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 (higher order expressions, limited polymorphism using type
+ 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
+ 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{}{\startboxed A program is in \emph{normal form} if none of the
+ \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
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 All \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 return values used must be first order.
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 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}
-
\startbuffer[MulSum]
alu :: Bit -> Word -> Word -> Word
alu = λa.λb.λc.
ncline(add)(sum);
\stopuseMPgraphic
- \placeexample[here][ex:MulSum]{Simple architecture consisting of a
+ \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
+ \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
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.}
As a more complete example, consider
\in{example}[ex:NormalComplete]. This example shows everything that
- is allowed in normal form, except for builtin higher order functions
+ 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. Also note that most synthesis tools will further optimize
- this architecture by removing the multiplexers at the register input
- and instead put some gates in front of the register's clock input,
- but we want to show the architecture as close to the description as
- possible.
+ 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
\stopuseMPgraphic
\todo{Don't split registers in this image?}
- \placeexample[here][ex:NormalComplete]{Simple architecture consisting of an adder and a
+ \placeexample[][ex:NormalComplete]{Simple architecture consisting of an adder and a
subtractor.}
\startcombination[2*1]
{\typebufferlam{NormalComplete}}{Core description in normal form.}
\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 following
- EBNF-like description captures most of the intended structure (and
- generates a subset of GHC's core format).
+ 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: 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
+ 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.
\italic{userarg} := var (lvar(var))
\italic{builtinapp} := \italic{builtinfunc}
| \italic{builtinapp} \italic{builtinarg}
- \italic{builtinfunc} := var (bvar(var))
- \italic{builtinarg} := var (representable(var) ∧ lvar(var))
+ \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}
{\defref{intended normal form definition}
\typebufferlam{IntendedNormal}}
- When looking at such a program from a hardware perspective, the
- top level lambda abstractions define the input ports. Lambda
- abstractions cannot appear anywhere else. The variable reference
- in the body of the recursive 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} expression) or call a
- builtin function (\eg \lam{+} or \lam{map}). For these, a
- hardcoded \small{VHDL} translation is available.
+ 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 hardcoded \small{VHDL} translation is available.
\section[sec:normalization:transformation]{Transformation notation}
To be able to concisely present transformations, we use a specific format
~
<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
+ 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:
against (subexpressions 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 letter (\eg
+ 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
+ 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
\stopdesc
To understand this notation better, the step by step application of
- the η-abstraction transformation to a simple \small{ALU} will be
- shown. Consider η-abstraction, described using above notation as
- follows:
+ the η-expansion transformation to a simple \small{ALU} will be
+ shown. Consider η-expansion, which is a common transformation from
+ labmda calculus, described using above notation as follows:
\starttrans
E \lam{E :: a -> b}
λx.E x \lam{E} is not a lambda abstraction.
\stoptrans
- η-abstraction is a well known transformation from lambda calculus. What
+ 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-abstraction).
+ 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
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
\stoplambda
The transformation does not apply to this lambda abstraction, so we
- look at its body. For brevity, we'll put the case expression on one line from
+ 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:
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'll skip to the components of the case expression: The scrutinee and
+ 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.
\stoplambda
The other alternative is left as an exercise to the reader. The final
- function, after applying η-abstraction until it does no longer apply is:
+ function, after applying η-expansion until it does no longer apply is:
\startlambda
alu :: Bit -> Word -> Word -> Word
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.
+ 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 runtime representable notably include (but are not limited to): Types,
+ 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}
+ 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{builtin hardware translation}, so calls
- to these functions can still be translated. These are functions like
- \lam{map}, \lam{hwor} 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 we do have a Cλash
- implementation available.
+ 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
\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,
+ 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
+ \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
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, the following points can be
- observed.
+ a bit harder to do (mostly due to implementation issues, the prototype
+ does not use \small{GHC}'s subsitution code). Also, the following points
+ can be observed.
\startitemize
\item Deshadowing does not guarantee overall uniqueness. For example, the
\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).
M
\stoptrans
- \todo{Example}
+ % And an example
+ \startbuffer[from]
+ let
+ in
+ 2
+ \stopbuffer
+
+ \startbuffer[to]
+ 2
+ \stopbuffer
+
+ \transexample{emptylet}{Empty let removal}{from}{to}
\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
+ 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
\todo{example}
- \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{Don't 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}
- ---------------------------- \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
-
- \todo{Example}
-
\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
Note that this transformation is completely optional. It is not
required to get any function into intended 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).
+ 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 \GHC,
whose normalized form contains only a single let binding.
\in{Section}[section:prototype:polymorphism].
Without this transformation, there would be a \lam{(+)} entity
- in the \VHDL which would just add its inputs. This generates a
+ 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 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
+ extended identifiers in filenames, so it effectively does not
work.}, so the entity would be called \quote{w7aA7f} or
- something similarly unreadable and autogenerated).
+ something similarly meaningless and autogenerated).
\subsection{Program structure}
These transformations are aimed at normalizing the overall structure
of the other value definitions in let bindings and making the final
return value a simple variable reference.
- \subsubsection[sec:normalization:eta]{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
False -> λy.id y) x
\stopbuffer
- \transexample{eta}{η-abstraction}{from}{to}
+ \transexample{eta}{η-expansion}{from}{to}
\subsubsection[sec:normalization:appprop]{Application propagation}
This transformation is meant to propagate application expressions downwards
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
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. If the return value is
- not representable because it has a function type, η-abstraction
+ 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
\stoptrans
\starttrans
- x = λv0 ... λvn.E
- ~ \lam{E} is representable
+ x = λv0 ... λvn.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
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
+ 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.
function without arguments, but also an argumentless dataconstructors
like \lam{True}) are also simplified. Only local variables generate
signals in the resulting architecture. Even though argumentless
- dataconstructors generate constants in generated \VHDL code and could be
+ dataconstructors 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.
\transexample{argsimpl}{Argument simplification}{from}{to}
- \subsection[sec:normalization:builtins]{Builtin functions}
- This section deals with (arguments to) builtin functions. In the
+ \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 builtin function
+ we can see that there are three sorts of arguments a built-in function
can receive.
\startitemize[KR]
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 builtin functions) is turned into a local variable
+ function (including built-in functions) is turned into a local variable
reference.
- \item (A partial application of) a top level function (either builtin on
+ \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 functiontyped argument into
(a partial application of) a top level function.
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 builtin function will be either
+ 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
categorie. In any case, the result is in normal form.
\stopitemize
As noted, the argument simplification will handle any representable
- arguments to a builtin function. The following transformation is needed
+ 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 don't need any special handling.
+ non-representable arguments do not need any special handling.
\subsubsection[sec:normalization:funextract]{Function extraction}
- This transform deals with function-typed arguments to builtin
+ This transform deals with function-typed arguments to built-in
functions.
- Since builtin functions cannot be specialized (see
+ 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 builtin functions, since they only need
+ translation rules needed for built-in functions, since they only need
to handle (partial applications of) top level functions.
Any free variables occuring in the extracted arguments will become
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.
+ M N \lam{M} is (a partial aplication 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
this.
\subsection{Case normalisation}
+ The transformations in this section ensure that case statements end up
+ in normal form.
+
\subsubsection{Scrutinee simplification}
This transform ensures that the scrutinee of a case expression is always
a simple variable reference.
\transexample{letflat}{Case normalisation}{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 everytime 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: \refdef{intended normal form definition}
+ will become one of:
+
\startitemize
\item An extractor case with a single alternative that picks a field
- from a datatype, \eg \lam{case x of (a, b) -> a}.
+ from a datatype, \eg\ \lam{case x of (a, b) -> a}.
\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}.
\stopitemize
For an arbitrary case, that has \lam{n} alternatives, with
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)
+ --------------------------------------------------- \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
Cn wn,0 ... wn,m -> yn
\stoptrans
- \refdef{wild binder}
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
\subsubsection[sec:transformation:caseremoval]{Case removal}
This transform removes any case expression with a single alternative and
- only wild binders.\refdef{wild binder}
+ only wild binders.\refdef{wild binders}
These "useless" case expressions are usually leftovers from case simplification
on extractor case (see the previous example).
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 don't 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).
+ 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.
take care of exactly this.
There is one case where polymorphism cannot be completely
- removed: Builtin functions are still allowed to be polymorphic
+ 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 builtin
+ specialize). However, the code that generates \VHDL\ for built-in
functions knows how to handle this, so this is not a problem.
- \subsubsection{Defunctionalization}
- These transformations remove higher order expressions from our
- program, making all values first-order.
+ \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
+ For example, the following expression is a higher-order expression
that is not a lambda expression itself:
\refdef{id function}
\stoplambda
The reference to the \lam{id} function shows that we can introduce a
- higher order expression in our program without using a lambda
+ 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
+ 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:builtinarg] (Partial applications of) top level functions can appear as an
- argument to a builtin 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 builtin function arguments),
+ 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:
allowed, since it is inside a complete application.
\stopitemize
- We will take a typical function with some higher order values as an
+ 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
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
+ 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} builtin function, whereas the second alternative is a
+ 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've already seen are used. The η-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-abstraction, our example
+ \in{item}[item:toplambda] above. After η-expansion, our example
becomes a bit bigger:
\startlambda
) q
\stoplambda
- η-abstraction also introduces extra applications (the application of
+ 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
High -> (λz. z) q
\stoplambda
- This propagation makes higher order values become applied (in
+ 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-abstraction. For our example,
- applying β-abstraction results in the following:
+ 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
\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
+ 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
+ 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:
High -> q
\stoplambda
- We've nearly eliminated all unsupported higher order values from this
- expressions. The one that's remaining is the first argument to the
- \lam{map} function. Having higher order arguments to a builtin
+ 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
intended normal form.
There is one case that has not been discussed yet. What if the
- \lam{map} function in the example above was not a builtin function
+ \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
+ can never have higher-order arguments. For example, the following
expression shows an example:
\startlambda
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've made the function monomorphic for clarity, even though
+ 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 epression twice.
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
+ 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:
main = λa.app' a
\stoplambda
- 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've specialized for (\lam{λx. x + x})
- and the other arguments. This complex expression can fortunately be
- effectively reduced by repeatedly applying β-reduction:
+ 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
\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
+ 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} typeclass, which offers a \hs{fromInteger} method
+ Haskell's \hs{Num} type class, which offers a \hs{fromInteger} method
that converts any \hs{Integer} to the Cλash datatypes.
- When \GHC sees integer literals, it will automatically insert calls to
+ 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.
+ there is no context for \GHC\ to determine the type from otherwise.
\starthaskell
1 :: 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 builtin
+ 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.
By inlining these let-bindings, we can ensure that unrepresentable
literals bound by a let binding end up in an application of the
- appropriate builtin function, where they are allowed. Since it is
+ 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
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
+ solves (part of) the polymorphism, higher-order values and
unrepresentable literals in an expression.
\refdef{substitution notation}
letrec x = M in E
\stoptrans
- This doesn't seem like much of an improvement, but it does get rid of
- the lambda expression (and the associated higher order value), while
+ 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
in y + z
\stoplambda
- Looking at this, we could imagine an alternative approach: Create a
+ 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:
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
+ 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 substate itself instead of passing it to
a subfunction).
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
+ 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
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 and should be improved (or extra conditions
- on the input hardware descriptions should be formulated).
+ 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 als the termination property. The soundness
+ properties, perhaps also the termination property. The soundness
property probably holds, since it is easier to manually verify (each
transformation can be reviewed separately).
possible proof strategies are shown below.
\subsection{Graph representation}
- Before looking into how to prove these properties, we'll look at
+ 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
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
- seems unlikely to actually happen in our system.
+ 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
\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 expressions to which
- no transformation applies anymore. We call this set of nodes the \emph{normal
- set}. The set of nodes containing expressions in intended normal
- form \refdef{intended normal form} is called the \emph{intended
- 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} 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, but can also happen without 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
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
+ 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.)
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. Currently there seems to be no formal definition of
- the meaning or semantics of \GHC's core language, only informal
- descriptions are available.
+ 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
+ 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
Since each of the transformations can be applied to any
subexpression as well, there is a constraint on our meaning
- definition: The meaning of an expression should depend only on the
+ definition: the meaning of an expression should depend only on the
meaning of subexpressions, 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
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}.
+ 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
+ \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
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
+ 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.
projects / matthijs / master-project / report.git / blobdiff