\stopcombination
}
- The first step in the core to \small{VHDL} translation process, is normalization. We
- aim to bring the core description into a simpler form, which we can
+ The first step in the Core to \small{VHDL} translation process, is normalization. We
+ aim to bring the Core description into a simpler form, which we can
subsequently translate into \small{VHDL} easily. This normal form is needed because
- the full core language is more expressive than \small{VHDL} in some areas and because
- core can describe expressions that do not have a direct hardware
- interpretation.
+ the full Core language is more expressive than \small{VHDL} in some
+ areas (higher-order expressions, limited polymorphism using type
+ classes, etc.) and because Core can describe expressions that do not
+ have a direct hardware interpretation.
\section{Normal form}
- The transformations described here have a well-defined goal: To bring the
- program in a well-defined form that is directly translatable to hardware,
- while fully preserving the semantics of the program. We refer to this form as
- the \emph{normal form} of the program. The formal definition of this normal
- form is quite simple:
+ The transformations described here have a well-defined goal: to bring the
+ program in a well-defined form that is directly translatable to
+ \VHDL, while fully preserving the semantics of the program. We refer
+ to this form as the \emph{normal form} of the program. The formal
+ definition of this normal form is quite simple:
- \placedefinition{}{A program is in \emph{normal form} if none of the
- transformations from this chapter apply.}
+ \placedefinition[force]{}{\startboxed A program is in \emph{normal form} if none of the
+ transformations from this chapter apply.\stopboxed}
Of course, this is an \quote{easy} definition of the normal form, since our
program will end up in normal form automatically. The more interesting part is
to see if this normal form actually has the properties we would like it to
have.
- But, before getting into more definitions and details about this normal form,
- let's try to get a feeling for it first. The easiest way to do this is by
- describing the things we want to not have in a normal form.
+ But, before getting into more definitions and details about this normal
+ form, let us try to get a feeling for it first. The easiest way to do this
+ is by describing the things that are unwanted in the intended normal form.
\startitemize
\item Any \emph{polymorphism} must be removed. When laying down hardware, we
- can't generate any signals that can have multiple types. All types must be
+ cannot generate any signals that can have multiple types. All types must be
completely known to generate hardware.
- \item Any \emph{higher order} constructions must be removed. We can't
+ \item All \emph{higher-order} constructions must be removed. We cannot
generate a hardware signal that contains a function, so all values,
- arguments and returns values used must be first order.
+ arguments and return values used must be first order.
- \item Any complex \emph{nested scopes} must be removed. In the \small{VHDL}
+ \item All complex \emph{nested scopes} must be removed. In the \small{VHDL}
description, every signal is in a single scope. Also, full expressions are
not supported everywhere (in particular port maps can only map signal
names and constants, not complete expressions). To make the \small{VHDL}
other expression.
\stopitemize
- \todo{Intermezzo: functions vs plain values}
-
- A very simple example of a program in normal form is given in
- \in{example}[ex:MulSum]. As you can see, all arguments to the function (which
- will become input ports in the final hardware) are at the 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 final hardware, bound to the output port
- of the \lam{*} and \lam{+} components.
-
- The final line (the \quote{return value} of the function) selects the
- \lam{sum} signal to be the output port of the function. This \quote{return
- value} can always only be a variable reference, never a more complex
- expression.
-
- \todo{Add generated VHDL}
-
\startbuffer[MulSum]
alu :: Bit -> Word -> Word -> Word
alu = λa.λb.λc.
newCircle.a(btex $a$ etex) "framed(false)";
newCircle.b(btex $b$ etex) "framed(false)";
newCircle.c(btex $c$ etex) "framed(false)";
- newCircle.sum(btex $res$ etex) "framed(false)";
+ newCircle.sum(btex $sum$ etex) "framed(false)";
% Components
newCircle.mul(btex * etex);
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
- The previous example described composing an architecture by calling other
- functions (operators), resulting in a simple architecture with components and
- connections. There is of course also some mechanism for choice in the normal
- form. In a normal Core program, the \emph{case} expression can be used in a
- few different ways to describe choice. In normal form, this is limited to a
- very specific form.
+ \todo{Intermezzo: functions vs plain values}
+
+ A very simple example of a program in normal form is given in
+ \in{example}[ex:MulSum]. As you can see, all arguments to the function (which
+ will become input ports in the generated \VHDL) are at the outer level.
+ This means that the body of the inner lambda abstraction is never a
+ function, but always a plain value.
+
+ As the body of the inner lambda abstraction, we see a single (recursive)
+ let expression, that binds two variables (\lam{mul} and \lam{sum}). These
+ variables will be signals in the generated \VHDL, bound to the output port
+ of the \lam{*} and \lam{+} components.
+
+ The final line (the \quote{return value} of the function) selects the
+ \lam{sum} signal to be the output port of the function. This \quote{return
+ value} can always only be a variable reference, never a more complex
+ expression.
+
+ \todo{Add generated VHDL}
+
+ \in{Example}[ex:MulSum] showed a function that just applied two
+ other functions (multiplication and addition), resulting in a simple
+ architecture with two components and some connections. There is of
+ course also some mechanism for choice in the normal form. In a
+ normal Core program, the \emph{case} expression can be used in a few
+ different ways to describe choice. In normal form, this is limited
+ to a very specific form.
\in{Example}[ex:AddSubAlu] shows an example describing a
simple \small{ALU}, which chooses between two operations based on an opcode
expression scrutinizes the variable \lam{opcode} (and scrutinizing more
complex expressions is not supported). The case expression can select a
different variable based on the constructor of \lam{opcode}.
+ \refdef{case expression}
\startbuffer[AddSubAlu]
alu :: Bit -> Word -> Word -> Word
ncline(mux)(res) "posA(out)";
\stopuseMPgraphic
- \placeexample[here][ex:AddSubAlu]{Simple \small{ALU} supporting two operations.}
+ \placeexample[][ex:AddSubAlu]{Simple \small{ALU} supporting two operations.}
\startcombination[2*1]
{\typebufferlam{AddSubAlu}}{Core description in normal form.}
{\boxedgraphic{AddSubAlu}}{The architecture described by the normal form.}
\stopcombination
- As a more complete example, consider \in{example}[ex:NormalComplete]. This
- example contains everything that is supported in normal form, with the
- exception of builtin higher order functions. The graphical version of the
- architecture contains a slightly simplified version, since the state tuple
- packing and unpacking have been left out. Instead, two seperate registers are
- drawn. Also note that most synthesis tools will further optimize this
- architecture by removing the multiplexers at the register input and
- 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.
+ As a more complete example, consider
+ \in{example}[ex:NormalComplete]. This example shows everything that
+ is allowed in normal form, except for built-in higher-order functions
+ (like \lam{map}). The graphical version of the architecture contains
+ a slightly simplified version, since the state tuple packing and
+ unpacking have been left out. Instead, two separate registers are
+ drawn. Most synthesis tools will further optimize this architecture by
+ removing the multiplexers at the register input and instead use the write
+ enable port of the register (when it is available), but we want to show
+ the architecture as close to the description as possible.
As you can see from the previous examples, the generation of the final
architecture from the normal form is straightforward. In each of the
-> State (Word, Word)
-> (State (Word, Word), Word)
- -- All arguments are an inital lambda (address, data, packed state)
+ -- All arguments are an initial lambda
+ -- (address, data, packed state)
regbank = λa.λd.λsp.
-- There are nested let expressions at top level
let
\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 completely captures the intended structure (and
- generates a subset of GHC's core format).
-
- Some clauses have an expression listed in parentheses. These are conditions
- that need to apply to the clause.
+ the normal form to look like in a slightly more formal manner. The
+ EBNF-like description in \in{definition}[def:IntendedNormal] captures
+ most of the intended structure (and generates a subset of \GHC's Core
+ format).
+
+ There are two things missing from this definition: cast expressions are
+ sometimes allowed by the prototype, but not specified here and the below
+ definition allows uses of state that cannot be translated to \VHDL\
+ properly. These two problems are discussed in
+ \in{section}[sec:normalization:castproblems] and
+ \in{section}[sec:normalization:stateproblems] respectively.
+
+ Some clauses have an expression listed behind them in parentheses.
+ These are conditions that need to apply to the clause. The
+ predicates used there (\lam{lvar()}, \lam{representable()},
+ \lam{gvar()}) will be defined in
+ \in{section}[sec:normalization:predicates].
+
+ An expression is in normal form if it matches the first
+ definition, \emph{normal}.
- \defref{intended normal form definition}
\todo{Fix indentation}
- \startlambda
+ \startbuffer[IntendedNormal]
\italic{normal} := \italic{lambda}
- \italic{lambda} := λvar.\italic{lambda} (representable(var))
+ \italic{lambda} := λvar.\italic{lambda} (representable(var))
| \italic{toplet}
- \italic{toplet} := letrec [\italic{binding}...] in var (representable(var))
- \italic{binding} := var = \italic{rhs} (representable(rhs))
+ \italic{toplet} := letrec [\italic{binding}...] in var (representable(var))
+ \italic{binding} := var = \italic{rhs} (representable(rhs))
-- State packing and unpacking by coercion
- | var0 = var1 ▶ State ty (lvar(var1))
- | var0 = var1 ▶ ty (var1 :: State ty ∧ lvar(var1))
- \italic{rhs} := userapp
- | builtinapp
+ | var0 = var1 ▶ State ty (lvar(var1))
+ | var0 = var1 ▶ ty (var1 :: State ty ∧ lvar(var1))
+ \italic{rhs} := \italic{userapp}
+ | \italic{builtinapp}
-- Extractor case
- | case var of C a0 ... an -> ai (lvar(var))
+ | case var of C a0 ... an -> ai (lvar(var))
-- Selector case
- | case var of (lvar(var))
- [ DEFAULT -> var ] (lvar(var))
+ | case var of (lvar(var))
+ [ DEFAULT -> var ] (lvar(var))
C0 w0,0 ... w0,n -> var0
\vdots
- Cm wm,0 ... wm,n -> varm (\forall{}i \forall{}j, wi,j \neq vari, lvar(vari))
+ Cm wm,0 ... wm,n -> varm (\forall{}i \forall{}j, wi,j \neq vari, lvar(vari))
\italic{userapp} := \italic{userfunc}
| \italic{userapp} {userarg}
- \italic{userfunc} := var (gvar(var))
- \italic{userarg} := var (lvar(var))
+ \italic{userfunc} := var (gvar(var))
+ \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{partapp} (partapp :: a -> b)
- | \italic{coreexpr} (¬representable(coreexpr) ∧ ¬(coreexpr :: a -> b))
- \italic{partapp} := \italic{userapp} | \italic{builtinapp}
- \stoplambda
-
- \todo{There can still be other casts around (which the code can handle,
- e.g., ignore), which still need to be documented here}
+ \italic{built-infunc} := var (bvar(var))
+ \italic{built-inarg} := var (representable(var) ∧ lvar(var))
+ | \italic{partapp} (partapp :: a -> b)
+ | \italic{coreexpr} (¬representable(coreexpr) ∧ ¬(coreexpr :: a -> b))
+ \italic{partapp} := \italic{userapp}
+ | \italic{builtinapp}
+ \stopbuffer
- When looking at such a program from a hardware perspective, the top level
- lambda's define the input ports. 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.
+ \placedefinition[][def:IntendedNormal]{Definition of the intended normal form using an \small{EBNF}-like syntax.}
+ {\defref{intended normal form definition}
+ \typebufferlam{IntendedNormal}}
+
+ When looking at such a program from a hardware perspective, the top
+ level lambda abstractions (\italic{lambda}) define the input ports.
+ Lambda abstractions cannot appear anywhere else. The variable reference
+ in the body of the recursive let expression (\italic{toplet}) is the
+ output port. Most binders bound by the let expression define a
+ component instantiation (\italic{userapp}), where the input and output
+ ports are mapped to local signals (\italic{userarg}). Some of the others
+ use a built-in construction (\eg\ the \lam{case} expression) or call a
+ built-in function (\italic{builtinapp}) such as \lam{+} or \lam{map}.
+ For these, a hard-coded \small{VHDL} translation is available.
\section[sec:normalization:transformation]{Transformation notation}
To be able to concisely present transformations, we use a specific format
~
<original expression>
-------------------------- <expression conditions>
- <transformed expresssion>
+ <transformed expression>
~
<context additions>
\stoptrans
- This format desribes a transformation that applies to \lam{<original
- expresssion>} and transforms it into \lam{<transformed expression>}, assuming
- that all conditions apply. In this format, there are a number of placeholders
+ This format describes a transformation that applies to \lam{<original
+ expression>} and transforms it into \lam{<transformed expression>}, assuming
+ that all conditions are satisfied. In this format, there are a number of placeholders
in pointy brackets, most of which should be rather obvious in their meaning.
Nevertheless, we will more precisely specify their meaning below:
\startdesc{<original expression>} The expression pattern that will be matched
- against (subexpressions of) the expression to be transformed. We call this a
+ against (sub-expressions of) the expression to be transformed. We call this a
pattern, because it can contain \emph{placeholders} (variables), which match
any expression or binder. Any such placeholder is said to be \emph{bound} to
- the expression it matches. It is convention to use an uppercase 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
transformation applies, commonly to prevent a transformation from
causing a loop with itself or another transformation.
- Only if these conditions are \emph{all} true, the transformation
+ Only if these conditions are \emph{all} satisfied, the transformation
applies.
\stopdesc
\startdesc{<transformed expression>}
This is the expression template that is the result of the transformation. If, looking
at the above three items, the transformation applies, the \lam{<original
- expression>} is completely replaced with the \lam{<transformed expression>}.
+ expression>} is completely replaced by the \lam{<transformed expression>}.
We call this a template, because it can contain placeholders, referring to
any placeholder bound by the \lam{<original expression>} or the
\lam{<context conditions>}. The resulting expression will have those
\stopdesc
\startdesc{<context additions>}
- These are templates for new functions to add to the context. This is a way
- to have a transformation create new top level functions.
+ These are templates for new functions to be added to the context.
+ This is a way to let a transformation create new top level
+ functions.
Each addition has the form \lam{binder = template}. As above, any
placeholder in the addition is replaced with the value bound to it, and any
replaced with) a fresh binder.
\stopdesc
- As an example, we'll look at η-abstraction:
+ To understand this notation better, the step by step application of
+ the η-expansion transformation to a simple \small{ALU} will be
+ shown. Consider η-expansion, which is a common transformation from
+ lambda calculus, described using above notation as follows:
\starttrans
E \lam{E :: a -> b}
λx.E x \lam{E} is not a lambda abstraction.
\stoptrans
- 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, which is a fairly obvious way to specify a
- simple ALU (Note that \in{example}[ex:AddSubAlu] shows the normal form of this
- function). The parentheses around the \lam{+} and \lam{-} operators are
- commonly used in Haskell to show that the operators are used as normal
- functions, instead of \emph{infix} operators (\eg, the operators appear
- before their arguments, instead of in between).
+ Consider the following function, in Core notation, which is a fairly obvious way to specify a
+ simple \small{ALU} (Note that it is not yet in normal form, but
+ \in{example}[ex:AddSubAlu] shows the normal form of this function).
+ The parentheses around the \lam{+} and \lam{-} operators are
+ commonly used in Haskell to show that the operators are used as
+ normal functions, instead of \emph{infix} operators (\eg, the
+ operators appear before their arguments, instead of in between).
\startlambda
alu :: Bit -> Word -> Word -> Word
High -> (-)
\stoplambda
- There are a few subexpressions in this function to which we could possibly
+ There are a few sub-expressions in this function to which we could possibly
apply the transformation. Since the pattern of the transformation is only
the placeholder \lam{E}, any expression will match that. Whether the
transformation applies to an expression is thus solely decided by the
By now, the placeholder \lam{E} is bound to the entire expression. The
placeholder \lam{x}, which occurs in the replacement template, is not bound
- yet, so we need to generate a fresh binder for that. Let's use the binder
+ yet, so we need to generate a fresh binder for that. Let us use the binder
\lam{a}. This results in the following replacement expression:
\startlambda
\stoplambda
Here, the transformation does apply, binding \lam{E} to the entire
- expression and \lam{x} to the fresh binder \lam{b}, resulting in the
- replacement:
+ expression (which has type \lam{Word -> Word}) and binding \lam{x}
+ to the fresh binder \lam{b}, resulting in the replacement:
\startlambda
λb.(case opcode of
High -> (-)) a b
\stoplambda
- Again, the transformation does not apply to this lambda abstraction, so we
- look at its body. For brevity, we'll put the case expression on one line from
+ The transformation does not apply to this lambda abstraction, so we
+ look at its body. For brevity, we will put the case expression on one line from
now on.
\startlambda
The type of this expression is \lam{Word}, so it does not match \lam{a -> b}
and the transformation does not apply. Next, we have two options for the
- next expression to look at: The function position and argument position of
+ next expression to look at: the function position and argument position of
the application. The expression in the argument position is \lam{b}, which
has type \lam{Word}, so the transformation does not apply. The expression in
the function position is:
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.
We look at the \lam{<original expression>} pattern, which is \lam{E}.
This means we bind \lam{E} to \lam{(+)}. We then replace the expression
- with the \lam{<transformed expression>}, replacing all occurences of
+ with the \lam{<transformed expression>}, replacing all occurrences of
\lam{E} with \lam{(+)}. In the \lam{<transformed expression>}, the This gives us the replacement expression:
\lam{λx.(+) x} (A lambda expression binding \lam{x}, with a body that
applies the addition operator to \lam{x}).
\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
keep the transformation descriptions concise and powerful.
\subsection{Definitions}
- In the following sections, we will be using a number of functions and
- notations, which we will define here.
-
- \subsubsection{Concepts}
- A \emph{global variable} is any variable (binder) that is bound at the
- top level of a program, or an external module. A \emph{local variable} is any
- other variable (\eg, variables local to a function, which can be bound by
- lambda abstractions, let expressions and pattern matches of case
- alternatives). Note that this is a slightly different notion of global versus
- local than what \small{GHC} uses internally.
- \defref{global variable} \defref{local variable}
-
- A \emph{hardware representable} (or just \emph{representable}) type or value
- is (a value of) a type that we can generate a signal for in hardware. For
- example, a bit, a vector of bits, a 32 bit unsigned word, etc. Values that are
- not runtime representable notably include (but are not limited to): Types,
- dictionaries, functions.
- \defref{representable}
-
- A \emph{builtin function} is a function supplied by the Cλash framework, whose
- implementation is not valid Cλash. The implementation is of course valid
- Haskell, for simulation, but it is not expressable in Cλash.
- \defref{builtin function} \defref{user-defined function}
-
- For these functions, Cλash has a \emph{builtin hardware translation}, so calls
- to these functions can still be translated. These are functions like
- \lam{map}, \lam{hwor} and \lam{length}.
-
- A \emph{user-defined} function is a function for which we do have a Cλash
- implementation available.
-
- \subsubsection{Predicates}
+ A \emph{global variable} is any variable (binder) that is bound at the
+ top level of a program, or an external module. A \emph{local variable} is any
+ other variable (\eg, variables local to a function, which can be bound by
+ lambda abstractions, let expressions and pattern matches of case
+ alternatives). This is a slightly different notion of global versus
+ local than what \small{GHC} uses internally, but for our purposes
+ the distinction \GHC\ makes is not useful.
+ \defref{global variable} \defref{local variable}
+
+ A \emph{hardware representable} (or just \emph{representable}) type or value
+ is (a value of) a type that we can generate a signal for in hardware. For
+ example, a bit, a vector of bits, a 32 bit unsigned word, etc. Values that are
+ not run-time representable notably include (but are not limited to): types,
+ dictionaries, functions.
+ \defref{representable}
+
+ A \emph{built-in function} is a function supplied by the Cλash
+ framework, whose implementation is not used to generate \VHDL. This is
+ either because it is no valid Cλash (like most list functions that need
+ recursion) or because a Cλash implementation would be unwanted (for the
+ addition operator, for example, we would rather use the \VHDL addition
+ operator to let the synthesis tool decide what kind of adder to use
+ instead of explicitly describing one in Cλash). \defref{built-in
+ function}
+
+ These are functions like \lam{map}, \lam{hwor}, \lam{+} and \lam{length}.
+
+ For these functions, Cλash has a \emph{built-in hardware translation},
+ so calls to these functions can still be translated. Built-in functions
+ must have a valid Haskell implementation, of course, to allow
+ simulation.
+
+ A \emph{user-defined} function is a function for which no built-in
+ translation is available and whose definition will thus need to be
+ translated to Cλash. \defref{user-defined function}
+
+ \subsubsection[sec:normalization:predicates]{Predicates}
Here, we define a number of predicates that can be used below to concisely
- specify conditions.\refdef{global variable}
+ specify conditions.
\emph{gvar(expr)} is true when \emph{expr} is a variable that references a
global variable. It is false when it references a local variable.
- \refdef{local variable}\emph{lvar(expr)} is the complement of \emph{gvar}; it is true when \emph{expr}
+ \emph{lvar(expr)} is the complement of \emph{gvar}; it is true when \emph{expr}
references a local variable, false when it references a global variable.
- \refdef{representable}\emph{representable(expr)} or \emph{representable(var)} is true when
- \emph{expr} or \emph{var} is \emph{representable}.
+ \emph{representable(expr)} is true when \emph{expr} is \emph{representable}.
\subsection[sec:normalization:uniq]{Binder uniqueness}
A common problem in transformation systems, is binder uniqueness. When not
considering this problem, it is easy to create transformations that mix up
- bindings and cause name collisions. Take for example, the following core
+ bindings and cause name collisions. Take for example, the following Core
expression:
\startlambda
\stoplambda
Now, we have replaced the \lam{a} binder with a reference to the \lam{x}
- binder. No harm done here. But note that we see multiple occurences of the
- \lam{c} binder. The first is a binding occurence, to which the second refers.
+ binder. No harm done here. But note that we see multiple occurrences of the
+ \lam{c} binder. The first is a binding occurrence, to which the second refers.
The last, however refers to \emph{another} instance of \lam{c}, which is
bound somewhere outside of this expression. Now, if we would apply beta
reduction without taking heed of binder uniqueness, we would get:
\stoplambda
This is obviously not what was supposed to happen! The root of this problem is
- the reuse of binders: Identical binders can be bound in different scopes, such
- that only the inner one is \quote{visible} in the inner expression. In the example
- above, the \lam{c} binder was bound outside of the expression and in the inner
- lambda expression. Inside that lambda expression, only the inner \lam{c} is
- visible.
+ the reuse of binders: identical binders can be bound in different,
+ but overlapping scopes. Any variable reference in those
+ overlapping scopes then refers to the variable bound in the inner
+ (smallest) scope. There is not way to refer to the variable in the
+ outer scope. This effect is usually referred to as
+ \emph{shadowing}: when a binder is bound in a scope where the
+ binder already had a value, the inner binding is said to
+ \emph{shadow} the outer binding. In the example above, the \lam{c}
+ binder was bound outside of the expression and in the inner lambda
+ expression. Inside that lambda expression, only the inner \lam{c}
+ can be accessed.
There are a number of ways to solve this. \small{GHC} has isolated this
- problem to their binder substitution code, which performs \emph{deshadowing}
+ problem to their binder substitution code, which performs \emph{de-shadowing}
during its expression traversal. This means that any binding that shadows
another binding on a higher level is replaced by a new binder that does not
shadow any other binding. This non-shadowing invariant is enough to prevent
binder uniqueness problems in \small{GHC}.
In our transformation system, maintaining this non-shadowing invariant is
- a bit harder to do (mostly due to implementation issues, the prototype doesn't
- use \small{GHC}'s subsitution code). Also, the following points can be
- observed.
+ a bit harder to do (mostly due to implementation issues, the prototype
+ does not use \small{GHC}'s substitution code). Also, the following points
+ can be observed.
\startitemize
- \item Deshadowing does not guarantee overall uniqueness. For example, the
+ \item De-shadowing does not guarantee overall uniqueness. For example, the
following (slightly contrived) expression shows the identifier \lam{x} bound in
- two seperate places (and to different values), even though no shadowing
+ two separate places (and to different values), even though no shadowing
occurs.
\startlambda
unique.
\item When we know that all binders in an expression are unique, moving around
- or removing a subexpression will never cause any binder conflicts. If we have
- some way to generate fresh binders, introducing new subexpressions will not
+ or removing a sub-expression will never cause any binder conflicts. If we have
+ some way to generate fresh binders, introducing new sub-expressions will not
cause any problems either. The only way to cause conflicts is thus to
- duplicate an existing subexpression.
+ duplicate an existing sub-expression.
\stopitemize
Given the above, our prototype maintains a unique binder invariant. This
unique. This is done by generating a fresh binder for every binder used. This
also replaces binders that did not cause any conflict, but it does ensure that
all binders within the function are generated by the same unique supply.
- \refdef{fresh binder}
\item Whenever a new binder must be generated, we generate a fresh binder that
is guaranteed to be different from \emph{all binders generated so far}. This
can thus never introduce duplication and will maintain the invariant.
\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
+ occurrence of some expression (usually a variable reference) with
+ another expression.
+
+ There have been a lot of different notations used in literature for
+ specifying substitution. The notation that will be used in this report
+ is the following:
- \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 occurrences 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
In our transformation system, this step helps to remove unwanted lambda
abstractions (basically all but the ones at the top level). Other
transformations (application propagation, non-representable inlining) make
- sure that most lambda abstractions will eventually be reducable by
+ sure that most lambda abstractions will eventually be reducible by
β-reduction.
Note that β-reduction also works on type lambda abstractions and type
applications as well. This means the substitution below also works on
- type variables, in the case that the binder is a type variable and teh
+ type variables, in the case that the binder is a type variable and the
expression applied to is a type.
\starttrans
\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.
+ \subsubsection{Cast propagation / simplification}
+ This transform pushes casts down into the expression as far as
+ possible. This transformation has been added to make a few
+ specific corner cases work, but it is not clear yet if this
+ transformation handles cast expressions completely or in the
+ right way. See \in{section}[sec:normalization:castproblems].
- This normalization pass should really be unneeded to get into intended normal form
- (since unused bindings are not forbidden by the normal form), but in practice
- the desugarer or simplifier emits some unused bindings that cannot be
- normalized (e.g., calls to a \type{PatError}\todo{Check this name}). Also,
- this transformation makes the resulting \small{VHDL} a lot shorter.
+ \starttrans
+ (let binds in E) ▶ T
+ -------------------------
+ let binds in (E ▶ T)
+ \stoptrans
- \todo{Don't use old-style numerals in transformations}
\starttrans
- letrec
- a0 = E0
+ (case S of
+ p0 -> E0
\vdots
- ai = Ei
+ pn -> En
+ ) ▶ T
+ -------------------------
+ case S of
+ p0 -> E0 ▶ T
\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
+ pn -> En ▶ T
\stoptrans
- \todo{Example}
-
- \subsubsection{Cast propagation / simplification}
- This transform pushes casts down into the expression as far as possible.
- Since its exact role and need is not clear yet, this transformation is
- not yet specified.
-
- \todo{Cast propagation}
-
\subsubsection{Top level binding inlining}
+ \refdef{top level binding}
This transform takes simple top level bindings generated by the
\small{GHC} compiler. \small{GHC} sometimes generates very simple
\quote{wrapper} bindings, which are bound to just a variable
- reference, or a partial application to constants or other variable
- references.
+ reference, or contain just a (partial) function application with
+ the type and dictionary arguments filled in (such as the
+ \lam{(+)} in the example below).
Note that this transformation is completely optional. It is not
required to get any function into 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 the compiler,
+ This transform takes any top level binding generated by \GHC,
whose normalized form contains only a single let binding.
\starttrans
\in{Example}[ex:trans:toplevelinline] shows a typical application of
the addition operator generated by \GHC. The type and dictionary
arguments used here are described in
- \in{Section}[section:prototype:polymorphism].
+ \in{Section}[sec:prototype:coretypes].
Without this transformation, there would be a \lam{(+)} entity
- in the \VHDL which would just add its inputs. This generates a
+ 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 file names, so it effectively does not
work.}, so the entity would be called \quote{w7aA7f} or
- something similarly unreadable and autogenerated).
+ something similarly meaningless and auto-generated).
\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
\starttrans
E \lam{E :: a -> b}
- -------------- \lam{E} is not the first argument of an application.
+ -------------- \lam{E} does not occur on a function position in an application
λx.E x \lam{E} is not a lambda abstraction.
- \lam{x} is a variable that does not occur free in \lam{E}.
\stoptrans
\startbuffer[from]
False -> λy.id y) x
\stopbuffer
- \transexample{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
\starttrans
(case x of
- p1 -> E1
+ p0 -> E0
\vdots
pn -> En) M
-----------------
case x of
- p1 -> E1 M
+ p0 -> E0 M
\vdots
pn -> En M
\stoptrans
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
This transformation ensures that the return value of a function is always a
simple local variable reference.
- Currently implemented using lambda simplification, let simplification, and
- top simplification. Should change into something like the following, which
- works only on the result of a function instead of any subexpression. This is
- achieved by the contexts, like \lam{x = E}, though this is strictly not
- correct (you could read this as "if there is any function \lam{x} that binds
- \lam{E}, any \lam{E} can be transformed, while we only mean the \lam{E} that
- is bound by \lam{x}. This might need some extra notes or something).
-
- Note that the return value is not simplified if its not representable.
- Otherwise, this would cause a direct loop with the inlining of
- unrepresentable bindings. 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
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.
\refdef{global variable}
Note that references to \emph{global variables} (like a top level
- function without arguments, but also an argumentless dataconstructors
+ function without arguments, but also an argumentless data-constructors
like \lam{True}) are also simplified. Only local variables generate
signals in the resulting architecture. Even though argumentless
- dataconstructors generate constants in generated \VHDL code and could be
+ data-constructors generate constants in generated \VHDL\ code and could be
mapped to an input port directly, they are still simplified to make the
normal form more regular.
\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
+ this section takes care of turning every function-typed argument into
(a partial application of) a top level function.
\item Any expression that is not representable and does not have a
function type. Since these can be any expression, there is no
transformation needed. Note that this category is exactly all
expressions that are not transformed by the transformations for the
- previous two categories. This means that \emph{any} core expression
- that is used as an argument to a builtin function will be either
+ previous two categories. This means that \emph{any} Core expression
+ that is used as an argument to a built-in function will be either
transformed into one of the above categories, or end up in this
- categorie. In any case, the result is in normal form.
+ category. 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
+ Any free variables occurring in the extracted arguments will become
parameters to the new global function. The original argument is replaced
with a reference to the new function, applied to any free variables from
the original argument.
- This transformation is useful when applying higher order builtin functions
+ This transformation is useful when applying higher-order built-in functions
like \hs{map} to a lambda abstraction, for example. In this case, the code
that generates \small{VHDL} for \hs{map} only needs to handle top level functions and
partial applications, not any other expression (such as lambda abstractions or
even more complicated expressions).
\starttrans
- M N \lam{M} is (a partial aplication of) a builtin function.
+ M N \lam{M} is (a partial application of) a built-in function.
--------------------- \lam{f0 ... fn} are all free local variables of \lam{N}
M (x f0 ... fn) \lam{N :: a -> b}
~ \lam{N} is not a (partial application of) a top level function
Note that the function \lam{f} will still need normalization after
this.
- \subsection{Case normalisation}
+ \subsection{Case normalization}
+ 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.
alts
----------------- \lam{E} is not a local variable reference
letrec x = E in
- case E of
+ case x of
alts
\stoptrans
False -> b
\stopbuffer
- \transexample{letflat}{Case normalisation}{from}{to}
+ \transexample{letflat}{Case normalization}{from}{to}
+
+
+ \placeintermezzo{}{
+ \defref{wild binders}
+ \startframedtext[width=7cm,background=box,frame=no]
+ \startalignment[center]
+ {\tfa Wild binders}
+ \stopalignment
+ \blank[medium]
+ In a functional expression, a \emph{wild binder} refers to any
+ binder that is never referenced. This means that even though it
+ will be bound to a particular value, that value is never used.
+
+ The Haskell syntax offers the underscore as a wild binder that
+ cannot even be referenced (It can be seen as introducing a new,
+ anonymous, binder every time it is used).
+
+ In these transformations, the term wild binder will sometimes be
+ used to indicate that a binder must not be referenced.
+ \stopframedtext
+ }
+
+ \subsubsection{Scrutinee binder removal}
+ This transformation removes (or rather, makes wild) the binder to
+ which the scrutinee is bound after evaluation. This is done by
+ replacing the bndr with the scrutinee in all alternatives. To prevent
+ duplication of work, this transformation is only applied when the
+ scrutinee is already a simple variable reference (but the previous
+ transformation ensures this will eventually be the case). The
+ scrutinee binder itself is replaced by a wild binder (which is no
+ longer displayed).
+
+ Note that one could argue that this transformation can change the
+ meaning of the Core expression. In the regular Core semantics, a case
+ expression forces the evaluation of its scrutinee and can be used to
+ implement strict evaluation. However, in the generated \VHDL,
+ evaluation is always strict. So the semantics we assign to the Core
+ expression (which differ only at this particular point), this
+ transformation is completely valid.
+
+ \starttrans
+ case x of bndr
+ alts
+ ----------------- \lam{x} is a local variable reference
+ case x of
+ alts[bndr=>x]
+ \stoptrans
+
+ \startbuffer[from]
+ case x of y
+ True -> y
+ False -> not y
+ \stopbuffer
+
+ \startbuffer[to]
+ case x of
+ True -> x
+ False -> not x
+ \stopbuffer
+
+ \transexample{scrutbndrremove}{Scrutinee binder removal}{from}{to}
+ \subsubsection{Case normalization}
+ This transformation ensures that all case expressions get a form
+ that is allowed by the intended normal form. This means they
+ will become one of:
- \subsubsection{Case simplification}
- This transformation ensures that all case expressions become normal form. This
- means they will become one of:
\startitemize
- \item An extractor case with a single alternative that picks a single field
- from a datatype, \eg \lam{case x of (a, b) -> a}.
+ \item An extractor case with a single alternative that picks a field
+ from a datatype, \eg\ \lam{case x of (a, b) ->
+ a}.\defref{extractor case}
\item A selector case with multiple alternatives and only wild binders, that
makes a choice between expressions based on the constructor of another
- expression, \eg \lam{case x of Low -> a; High -> b}.
+ expression, \eg\ \lam{case x of Low -> a; High ->
+ b}.\defref{selector case}
\stopitemize
+
+ For an arbitrary case, that has \lam{n} alternatives, with
+ \lam{m} binders in each alternatives, this will result in \lam{m
+ * n} extractor case expression to get at each variable, \lam{n}
+ let bindings for each of the alternatives' value and a single
+ selector case to select the right value out of these.
+
+ Technically, the definition of this transformation would require
+ that the constructor for every alternative has exactly the same
+ amount (\lam{m}) of arguments, but of course this transformation
+ also applies when this is not the case.
- \defref{wild binder}
\starttrans
case E of
C0 v0,0 ... v0,m -> E0
\vdots
Cn vn,0 ... vn,m -> En
- --------------------------------------------------- \forall i \forall j, 0 ≤ i ≤ n, 0 ≤ i < m (\lam{wi,j} is a wild (unused) binder)
- letrec
- v0,0 = case E of C0 v0,0 .. v0,m -> v0,0
+ --------------------------------------------------- \lam{\forall i \forall j, 0 ≤ i ≤ n, 0 ≤ i < m} (\lam{wi,j} is a wild (unused) binder)
+ letrec The case expression is not an extractor case
+ v0,0 = case E of C0 x0,0 .. x0,m -> x0,0 The case expression is not a selector case
\vdots
- v0,m = case E of C0 v0,0 .. v0,m -> v0,m
+ v0,m = case E of C0 x0,0 .. x0,m -> x0,m
\vdots
- vn,m = case E of Cn vn,0 .. vn,m -> vn,m
- x0 = E0
+ vn,m = case E of Cn xn,0 .. xn,m -> xn,m
+ y0 = E0
\vdots
- xn = En
+ yn = En
in
case E of
- C0 w0,0 ... w0,m -> x0
+ C0 w0,0 ... w0,m -> y0
\vdots
- Cn wn,0 ... wn,m -> xn
+ Cn wn,0 ... wn,m -> yn
\stoptrans
- \todo{Check the subscripts of this transformation}
Note that this transformation applies to case expressions with any
- scrutinee. If the scrutinee is a complex expression, this might result
- in duplicate hardware. An extra condition to only apply this
- transformation when the scrutinee is already simple (effectively
- causing this transformation to be only applied after the scrutinee
- simplification transformation) might be in order.
-
- \fxnote{This transformation specified like this is complicated and misses
- conditions to prevent looping with itself. Perhaps it should be split here for
- discussion?}
+ scrutinee. If the scrutinee is a complex expression, this might
+ result in duplication of work (hardware). An extra condition to
+ only apply this transformation when the scrutinee is already
+ simple (effectively causing this transformation to be only
+ applied after the scrutinee simplification transformation) might
+ be in order.
\startbuffer[from]
case a of
\stopbuffer
\startbuffer[to]
- letnonrec
+ letrec
x0 = add b 1
x1 = add b 2
in
\subsubsection[sec:transformation:caseremoval]{Case removal}
This transform removes any case expression with a single alternative and
- only wild binders.
+ 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.
\subsubsection{Removing Polymorphism}
- As noted in \in{section}[sec:prototype:polymporphism],
+ As noted in \in{section}[sec:prototype:coretypes],
polymorphism is made explicit in Core through type and
dictionary arguments. To remove the polymorphism from a
function, we can simply specialize the polymorphic function for
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:
\startlambda
f 1 2
\stoplambda
- the subexpression \lam{f 1} has a function type. But this is
+ the sub-expression \lam{f 1} has a function type. But this is
allowed, since it is inside a complete application.
\stopitemize
- We will take a typical function with some higher order values as an
+ 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.
+ function \lam{main} uses \lam{twice} to apply a lambda expression 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
\subsubsection{Literals}
There are a limited number of literals available in Haskell and Core.
\refdef{enumerated types} When using (enumerating) algebraic
- datatypes, a literal is just a reference to the corresponding data
+ data-types, a literal is just a reference to the corresponding data
constructor, which has a representable type (the algebraic datatype)
and can be translated directly. This also holds for literals of the
\hs{Bool} Haskell type, which is just an enumerated type.
There is, however, a second type of literal that does not have a
- representable type: Integer literals. Cλash supports using integer
+ 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
- that converts any \hs{Integer} to the Cλash datatypes.
+ Haskell's \hs{Num} type class, which offers a \hs{fromInteger} method
+ that converts any \hs{Integer} to the Cλash data-types.
- When \GHC sees integer literals, it will automatically insert calls to
+ 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
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
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}
(λb -> add b 1) (add 1 x)
\stopbuffer
- \transexample{nonrepinline}{Nonrepresentable binding inlining}{from}{to}
+ \transexample{nonrepinline}{Non-representable binding inlining}{from}{to}
\subsubsection[sec:normalization:specialize]{Function specialization}
This transform removes arguments to user-defined functions that are
- not representable at runtime. This is done by creating a
+ not representable at run-time. This is done by creating a
\emph{specialized} version of the function that only works for one
particular value of that argument (in other words, the argument can be
removed).
exactly one free variable, itself, we would only replace that argument
with itself).
- This shows that any free local variables that are not runtime
+ This shows that any free local variables that are not run-time
representable cannot be brought into normal form by this transform. We
rely on an inlining or β-reduction transformation to replace such a
variable with an expression we can propagate again.
~
x Y0 ... Yi ... Yn \lam{Yi} is not representable
--------------------------------------------- \lam{Yi} is not a local variable reference
- x' y0 ... yi-1 f0 ... fm Yi+1 ... Yn \lam{f0 ... fm} are all free local vars of \lam{Yi}
+ x' Y0 ... Yi-1 f0 ... fm Yi+1 ... Yn \lam{f0 ... fm} are all free local vars of \lam{Yi}
~ \lam{T0 ... Tn} are the types of \lam{Y0 ... Yn}
- x' = λ(y0 :: T0) ... λ(yi-1 :: Ty-1). λf0 ... λfm. λ(yi+1 :: Ty+1) ... λ(yn :: Tn).
- E y0 ... yi-1 Yi yi+1 ... yn
+ x' = λ(y0 :: T0) ... λ(yi-1 :: Ty-1).
+ λf0 ... λfm.
+ λ(yi+1 :: Ty+1) ... λ(yn :: Tn).
+ E y0 ... yi-1 Yi yi+1 ... yn
\stoptrans
This is a bit of a complex transformation. It transforms an
The argument that we are specializing for, \lam{Y_i}, is put inside
the new function body. The old function body is applied to it. Since
we use this new function only in place of an application with that
- particular argument \lam{Y_i}, behaviour should not change.
+ particular argument \lam{Y_i}, behavior should not change.
Note that the types of the arguments of our new function are taken
from the types of the \emph{actual} arguments (\lam{T0 ... Tn}). This
means that any polymorphism in the arguments is removed, even when the
corresponding explicit type lambda is not removed
- yet.\refdef{type lambda}
+ yet.
\todo{Examples. Perhaps reference the previous sections}
A possible solution to this would be to use the following alternative
transformation, which is of course no longer normal β-reduction. The
- followin transformation has not been tested in the prototype, but is
+ following transformation has not been tested in the prototype, but is
given here for future reference:
\starttrans
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:
For this particular problem, the solutions for duplication of work
seem from the previous section seem to fix the determinism of our
transformation system as well. However, it is likely that there are
- other occurences of this problem.
+ other occurrences of this problem.
- \subsection{Casts}
+ \subsection[sec:normalization:castproblems]{Casts}
We do not fully understand the use of cast expressions in Core, so
there are probably expressions involving cast expressions that cannot
be brought into intended normal form by this transformation system.
- The uses of casts in the core system should be investigated more and
+ The uses of casts in the Core system should be investigated more and
transformations will probably need updating to handle them in all
cases.
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).
+ function that changes a sub-state itself instead of passing it to
+ a sub-function).
It is now up to the programmer to not do anything funny with
these state values, whereas the normalization just tries not to
mess up the flow of state values. In practice, there are
situations where a Core program that \emph{could} be a valid
- stateful description is not translateable by the prototype. This
+ stateful description is not translatable by the prototype. This
most often happens when statefulness is mixed with pattern
matching, causing a state input to be unpacked multiple times or
be unpacked and repacked only in some of the code paths.
Without going into detail about the exact problems (of which
there are probably more than have shown up so far), it seems
unlikely that these problems can be solved entirely by just
- improving the \VHDL state generation in the final stage. The
+ 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
When looking at the system of transformations outlined above, there are a
number of questions that we can ask ourselves. The main question is of course:
\quote{Does our system work as intended?}. We can split this question into a
- number of subquestions:
+ number of sub-questions:
\startitemize[KR]
\item[q:termination] Does our system \emph{terminate}? Since our system will
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 expression 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 definition} is called the \emph{intended normal set}.
From such a graph, we can derive some properties easily:
\startitemize[KR]
- \item A system will \emph{terminate} if there is no path of infinite length
- in the graph (this includes cycles, 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.)
will of course leave the meaning unchanged and is thus
\emph{sound}.
- The current prototype has only been verified in an ad-hoc fashion
+ The current prototype has only been verified in an ad hoc fashion
by inspecting (the code for) each transformation. A more formal
verification would be more appropriate.
To be able to formally show that each transformation properly
preserves the meaning of every expression, we require an exact
definition of the \emph{meaning} of every expression, so we can
- compare them. 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
- assign a meaning to core expressions that assumes neither.
+ assign a meaning to Core expressions that assumes neither.
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
- meaning of subexpressions, not on the expressions themselves. For
+ sub-expression as well, there is a constraint on our meaning
+ definition: the meaning of an expression should depend only on the
+ meaning of sub-expressions, not on the expressions themselves. For
example, the meaning of the application in \lam{f (let x = 4 in
x)} should be the same as the meaning of the application in \lam{f
- 4}, since the argument subexpression has the same meaning (though
+ 4}, since the argument sub-expression has the same meaning (though
the actual expression is different).
\subsection{Completeness}
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
By systematically reviewing the entire Core language definition
along with the intended normal form definition (both of which have
a similar structure), it should be possible to identify all
- possible (sets of) core expressions that are not in intended
+ possible (sets of) Core expressions that are not in intended
normal form and identify a transformation that applies to it.
This approach is especially useful for proving completeness of our
system, since if expressions exist to which none of the
- transformations apply (\ie if the system is not yet complete), it
+ 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.
we need to check every (set of) transformation(s) separately.
\todo{Perhaps do a few steps of the proofs as proof-of-concept}
+ \subsection{Determinism}
+ A well-known technique for proving determinism in lambda calculus
+ and other reduction systems, is using the Church-Rosser property
+ \cite[church36]. A reduction system has the CR property if and only if:
+
+ \placedefinition[here]{Church-Rosser theorem}
+ {\lam{\forall A, B, C \exists D (A ->> B ∧ A ->> C => B ->> D ∧ C ->> D)}}
+
+ Here, \lam{A ->> B} means \lam{A} \emph{reduces to} \lam{B}. In
+ other words, there is a set of transformations that can transform
+ \lam{A} to \lam{B}. \lam{=>} is used to mean \emph{implies}.
+
+ For a transformation system holding the Church-Rosser property, it
+ is easy to show that it is in fact deterministic. Showing that this
+ property actually holds is a harder problem, but has been
+ done for some reduction systems in the lambda calculus
+ \cite[klop80]\ \cite[barendregt84]. Doing the same for our
+ transformation system is probably more complicated, but not
+ impossible.
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projects / matthijs / master-project / report.git / blobdiff