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
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.}
\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
+ 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
{\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
- built-in construction (\eg\ the \lam{case} expression) or call a
- built-in 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
\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, which is a common transformation from
+ 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
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
\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
dictionaries, functions.
\defref{representable}
- A \emph{built-in 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{built-in 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}
- For these functions, Cλash has a \emph{built-in hardware translation}, so calls
- to these functions can still be translated. These are functions like
- \lam{map}, \lam{hwor} and \lam{length}.
+ These are functions like \lam{map}, \lam{hwor}, \lam{+} and \lam{length}.
- A \emph{user-defined} function is a function for which we do have a Cλash
- implementation available.
+ 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
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
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
\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
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-abstraction
+ 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
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
\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]
projects / matthijs / master-project / report.git / blobdiff