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
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
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 will 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
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{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}
+
+ 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. 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
\in{section}[sec:normalization:transformation].
\subsection{General cleanup}
- 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.
-
- 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.
+ \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
+ occurence 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:
- \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
- occurence of some expression (usually a variable reference) with
- another expression.
+ \startlambda
+ E[A=>B]
+ \stoplambda
- 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:
+ This means expression \lam{E} with all occurences of \lam{A} replaced
+ with \lam{B}.
+ \stopframedtext
+ }
- \startlambda
- E[A=>B]
- \stoplambda
+ 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.
- This means expression \lam{E} with all occurences of \lam{A} replaced
- with \lam{B}.
- \stopframedtext
- }
+ 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[sec:normalization:beta]{β-reduction}
β-reduction is a well known transformation from lambda calculus, where it is
\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).
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 only need to be specified for 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).
This transformation ensures that the return value of a function is always a
simple local variable reference.
- This transformation only applies to the entire body of a
- function instead of any subexpression in a function. 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}).
-
- 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
- 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.
+ 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{retvalsimpl}{Return value simplification}{from}{to}
+
+ \startbuffer[from]
+ x = λa. add 1 a
+ \stopbuffer
+
+ \startbuffer[to]
+ x = λa. letrec
+ y = add 1 a
+ in
+ y
+ \stopbuffer
+
+ \transexample{retvalsimpllam}{Return value simplification with a lambda abstraction}{from}{to}
- \todo{More examples}
+ \startbuffer[from]
+ x = letrec
+ a = add 1 2
+ in
+ add a 3
+ \stopbuffer
+
+ \startbuffer[to]
+ x = letrec
+ a = add 1 2
+ y = add a 3
+ in
+ y
+ \stopbuffer
+
+ \transexample{retvalsimpllet}{Return value simplification with a let expression}{from}{to}
\subsection[sec:normalization:argsimpl]{Representable arguments simplification}
This section contains just a single transformation that deals with
\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
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,
+ 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,
+ 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
take care of exactly this.
There is one case where polymorphism cannot be completely
- removed: Built-in 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 built-in
functions knows how to handle this, so this is not a problem.
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
\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} type class, which offers a \hs{fromInteger} method
to specialize away any unrepresentable literals that are used as
function arguments. The following two transformations do exactly this.
- \subsubsection{Non-representable binding inlining}
+ \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
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:
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).
\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]
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
projects / matthijs / master-project / report.git / blobdiff