builtinapp} := \italic{builtinfunc}
| \italic{builtinapp} \italic{builtinarg}
- \italic{builtinfunc} = var (bvar(var))
- \italic{builtinarg} = \italic{coreexpr}
- \stoplambda
-
- -- TODO: Limit builtinarg further
-
- -- TODO: There can still be other casts around (which the code can handle,
- e.g., ignore), which still need to be documented here.
-
- -- TODO: Note about the selector case. It just supports Bit and Bool
- currently, perhaps it should be generalized in the normal form?
-
- When looking at such a program from a hardware perspective, the top level
- lambda's define the input ports. The value produced by the let expression is
- the output port. Most function applications bound by the let expression
- define a component instantiation, where the input and output ports are mapped
- to local signals or arguments. Some of the others use a builtin
- construction (\eg the \lam{case} statement) or call a builtin function
- (\eg \lam{add} or \lam{sub}). For these, a hardcoded \small{VHDL} translation is
- available.
-
- \section{Transformation notation}
- To be able to concisely present transformations, we use a specific format to
- them. It is a simple format, similar to one used in logic reasoning.
+ \italic{built-infunc} := var (bvar(var))
+ \italic{built-inarg} := var (representable(var) ∧ lvar(var))
+ | \italic{partapp} (partapp :: a -> b)
+ | \italic{coreexpr} (¬representable(coreexpr) ∧ ¬(coreexpr :: a -> b))
+ \italic{partapp} := \italic{userapp}
+ | \italic{builtinapp}
+ \stopbuffer
+
+ \placedefinition[][def:IntendedNormal]{Definition of the intended normal form using an \small{EBNF}-like syntax.}
+ {\defref{intended normal form definition}
+ \typebufferlam{IntendedNormal}}
+
+ When looking at such a program from a hardware perspective, the top
+ level lambda abstractions (\italic{lambda}) define the input ports.
+ Lambda abstractions cannot appear anywhere else. The variable reference
+ in the body of the recursive let expression (\italic{toplet}) is the
+ output port. Most binders bound by the let expression define a
+ component instantiation (\italic{userapp}), where the input and output
+ ports are mapped to local signals (\italic{userarg}). Some of the others
+ use a built-in construction (\eg\ the \lam{case} expression) or call a
+ built-in function (\italic{builtinapp}) such as \lam{+} or \lam{map}.
+ For these, a hard-coded \small{VHDL} translation is available.
+
+ \section[sec:normalization:transformation]{Transformation notation}
+ To be able to concisely present transformations, we use a specific format
+ for them. It is a simple format, similar to one used in logic reasoning.
Such a transformation description looks like the following.
~
<original expression>
-------------------------- <expression conditions>
- <transformed expresssion>
+ <transformed expression>
~
<context additions>
\stoptrans
- This format desribes a transformation that applies to \lam{original
- expresssion} and transforms it into \lam{transformed expression}, assuming
- that all conditions apply. In this format, there are a number of placeholders
+ This format describes a transformation that applies to \lam{<original
+ expression>} and transforms it into \lam{<transformed expression>}, assuming
+ that all conditions are satisfied. In this format, there are a number of placeholders
in pointy brackets, most of which should be rather obvious in their meaning.
Nevertheless, we will more precisely specify their meaning below:
\startdesc{<original expression>} The expression pattern that will be matched
- against (subexpressions of) the expression to be transformed. We call this a
+ against (sub-expressions of) the expression to be transformed. We call this a
pattern, because it can contain \emph{placeholders} (variables), which match
any expression or binder. Any such placeholder is said to be \emph{bound} to
- the expression it matches. It is convention to use an uppercase latter (\eg
- \lam{M} or \lam{E} to refer to any expression (including a simple variable
- reference) and lowercase letters (\eg \lam{v} or \lam{b}) to refer to
+ the expression it matches. It is convention to use an uppercase letter (\eg\
+ \lam{M} or \lam{E}) to refer to any expression (including a simple variable
+ reference) and lowercase letters (\eg\ \lam{v} or \lam{b}) to refer to
(references to) binders.
For example, the pattern \lam{a + B} will match the expression
- \lam{v + (2 * w)} (and bind \lam{a} to \lam{v} and \lam{B} to
- \lam{(2 * 2)}), but not \lam{v + (2 * w)}.
+ \lam{v + (2 * w)} (binding \lam{a} to \lam{v} and \lam{B} to
+ \lam{(2 * w)}), but not \lam{(2 * w) + v}.
\stopdesc
\startdesc{<expression conditions>}
These are extra conditions on the expression that is matched. These
conditions can be used to further limit the cases in which the
- transformation applies, in particular to prevent a transformation from
+ transformation applies, commonly to prevent a transformation from
causing a loop with itself or another transformation.
- Only if these if these conditions are \emph{all} true, this transformation
+ Only if these conditions are \emph{all} satisfied, the transformation
applies.
\stopdesc
\startdesc{<context conditions>}
These are a number of extra conditions on the context of the function. In
- particular, these conditions can require some other top level function to be
+ particular, these conditions can require some (other) top level function to be
present, whose value matches the pattern given here. The format of each of
these conditions is: \lam{binder = <pattern>}.
expression>}, while the pattern contains some placeholders that are used in
the \lam{transformed expression}.
- Only if a top level binder exists that matches each binder and pattern, this
- transformation applies.
+ Only if a top level binder exists that matches each binder and pattern,
+ the transformation applies.
\stopdesc
\startdesc{<transformed expression>}
This is the expression template that is the result of the transformation. If, looking
- at the above three items, the transformation applies, the \lam{original
- expression} is completely replaced with the \lam{<transformed expression>}.
+ at the above three items, the transformation applies, the \lam{<original
+ expression>} is completely replaced by the \lam{<transformed expression>}.
We call this a template, because it can contain placeholders, referring to
any placeholder bound by the \lam{<original expression>} or the
\lam{<context conditions>}. The resulting expression will have those
\stopdesc
\startdesc{<context additions>}
- These are templates for new functions to add to the context. This is a way
- to have a transformation create new top level functiosn.
+ These are templates for new functions to be added to the context.
+ This is a way to let a transformation create new top level
+ functions.
Each addition has the form \lam{binder = template}. As above, any
placeholder in the addition is replaced with the value bound to it, and any
replaced with) a fresh binder.
\stopdesc
- As an example, we'll look at η-abstraction:
+ To understand this notation better, the step by step application of
+ the η-expansion transformation to a simple \small{ALU} will be
+ shown. Consider η-expansion, which is a common transformation from
+ lambda calculus, described using above notation as follows:
\starttrans
E \lam{E :: a -> b}
λx.E x \lam{E} is not a lambda abstraction.
\stoptrans
- Consider the following function, which is a fairly obvious way to specify a
- simple ALU (Note \at{example}[ex:AddSubAlu] is the normal form of this
- function):
+ η-expansion is a well known transformation from lambda calculus. What
+ this transformation does, is take any expression that has a function type
+ and turn it into a lambda expression (giving an explicit name to the
+ argument). There are some extra conditions that ensure that this
+ transformation does not apply infinitely (which are not necessarily part
+ of the conventional definition of η-expansion).
+
+ Consider the following function, in Core notation, which is a fairly obvious way to specify a
+ simple \small{ALU} (Note that it is not yet in normal form, but
+ \in{example}[ex:AddSubAlu] shows the normal form of this function).
+ The parentheses around the \lam{+} and \lam{-} operators are
+ commonly used in Haskell to show that the operators are used as
+ normal functions, instead of \emph{infix} operators (\eg, the
+ operators appear before their arguments, instead of in between).
\startlambda
alu :: Bit -> Word -> Word -> Word
High -> (-)
\stoplambda
- There are a few subexpressions in this function to which we could possibly
+ There are a few sub-expressions in this function to which we could possibly
apply the transformation. Since the pattern of the transformation is only
the placeholder \lam{E}, any expression will match that. Whether the
transformation applies to an expression is thus solely decided by the
By now, the placeholder \lam{E} is bound to the entire expression. The
placeholder \lam{x}, which occurs in the replacement template, is not bound
- yet, so we need to generate a fresh binder for that. Let's use the binder
+ yet, so we need to generate a fresh binder for that. Let us use the binder
\lam{a}. This results in the following replacement expression:
\startlambda
Continuing with this expression, we see that the transformation does not
apply again (it is a lambda expression). Next we look at the body of this
- labmda abstraction:
+ lambda abstraction:
\startlambda
(case opcode of
\stoplambda
Here, the transformation does apply, binding \lam{E} to the entire
- expression and \lam{x} to the fresh binder \lam{b}, resulting in the
- replacement:
+ expression (which has type \lam{Word -> Word}) and binding \lam{x}
+ to the fresh binder \lam{b}, resulting in the replacement:
\startlambda
λb.(case opcode of
High -> (-)) a b
\stoplambda
- Again, the transformation does not apply to this lambda abstraction, so we
- look at its body. For brevity, we'll put the case statement on one line from
+ The transformation does not apply to this lambda abstraction, so we
+ look at its body. For brevity, we will put the case expression on one line from
now on.
\startlambda
The type of this expression is \lam{Word}, so it does not match \lam{a -> b}
and the transformation does not apply. Next, we have two options for the
- next expression to look at: The function position and argument position of
+ next expression to look at: the function position and argument position of
the application. The expression in the argument position is \lam{b}, which
has type \lam{Word}, so the transformation does not apply. The expression in
the function position is:
\stoplambda
Obviously, the transformation does not apply here, since it occurs in
- function position. In the same way the transformation does not apply to both
- components of this expression (\lam{case opcode of Low -> (+); High -> (-)}
- and \lam{a}), so we'll skip to the components of the case expression: The
- scrutinee and both alternatives. Since the opcode is not a function, it does
- not apply here, and we'll leave both alternatives as an exercise to the
- reader. The final function, after all these transformations becomes:
+ function position (which makes the second condition false). In the same
+ way the transformation does not apply to both components of this
+ expression (\lam{case opcode of Low -> (+); High -> (-)} and \lam{a}), so
+ we will skip to the components of the case expression: the scrutinee and
+ both alternatives. Since the opcode is not a function, it does not apply
+ here.
+
+ The first alternative is \lam{(+)}. This expression has a function type
+ (the operator still needs two arguments). It does not occur in function
+ position of an application and it is not a lambda expression, so the
+ transformation applies.
+
+ We look at the \lam{<original expression>} pattern, which is \lam{E}.
+ This means we bind \lam{E} to \lam{(+)}. We then replace the expression
+ with the \lam{<transformed expression>}, replacing all occurrences of
+ \lam{E} with \lam{(+)}. In the \lam{<transformed expression>}, the This gives us the replacement expression:
+ \lam{λx.(+) x} (A lambda expression binding \lam{x}, with a body that
+ applies the addition operator to \lam{x}).
+
+ The complete function then becomes:
+ \startlambda
+ (case opcode of Low -> λa1.(+) a1; High -> (-)) a
+ \stoplambda
+
+ Now the transformation no longer applies to the complete first alternative
+ (since it is a lambda expression). It does not apply to the addition
+ operator again, since it is now in function position in an application. It
+ does, however, apply to the application of the addition operator, since
+ that is neither a lambda expression nor does it occur in function
+ position. This means after one more application of the transformation, the
+ function becomes:
+
+ \startlambda
+ (case opcode of Low -> λa1.λb1.(+) a1 b1; High -> (-)) a
+ \stoplambda
+
+ The other alternative is left as an exercise to the reader. The final
+ function, after applying η-expansion until it does no longer apply is:
\startlambda
alu :: Bit -> Word -> Word -> Word
High -> λa2.λb2 (-) a2 b2) a b
\stoplambda
- In this case, the transformation does not apply anymore, though this might
- not always be the case (e.g., the application of a transformation on a
- subexpression might open up possibilities to apply the transformation
- further up in the expression).
-
\subsection{Transformation application}
In this chapter we define a number of transformations, but how will we apply
these? As stated before, our normal form is reached as soon as no
the result of each transformation.
In particular, we define no particular order of transformations. Since
- transformation order should not influence the resulting normal form (see TODO
- ref), this leaves the implementation free to choose any application order that
- results in an efficient implementation.
+ transformation order should not influence the resulting normal form,
+ this leaves the implementation free to choose any application order that
+ results in an efficient implementation. Unfortunately this is not
+ entirely true for the current set of transformations. See
+ \in{section}[sec:normalization:non-determinism] for a discussion of this
+ problem.
When applying a single transformation, we try to apply it to every (sub)expression
- in a function, not just the top level function. This allows us to keep the
- transformation descriptions concise and powerful.
+ in a function, not just the top level function body. This allows us to
+ keep the transformation descriptions concise and powerful.
\subsection{Definitions}
- In the following sections, we will be using a number of functions and
- notations, which we will define here.
-
- TODO: Define substitution
-
- \subsubsection{Other concepts}
- A \emph{global variable} is any variable that is bound at the
- top level of a program, or an external module. A \emph{local variable} is any
- other variable (\eg, variables local to a function, which can be bound by
- lambda abstractions, let expressions and pattern matches of case
- alternatives). Note that this is a slightly different notion of global versus
- local than what \small{GHC} uses internally.
- \defref{global variable} \defref{local variable}
-
- A \emph{hardware representable} (or just \emph{representable}) type or value
- is (a value of) a type that we can generate a signal for in hardware. For
- example, a bit, a vector of bits, a 32 bit unsigned word, etc. Types that are
- not runtime representable notably include (but are not limited to): Types,
- dictionaries, functions.
- \defref{representable}
-
- A \emph{builtin function} is a function supplied by the Cλash framework, whose
- implementation is not valid Cλash. The implementation is of course valid
- Haskell, for simulation, but it is not expressable in Cλash.
- \defref{builtin function} \defref{user-defined function}
-
- For these functions, Cλash has a \emph{builtin hardware translation}, so calls
- to these functions can still be translated. These are functions like
- \lam{map}, \lam{hwor} and \lam{length}.
-
- A \emph{user-defined} function is a function for which we do have a Cλash
- implementation available.
-
- \subsubsection{Functions}
- Here, we define a number of functions that can be used below to concisely
+ A \emph{global variable} is any variable (binder) that is bound at the
+ top level of a program, or an external module. A \emph{local variable} is any
+ other variable (\eg, variables local to a function, which can be bound by
+ lambda abstractions, let expressions and pattern matches of case
+ alternatives). This is a slightly different notion of global versus
+ local than what \small{GHC} uses internally, but for our purposes
+ the distinction \GHC\ makes is not useful.
+ \defref{global variable} \defref{local variable}
+
+ A \emph{hardware representable} (or just \emph{representable}) type or value
+ is (a value of) a type that we can generate a signal for in hardware. For
+ example, a bit, a vector of bits, a 32 bit unsigned word, etc. Values that are
+ not run-time representable notably include (but are not limited to): types,
+ dictionaries, functions.
+ \defref{representable}
+
+ A \emph{built-in function} is a function supplied by the Cλash
+ framework, whose implementation is not used to generate \VHDL. This is
+ either because it is no valid Cλash (like most list functions that need
+ recursion) or because a Cλash implementation would be unwanted (for the
+ addition operator, for example, we would rather use the \VHDL addition
+ operator to let the synthesis tool decide what kind of adder to use
+ instead of explicitly describing one in Cλash). \defref{built-in
+ function}
+
+ These are functions like \lam{map}, \lam{hwor}, \lam{+} and \lam{length}.
+
+ For these functions, Cλash has a \emph{built-in hardware translation},
+ so calls to these functions can still be translated. Built-in functions
+ must have a valid Haskell implementation, of course, to allow
+ simulation.
+
+ A \emph{user-defined} function is a function for which no built-in
+ translation is available and whose definition will thus need to be
+ translated to Cλash. \defref{user-defined function}
+
+ \subsubsection[sec:normalization:predicates]{Predicates}
+ Here, we define a number of predicates that can be used below to concisely
specify conditions.
- \refdef{global variable}\emph{gvar(expr)} is true when \emph{expr} is a variable that references a
+ \emph{gvar(expr)} is true when \emph{expr} is a variable that references a
global variable. It is false when it references a local variable.
- \refdef{local variable}\emph{lvar(expr)} is the complement of \emph{gvar}; it is true when \emph{expr}
+ \emph{lvar(expr)} is the complement of \emph{gvar}; it is true when \emph{expr}
references a local variable, false when it references a global variable.
- \refdef{representable}\emph{representable(expr)} or \emph{representable(var)} is true when
- \emph{expr} or \emph{var} is \emph{representable}.
+ \emph{representable(expr)} is true when \emph{expr} is \emph{representable}.
- \subsection{Binder uniqueness}
+ \subsection[sec:normalization:uniq]{Binder uniqueness}
A common problem in transformation systems, is binder uniqueness. When not
considering this problem, it is easy to create transformations that mix up
- bindings and cause name collisions. Take for example, the following core
+ bindings and cause name collisions. Take for example, the following Core
expression:
\startlambda
(λa.λb.λc. a * b * c) x c
\stoplambda
- By applying β-reduction (see below) once, we can simplify this expression to:
+ By applying β-reduction (see \in{section}[sec:normalization:beta]) once,
+ we can simplify this expression to:
\startlambda
(λb.λc. x * b * c) c
\stoplambda
Now, we have replaced the \lam{a} binder with a reference to the \lam{x}
- binder. No harm done here. But note that we see multiple occurences of the
- \lam{c} binder. The first is a binding occurence, to which the second refers.
+ binder. No harm done here. But note that we see multiple occurrences of the
+ \lam{c} binder. The first is a binding occurrence, to which the second refers.
The last, however refers to \emph{another} instance of \lam{c}, which is
bound somewhere outside of this expression. Now, if we would apply beta
reduction without taking heed of binder uniqueness, we would get:
\stoplambda
This is obviously not what was supposed to happen! The root of this problem is
- the reuse of binders: Identical binders can be bound in different scopes, such
- that only the inner one is \quote{visible} in the inner expression. In the example
- above, the \lam{c} binder was bound outside of the expression and in the inner
- lambda expression. Inside that lambda expression, only the inner \lam{c} is
- visible.
+ the reuse of binders: identical binders can be bound in different,
+ but overlapping scopes. Any variable reference in those
+ overlapping scopes then refers to the variable bound in the inner
+ (smallest) scope. There is not way to refer to the variable in the
+ outer scope. This effect is usually referred to as
+ \emph{shadowing}: when a binder is bound in a scope where the
+ binder already had a value, the inner binding is said to
+ \emph{shadow} the outer binding. In the example above, the \lam{c}
+ binder was bound outside of the expression and in the inner lambda
+ expression. Inside that lambda expression, only the inner \lam{c}
+ can be accessed.
There are a number of ways to solve this. \small{GHC} has isolated this
- problem to their binder substitution code, which performs \emph{deshadowing}
+ problem to their binder substitution code, which performs \emph{de-shadowing}
during its expression traversal. This means that any binding that shadows
another binding on a higher level is replaced by a new binder that does not
shadow any other binding. This non-shadowing invariant is enough to prevent
binder uniqueness problems in \small{GHC}.
In our transformation system, maintaining this non-shadowing invariant is
- a bit harder to do (mostly due to implementation issues, the prototype doesn't
- use \small{GHC}'s subsitution code). Also, we can observe the following
- points.
+ a bit harder to do (mostly due to implementation issues, the prototype
+ does not use \small{GHC}'s substitution code). Also, the following points
+ can be observed.
\startitemize
- \item Deshadowing does not guarantee overall uniqueness. For example, the
+ \item De-shadowing does not guarantee overall uniqueness. For example, the
following (slightly contrived) expression shows the identifier \lam{x} bound in
- two seperate places (and to different values), even though no shadowing
+ two separate places (and to different values), even though no shadowing
occurs.
\startlambda
\stoplambda
\item In our normal form (and the resulting \small{VHDL}), all binders
- (signals) will end up in the same scope. To allow this, all binders within the
- same function should be unique.
+ (signals) within the same function (entity) will end up in the same
+ scope. To allow this, all binders within the same function should be
+ unique.
\item When we know that all binders in an expression are unique, moving around
- or removing a subexpression will never cause any binder conflicts. If we have
- some way to generate fresh binders, introducing new subexpressions will not
+ or removing a sub-expression will never cause any binder conflicts. If we have
+ some way to generate fresh binders, introducing new sub-expressions will not
cause any problems either. The only way to cause conflicts is thus to
- duplicate an existing subexpression.
+ duplicate an existing sub-expression.
\stopitemize
Given the above, our prototype maintains a unique binder invariant. This
- meanst that in any given moment during normalization, all binders \emph{within
+ means that in any given moment during normalization, all binders \emph{within
a single function} must be unique. To achieve this, we apply the following
technique.
- TODO: Define fresh binders and unique supplies
+ \todo{Define fresh binders and unique supplies}
\startitemize
\item Before starting normalization, all binders in the function are made
unique. This is done by generating a fresh binder for every binder used. This
- also replaces binders that did not pose any conflict, but it does ensure that
- all binders within the function are generated by the same unique supply. See
- (TODO: ref fresh binder).
+ also replaces binders that did not cause any conflict, but it does ensure that
+ all binders within the function are generated by the same unique supply.
\item Whenever a new binder must be generated, we generate a fresh binder that
is guaranteed to be different from \emph{all binders generated so far}. This
can thus never introduce duplication and will maintain the invariant.
- \item Whenever (part of) an expression is duplicated (for example when
+ \item Whenever (a part of) an expression is duplicated (for example when
inlining), all binders in the expression are replaced with fresh binders
(using the same method as at the start of normalization). These fresh binders
can never introduce duplication, so this will maintain the invariant.
\stopitemize
\section{Transform passes}
- In this section we describe the actual transforms. Here we're using
- the core language in a notation that resembles lambda calculus.
-
- Each of these transforms is meant to be applied to every (sub)expression
- in a program, for as long as it applies. Only when none of the
- transformations can be applied anymore, the program is in normal form (by
- definition). We hope to be able to prove that this form will obey all of the
- constraints defined above, but this has yet to happen (though it seems likely
- that it will).
-
- Each of the transforms will be described informally first, explaining
- the need for and goal of the transform. Then, a formal definition is
- given, using a familiar syntax from the world of logic. Each transform
- is specified as a number of conditions (above the horizontal line) and a
- number of conclusions (below the horizontal line). The details of using
- this notation are still a bit fuzzy, so comments are welcom.
+ In this section we describe the actual transforms.
+
+ Each transformation will be described informally first, explaining
+ the need for and goal of the transformation. Then, we will formally define
+ the transformation using the syntax introduced in
+ \in{section}[sec:normalization:transformation].
\subsection{General cleanup}
+ \placeintermezzo{}{
+ \defref{substitution notation}
+ \startframedtext[width=8cm,background=box,frame=no]
+ \startalignment[center]
+ {\tfa Substitution notation}
+ \stopalignment
+ \blank[medium]
+
+ In some of the transformations in this chapter, we need to perform
+ substitution on an expression. Substitution means replacing every
+ occurrence of some expression (usually a variable reference) with
+ another expression.
+
+ There have been a lot of different notations used in literature for
+ specifying substitution. The notation that will be used in this report
+ is the following:
+
+ \startlambda
+ E[A=>B]
+ \stoplambda
+
+ This means expression \lam{E} with all occurrences of \lam{A} replaced
+ with \lam{B}.
+ \stopframedtext
+ }
+
These transformations are general cleanup transformations, that aim to
make expressions simpler. These transformations usually clean up the
- mess left behind by other transformations or clean up expressions to
- expose new transformation opportunities for other transformations.
+ mess left behind by other transformations or clean up expressions to
+ expose new transformation opportunities for other transformations.
- Most of these transformations are standard optimizations in other
- compilers as well. However, in our compiler, most of these are not just
- optimizations, but they are required to get our program into normal
- form.
+ Most of these transformations are standard optimizations in other
+ compilers as well. However, in our compiler, most of these are not just
+ optimizations, but they are required to get our program into intended
+ normal form.
- \subsubsection{β-reduction}
+ \subsubsection[sec:normalization:beta]{β-reduction}
β-reduction is a well known transformation from lambda calculus, where it is
- the main reduction step. It reduces applications of labmda abstractions,
+ the main reduction step. It reduces applications of lambda abstractions,
removing both the lambda abstraction and the application.
In our transformation system, this step helps to remove unwanted lambda
abstractions (basically all but the ones at the top level). Other
transformations (application propagation, non-representable inlining) make
- sure that most lambda abstractions will eventually be reducable by
+ sure that most lambda abstractions will eventually be reducible by
β-reduction.
+ Note that β-reduction also works on type lambda abstractions and type
+ applications as well. This means the substitution below also works on
+ type variables, in the case that the binder is a type variable and the
+ expression applied to is a type.
+
\starttrans
(λx.E) M
-----------------
- E[M/x]
+ E[x=>M]
\stoptrans
% And an example
2 * (2 * b)
\stopbuffer
- \transexample{β-reduction}{from}{to}
+ \transexample{beta}{β-reduction}{from}{to}
+
+ \startbuffer[from]
+ (λt.λa::t. a) @Int
+ \stopbuffer
+
+ \startbuffer[to]
+ (λa::Int. a)
+ \stopbuffer
+
+ \transexample{beta-type}{β-reduction for type abstractions}{from}{to}
+
+ \subsubsection{Unused let binding removal}
+ This transformation removes let bindings that are never used.
+ Occasionally, \GHC's desugarer introduces some unused let bindings.
+
+ This normalization pass should really be not be necessary to get
+ into intended normal form (since the intended normal form
+ definition \refdef{intended normal form definition} does not
+ require that every binding is used), but in practice the
+ desugarer or simplifier emits some bindings that cannot be
+ normalized (e.g., calls to a
+ \hs{Control.Exception.Base.patError}) but are not used anywhere
+ either. To prevent the \VHDL\ generation from breaking on these
+ artifacts, this transformation removes them.
+
+ \todo{Do not use old-style numerals in transformations}
+ \starttrans
+ letrec
+ a0 = E0
+ \vdots
+ ai = Ei
+ \vdots
+ an = En
+ in
+ M \lam{ai} does not occur free in \lam{M}
+ ---------------------------- \lam{\forall j, 0 ≤ j ≤ n, j ≠ i} (\lam{ai} does not occur free in \lam{Ej})
+ letrec
+ a0 = E0
+ \vdots
+ ai-1 = Ei-1
+ ai+1 = Ei+1
+ \vdots
+ an = En
+ in
+ M
+ \stoptrans
+
+ % And an example
+ \startbuffer[from]
+ let
+ x = 1
+ in
+ 2
+ \stopbuffer
+
+ \startbuffer[to]
+ let
+ in
+ 2
+ \stopbuffer
+
+ \transexample{unusedlet}{Unused let binding removal}{from}{to}
\subsubsection{Empty let removal}
- This transformation is simple: It removes recursive lets that have no bindings
+ This transformation is simple: it removes recursive lets that have no bindings
(which usually occurs when unused let binding removal removes the last
binding from it).
+ Note that there is no need to define this transformation for
+ non-recursive lets, since they always contain exactly one binding.
+
\starttrans
letrec in M
--------------
M
\stoptrans
- TODO: Example
+ % And an example
+ \startbuffer[from]
+ let
+ in
+ 2
+ \stopbuffer
- \subsubsection{Simple let binding removal}
- This transformation inlines simple let bindings (\eg a = b).
+ \startbuffer[to]
+ 2
+ \stopbuffer
+
+ \transexample{emptylet}{Empty let removal}{from}{to}
- This transformation is not needed to get into normal form, but makes the
- resulting \small{VHDL} a lot shorter.
+ \subsubsection[sec:normalization:simplelet]{Simple let binding removal}
+ This transformation inlines simple let bindings, that bind some
+ binder to some other binder instead of a more complex expression (\ie\
+ a = b).
+ This transformation is not needed to get an expression into intended
+ normal form (since these bindings are part of the intended normal
+ form), but makes the resulting \small{VHDL} a lot shorter.
+
+ \refdef{substitution notation}
\starttrans
letrec
a0 = E0
in
M
----------------------------- \lam{b} is a variable reference
- letrec
- a0 = E0 [b/ai]
+ letrec \lam{ai} ≠ \lam{b}
+ a0 = E0 [ai=>b]
\vdots
- ai-1 = Ei-1 [b/ai]
- ai+1 = Ei+1 [b/ai]
+ ai-1 = Ei-1 [ai=>b]
+ ai+1 = Ei+1 [ai=>b]
\vdots
- an = En [b/ai]
+ an = En [ai=>b]
in
- M[b/ai]
+ M[ai=>b]
\stoptrans
- TODO: Example
+ \todo{example}
- \subsubsection{Unused let binding removal}
- This transformation removes let bindings that are never used. Usually,
- the desugarer introduces some unused let bindings.
+ \subsubsection{Cast propagation / simplification}
+ This transform pushes casts down into the expression as far as
+ possible. This transformation has been added to make a few
+ specific corner cases work, but it is not clear yet if this
+ transformation handles cast expressions completely or in the
+ right way. See \in{section}[sec:normalization:castproblems].
- This normalization pass should really be unneeded to get into normal form
- (since unused bindings are not forbidden by the normal form), but in practice
- the desugarer or simplifier emits some unused bindings that cannot be
- normalized (e.g., calls to a \type{PatError} (TODO: Check this name)). Also,
- this transformation makes the resulting \small{VHDL} a lot shorter.
+ \starttrans
+ (let binds in E) ▶ T
+ -------------------------
+ let binds in (E ▶ T)
+ \stoptrans
\starttrans
- letrec
- a0 = E0
- \vdots
- ai = Ei
- \vdots
- an = En
- in
- M \lam{a} does not occur free in \lam{M}
- ---------------------------- \forall j, 0 <= j <= n, j ≠ i (\lam{a} does not occur free in \lam{Ej})
- letrec
- a0 = E0
+ (case S of
+ p0 -> E0
\vdots
- ai-1 = Ei-1
- ai+1 = Ei+1
+ pn -> En
+ ) ▶ T
+ -------------------------
+ case S of
+ p0 -> E0 ▶ T
\vdots
- an = En
- in
- M
+ pn -> En ▶ T
\stoptrans
- TODO: Example
-
- \subsubsection{Cast propagation / simplification}
- This transform pushes casts down into the expression as far as possible.
- Since its exact role and need is not clear yet, this transformation is
- not yet specified.
-
\subsubsection{Top level binding inlining}
+ \refdef{top level binding}
This transform takes simple top level bindings generated by the
\small{GHC} compiler. \small{GHC} sometimes generates very simple
\quote{wrapper} bindings, which are bound to just a variable
- reference, or a partial application to constants or other variable
- references.
+ reference, or contain just a (partial) function application with
+ the type and dictionary arguments filled in (such as the
+ \lam{(+)} in the example below).
Note that this transformation is completely optional. It is not
- required to get any function into normal form, but it does help making
- the resulting VHDL output easier to read (since it removes a bunch of
- components that are really boring).
+ required to get any function into intended normal form, but it does help making
+ the resulting VHDL output easier to read (since it removes components
+ that do not add any real structure, but do hide away operations and
+ cause extra clutter).
- This transform takes any top level binding generated by the compiler,
+ This transform takes any top level binding generated by \GHC,
whose normalized form contains only a single let binding.
\starttrans
\startbuffer[from]
(+) :: Word -> Word -> Word
- (+) = GHC.Num.(+) @Word $dNum
+ (+) = GHC.Num.(+) @Word \$dNum
~
(+) a b
\stopbuffer
\startbuffer[to]
- GHC.Num.(+) @ Alu.Word $dNum a b
+ GHC.Num.(+) @ Alu.Word \$dNum a b
\stopbuffer
- \transexample{Top level binding inlining}{from}{to}
+ \transexample{toplevelinline}{Top level binding inlining}{from}{to}
- Without this transformation, the (+) function would generate an
- architecture which would just add its inputs. This generates a lot of
- overhead in the VHDL, which is particularly annoying when browsing the
- generated RTL schematic (especially since + is not allowed in VHDL
- architecture names\footnote{Technically, it is allowed to use
- non-alphanumerics when using extended identifiers, but it seems that
- none of the tooling likes extended identifiers in filenames, so it
- effectively doesn't work}, so the entity would be called
- \quote{w7aA7f} or something similarly unreadable and autogenerated).
+ \in{Example}[ex:trans:toplevelinline] shows a typical application of
+ the addition operator generated by \GHC. The type and dictionary
+ arguments used here are described in
+ \in{Section}[sec:prototype:coretypes].
+
+ Without this transformation, there would be a \lam{(+)} entity
+ in the \VHDL\ which would just add its inputs. This generates a
+ lot of overhead in the \VHDL, which is particularly annoying
+ when browsing the generated RTL schematic (especially since most
+ non-alphanumerics, like all characters in \lam{(+)}, are not
+ allowed in \VHDL\ architecture names\footnote{Technically, it is
+ allowed to use non-alphanumerics when using extended
+ identifiers, but it seems that none of the tooling likes
+ extended identifiers in file names, so it effectively does not
+ work.}, so the entity would be called \quote{w7aA7f} or
+ something similarly meaningless and auto-generated).
\subsection{Program structure}
These transformations are aimed at normalizing the overall structure
into the intended form. This means ensuring there is a lambda abstraction
- at the top for every argument (input port), putting all of the other
- value definitions in let bindings and making the final return value a
- simple variable reference.
+ at the top for every argument (input port or current state), putting all
+ of the other value definitions in let bindings and making the final
+ return value a simple variable reference.
- \subsubsection{η-abstraction}
+ \subsubsection[sec:normalization:eta]{η-expansion}
This transformation makes sure that all arguments of a function-typed
expression are named, by introducing lambda expressions. When combined with
β-reduction and non-representable binding inlining, all function-typed
\starttrans
E \lam{E :: a -> b}
- -------------- \lam{E} is not the first argument of an application.
+ -------------- \lam{E} does not occur on a function position in an application
λx.E x \lam{E} is not a lambda abstraction.
- \lam{x} is a variable that does not occur free in \lam{E}.
\stoptrans
\startbuffer[from]
False -> λy.id y) x
\stopbuffer
- \transexample{η-abstraction}{from}{to}
+ \transexample{eta}{η-expansion}{from}{to}
- \subsubsection{Application propagation}
+ \subsubsection[sec:normalization:appprop]{Application propagation}
This transformation is meant to propagate application expressions downwards
into expressions as far as possible. This allows partial applications inside
expressions to become fully applied and exposes new transformation
opportunities for other transformations (like β-reduction and
specialization).
+ Since all binders in our expression are unique (see
+ \in{section}[sec:normalization:uniq]), there is no risk that we will
+ introduce unintended shadowing by moving an expression into a lower
+ scope. Also, since only move expression into smaller scopes (down into
+ our expression), there is no risk of moving a variable reference out
+ of the scope in which it is defined.
+
\starttrans
(letrec binds in E) M
- -----------------
+ ------------------------
letrec binds in E M
\stoptrans
add val 3
\stopbuffer
- \transexample{Application propagation for a let expression}{from}{to}
+ \transexample{appproplet}{Application propagation for a let expression}{from}{to}
\starttrans
(case x of
- p1 -> E1
+ p0 -> E0
\vdots
pn -> En) M
-----------------
case x of
- p1 -> E1 M
+ p0 -> E0 M
\vdots
pn -> En M
\stoptrans
False -> neg 1
\stopbuffer
- \transexample{Application propagation for a case expression}{from}{to}
+ \transexample{apppropcase}{Application propagation for a case expression}{from}{to}
- \subsubsection{Let recursification}
+ \subsubsection[sec:normalization:letrecurse]{Let recursification}
This transformation makes all non-recursive lets recursive. In the
end, we want a single recursive let in our normalized program, so all
non-recursive lets can be converted. This also makes other
- transformations simpler: They can simply assume all lets are
- recursive.
+ transformations simpler: they only need to be specified for recursive
+ let expressions (and simply will not apply to non-recursive let
+ expressions until this transformation has been applied).
\starttrans
let
\subsubsection{Let flattening}
This transformation puts nested lets in the same scope, by lifting the
- binding(s) of the inner let into a new let around the outer let. Eventually,
- this will cause all let bindings to appear in the same scope (they will all be
- in scope for the function return value).
+ binding(s) of the inner let into the outer let. Eventually, this will
+ cause all let bindings to appear in the same scope.
+
+ This transformation only applies to recursive lets, since all
+ non-recursive lets will be made recursive (see
+ \in{section}[sec:normalization:letrecurse]).
+
+ Since we are joining two scopes together, there is no risk of moving a
+ variable reference out of the scope where it is defined.
\starttrans
letrec
+ a0 = E0
\vdots
- x = (letrec bindings in M)
+ ai = (letrec bindings in M)
\vdots
+ an = En
in
N
------------------------------------------
letrec
+ a0 = E0
\vdots
- bindings
- x = M
+ ai = M
\vdots
+ an = En
+ bindings
in
N
\stoptrans
\startbuffer[from]
letrec
- a = letrec
- x = 1
- y = 2
+ a = 1
+ b = letrec
+ x = a
+ y = c
in
x + y
+ c = 2
in
- a
+ b
\stopbuffer
\startbuffer[to]
letrec
- x = 1
- y = 2
- a = x + y
+ a = 1
+ b = x + y
+ c = 2
+ x = a
+ y = c
in
- a
+ b
\stopbuffer
- \transexample{Let flattening}{from}{to}
+ \transexample{letflat}{Let flattening}{from}{to}
\subsubsection{Return value simplification}
This transformation ensures that the return value of a function is always a
simple local variable reference.
- Currently implemented using lambda simplification, let simplification, and
- top simplification. Should change into something like the following, which
- works only on the result of a function instead of any subexpression. This is
- achieved by the contexts, like \lam{x = E}, though this is strictly not
- correct (you could read this as "if there is any function \lam{x} that binds
- \lam{E}, any \lam{E} can be transformed, while we only mean the \lam{E} that
- is bound by \lam{x}. This might need some extra notes or something).
-
- Note that the return value is not simplified if its not representable.
- Otherwise, this would cause a direct loop with the inlining of
- unrepresentable bindings, of course. If the return value is not
- representable because it has a function type, η-abstraction should
- make sure that this transformation will eventually apply. If the value
- is not representable for other reasons, the function result itself is
- not representable, meaning this function is not representable anyway!
+ The basic idea of this transformation is to take the body of a
+ function and bind it with a let expression (so the body of that let
+ expression becomes a variable reference that can be used as the output
+ port). If the body of the function happens to have lambda abstractions
+ at the top level (which is allowed by the intended normal
+ form\refdef{intended normal form definition}), we take the body of the
+ inner lambda instead. If that happens to be a let expression already
+ (which is allowed by the intended normal form), we take the body of
+ that let (which is not allowed to be anything but a variable reference
+ according the the intended normal form).
+
+ This transformation uses the context conditions in a special way.
+ These contexts, like \lam{x = λv1 ... λvn.E}, are above the dotted
+ line and provide a condition on the environment (\ie\ they require a
+ certain top level binding to be present). These ensure that
+ expressions are only transformed when they are in the functions
+ \quote{return value} directly. This means the context conditions have
+ to interpreted in the right way: not \quote{if there is any function
+ \lam{x} that binds \lam{E}, any \lam{E} can be transformed}, but we
+ mean only the \lam{E} that is bound by \lam{x}).
+
+ Be careful when reading the transformations: Not the entire function
+ from the context is transformed, just a part of it.
+
+ Note that the return value is not simplified if it is not representable.
+ Otherwise, this would cause a loop with the inlining of
+ unrepresentable bindings in
+ \in{section}[sec:normalization:nonrepinline]. If the return value is
+ not representable because it has a function type, η-expansion should
+ make sure that this transformation will eventually apply. If the
+ value is not representable for other reasons, the function result
+ itself is not representable, meaning this function is not translatable
+ anyway.
\starttrans
- x = E \lam{E} is representable
- ~ \lam{E} is not a lambda abstraction
- E \lam{E} is not a let expression
- --------------------------- \lam{E} is not a local variable reference
- letrec x = E in x
- \stoptrans
-
- \starttrans
- x = λv0 ... λvn.E
+ x = λv1 ... λvn.E \lam{n} can be zero
~ \lam{E} is representable
- E \lam{E} is not a let expression
- --------------------------- \lam{E} is not a local variable reference
- letrec x = E in x
+ E \lam{E} is not a lambda abstraction
+ --------------------------- \lam{E} is not a let expression
+ letrec y = E in y \lam{E} is not a local variable reference
\stoptrans
\starttrans
- x = λv0 ... λvn.let ... in E
- ~ \lam{E} is representable
- E \lam{E} is not a local variable reference
- ---------------------------
- letrec x = E in x
+ x = λv1 ... λvn.letrec binds in E \lam{n} can be zero
+ ~ \lam{E} is representable
+ letrec binds in E \lam{E} is not a local variable reference
+ ------------------------------------
+ letrec binds; y = E in y
\stoptrans
\startbuffer[from]
\stopbuffer
\startbuffer[to]
- x = letrec x = add 1 2 in x
+ x = letrec y = add 1 2 in y
\stopbuffer
- \transexample{Return value simplification}{from}{to}
+ \transexample{retvalsimpl}{Return value simplification}{from}{to}
- \subsection{Argument simplification}
- The transforms in this section deal with simplifying application
- arguments into normal form. The goal here is to:
-
- \startitemize
- \item Make all arguments of user-defined functions (\eg, of which
- we have a function body) simple variable references of a runtime
- representable type. This is needed, since these applications will be turned
- into component instantiations.
- \item Make all arguments of builtin functions one of:
- \startitemize
- \item A type argument.
- \item A dictionary argument.
- \item A type level expression.
- \item A variable reference of a runtime representable type.
- \item A variable reference or partial application of a function type.
- \stopitemize
- \stopitemize
-
- When looking at the arguments of a user-defined function, we can
- divide them into two categories:
- \startitemize
- \item Arguments of a runtime representable type (\eg bits or vectors).
-
- These arguments can be preserved in the program, since they can
- be translated to input ports later on. However, since we can
- only connect signals to input ports, these arguments must be
- reduced to simple variables (for which signals will be
- produced). This is taken care of by the argument extraction
- transform.
- \item Non-runtime representable typed arguments.
-
- These arguments cannot be preserved in the program, since we
- cannot represent them as input or output ports in the resulting
- \small{VHDL}. To remove them, we create a specialized version of the
- called function with these arguments filled in. This is done by
- the argument propagation transform.
-
- Typically, these arguments are type and dictionary arguments that are
- used to make functions polymorphic. By propagating these arguments, we
- are essentially doing the same which GHC does when it specializes
- functions: Creating multiple variants of the same function, one for
- each type for which it is used. Other common non-representable
- arguments are functions, e.g. when calling a higher order function
- with another function or a lambda abstraction as an argument.
-
- The reason for doing this is similar to the reasoning provided for
- the inlining of non-representable let bindings above. In fact, this
- argument propagation could be viewed as a form of cross-function
- inlining.
- \stopitemize
-
- TODO: Check the following itemization.
-
- When looking at the arguments of a builtin function, we can divide them
- into categories:
-
- \startitemize
- \item Arguments of a runtime representable type.
-
- As we have seen with user-defined functions, these arguments can
- always be reduced to a simple variable reference, by the
- argument extraction transform. Performing this transform for
- builtin functions as well, means that the translation of builtin
- functions can be limited to signal references, instead of
- needing to support all possible expressions.
-
- \item Arguments of a function type.
-
- These arguments are functions passed to higher order builtins,
- like \lam{map} and \lam{foldl}. Since implementing these
- functions for arbitrary function-typed expressions (\eg, lambda
- expressions) is rather comlex, we reduce these arguments to
- (partial applications of) global functions.
-
- We can still support arbitrary expressions from the user code,
- by creating a new global function containing that expression.
- This way, we can simply replace the argument with a reference to
- that new function. However, since the expression can contain any
- number of free variables we also have to include partial
- applications in our normal form.
-
- This category of arguments is handled by the function extraction
- transform.
- \item Other unrepresentable arguments.
-
- These arguments can take a few different forms:
- \startdesc{Type arguments}
- In the core language, type arguments can only take a single
- form: A type wrapped in the Type constructor. Also, there is
- nothing that can be done with type expressions, except for
- applying functions to them, so we can simply leave type
- arguments as they are.
- \stopdesc
- \startdesc{Dictionary arguments}
- In the core language, dictionary arguments are used to find
- operations operating on one of the type arguments (mostly for
- finding class methods). Since we will not actually evaluatie
- the function body for builtin functions and can generate
- code for builtin functions by just looking at the type
- arguments, these arguments can be ignored and left as they
- are.
- \stopdesc
- \startdesc{Type level arguments}
- Sometimes, we want to pass a value to a builtin function, but
- we need to know the value at compile time. Additionally, the
- value has an impact on the type of the function. This is
- encoded using type-level values, where the actual value of the
- argument is not important, but the type encodes some integer,
- for example. Since the value is not important, the actual form
- of the expression does not matter either and we can leave
- these arguments as they are.
- \stopdesc
- \startdesc{Other arguments}
- Technically, there is still a wide array of arguments that can
- be passed, but does not fall into any of the above categories.
- However, none of the supported builtin functions requires such
- an argument. This leaves use with passing unsupported types to
- a function, such as calling \lam{head} on a list of functions.
-
- In these cases, it would be impossible to generate hardware
- for such a function call anyway, so we can ignore these
- arguments.
-
- The only way to generate hardware for builtin functions with
- arguments like these, is to expand the function call into an
- equivalent core expression (\eg, expand map into a series of
- function applications). But for now, we choose to simply not
- support expressions like these.
- \stopdesc
-
- From the above, we can conclude that we can simply ignore these
- other unrepresentable arguments and focus on the first two
- categories instead.
- \stopitemize
-
- \subsubsection{Argument simplification}
- This transform deals with arguments to functions that
- are of a runtime representable type. It ensures that they will all become
- references to global variables, or local signals in the resulting \small{VHDL}.
-
- TODO: It seems we can map an expression to a port, not only a signal.
- Perhaps this makes this transformation not needed?
- TODO: Say something about dataconstructors (without arguments, like True
- or False), which are variable references of a runtime representable
- type, but do not result in a signal.
-
- To reduce a complex expression to a simple variable reference, we create
- a new let expression around the application, which binds the complex
- expression to a new variable. The original function is then applied to
- this variable.
+ \startbuffer[from]
+ x = λa. add 1 a
+ \stopbuffer
- \starttrans
- M N
- -------------------- \lam{N} is of a representable type
- letrec x = N in M x \lam{N} is not a local variable reference
- \stoptrans
+ \startbuffer[to]
+ x = λa. letrec
+ y = add 1 a
+ in
+ y
+ \stopbuffer
+ \transexample{retvalsimpllam}{Return value simplification with a lambda abstraction}{from}{to}
+
\startbuffer[from]
- add (add a 1) 1
+ x = letrec
+ a = add 1 2
+ in
+ add a 3
\stopbuffer
\startbuffer[to]
- letrec x = add a 1 in add x 1
+ x = letrec
+ a = add 1 2
+ y = add a 3
+ in
+ y
\stopbuffer
- \transexample{Argument extraction}{from}{to}
+ \transexample{retvalsimpllet}{Return value simplification with a let expression}{from}{to}
- \subsubsection{Function extraction}
- This transform deals with function-typed arguments to builtin functions.
- Since these arguments cannot be propagated, we choose to extract them
- into a new global function instead.
+ \subsection[sec:normalization:argsimpl]{Representable arguments simplification}
+ This section contains just a single transformation that deals with
+ representable arguments in applications. Non-representable arguments are
+ handled by the transformations in
+ \in{section}[sec:normalization:nonrep].
+
+ This transformation ensures that all representable arguments will become
+ references to local variables. This ensures they will become references
+ to local signals in the resulting \small{VHDL}, which is required due to
+ limitations in the component instantiation code in \VHDL\ (one can only
+ assign a signal or constant to an input port). By ensuring that all
+ arguments are always simple variable references, we always have a signal
+ available to map to the input ports.
+
+ To reduce a complex expression to a simple variable reference, we create
+ a new let expression around the application, which binds the complex
+ expression to a new variable. The original function is then applied to
+ this variable.
+
+ \refdef{global variable}
+ Note that references to \emph{global variables} (like a top level
+ function without arguments, but also an argumentless data-constructors
+ like \lam{True}) are also simplified. Only local variables generate
+ signals in the resulting architecture. Even though argumentless
+ data-constructors generate constants in generated \VHDL\ code and could be
+ mapped to an input port directly, they are still simplified to make the
+ normal form more regular.
+
+ \refdef{representable}
+ \starttrans
+ M N
+ -------------------- \lam{N} is representable
+ letrec x = N in M x \lam{N} is not a local variable reference
+ \stoptrans
+ \refdef{local variable}
+
+ \startbuffer[from]
+ add (add a 1) 1
+ \stopbuffer
+
+ \startbuffer[to]
+ letrec x = add a 1 in add x 1
+ \stopbuffer
+
+ \transexample{argsimpl}{Argument simplification}{from}{to}
+
+ \subsection[sec:normalization:built-ins]{Built-in functions}
+ This section deals with (arguments to) built-in functions. In the
+ intended normal form definition\refdef{intended normal form definition}
+ we can see that there are three sorts of arguments a built-in function
+ can receive.
+
+ \startitemize[KR]
+ \item A representable local variable reference. This is the most
+ common argument to any function. The argument simplification
+ transformation described in \in{section}[sec:normalization:argsimpl]
+ makes sure that \emph{any} representable argument to \emph{any}
+ function (including built-in functions) is turned into a local variable
+ reference.
+ \item (A partial application of) a top level function (either built-in on
+ user-defined). The function extraction transformation described in
+ this section takes care of turning every function-typed argument into
+ (a partial application of) a top level function.
+ \item Any expression that is not representable and does not have a
+ function type. Since these can be any expression, there is no
+ transformation needed. Note that this category is exactly all
+ expressions that are not transformed by the transformations for the
+ previous two categories. This means that \emph{any} Core expression
+ that is used as an argument to a built-in function will be either
+ transformed into one of the above categories, or end up in this
+ category. In any case, the result is in normal form.
+ \stopitemize
- Any free variables occuring in the extracted arguments will become
+ As noted, the argument simplification will handle any representable
+ arguments to a built-in function. The following transformation is needed
+ to handle non-representable arguments with a function type, all other
+ non-representable arguments do not need any special handling.
+
+ \subsubsection[sec:normalization:funextract]{Function extraction}
+ This transform deals with function-typed arguments to built-in
+ functions.
+ Since built-in functions cannot be specialized (see
+ \in{section}[sec:normalization:specialize]) to remove the arguments,
+ these arguments are extracted into a new global function instead. In
+ other words, we create a new top level function that has exactly the
+ extracted argument as its body. This greatly simplifies the
+ translation rules needed for built-in functions, since they only need
+ to handle (partial applications of) top level functions.
+
+ Any free variables occurring in the extracted arguments will become
parameters to the new global function. The original argument is replaced
with a reference to the new function, applied to any free variables from
the original argument.
- This transformation is useful when applying higher order builtin functions
+ This transformation is useful when applying higher-order built-in functions
like \hs{map} to a lambda abstraction, for example. In this case, the code
that generates \small{VHDL} for \hs{map} only needs to handle top level functions and
partial applications, not any other expression (such as lambda abstractions or
even more complicated expressions).
\starttrans
- M N \lam{M} is a (partial aplication of a) builtin function.
- --------------------- \lam{f0 ... fn} = free local variables of \lam{N}
+ M N \lam{M} is (a partial application of) a built-in function.
+ --------------------- \lam{f0 ... fn} are all free local variables of \lam{N}
M (x f0 ... fn) \lam{N :: a -> b}
~ \lam{N} is not a (partial application of) a top level function
x = λf0 ... λfn.N
\stoptrans
\startbuffer[from]
- map (λa . add a b) xs
-
- map (add b) ys
+ addList = λb.λxs.map (λa . add a b) xs
\stopbuffer
\startbuffer[to]
- map (x0 b) xs
-
- map x1 ys
+ addList = λb.λxs.map (f b) xs
~
- x0 = λb.λa.add a b
- x1 = λb.add b
+ f = λb.λa.add a b
\stopbuffer
- \transexample{Function extraction}{from}{to}
-
- Note that \lam{x0} and {x1} will still need normalization after this.
-
- \subsubsection{Argument propagation}
- This transform deals with arguments to user-defined functions that are
- not representable at runtime. This means these arguments cannot be
- preserved in the final form and most be {\em propagated}.
+ \transexample{funextract}{Function extraction}{from}{to}
- Propagation means to create a specialized version of the called
- function, with the propagated argument already filled in. As a simple
- example, in the following program:
-
- \startlambda
- f = λa.λb.a + b
- inc = λa.f a 1
- \stoplambda
+ Note that the function \lam{f} will still need normalization after
+ this.
- We could {\em propagate} the constant argument 1, with the following
- result:
+ \subsection{Case normalization}
+ The transformations in this section ensure that case statements end up
+ in normal form.
- \startlambda
- f' = λa.a + 1
- inc = λa.f' a
- \stoplambda
-
- Special care must be taken when the to-be-propagated expression has any
- free variables. If this is the case, the original argument should not be
- removed alltogether, but replaced by all the free variables of the
- expression. In this way, the original expression can still be evaluated
- inside the new function. Also, this brings us closer to our goal: All
- these free variables will be simple variable references.
-
- To prevent us from propagating the same argument over and over, a simple
- local variable reference is not propagated (since is has exactly one
- free variable, itself, we would only replace that argument with itself).
-
- This shows that any free local variables that are not runtime representable
- cannot be brought into normal form by this transform. We rely on an
- inlining transformation to replace such a variable with an expression we
- can propagate again.
-
- \starttrans
- x = E
- ~
- x Y0 ... Yi ... Yn \lam{Yi} is not of a runtime representable type
- --------------------------------------------- \lam{Yi} is not a local variable reference
- x' y0 ... yi-1 f0 ... fm Yi+1 ... Yn \lam{f0 ... fm} = free local vars of \lam{Yi}
- ~
- x' = λy0 ... yi-1 f0 ... fm yi+1 ... yn .
- E y0 ... yi-1 Yi yi+1 ... yn
-
- \stoptrans
-
- TODO: Example
-
- \subsection{Case simplification}
\subsubsection{Scrutinee simplification}
This transform ensures that the scrutinee of a case expression is always
a simple variable reference.
alts
----------------- \lam{E} is not a local variable reference
letrec x = E in
- case E of
+ case x of
alts
\stoptrans
False -> b
\stopbuffer
- \transexample{Let flattening}{from}{to}
+ \transexample{letflat}{Case normalization}{from}{to}
+
+
+ \placeintermezzo{}{
+ \defref{wild binders}
+ \startframedtext[width=7cm,background=box,frame=no]
+ \startalignment[center]
+ {\tfa Wild binders}
+ \stopalignment
+ \blank[medium]
+ In a functional expression, a \emph{wild binder} refers to any
+ binder that is never referenced. This means that even though it
+ will be bound to a particular value, that value is never used.
+
+ The Haskell syntax offers the underscore as a wild binder that
+ cannot even be referenced (It can be seen as introducing a new,
+ anonymous, binder every time it is used).
+
+ In these transformations, the term wild binder will sometimes be
+ used to indicate that a binder must not be referenced.
+ \stopframedtext
+ }
+
+ \subsubsection{Scrutinee binder removal}
+ This transformation removes (or rather, makes wild) the binder to
+ which the scrutinee is bound after evaluation. This is done by
+ replacing the bndr with the scrutinee in all alternatives. To prevent
+ duplication of work, this transformation is only applied when the
+ scrutinee is already a simple variable reference (but the previous
+ transformation ensures this will eventually be the case). The
+ scrutinee binder itself is replaced by a wild binder (which is no
+ longer displayed).
+
+ Note that one could argue that this transformation can change the
+ meaning of the Core expression. In the regular Core semantics, a case
+ expression forces the evaluation of its scrutinee and can be used to
+ implement strict evaluation. However, in the generated \VHDL,
+ evaluation is always strict. So the semantics we assign to the Core
+ expression (which differ only at this particular point), this
+ transformation is completely valid.
+
+ \starttrans
+ case x of bndr
+ alts
+ ----------------- \lam{x} is a local variable reference
+ case x of
+ alts[bndr=>x]
+ \stoptrans
+
+ \startbuffer[from]
+ case x of y
+ True -> y
+ False -> not y
+ \stopbuffer
+
+ \startbuffer[to]
+ case x of
+ True -> x
+ False -> not x
+ \stopbuffer
+
+ \transexample{scrutbndrremove}{Scrutinee binder removal}{from}{to}
+ \subsubsection{Case normalization}
+ This transformation ensures that all case expressions get a form
+ that is allowed by the intended normal form. This means they
+ will become one of:
- \subsubsection{Case simplification}
- This transformation ensures that all case expressions become normal form. This
- means they will become one of:
\startitemize
- \item An extractor case with a single alternative that picks a single field
- from a datatype, \eg \lam{case x of (a, b) -> a}.
+ \item An extractor case with a single alternative that picks a field
+ from a datatype, \eg\ \lam{case x of (a, b) ->
+ a}.\defref{extractor case}
\item A selector case with multiple alternatives and only wild binders, that
makes a choice between expressions based on the constructor of another
- expression, \eg \lam{case x of Low -> a; High -> b}.
+ expression, \eg\ \lam{case x of Low -> a; High ->
+ b}.\defref{selector case}
\stopitemize
+ For an arbitrary case, that has \lam{n} alternatives, with
+ \lam{m} binders in each alternatives, this will result in \lam{m
+ * n} extractor case expression to get at each variable, \lam{n}
+ let bindings for each of the alternatives' value and a single
+ selector case to select the right value out of these.
+
+ Technically, the definition of this transformation would require
+ that the constructor for every alternative has exactly the same
+ amount (\lam{m}) of arguments, but of course this transformation
+ also applies when this is not the case.
+
\starttrans
case E of
C0 v0,0 ... v0,m -> E0
\vdots
Cn vn,0 ... vn,m -> En
- --------------------------------------------------- \forall i \forall j, 0 <= i <= n, 0 <= i < m (\lam{wi,j} is a wild (unused) binder)
- letrec
- v0,0 = case x of C0 v0,0 .. v0,m -> v0,0
+ --------------------------------------------------- \lam{\forall i \forall j, 0 ≤ i ≤ n, 0 ≤ i < m} (\lam{wi,j} is a wild (unused) binder)
+ letrec The case expression is not an extractor case
+ v0,0 = case E of C0 x0,0 .. x0,m -> x0,0 The case expression is not a selector case
\vdots
- v0,m = case x of C0 v0,0 .. v0,m -> v0,m
- x0 = E0
- \dots
- vn,m = case x of Cn vn,0 .. vn,m -> vn,m
- xn = En
+ v0,m = case E of C0 x0,0 .. x0,m -> x0,m
+ \vdots
+ vn,m = case E of Cn xn,0 .. xn,m -> xn,m
+ y0 = E0
+ \vdots
+ yn = En
in
case E of
- C0 w0,0 ... w0,m -> x0
+ C0 w0,0 ... w0,m -> y0
\vdots
- Cn wn,0 ... wn,m -> xn
+ Cn wn,0 ... wn,m -> yn
\stoptrans
- TODO: This transformation specified like this is complicated and misses
- conditions to prevent looping with itself. Perhaps we should split it here for
- discussion?
+ Note that this transformation applies to case expressions with any
+ scrutinee. If the scrutinee is a complex expression, this might
+ result in duplication of work (hardware). An extra condition to
+ only apply this transformation when the scrutinee is already
+ simple (effectively causing this transformation to be only
+ applied after the scrutinee simplification transformation) might
+ be in order.
\startbuffer[from]
case a of
\stopbuffer
\startbuffer[to]
- letnonrec
+ letrec
x0 = add b 1
x1 = add b 2
in
False -> x1
\stopbuffer
- \transexample{Selector case simplification}{from}{to}
+ \transexample{selcasesimpl}{Selector case simplification}{from}{to}
\startbuffer[from]
case a of
(,) w0 w1 -> x0
\stopbuffer
- \transexample{Extractor case simplification}{from}{to}
+ \transexample{excasesimpl}{Extractor case simplification}{from}{to}
- \subsubsection{Case removal}
- This transform removes any case statements with a single alternative and
- only wild binders.
+ \refdef{selector case}
+ In \in{example}[ex:trans:excasesimpl] the case expression is expanded
+ into multiple case expressions, including a pretty useless expression
+ (that is neither a selector or extractor case). This case can be
+ removed by the Case removal transformation in
+ \in{section}[sec:transformation:caseremoval].
- These "useless" case statements are usually leftovers from case simplification
+ \subsubsection[sec:transformation:caseremoval]{Case removal}
+ This transform removes any case expression with a single alternative and
+ only wild binders.\refdef{wild binders}
+
+ These "useless" case expressions are usually leftovers from case simplification
on extractor case (see the previous example).
\starttrans
case x of
C v0 ... vm -> E
- ---------------------- \lam{\forall i, 0 <= i <= m} (\lam{vi} does not occur free in E)
+ ---------------------- \lam{\forall i, 0 ≤ i ≤ m} (\lam{vi} does not occur free in E)
E
\stoptrans
x0
\stopbuffer
- \transexample{Case removal}{from}{to}
+ \transexample{caserem}{Case removal}{from}{to}
+
+ \subsection[sec:normalization:nonrep]{Removing unrepresentable values}
+ The transformations in this section are aimed at making all the
+ values used in our expression representable. There are two main
+ transformations that are applied to \emph{all} unrepresentable let
+ bindings and function arguments. These are meant to address three
+ different kinds of unrepresentable values: polymorphic values,
+ higher-order values and literals. The transformation are described
+ generically: they apply to all non-representable values. However,
+ non-representable values that do not fall into one of these three
+ categories will be moved around by these transformations but are
+ unlikely to completely disappear. They usually mean the program was not
+ valid in the first place, because unsupported types were used (for
+ example, a program using strings).
+
+ Each of these three categories will be detailed below, followed by the
+ actual transformations.
+
+ \subsubsection{Removing Polymorphism}
+ As noted in \in{section}[sec:prototype:coretypes],
+ polymorphism is made explicit in Core through type and
+ dictionary arguments. To remove the polymorphism from a
+ function, we can simply specialize the polymorphic function for
+ the particular type applied to it. The same goes for dictionary
+ arguments. To remove polymorphism from let bound values, we
+ simply inline the let bindings that have a polymorphic type,
+ which should (eventually) make sure that the polymorphic
+ expression is applied to a type and/or dictionary, which can
+ then be removed by β-reduction (\in{section}[sec:normalization:beta]).
+
+ Since both type and dictionary arguments are not representable,
+ \refdef{representable}
+ the non-representable argument specialization and
+ non-representable let binding inlining transformations below
+ take care of exactly this.
+
+ There is one case where polymorphism cannot be completely
+ removed: built-in functions are still allowed to be polymorphic
+ (Since we have no function body that we could properly
+ specialize). However, the code that generates \VHDL\ for built-in
+ functions knows how to handle this, so this is not a problem.
+
+ \subsubsection[sec:normalization:defunctionalization]{Defunctionalization}
+ These transformations remove higher-order expressions from our
+ program, making all values first-order. The approach used for
+ defunctionalization uses a combination of specialization, inlining and
+ some cleanup transformations, was also proposed in parallel research
+ by Neil Mitchell \cite[mitchell09].
+
+ Higher order values are always introduced by lambda abstractions, none
+ of the other Core expression elements can introduce a function type.
+ However, other expressions can \emph{have} a function type, when they
+ have a lambda expression in their body.
+
+ For example, the following expression is a higher-order expression
+ that is not a lambda expression itself:
+
+ \refdef{id function}
+ \startlambda
+ case x of
+ High -> id
+ Low -> λx.x
+ \stoplambda
+
+ The reference to the \lam{id} function shows that we can introduce a
+ higher-order expression in our program without using a lambda
+ expression directly. However, inside the definition of the \lam{id}
+ function, we can be sure that a lambda expression is present.
+
+ Looking closely at the definition of our normal form in
+ \in{section}[sec:normalization:intendednormalform], we can see that
+ there are three possibilities for higher-order values to appear in our
+ intended normal form:
+
+ \startitemize[KR]
+ \item[item:toplambda] Lambda abstractions can appear at the highest level of a
+ top level function. These lambda abstractions introduce the
+ arguments (input ports / current state) of the function.
+ \item[item:built-inarg] (Partial applications of) top level functions can appear as an
+ argument to a built-in function.
+ \item[item:completeapp] (Partial applications of) top level functions can appear in
+ function position of an application. Since a partial application
+ cannot appear anywhere else (except as built-in function arguments),
+ all partial applications are applied, meaning that all applications
+ will become complete applications. However, since application of
+ arguments happens one by one, in the expression:
+ \startlambda
+ f 1 2
+ \stoplambda
+ the sub-expression \lam{f 1} has a function type. But this is
+ allowed, since it is inside a complete application.
+ \stopitemize
+
+ We will take a typical function with some higher-order values as an
+ example. The following function takes two arguments: a \lam{Bit} and a
+ list of numbers. Depending on the first argument, each number in the
+ list is doubled, or the list is returned unmodified. For the sake of
+ the example, no polymorphism is shown. In reality, at least map would
+ be polymorphic.
+
+ \startlambda
+ λy.let double = λx. x + x in
+ case y of
+ Low -> map double
+ High -> λz. z
+ \stoplambda
+
+ This example shows a number of higher-order values that we cannot
+ translate to \VHDL\ directly. The \lam{double} binder bound in the let
+ expression has a function type, as well as both of the alternatives of
+ the case expression. The first alternative is a partial application of
+ the \lam{map} built-in function, whereas the second alternative is a
+ lambda abstraction.
+
+ To reduce all higher-order values to one of the above items, a number
+ of transformations we have already seen are used. The η-expansion
+ transformation from \in{section}[sec:normalization:eta] ensures all
+ function arguments are introduced by lambda abstraction on the highest
+ level of a function. These lambda arguments are allowed because of
+ \in{item}[item:toplambda] above. After η-expansion, our example
+ becomes a bit bigger:
+
+ \startlambda
+ λy.λq.(let double = λx. x + x in
+ case y of
+ Low -> map double
+ High -> λz. z
+ ) q
+ \stoplambda
+
+ η-expansion also introduces extra applications (the application of
+ the let expression to \lam{q} in the above example). These
+ applications can then propagated down by the application propagation
+ transformation (\in{section}[sec:normalization:appprop]). In our
+ example, the \lam{q} and \lam{r} variable will be propagated into the
+ let expression and then into the case expression:
+
+ \startlambda
+ λy.λq.let double = λx. x + x in
+ case y of
+ Low -> map double q
+ High -> (λz. z) q
+ \stoplambda
+
+ This propagation makes higher-order values become applied (in
+ particular both of the alternatives of the case now have a
+ representable type). Completely applied top level functions (like the
+ first alternative) are now no longer invalid (they fall under
+ \in{item}[item:completeapp] above). (Completely) applied lambda
+ abstractions can be removed by β-expansion. For our example,
+ applying β-expansion results in the following:
+
+ \startlambda
+ λy.λq.let double = λx. x + x in
+ case y of
+ Low -> map double q
+ High -> q
+ \stoplambda
+
+ As you can see in our example, all of this moves applications towards
+ the higher-order values, but misses higher-order functions bound by
+ let expressions. The applications cannot be moved towards these values
+ (since they can be used in multiple places), so the values will have
+ to be moved towards the applications. This is achieved by inlining all
+ higher-order values bound by let applications, by the
+ non-representable binding inlining transformation below. When applying
+ it to our example, we get the following:
+
+ \startlambda
+ λy.λq.case y of
+ Low -> map (λx. x + x) q
+ High -> q
+ \stoplambda
+
+ We have nearly eliminated all unsupported higher-order values from this
+ expressions. The one that is remaining is the first argument to the
+ \lam{map} function. Having higher-order arguments to a built-in
+ function like \lam{map} is allowed in the intended normal form, but
+ only if the argument is a (partial application) of a top level
+ function. This is easily done by introducing a new top level function
+ and put the lambda abstraction inside. This is done by the function
+ extraction transformation from
+ \in{section}[sec:normalization:funextract].
+
+ \startlambda
+ λy.λq.case y of
+ Low -> map func q
+ High -> q
+ \stoplambda
+
+ This also introduces a new function, that we have called \lam{func}:
+
+ \startlambda
+ func = λx. x + x
+ \stoplambda
+
+ Note that this does not actually remove the lambda, but now it is a
+ lambda at the highest level of a function, which is allowed in the
+ intended normal form.
- \subsection{Removing polymorphism}
- Reference type-specialization (== argument propagation)
+ There is one case that has not been discussed yet. What if the
+ \lam{map} function in the example above was not a built-in function
+ but a user-defined function? Then extracting the lambda expression
+ into a new function would not be enough, since user-defined functions
+ can never have higher-order arguments. For example, the following
+ expression shows an example:
- Reference polymporphic binding inlining (== non-representable binding
- inlining).
+ \startlambda
+ twice :: (Word -> Word) -> Word -> Word
+ twice = λf.λa.f (f a)
+
+ main = λa.app (λx. x + x) a
+ \stoplambda
- \subsection{Defunctionalization}
- These transformations remove most higher order expressions from our
- program, making it completely first-order (the only exception here is for
- arguments to builtin functions, since we can't specialize builtin
- function. TODO: Talk more about this somewhere).
+ This example shows a function \lam{twice} that takes a function as a
+ first argument and applies that function twice to the second argument.
+ Again, we have made the function monomorphic for clarity, even though
+ this function would be a lot more useful if it was polymorphic. The
+ function \lam{main} uses \lam{twice} to apply a lambda expression twice.
- Reference higher-order-specialization (== argument propagation)
+ When faced with a user defined function, a body is available for that
+ function. This means we could create a specialized version of the
+ function that only works for this particular higher-order argument
+ (\ie, we can just remove the argument and call the specialized
+ function without the argument). This transformation is detailed below.
+ Applying this transformation to the example gives:
- \subsubsection{Non-representable binding inlining}
- This transform inlines let bindings that have a non-representable type. Since
- we can never generate a signal assignment for these bindings (we cannot
- declare a signal assignment with a non-representable type, for obvious
- reasons), we have no choice but to inline the binding to remove it.
+ \startlambda
+ twice' :: Word -> Word
+ twice' = λb.(λf.λa.f (f a)) (λx. x + x) b
- If the binding is non-representable because it is a lambda abstraction, it is
- likely that it will inlined into an application and β-reduction will remove
- the lambda abstraction and turn it into a representable expression at the
- inline site. The same holds for partial applications, which can be turned into
- full applications by inlining.
+ main = λa.app' a
+ \stoplambda
- Other cases of non-representable bindings we see in practice are primitive
- Haskell types. In most cases, these will not result in a valid normalized
- output, but then the input would have been invalid to start with. There is one
- exception to this: When a builtin function is applied to a non-representable
- expression, things might work out in some cases. For example, when you write a
- literal \hs{SizedInt} in Haskell, like \hs{1 :: SizedInt D8}, this results in
- the following core: \lam{fromInteger (smallInteger 10)}, where for example
- \lam{10 :: GHC.Prim.Int\#} and \lam{smallInteger 10 :: Integer} have
- non-representable types. TODO: This/these paragraph(s) should probably become a
- separate discussion somewhere else.
+ The \lam{main} function is now in normal form, since the only
+ higher-order value there is the top level lambda expression. The new
+ \lam{twice'} function is a bit complex, but the entire original body
+ of the original \lam{twice} function is wrapped in a lambda
+ abstraction and applied to the argument we have specialized for
+ (\lam{λx. x + x}) and the other arguments. This complex expression can
+ fortunately be effectively reduced by repeatedly applying β-reduction:
+ \startlambda
+ twice' :: Word -> Word
+ twice' = λb.(b + b) + (b + b)
+ \stoplambda
+
+ This example also shows that the resulting normal form might not be as
+ efficient as we might hope it to be (it is calculating \lam{(b + b)}
+ twice). This is discussed in more detail in
+ \in{section}[sec:normalization:duplicatework].
+
+ \subsubsection[sec:normalization:literals]{Literals}
+ There are a limited number of literals available in Haskell and Core.
+ \refdef{enumerated types} When using (enumerating) algebraic
+ data-types, a literal is just a reference to the corresponding data
+ constructor, which has a representable type (the algebraic datatype)
+ and can be translated directly. This also holds for literals of the
+ \hs{Bool} Haskell type, which is just an enumerated type.
+
+ There is, however, a second type of literal that does not have a
+ representable type: integer literals. Cλash supports using integer
+ literals for all three integer types supported (\hs{SizedWord},
+ \hs{SizedInt} and \hs{RangedWord}). This is implemented using
+ Haskell's \hs{Num} type class, which offers a \hs{fromInteger} method
+ that converts any \hs{Integer} to the Cλash data-types.
+
+ When \GHC\ sees integer literals, it will automatically insert calls to
+ the \hs{fromInteger} method in the resulting Core expression. For
+ example, the following expression in Haskell creates a 32 bit unsigned
+ word with the value 1. The explicit type signature is needed, since
+ there is no context for \GHC\ to determine the type from otherwise.
+
+ \starthaskell
+ 1 :: SizedWord D32
+ \stophaskell
+
+ This Haskell code results in the following Core expression:
+
+ \startlambda
+ fromInteger @(SizedWord D32) \$dNum (smallInteger 10)
+ \stoplambda
+ The literal 10 will have the type \lam{GHC.Prim.Int\#}, which is
+ converted into an \lam{Integer} by \lam{smallInteger}. Finally, the
+ \lam{fromInteger} function will finally convert this into a
+ \lam{SizedWord D32}.
+
+ Both the \lam{GHC.Prim.Int\#} and \lam{Integer} types are not
+ representable, and cannot be translated directly. Fortunately, there
+ is no need to translate them, since \lam{fromInteger} is a built-in
+ function that knows how to handle these values. However, this does
+ require that the \lam{fromInteger} function is directly applied to
+ these non-representable literal values, otherwise errors will occur.
+ For example, the following expression is not in the intended normal
+ form, since one of the let bindings has an unrepresentable type
+ (\lam{Integer}):
+
+ \startlambda
+ let l = smallInteger 10 in fromInteger @(SizedWord D32) \$dNum l
+ \stoplambda
+
+ By inlining these let-bindings, we can ensure that unrepresentable
+ literals bound by a let binding end up in an application of the
+ appropriate built-in function, where they are allowed. Since it is
+ possible that the application of that function is in a different
+ function than the definition of the literal value, we will always need
+ to specialize away any unrepresentable literals that are used as
+ function arguments. The following two transformations do exactly this.
+
+ \subsubsection[sec:normalization:nonrepinline]{Non-representable binding inlining}
+ This transform inlines let bindings that are bound to a
+ non-representable value. Since we can never generate a signal
+ assignment for these bindings (we cannot declare a signal assignment
+ with a non-representable type, for obvious reasons), we have no choice
+ but to inline the binding to remove it.
+
+ As we have seen in the previous sections, inlining these bindings
+ solves (part of) the polymorphism, higher-order values and
+ unrepresentable literals in an expression.
+
+ \refdef{substitution notation}
\starttrans
letrec
a0 = E0
M
-------------------------- \lam{Ei} has a non-representable type.
letrec
- a0 = E0 [Ei/ai]
- \vdots
- ai-1 = Ei-1 [Ei/ai]
- ai+1 = Ei+1 [Ei/ai]
+ a0 = E0 [ai=>Ei] \vdots
+ ai-1 = Ei-1 [ai=>Ei]
+ ai+1 = Ei+1 [ai=>Ei]
\vdots
- an = En [Ei/ai]
+ an = En [ai=>Ei]
in
- M[Ei/ai]
+ M[ai=>Ei]
\stoptrans
\startbuffer[from]
(λb -> add b 1) (add 1 x)
\stopbuffer
- \transexample{None representable binding inlining}{from}{to}
+ \transexample{nonrepinline}{Non-representable binding inlining}{from}{to}
+
+ \subsubsection[sec:normalization:specialize]{Function specialization}
+ This transform removes arguments to user-defined functions that are
+ not representable at run-time. This is done by creating a
+ \emph{specialized} version of the function that only works for one
+ particular value of that argument (in other words, the argument can be
+ removed).
+
+ Specialization means to create a specialized version of the called
+ function, with one argument already filled in. As a simple example, in
+ the following program (this is not actual Core, since it directly uses
+ a literal with the unrepresentable type \lam{GHC.Prim.Int\#}).
+
+ \startlambda
+ f = λa.λb.a + b
+ inc = λa.f a 1
+ \stoplambda
+
+ We could specialize the function \lam{f} against the literal argument
+ 1, with the following result:
+
+ \startlambda
+ f' = λa.a + 1
+ inc = λa.f' a
+ \stoplambda
+
+ In some way, this transformation is similar to β-reduction, but it
+ operates across function boundaries. It is also similar to
+ non-representable let binding inlining above, since it sort of
+ \quote{inlines} an expression into a called function.
+
+ Special care must be taken when the argument has any free variables.
+ If this is the case, the original argument should not be removed
+ completely, but replaced by all the free variables of the expression.
+ In this way, the original expression can still be evaluated inside the
+ new function.
+
+ To prevent us from propagating the same argument over and over, a
+ simple local variable reference is not propagated (since is has
+ exactly one free variable, itself, we would only replace that argument
+ with itself).
+
+ This shows that any free local variables that are not run-time
+ representable cannot be brought into normal form by this transform. We
+ rely on an inlining or β-reduction transformation to replace such a
+ variable with an expression we can propagate again.
+
+ \starttrans
+ x = E
+ ~
+ x Y0 ... Yi ... Yn \lam{Yi} is not representable
+ --------------------------------------------- \lam{Yi} is not a local variable reference
+ x' Y0 ... Yi-1 f0 ... fm Yi+1 ... Yn \lam{f0 ... fm} are all free local vars of \lam{Yi}
+ ~ \lam{T0 ... Tn} are the types of \lam{Y0 ... Yn}
+ x' = λ(y0 :: T0) ... λ(yi-1 :: Ty-1).
+ λf0 ... λfm.
+ λ(yi+1 :: Ty+1) ... λ(yn :: Tn).
+ E y0 ... yi-1 Yi yi+1 ... yn
+ \stoptrans
+
+ This is a bit of a complex transformation. It transforms an
+ application of the function \lam{x}, where one of the arguments
+ (\lam{Y_i}) is not representable. A new
+ function \lam{x'} is created that wraps the body of the old function.
+ The body of the new function becomes a number of nested lambda
+ abstractions, one for each of the original arguments that are left
+ unchanged.
+
+ The ith argument is replaced with the free variables of
+ \lam{Y_i}. Note that we reuse the same binders as those used in
+ \lam{Y_i}, since we can then just use \lam{Y_i} inside the new
+ function body and have all of the variables it uses be in scope.
+
+ The argument that we are specializing for, \lam{Y_i}, is put inside
+ the new function body. The old function body is applied to it. Since
+ we use this new function only in place of an application with that
+ particular argument \lam{Y_i}, behavior should not change.
+
+ Note that the types of the arguments of our new function are taken
+ from the types of the \emph{actual} arguments (\lam{T0 ... Tn}). This
+ means that any polymorphism in the arguments is removed, even when the
+ corresponding explicit type lambda is not removed
+ yet.
+
+ \todo{Examples. Perhaps reference the previous sections}
+
+ \section{Unsolved problems}
+ The above system of transformations has been implemented in the prototype
+ and seems to work well to compile simple and more complex examples of
+ hardware descriptions. \todo{Ref christiaan?} However, this normalization
+ system has not seen enough review and work to be complete and work for
+ every Core expression that is supplied to it. A number of problems
+ have already been identified and are discussed in this section.
+
+ \subsection[sec:normalization:duplicatework]{Work duplication}
+ A possible problem of β-reduction is that it could duplicate work.
+ When the expression applied is not a simple variable reference, but
+ requires calculation and the binder the lambda abstraction binds to
+ is used more than once, more hardware might be generated than strictly
+ needed.
+
+ As an example, consider the expression:
+
+ \startlambda
+ (λx. x + x) (a * b)
+ \stoplambda
+
+ When applying β-reduction to this expression, we get:
+
+ \startlambda
+ (a * b) + (a * b)
+ \stoplambda
+ which of course calculates \lam{(a * b)} twice.
+
+ A possible solution to this would be to use the following alternative
+ transformation, which is of course no longer normal β-reduction. The
+ following transformation has not been tested in the prototype, but is
+ given here for future reference:
+
+ \starttrans
+ (λx.E) M
+ -----------------
+ letrec x = M in E
+ \stoptrans
+
+ This does not seem like much of an improvement, but it does get rid of
+ the lambda expression (and the associated higher-order value), while
+ at the same time introducing a new let binding. Since the result of
+ every application or case expression must be bound by a let expression
+ in the intended normal form anyway, this is probably not a problem. If
+ the argument happens to be a variable reference, then simple let
+ binding removal (\in{section}[sec:normalization:simplelet]) will
+ remove it, making the result identical to that of the original
+ β-reduction transformation.
+
+ When also applying argument simplification to the above example, we
+ get the following expression:
- \section{Provable properties}
+ \startlambda
+ let y = (a * b)
+ z = (a * b)
+ in y + z
+ \stoplambda
+
+ Looking at this, we could imagine an alternative approach: create a
+ transformation that removes let bindings that bind identical values.
+ In the above expression, the \lam{y} and \lam{z} variables could be
+ merged together, resulting in the more efficient expression:
+
+ \startlambda
+ let y = (a * b) in y + y
+ \stoplambda
+
+ \subsection[sec:normalization:non-determinism]{Non-determinism}
+ As an example, again consider the following expression:
+
+ \startlambda
+ (λx. x + x) (a * b)
+ \stoplambda
+
+ We can apply both β-reduction (\in{section}[sec:normalization:beta])
+ as well as argument simplification
+ (\in{section}[sec:normalization:argsimpl]) to this expression.
+
+ When applying argument simplification first and then β-reduction, we
+ get the following expression:
+
+ \startlambda
+ let y = (a * b) in y + y
+ \stoplambda
+
+ When applying β-reduction first and then argument simplification, we
+ get the following expression:
+
+ \startlambda
+ let y = (a * b)
+ z = (a * b)
+ in y + z
+ \stoplambda
+
+ As you can see, this is a different expression. This means that the
+ order of expressions, does in fact change the resulting normal form,
+ which is something that we would like to avoid. In this particular
+ case one of the alternatives is even clearly more efficient, so we
+ would of course like the more efficient form to be the normal form.
+
+ For this particular problem, the solutions for duplication of work
+ seem from the previous section seem to fix the determinism of our
+ transformation system as well. However, it is likely that there are
+ other occurrences of this problem.
+
+ \subsection[sec:normalization:castproblems]{Casts}
+ We do not fully understand the use of cast expressions in Core, so
+ there are probably expressions involving cast expressions that cannot
+ be brought into intended normal form by this transformation system.
+
+ The uses of casts in the Core system should be investigated more and
+ transformations will probably need updating to handle them in all
+ cases.
+
+ \subsection[sec:normalization:stateproblems]{Normalization of stateful descriptions}
+ Currently, the intended normal form definition\refdef{intended
+ normal form definition} offers enough freedom to describe all
+ valid stateful descriptions, but is not limiting enough. It is
+ possible to write descriptions which are in intended normal
+ form, but cannot be translated into \VHDL\ in a meaningful way
+ (\eg, a function that swaps two substates in its result, or a
+ function that changes a sub-state itself instead of passing it to
+ a sub-function).
+
+ It is now up to the programmer to not do anything funny with
+ these state values, whereas the normalization just tries not to
+ mess up the flow of state values. In practice, there are
+ situations where a Core program that \emph{could} be a valid
+ stateful description is not translatable by the prototype. This
+ most often happens when statefulness is mixed with pattern
+ matching, causing a state input to be unpacked multiple times or
+ be unpacked and repacked only in some of the code paths.
+
+ Without going into detail about the exact problems (of which
+ there are probably more than have shown up so far), it seems
+ unlikely that these problems can be solved entirely by just
+ improving the \VHDL\ state generation in the final stage. The
+ normalization stage seems the best place to apply the rewriting
+ needed to support more complex stateful descriptions. This does
+ of course mean that the intended normal form definition must be
+ extended as well to be more specific about how state handling
+ should look like in normal form.
+ \in{Section}[sec:prototype:statelimits] already contains a
+ tight description of the limitations on the use of state
+ variables, which could be adapted into the intended normal form.
+
+ \section[sec:normalization:properties]{Provable properties}
When looking at the system of transformations outlined above, there are a
number of questions that we can ask ourselves. The main question is of course:
\quote{Does our system work as intended?}. We can split this question into a
- number of subquestions:
+ number of sub-questions:
\startitemize[KR]
\item[q:termination] Does our system \emph{terminate}? Since our system will
keep running as long as transformations apply, there is an obvious risk that
- it will keep running indefinitely. One transformation produces a result that
- is transformed back to the original by another transformation, for example.
+ it will keep running indefinitely. This typically happens when one
+ transformation produces a result that is transformed back to the original
+ by another transformation, or when one or more transformations keep
+ expanding some expression.
\item[q:soundness] Is our system \emph{sound}? Since our transformations
continuously modify the expression, there is an obvious risk that the final
- normal form will not be equivalent to the original program: Its meaning could
+ normal form will not be equivalent to the original program: its meaning could
have changed.
\item[q:completeness] Is our system \emph{complete}? Since we have a complex
system of transformations, there is an obvious risk that some expressions will
not end up in our intended normal form, because we forgot some transformation.
- In other words: Does our transformation system result in our intended normal
+ In other words: does our transformation system result in our intended normal
form for all possible inputs?
\item[q:determinism] Is our system \emph{deterministic}? Since we have defined
no particular order in which the transformation should be applied, there is an
\emph{different} normal forms. They might still both be intended normal forms
(if our system is \emph{complete}) and describe correct hardware (if our
system is \emph{sound}), so this property is less important than the previous
- three: The translator would still function properly without it.
+ three: the translator would still function properly without it.
\stopitemize
+ Unfortunately, the final transformation system has only been
+ developed in the final part of the research, leaving no more time
+ for verifying these properties. In fact, it is likely that the
+ current transformation system still violates some of these
+ properties in some cases (see
+ \in{section}[sec:normalization:non-determinism] and
+ \in{section}[sec:normalization:stateproblems]) and should be improved (or
+ extra conditions on the input hardware descriptions should be formulated).
+
+ This is most likely the case with the completeness and determinism
+ properties, perhaps also the termination property. The soundness
+ property probably holds, since it is easier to manually verify (each
+ transformation can be reviewed separately).
+
+ Even though no complete proofs have been made, some ideas for
+ possible proof strategies are shown below.
+
\subsection{Graph representation}
- Before looking into how to prove these properties, we'll look at our
- transformation system from a graph perspective. The nodes of the graph are
- all possible Core expressions. The (directed) edges of the graph are
- transformations. When a transformation α applies to an expression \lam{A} to
- produce an expression \lam{B}, we add an edge from the node for \lam{A} to the
- node for \lam{B}, labeled α.
+ Before looking into how to prove these properties, we will look at
+ transformation systems from a graph perspective. We will first define
+ the graph view and then illustrate it using a simple example from lambda
+ calculus (which is a different system than the Cλash normalization
+ system). The nodes of the graph are all possible Core expressions. The
+ (directed) edges of the graph are transformations. When a transformation
+ α applies to an expression \lam{A} to produce an expression \lam{B}, we
+ add an edge from the node for \lam{A} to the node for \lam{B}, labeled
+ α.
\startuseMPgraphic{TransformGraph}
save a, b, c, d;
drawObj(a, b, c, d);
\stopuseMPgraphic
- \placeexample[right][ex:TransformGraph]{Partial graph of a labmda calculus
+ \placeexample[right][ex:TransformGraph]{Partial graph of a lambda calculus
system with β and η reduction (solid lines) and expansion (dotted lines).}
\boxedgraphic{TransformGraph}
- Of course our graph is unbounded, since we can construct an infinite amount of
- Core expressions. Also, there might potentially be multiple edges between two
- given nodes (with different labels), though seems unlikely to actually happen
- in our system.
+ Of course the graph for Cλash is unbounded, since we can construct an
+ infinite amount of Core expressions. Also, there might potentially be
+ multiple edges between two given nodes (with different labels), though
+ this seems unlikely to actually happen in our system.
See \in{example}[ex:TransformGraph] for the graph representation of a very
simple lambda calculus that contains just the expressions \lam{(λx.λy. (+) x
y) 1}, \lam{λy. (+) 1 y}, \lam{(λx.(+) x) 1} and \lam{(+) 1}. The
transformation system consists of β-reduction and η-reduction (solid edges) or
- β-reduction and η-reduction (dotted edges).
+ β-expansion and η-expansion (dotted edges).
- TODO: Define β-reduction and η-reduction?
+ \todo{Define β-reduction and η-reduction?}
- Note that the normal form of such a system consists of the set of nodes
- (expressions) without outgoing edges, since those are the expression to which
- no transformation applies anymore. We call this set of nodes the \emph{normal
- set}.
+ In such a graph a node (expression) is in normal form if it has no
+ outgoing edges (meaning no transformation applies to it). The set of
+ nodes without outgoing edges is called the \emph{normal set}. Similarly,
+ the set of nodes containing expressions in intended normal form
+ \refdef{intended normal form definition} is called the \emph{intended normal set}.
From such a graph, we can derive some properties easily:
\startitemize[KR]
- \item A system will \emph{terminate} if there is no path of infinite length
- in the graph (this includes cycles).
+ \item A system will \emph{terminate} if there is no walk (sequence of
+ edges, or transformations) of infinite length in the graph (this
+ includes cycles, but can also happen without cycles).
\item Soundness is not easily represented in the graph.
\item A system is \emph{complete} if all of the nodes in the normal set have
the intended normal form. The inverse (that all of the nodes outside of
the normal set are \emph{not} in the intended normal form) is not
- strictly required.
- \item A system is deterministic if all paths from a node, which end in a node
- in the normal set, end at the same node.
+ strictly required. In other words, our normal set must be a
+ subset of the intended normal form, but they do not need to be
+ the same set.
+ form.
+ \item A system is deterministic if all paths starting at a particular
+ node, which end in a node in the normal set, end at the same node.
\stopitemize
When looking at the \in{example}[ex:TransformGraph], we see that the system
terminates for both the reduction and expansion systems (but note that, for
- expansion, this is only true because we've limited the possible expressions!
- In comlete lambda calculus, there would be a path from \lam{(λx.λy. (+) x y)
- 1} to \lam{(λx.λy.(λz.(+) z) x y) 1} to \lam{(λx.λy.(λz.(λq.(+) q) z) x y) 1}
- etc.)
+ expansion, this is only true because we have limited the possible
+ expressions. In complete lambda calculus, there would be a path from
+ \lam{(λx.λy. (+) x y) 1} to \lam{(λx.λy.(λz.(+) z) x y) 1} to
+ \lam{(λx.λy.(λz.(λq.(+) q) z) x y) 1} etc.)
- If we would consider the system with both expansion and reduction, there would
- no longer be termination, since there would be cycles all over the place.
+ If we would consider the system with both expansion and reduction, there
+ would no longer be termination either, since there would be cycles all
+ over the place.
The reduction and expansion systems have a normal set of containing just
\lam{(+) 1} or \lam{(λx.λy. (+) x y) 1} respectively. Since all paths in
either system end up in these normal forms, both systems are \emph{complete}.
- Also, since there is only one normal form, it must obviously be
+ Also, since there is only one node in the normal set, it must obviously be
\emph{deterministic} as well.
\subsection{Termination}
- Approach: Counting.
-
- Church-Rosser?
+ In general, proving termination of an arbitrary program is a very
+ hard problem. \todo{Ref about arbitrary termination} Fortunately,
+ we only have to prove termination for our specific transformation
+ system.
+
+ A common approach for these kinds of proofs is to associate a
+ measure with each possible expression in our system. If we can
+ show that each transformation strictly decreases this measure
+ (\ie, the expression transformed to has a lower measure than the
+ expression transformed from). \todo{ref about measure-based
+ termination proofs / analysis}
+
+ A good measure for a system consisting of just β-reduction would
+ be the number of lambda expressions in the expression. Since every
+ application of β-reduction removes a lambda abstraction (and there
+ is always a bounded number of lambda abstractions in every
+ expression) we can easily see that a transformation system with
+ just β-reduction will always terminate.
+
+ For our complete system, this measure would be fairly complex
+ (probably the sum of a lot of things). Since the (conditions on)
+ our transformations are pretty complex, we would need to include
+ both simple things like the number of let expressions as well as
+ more complex things like the number of case expressions that are
+ not yet in normal form.
+
+ No real attempt has been made at finding a suitable measure for
+ our system yet.
\subsection{Soundness}
- Needs formal definition of semantics.
- Prove for each transformation seperately, implies soundness of the system.
-
+ Soundness is a property that can be proven for each transformation
+ separately. Since our system only runs separate transformations
+ sequentially, if each of our transformations leaves the
+ \emph{meaning} of the expression unchanged, then the entire system
+ will of course leave the meaning unchanged and is thus
+ \emph{sound}.
+
+ The current prototype has only been verified in an ad hoc fashion
+ by inspecting (the code for) each transformation. A more formal
+ verification would be more appropriate.
+
+ To be able to formally show that each transformation properly
+ preserves the meaning of every expression, we require an exact
+ definition of the \emph{meaning} of every expression, so we can
+ compare them. A definition of the operational semantics of \GHC's Core
+ language is available \cite[sulzmann07], but this does not seem
+ sufficient for our goals (but it is a good start).
+
+ It should be possible to have a single formal definition of
+ meaning for Core for both normal Core compilation by \GHC\ and for
+ our compilation to \VHDL. The main difference seems to be that in
+ hardware every expression is always evaluated, while in software
+ it is only evaluated if needed, but it should be possible to
+ assign a meaning to Core expressions that assumes neither.
+
+ Since each of the transformations can be applied to any
+ sub-expression as well, there is a constraint on our meaning
+ definition: the meaning of an expression should depend only on the
+ meaning of sub-expressions, not on the expressions themselves. For
+ example, the meaning of the application in \lam{f (let x = 4 in
+ x)} should be the same as the meaning of the application in \lam{f
+ 4}, since the argument sub-expression has the same meaning (though
+ the actual expression is different).
+
\subsection{Completeness}
- Show that any transformation applies to every Core expression that is not
- in normal form. To prove: no transformation applies => in intended form.
- Show the reverse: Not in intended form => transformation applies.
-
- \subsection{Determinism}
- How to prove this?
+ Proving completeness is probably not hard, but it could be a lot
+ of work. We have seen above that to prove completeness, we must
+ show that the normal set of our graph representation is a subset
+ of the intended normal set.
+
+ However, it is hard to systematically generate or reason about the
+ normal set, since it is defined as any nodes to which no
+ transformation applies. To determine this set, each transformation
+ must be considered and when a transformation is added, the entire
+ set should be re-evaluated. This means it is hard to show that
+ each node in the normal set is also in the intended normal set.
+ Reasoning about our intended normal set is easier, since we know
+ how to generate it from its definition. \refdef{intended normal
+ form definition}
+
+ Fortunately, we can also prove the complement (which is
+ equivalent, since $A \subseteq B \Leftrightarrow \overline{B}
+ \subseteq \overline{A}$): show that the set of nodes not in
+ intended normal form is a subset of the set of nodes not in normal
+ form. In other words, show that for every expression that is not
+ in intended normal form, that there is at least one transformation
+ that applies to it (since that means it is not in normal form
+ either and since $A \subseteq C \Leftrightarrow \forall x (x \in A
+ \rightarrow x \in C)$).
+
+ By systematically reviewing the entire Core language definition
+ along with the intended normal form definition (both of which have
+ a similar structure), it should be possible to identify all
+ possible (sets of) Core expressions that are not in intended
+ normal form and identify a transformation that applies to it.
+
+ This approach is especially useful for proving completeness of our
+ system, since if expressions exist to which none of the
+ transformations apply (\ie\ if the system is not yet complete), it
+ is immediately clear which expressions these are and adding
+ (or modifying) transformations to fix this should be relatively
+ easy.
+
+ As observed above, applying this approach is a lot of work, since
+ we need to check every (set of) transformation(s) separately.
+
+ \todo{Perhaps do a few steps of the proofs as proof-of-concept}
+ \subsection{Determinism}
+ A well-known technique for proving determinism in lambda calculus
+ and other reduction systems, is using the Church-Rosser property
+ \cite[church36]. A reduction system has the CR property if and only if:
+
+ \placedefinition[here]{Church-Rosser theorem}
+ {\lam{\forall A, B, C \exists D (A ->> B ∧ A ->> C => B ->> D ∧ C ->> D)}}
+
+ Here, \lam{A ->> B} means \lam{A} \emph{reduces to} \lam{B}. In
+ other words, there is a set of transformations that can transform
+ \lam{A} to \lam{B}. \lam{=>} is used to mean \emph{implies}.
+
+ For a transformation system holding the Church-Rosser property, it
+ is easy to show that it is in fact deterministic. Showing that this
+ property actually holds is a harder problem, but has been
+ done for some reduction systems in the lambda calculus
+ \cite[klop80]\ \cite[barendregt84]. Doing the same for our
+ transformation system is probably more complicated, but not
+ impossible.
+
+% vim: set sw=2 sts=2 expandtab:
projects / matthijs / master-project / report.git / blobdiff