X-Git-Url: https://git.stderr.nl/gitweb?p=matthijs%2Fmaster-project%2Freport.git;a=blobdiff_plain;f=Chapters%2FNormalization.tex;h=d8f2fe9723182bdc4d66236e1b79b48a63454470;hp=f64f7909aab0fbedf59e4d5c5a0c7d2777a5f927;hb=25d228c0ea102570073ccf5dd642615cc2ebe7a1;hpb=21c8a86147c874c29156d1859a670c3f7c6df7ee diff --git a/Chapters/Normalization.tex b/Chapters/Normalization.tex index f64f790..d8f2fe9 100644 --- a/Chapters/Normalization.tex +++ b/Chapters/Normalization.tex @@ -14,11 +14,6 @@ } } - -% A transformation example -\definefloat[example][examples] -\setupcaption[example][location=top] % Put captions on top - \define[3]\transexample{ \placeexample[here]{#1} \startcombination[2*1] @@ -26,24 +21,16 @@ {\example{#3}}{Transformed program} \stopcombination } -% -%\define[3]\transexampleh{ -%% \placeexample[here]{#1} -%% \startcombination[1*2] -%% {\example{#2}}{Original program} -%% {\example{#3}}{Transformed program} -%% \stopcombination -%} - -The first step in the core to VHDL translation process, is normalization. We + +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 VHDL easily. This normal form is needed because -the full core language is more expressive than VHDL in some areas and because +subsequently translate into \small{VHDL} easily. This normal form is needed because +the full core language is more expressive than \small{VHDL} in some areas and because core can describe expressions that do not have a direct hardware interpretation. -TODO: Describe core properties not supported in VHDL, and describe how the -VHDL we want to generate should look like. +TODO: Describe core properties not supported in \small{VHDL}, and describe how the +\small{VHDL} we want to generate should look like. \section{Normal form} The transformations described here have a well-defined goal: To bring the @@ -73,10 +60,10 @@ describing the things we want to not have in a normal form. generate a hardware signal that contains a function, so all values, arguments and returns values used must be first order. - \item Any complex \emph{nested scopes} must be removed. In the VHDL + \item Any complex \emph{nested scopes} must be removed. In the \small{VHDL} description, every signal is in a single scope. Also, full expressions are not supported everywhere (in particular port maps can only map signal names, - not expressions). To make the VHDL generation easy, all values must be bound + not expressions). To make the \small{VHDL} generation easy, all values must be bound on the \quote{top level}. \stopitemize @@ -215,7 +202,13 @@ alu = λopcode.λa.λb. 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. +exception of builtin higher order functions. The graphical version of the +architecture contains a slightly simplified version, since the state tuple +packing and unpacking have been left out. Instead, two seperate registers are +drawn. Also note that most synthesis tools will further optimize this +architecture by removing the multiplexers at the register input and replace +them with some logic in the clock inputs, but we want to show the architecture +as close to the description as possible. \startbuffer[NormalComplete] regbank :: Bit @@ -312,7 +305,7 @@ subtractor.} {\boxedgraphic{NormalComplete}}{The architecture described by the normal form.} \stopcombination -\subsection{Normal form definition} +\subsection{Intended normal form definition} Now we have some intuition for the normal form, we can describe how we want the normal form to look like in a slightly more formal manner. The following EBNF-like description completely captures the intended structure (and @@ -364,826 +357,1310 @@ the output port. Most function applications bound by the let expression define a component instantiation, where the input and output ports are mapped to local signals or arguments. Some of the others use a builtin construction (\eg the \lam{case} statement) or call a builtin function -(\eg \lam{add} or \lam{sub}). For these, a hardcoded VHDL translation is +(\eg \lam{add} or \lam{sub}). For these, a hardcoded \small{VHDL} translation is available. -\subsection{Definitions} -In the following sections, we will be using a number of functions and -notations, which we will define here. +\section{Transformation notation} +To be able to concisely present transformations, we use a specific format to +them. It is a simple format, similar to one used in logic reasoning. -\subsubsection{Transformations} -The most important notation is the one for transformation, which looks like -the following: +Such a transformation description looks like the following. \starttrans -context conditions + ~ -from ------------------------- expression conditions -to + +-------------------------- + ~ -context additions + \stoptrans -Here, we describe a transformation. The most import parts are \lam{from} and -\lam{to}, which describe the Core expresssion that should be matched and the -expression that it should be replaced with. This matching can occur anywhere -in function that is being normalized, so it applies to any subexpression as -well. - -The \lam{expression conditions} list a number of conditions on the \lam{from} -expression that must hold for the transformation to apply. - -Furthermore, there is some way to look into the environment (\eg, other top -level bindings). The \lam{context conditions} part specifies any number of -top level bindings that must be present for the transformation to apply. -Usually, this lists a top level binding that binds an identfier that is also -used in the \lam{from} expression, allowing us to "access" the value of a top -level binding in the \lam{to} expression (\eg, for inlining). - -Finally, there is a way to influence the environment. The \lam{context -additions} part lists any number of new top level bindings that should be -added. - -If there are no \lam{context conditions} or \lam{context additions}, they can -be left out alltogether, along with the separator \lam{~}. - -TODO: Example - -\subsubsection{Other concepts} -A \emph{global variable} is any variable that is bound at the -top level of a program, or an external module. A local variable is any other -variable (\eg, variables local to a function, which can be bound by lambda -abstractions, let expressions and case expressions). - -A \emph{hardware representable} type is a type that we can generate -a signal for in hardware. For example, a bit, a vector of bits, a 32 bit -unsigned word, etc. Types that are not runtime representable notably -include (but are not limited to): Types, dictionaries, functions. - -A \emph{builtin function} is a function for which a builtin -hardware translation is available, because its actual definition is not -translatable. A user-defined function is any other function. - -\subsubsection{Functions} -Here, we define a number of functions that can be used below to concisely -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. - -\emph{lvar(expr)} is the inverse of \emph{gvar}; it is true when \emph{expr} -references a local variable, false when it references a global variable. - -\emph{representable(expr)} or \emph{representable(var)} is true when -\emph{expr} or \emph{var} has a type that is representable at runtime. - -\section{Transform passes} -In this section we describe the actual transforms. Here we're using -the core language in a notation that resembles lambda calculus. - -Each of these transforms is meant to be applied to every (sub)expression -in a program, for as long as it applies. Only when none of the -transformations can be applied anymore, the program is in normal form (by -definition). We hope to be able to prove that this form will obey all of the -constraints defined above, but this has yet to happen (though it seems likely -that it will). - -Each of the transforms will be described informally first, explaining -the need for and goal of the transform. Then, a formal definition is -given, using a familiar syntax from the world of logic. Each transform -is specified as a number of conditions (above the horizontal line) and a -number of conclusions (below the horizontal line). The details of using -this notation are still a bit fuzzy, so comments are welcom. - -TODO: Formally describe the "apply to every (sub)expression" in terms of -rules with full transformations in the conditions. - -\subsection{η-abstraction} -This transformation makes sure that all arguments of a function-typed -expression are named, by introducing lambda expressions. When combined with -β-reduction and function inlining below, all function-typed expressions should -be lambda abstractions or global identifiers. +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 +in pointy brackets, most of which should be rather obvious in their meaning. +Nevertheless, we will more precisely specify their meaning below: + + \startdesc{} The expression pattern that will be matched + against (subexpressions of) the expression to be transformed. We call this a + pattern, because it can contain \emph{placeholders} (variables), which match + any expression or binder. Any such placeholder is said to be \emph{bound} to + the expression it matches. It is convention to use an uppercase latter (\eg + \lam{M} or \lam{E} to refer to any expression (including a simple variable + reference) and lowercase letters (\eg \lam{v} or \lam{b}) to refer to + (references to) binders. + + For example, the pattern \lam{a + B} will match the expression + \lam{v + (2 * w)} (and bind \lam{a} to \lam{v} and \lam{B} to + \lam{(2 * 2)}), but not \lam{v + (2 * w)}. + \stopdesc + + \startdesc{} + These are extra conditions on the expression that is matched. These + conditions can be used to further limit the cases in which the + transformation applies, in particular to prevent a transformation from + causing a loop with itself or another transformation. + + Only if these if these conditions are \emph{all} true, this transformation + applies. + \stopdesc + + \startdesc{} + These are a number of extra conditions on the context of the function. In + particular, these conditions can require some other top level function to be + present, whose value matches the pattern given here. The format of each of + these conditions is: \lam{binder = }. + + Typically, the binder is some placeholder bound in the \lam{}, while the pattern contains some placeholders that are used in + the \lam{transformed expression}. + + Only if a top level binder exists that matches each binder and pattern, this + transformation applies. + \stopdesc + + \startdesc{} + 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{}. + We call this a template, because it can contain placeholders, referring to + any placeholder bound by the \lam{} or the + \lam{}. The resulting expression will have those + placeholders replaced by the values bound to them. + + Any binder (lowercase) placeholder that has no value bound to it yet will be + bound to (and replaced with) a fresh binder. + \stopdesc + + \startdesc{} + These are templates for new functions to add to the context. This is a way + to have a transformation create new top level functiosn. + + 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 + binder placeholder that has no value bound to it yet will be bound to (and + replaced with) a fresh binder. + \stopdesc + + As an example, we'll look at η-abstraction: \starttrans -E \lam{E :: * -> *} --------------- \lam{E} is not the first argument of an application. +E \lam{E :: a -> b} +-------------- \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] -foo = λa.case a of - True -> λb.mul b b - False -> id -\stopbuffer - -\startbuffer[to] -foo = λa.λx.(case a of - True -> λb.mul b b - False -> λy.id y) x -\stopbuffer - -\transexample{η-abstraction}{from}{to} - -\subsection{Extended β-reduction} -This transformation is meant to propagate application expressions downwards -into expressions as far as possible. In lambda calculus, this reduction -is known as β-reduction, but it is of course only defined for -applications of lambda abstractions. We extend this reduction to also -work for the rest of core (case and let expressions). - -For let expressions: -\starttrans -let binds in E) M ------------------ -let binds in E M -\stoptrans - -For case statements: -\starttrans -(case x of - p1 -> E1 - \vdots - pn -> En) M ------------------ -case x of - p1 -> E1 M - \vdots - pn -> En M -\stoptrans - -For lambda expressions: -\starttrans -(λx.E) M ------------------ -E[M/x] \stoptrans -% And an example -\startbuffer[from] -( let a = (case x of - True -> id - False -> neg - ) 1 - b = (let y = 3 in add y) 2 - in - (λz.add 1 z) -) 3 -\stopbuffer - -\startbuffer[to] -let a = case x of - True -> id 1 - False -> neg 1 - b = let y = 3 in add y 2 -in - add 1 3 -\stopbuffer - -\transexample{Extended β-reduction}{from}{to} - -\subsection{Let derecursification} -This transformation is meant to make lets non-recursive whenever possible. -This might allow other optimizations to do their work better. TODO: Why is -this needed exactly? - -\subsection{Let flattening} -This transformation puts nested lets in the same scope, by lifting the -binding(s) of the inner let into a new let around the outer let. Eventually, -this will cause all let bindings to appear in the same scope (they will all be -in scope for the function return value). - -Note that this transformation does not try to be smart when faced with -recursive lets, it will just leave the lets recursive (possibly joining a -recursive and non-recursive let into a single recursive let). The let -rederursification transformation will do this instead. - -\starttrans -letnonrec x = (let bindings in M) in N ------------------------------------------- -let bindings in (letnonrec x = M) in N -\stoptrans - -\starttrans -letrec - \vdots - x = (let bindings in M) - \vdots -in - N ------------------------------------------- -letrec - \vdots - bindings - x = M - \vdots -in - N -\stoptrans - -\startbuffer[from] -let - a = letrec - x = 1 - y = 2 - in - x + y -in - letrec - b = let c = 3 in a + c - d = 4 - in - d + b -\stopbuffer -\startbuffer[to] -letrec - x = 1 - y = 2 -in - let - a = x + y - in - letrec - c = 3 - b = a + c - d = 4 - in - d + b -\stopbuffer - -\transexample{Let flattening}{from}{to} - -\subsection{Empty let removal} -This transformation is simple: It removes recursive lets that have no bindings -(which usually occurs when let derecursification removes the last binding from -it). - -\starttrans -letrec in M --------------- -M -\stoptrans - -\subsection{Simple let binding removal} -This transformation inlines simple let bindings (\eg a = b). - -This transformation is not needed to get into normal form, but makes the -resulting VHDL a lot shorter. - -\starttrans -letnonrec - a = b -in - M ------------------ -M[b/a] -\stoptrans - -\starttrans -letrec - \vdots - a = b - \vdots -in - M ------------------ -let - \vdots [b/a] - \vdots [b/a] -in - M[b/a] -\stoptrans - -\subsection{Unused let binding removal} -This transformation removes let bindings that are never used. Usually, -the desugarer introduces some unused let bindings. - -This normalization pass should really be unneeded to get into normal form -(since ununsed 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 VHDL a lot shorter. - -\starttrans -let a = E in M ----------------------------- \lam{a} does not occur free in \lam{M} -M -\stoptrans - -\starttrans -letrec - \vdots - a = E - \vdots -in - M ----------------------------- \lam{a} does not occur free in \lam{M} -letrec - \vdots - \vdots -in - M -\stoptrans - -\subsection{Non-representable binding inlining} -This transform inlines let bindings that have a non-representable type. Since -we can never generate a signal assignment for these bindings (we cannot -declare a signal assignment with a non-representable type, for obvious -reasons), we have no choice but to inline the binding to remove it. - -If the binding is non-representable because it is a lambda abstraction, it is -likely that it will inlined into an application and β-reduction will remove -the lambda abstraction and turn it into a representable expression at the -inline site. The same holds for partial applications, which can be turned into -full applications by inlining. - -Other cases of non-representable bindings we see in practice are primitive -Haskell types. In most cases, these will not result in a valid normalized -output, but then the input would have been invalid to start with. There is one -exception to this: When a builtin function is applied to a non-representable -expression, things might work out in some cases. For example, when you write a -literal \hs{SizedInt} in Haskell, like \hs{1 :: SizedInt D8}, this results in -the following core: \lam{fromInteger (smallInteger 10)}, where for example -\lam{10 :: GHC.Prim.Int\#} and \lam{smallInteger 10 :: Integer} have -non-representable types. TODO: This/these paragraph(s) should probably become a -separate discussion somewhere else. - -\starttrans -letnonrec a = E in M --------------------------- \lam{E} has a non-representable type. -M[E/a] -\stoptrans - -\starttrans -letrec - \vdots - a = E - \vdots -in - M --------------------------- \lam{E} has a non-representable type. -letrec - \vdots [E/a] - \vdots [E/a] -in - M[E/a] -\stoptrans - -\startbuffer[from] -letrec - a = smallInteger 10 - inc = λa -> add a 1 - inc' = add 1 - x = fromInteger a -in - inc (inc' x) -\stopbuffer + Consider the following function, which is a fairly obvious way to specify a + simple ALU (Note \at{example}[ex:AddSubAlu] is the normal form of this + function): -\startbuffer[to] -letrec - x = fromInteger (smallInteger 10) -in - (λa -> add a 1) (add 1 x) -\stopbuffer - -\transexample{Let flattening}{from}{to} - -\subsection{Compiler generated top level binding inlining} -TODO - -\subsection{Scrutinee simplification} -This transform ensures that the scrutinee of a case expression is always -a simple variable reference. - -\starttrans -case E of - alts ------------------ \lam{E} is not a local variable reference -let x = E in - case E of - alts -\stoptrans - -\startbuffer[from] -case (foo a) of - True -> a - False -> b -\stopbuffer - -\startbuffer[to] -let x = foo a in - case x of - True -> a - False -> b -\stopbuffer - -\transexample{Let flattening}{from}{to} - - -\subsection{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 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}. -\stopitemize - -\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) -letnonrec - v0,0 = case x of C0 v0,0 .. v0,m -> v0,0 - \vdots - v0,m = case x of C0 v0,0 .. v0,m -> v0,m - x0 = E0 - \dots - vn,m = case x of Cn vn,0 .. vn,m -> vn,m - xn = En -in - case E of - C0 w0,0 ... w0,m -> x0 - \vdots - Cn wn,0 ... wn,m -> xn -\stoptrans - -TODO: This transformation specified like this is complicated and misses -conditions to prevent looping with itself. Perhaps we should split it here for -discussion? - -\startbuffer[from] -case a of - True -> add b 1 - False -> add b 2 -\stopbuffer - -\startbuffer[to] -letnonrec - x0 = add b 1 - x1 = add b 2 -in - case a of - True -> x0 - False -> x1 -\stopbuffer - -\transexample{Selector case simplification}{from}{to} - -\startbuffer[from] -case a of - (,) b c -> add b c -\stopbuffer -\startbuffer[to] -letnonrec - b = case a of (,) b c -> b - c = case a of (,) b c -> c - x0 = add b c -in - case a of - (,) w0 w1 -> x0 -\stopbuffer - -\transexample{Extractor case simplification}{from}{to} +\startlambda +alu :: Bit -> Word -> Word -> Word +alu = λopcode. case opcode of + Low -> (+) + High -> (-) +\stoplambda -\subsection{Case removal} -This transform removes any case statements with a single alternative and -only wild binders. + There are a few subexpressions 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 + conditions to the right of the transformation. -These "useless" case statements are usually leftovers from case simplification -on extractor case (see the previous example). + We will look at each expression in the function in a top down manner. The + first expression is the entire expression the function is bound to. -\starttrans -case x of - C v0 ... vm -> E ----------------------- \lam{\forall i, 0 <= i <= m} (\lam{vi} does not occur free in E) -E -\stoptrans +\startlambda +λopcode. case opcode of + Low -> (+) + High -> (-) +\stoplambda -\startbuffer[from] -case a of - (,) w0 w1 -> x0 -\stopbuffer + As said, the expression pattern matches this. The type of this expression is + \lam{Bit -> Word -> Word -> Word}, which matches \lam{a -> b} (Note that in + this case \lam{a = Bit} and \lam{b = Word -> Word -> Word}). -\startbuffer[to] -x0 -\stopbuffer + Since this expression is at top level, it does not occur at a function + position of an application. However, The expression is a lambda abstraction, + so this transformation does not apply. -\transexample{Case removal}{from}{to} + The next expression we could apply this transformation to, is the body of + the lambda abstraction: -\subsection{Argument simplification} -The transforms in this section deal with simplifying application -arguments into normal form. The goal here is to: +\startlambda +case opcode of + Low -> (+) + High -> (-) +\stoplambda -\startitemize - \item Make all arguments of user-defined functions (\eg, of which - we have a function body) simple variable references of a runtime - representable type. This is needed, since these applications will be turned - into component instantiations. - \item Make all arguments of builtin functions one of: - \startitemize - \item A type argument. - \item A dictionary argument. - \item A type level expression. - \item A variable reference of a runtime representable type. - \item A variable reference or partial application of a function type. - \stopitemize -\stopitemize + The type of this expression is \lam{Word -> Word -> Word}, which again + matches \lam{a -> b}. The expression is the body of a lambda expression, so + it does not occur at a function position of an application. Finally, the + expression is not a lambda abstraction but a case expression, so all the + conditions match. There are no context conditions to match, so the + transformation applies. -When looking at the arguments of a user-defined function, we can -divide them into two categories: -\startitemize - \item Arguments of a runtime representable type (\eg bits or vectors). - - These arguments can be preserved in the program, since they can - be translated to input ports later on. However, since we can - only connect signals to input ports, these arguments must be - reduced to simple variables (for which signals will be - produced). This is taken care of by the argument extraction - transform. - \item Non-runtime representable typed arguments. - - These arguments cannot be preserved in the program, since we - cannot represent them as input or output ports in the resulting - VHDL. To remove them, we create a specialized version of the - called function with these arguments filled in. This is done by - the argument propagation transform. - - Typically, these arguments are type and dictionary arguments that are - used to make functions polymorphic. By propagating these arguments, we - are essentially doing the same which GHC does when it specializes - functions: Creating multiple variants of the same function, one for - each type for which it is used. Other common non-representable - arguments are functions, e.g. when calling a higher order function - with another function or a lambda abstraction as an argument. - - The reason for doing this is similar to the reasoning provided for - the inlining of non-representable let bindings above. In fact, this - argument propagation could be viewed as a form of cross-function - inlining. -\stopitemize + 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 + \lam{a}. This results in the following replacement expression: -TODO: Check the following itemization. +\startlambda +λa.(case opcode of + Low -> (+) + High -> (-)) a +\stoplambda -When looking at the arguments of a builtin function, we can divide them -into categories: + Continuing with this expression, we see that the transformation does not + apply again (it is a lambda expression). Next we look at the body of this + labmda abstraction: -\startitemize - \item Arguments of a runtime representable type. - - As we have seen with user-defined functions, these arguments can - always be reduced to a simple variable reference, by the - argument extraction transform. Performing this transform for - builtin functions as well, means that the translation of builtin - functions can be limited to signal references, instead of - needing to support all possible expressions. - - \item Arguments of a function type. - - These arguments are functions passed to higher order builtins, - like \lam{map} and \lam{foldl}. Since implementing these - functions for arbitrary function-typed expressions (\eg, lambda - expressions) is rather comlex, we reduce these arguments to - (partial applications of) global functions. - - We can still support arbitrary expressions from the user code, - by creating a new global function containing that expression. - This way, we can simply replace the argument with a reference to - that new function. However, since the expression can contain any - number of free variables we also have to include partial - applications in our normal form. - - This category of arguments is handled by the function extraction - transform. - \item Other unrepresentable arguments. - - These arguments can take a few different forms: - \startdesc{Type arguments} - In the core language, type arguments can only take a single - form: A type wrapped in the Type constructor. Also, there is - nothing that can be done with type expressions, except for - applying functions to them, so we can simply leave type - arguments as they are. - \stopdesc - \startdesc{Dictionary arguments} - In the core language, dictionary arguments are used to find - operations operating on one of the type arguments (mostly for - finding class methods). Since we will not actually evaluatie - the function body for builtin functions and can generate - code for builtin functions by just looking at the type - arguments, these arguments can be ignored and left as they - are. - \stopdesc - \startdesc{Type level arguments} - Sometimes, we want to pass a value to a builtin function, but - we need to know the value at compile time. Additionally, the - value has an impact on the type of the function. This is - encoded using type-level values, where the actual value of the - argument is not important, but the type encodes some integer, - for example. Since the value is not important, the actual form - of the expression does not matter either and we can leave - these arguments as they are. - \stopdesc - \startdesc{Other arguments} - Technically, there is still a wide array of arguments that can - be passed, but does not fall into any of the above categories. - However, none of the supported builtin functions requires such - an argument. This leaves use with passing unsupported types to - a function, such as calling \lam{head} on a list of functions. - - In these cases, it would be impossible to generate hardware - for such a function call anyway, so we can ignore these - arguments. - - The only way to generate hardware for builtin functions with - arguments like these, is to expand the function call into an - equivalent core expression (\eg, expand map into a series of - function applications). But for now, we choose to simply not - support expressions like these. - \stopdesc - - From the above, we can conclude that we can simply ignore these - other unrepresentable arguments and focus on the first two - categories instead. -\stopitemize +\startlambda +(case opcode of + Low -> (+) + High -> (-)) a +\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: -\subsubsection{Argument simplification} -This transform deals with arguments to functions that -are of a runtime representable type. It ensures that they will all become -references to global variables, or local signals in the resulting VHDL. +\startlambda +λb.(case opcode of + Low -> (+) + High -> (-)) a b +\stoplambda -TODO: It seems we can map an expression to a port, not only a signal. -Perhaps this makes this transformation not needed? -TODO: Say something about dataconstructors (without arguments, like True -or False), which are variable references of a runtime representable -type, but do not result in a signal. + Again, the transformation does not apply to this lambda abstraction, so we + look at its body. For brevity, we'll put the case statement on one line from + now on. -To reduce a complex expression to a simple variable reference, we create -a new let expression around the application, which binds the complex -expression to a new variable. The original function is then applied to -this variable. +\startlambda +(case opcode of Low -> (+); High -> (-)) a b +\stoplambda -\starttrans -M N --------------------- \lam{N} is of a representable type -let x = N in M x \lam{N} is not a local variable reference -\stoptrans + 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 + 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: -\startbuffer[from] -add (add a 1) 1 -\stopbuffer +\startlambda +(case opcode of Low -> (+); High -> (-)) a +\stoplambda -\startbuffer[to] -let x = add a 1 in add x 1 -\stopbuffer + Obviously, the transformation does not apply here, since it occurs in + function position. In the same way the transformation does not apply to both + components of this expression (\lam{case opcode of Low -> (+); High -> (-)} + and \lam{a}), so we'll skip to the components of the case expression: The + scrutinee and both alternatives. Since the opcode is not a function, it does + not apply here, and we'll leave both alternatives as an exercise to the + reader. The final function, after all these transformations becomes: -\transexample{Argument extraction}{from}{to} +\startlambda +alu :: Bit -> Word -> Word -> Word +alu = λopcode.λa.b. (case opcode of + Low -> λa1.λb1 (+) a1 b1 + High -> λa2.λb2 (-) a2 b2) a b +\stoplambda -\subsubsection{Function extraction} -This transform deals with function-typed arguments to builtin functions. -Since these arguments cannot be propagated, we choose to extract them -into a new global function instead. + In this case, the transformation does not apply anymore, though this might + not always be the case (e.g., the application of a transformation on a + subexpression might open up possibilities to apply the transformation + further up in the expression). -Any free variables occuring 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. +\subsection{Transformation application} +In this chapter we define a number of transformations, but how will we apply +these? As stated before, our normal form is reached as soon as no +transformation applies anymore. This means our application strategy is to +simply apply any transformation that applies, and continuing to do that with +the result of each transformation. -This transformation is useful when applying higher order builtin functions -like \hs{map} to a lambda abstraction, for example. In this case, the code -that generates 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). +In particular, we define no particular order of transformations. Since +transformation order should not influence the resulting normal form (see TODO +ref), this leaves the implementation free to choose any application order that +results in an efficient implementation. -\starttrans -M N \lam{M} is a (partial aplication of a) builtin function. ---------------------- \lam{f0 ... fn} = free local variables of \lam{N} -M x f0 ... fn \lam{N :: a -> b} -~ \lam{N} is not a (partial application of) a top level function -x = λf0 ... λfn.N -\stoptrans +When applying a single transformation, we try to apply it to every (sub)expression +in a function, not just the top level function. This allows us to keep the +transformation descriptions concise and powerful. -\startbuffer[from] -map (λa . add a b) xs +\subsection{Definitions} +In the following sections, we will be using a number of functions and +notations, which we will define here. -map (add b) ys -\stopbuffer +\subsubsection{Other concepts} +A \emph{global variable} is any variable that is bound at the +top level of a program, or an external module. A \emph{local variable} is any +other variable (\eg, variables local to a function, which can be bound by +lambda abstractions, let expressions and pattern matches of case +alternatives). Note that this is a slightly different notion of global versus +local than what \small{GHC} uses internally. +\defref{global variable} \defref{local variable} + +A \emph{hardware representable} (or just \emph{representable}) type or value +is (a value of) a type that we can generate a signal for in hardware. For +example, a bit, a vector of bits, a 32 bit unsigned word, etc. Types that are +not runtime representable notably include (but are not limited to): Types, +dictionaries, functions. +\defref{representable} + +A \emph{builtin function} is a function supplied by the Cλash framework, whose +implementation is not valid Cλash. The implementation is of course valid +Haskell, for simulation, but it is not expressable in Cλash. +\defref{builtin function} \defref{user-defined function} + +For these functions, Cλash has a \emph{builtin hardware translation}, so calls +to these functions can still be translated. These are functions like +\lam{map}, \lam{hwor} and \lam{length}. + +A \emph{user-defined} function is a function for which we do have a Cλash +implementation available. -\startbuffer[to] -x0 = λb.λa.add a b -~ -map x0 xs +\subsubsection{Functions} +Here, we define a number of functions that can be used below to concisely +specify conditions. -x1 = λb.add b -map x1 ys -\stopbuffer +\refdef{global variable}\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. -\transexample{Function extraction}{from}{to} +\refdef{local variable}\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. -\subsubsection{Argument propagation} -This transform deals with arguments to user-defined functions that are -not representable at runtime. This means these arguments cannot be -preserved in the final form and most be {\em propagated}. +\refdef{representable}\emph{representable(expr)} or \emph{representable(var)} is true when +\emph{expr} or \emph{var} is \emph{representable}. -Propagation means to create a specialized version of the called -function, with the propagated argument already filled in. As a simple -example, in the following program: +\subsection{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 +expression: \startlambda -f = λa.λb.a + b -inc = λa.f a 1 +(λa.λb.λc. a * b * c) x c \stoplambda -we could {\em propagate} the constant argument 1, with the following -result: +By applying β-reduction (see below) once, we can simplify this expression to: \startlambda -f' = λa.a + 1 -inc = λa.f' a +(λb.λc. x * b * c) c \stoplambda -Special care must be taken when the to-be-propagated expression has any -free variables. If this is the case, the original argument should not be -removed alltogether, but replaced by all the free variables of the -expression. In this way, the original expression can still be evaluated -inside the new function. Also, this brings us closer to our goal: All -these free variables will be simple variable references. - -To prevent us from propagating the same argument over and over, a simple -local variable reference is not propagated (since is has exactly one -free variable, itself, we would only replace that argument with itself). +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. +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: -This shows that any free local variables that are not runtime representable -cannot be brought into normal form by this transform. We rely on an -inlining transformation to replace such a variable with an expression we -can propagate again. +\startlambda +λc. x * c * c +\stoplambda -\starttrans -x = E -~ -x Y0 ... Yi ... Yn \lam{Yi} is not of a runtime representable type ---------------------------------------------- \lam{Yi} is not a local variable reference -x' y0 ... yi-1 f0 ... fm Yi+1 ... Yn \lam{f0 ... fm} = free local vars of \lam{Yi} -~ -x' = λy0 ... yi-1 f0 ... fm yi+1 ... yn . - E y0 ... yi-1 Yi yi+1 ... yn +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. + +There are a number of ways to solve this. \small{GHC} has isolated this +problem to their binder substitution code, which performs \emph{deshadowing} +during its expression traversal. This means that any binding that shadows +another binding on a higher level is replaced by a new binder that does not +shadow any other binding. This non-shadowing invariant is enough to prevent +binder uniqueness problems in \small{GHC}. + +In our transformation system, maintaining this non-shadowing invariant is +a bit harder to do (mostly due to implementation issues, the prototype doesn't +use \small{GHC}'s subsitution code). Also, we can observe the following +points. -\stoptrans +\startitemize +\item Deshadowing 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 +occurs. -TODO: Example +\startlambda +(let x = 1 in x) + (let x = 2 in x) +\stoplambda -\subsection{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. +\item In our normal form (and the resulting \small{VHDL}), all binders +(signals) will end up in the same scope. To allow this, all binders within the +same function should be unique. -\subsection{Return value simplification} -This transformation ensures that the return value of a function is always a -simple local variable reference. +\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 +cause any problems either. The only way to cause conflicts is thus to +duplicate an existing subexpression. +\stopitemize -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). +Given the above, our prototype maintains a unique binder invariant. This +meanst that in any given moment during normalization, all binders \emph{within +a single function} must be unique. To achieve this, we apply the following +technique. -\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 -let x = E in x -\stoptrans +TODO: Define fresh binders and unique supplies -\starttrans -x = λv0 ... λvn.E -~ \lam{E} is representable -E \lam{E} is not a let expression ---------------------------- \lam{E} is not a local variable reference -let x = E in x -\stoptrans +\startitemize +\item Before starting normalization, all binders in the function are made +unique. This is done by generating a fresh binder for every binder used. This +also replaces binders that did not pose any conflict, but it does ensure that +all binders within the function are generated by the same unique supply. See +(TODO: ref fresh binder). +\item Whenever a new binder must be generated, we generate a fresh binder that +is guaranteed to be different from \emph{all binders generated so far}. This +can thus never introduce duplication and will maintain the invariant. +\item Whenever (part of) an expression is duplicated (for example when +inlining), all binders in the expression are replaced with fresh binders +(using the same method as at the start of normalization). These fresh binders +can never introduce duplication, so this will maintain the invariant. +\item Whenever we move part of an expression around within the function, there +is no need to do anything special. There is obviously no way to introduce +duplication by moving expressions around. Since we know that each of the +binders is already unique, there is no way to introduce (incorrect) shadowing +either. +\stopitemize -\starttrans -x = λv0 ... λvn.let ... in E -~ \lam{E} is representable -E \lam{E} is not a local variable reference ---------------------------- -let x = E in x -\stoptrans +\section{Transform passes} +In this section we describe the actual transforms. Here we're using +the core language in a notation that resembles lambda calculus. -\startbuffer[from] -x = add 1 2 -\stopbuffer +Each of these transforms is meant to be applied to every (sub)expression +in a program, for as long as it applies. Only when none of the +transformations can be applied anymore, the program is in normal form (by +definition). We hope to be able to prove that this form will obey all of the +constraints defined above, but this has yet to happen (though it seems likely +that it will). -\startbuffer[to] -x = let x = add 1 2 in x -\stopbuffer +Each of the transforms will be described informally first, explaining +the need for and goal of the transform. Then, a formal definition is +given, using a familiar syntax from the world of logic. Each transform +is specified as a number of conditions (above the horizontal line) and a +number of conclusions (below the horizontal line). The details of using +this notation are still a bit fuzzy, so comments are welcom. -\transexample{Return value simplification}{from}{to} + \subsection{General cleanup} + + \subsubsection{β-reduction} + β-reduction is a well known transformation from lambda calculus, where it is + the main reduction step. It reduces applications of labmda abstractions, + removing both the lambda abstraction and the application. + + In our transformation system, this step helps to remove unwanted lambda + abstractions (basically all but the ones at the top level). Other + transformations (application propagation, non-representable inlining) make + sure that most lambda abstractions will eventually be reducable by + β-reduction. + + TODO: Define substitution syntax + + \starttrans + (λx.E) M + ----------------- + E[M/x] + \stoptrans + + % And an example + \startbuffer[from] + (λa. 2 * a) (2 * b) + \stopbuffer + + \startbuffer[to] + 2 * (2 * b) + \stopbuffer + + \transexample{β-reduction}{from}{to} + + \subsubsection{Application propagation} + This transformation is meant to propagate application expressions downwards + into expressions as far as possible. This allows partial applications inside + expressions to become fully applied and exposes new transformation + possibilities for other transformations (like β-reduction). + + \starttrans + let binds in E) M + ----------------- + let binds in E M + \stoptrans + + % And an example + \startbuffer[from] + ( let + val = 1 + in + add val + ) 3 + \stopbuffer + + \startbuffer[to] + let + val = 1 + in + add val 3 + \stopbuffer + + \transexample{Application propagation for a let expression}{from}{to} + + \starttrans + (case x of + p1 -> E1 + \vdots + pn -> En) M + ----------------- + case x of + p1 -> E1 M + \vdots + pn -> En M + \stoptrans + + % And an example + \startbuffer[from] + ( case x of + True -> id + False -> neg + ) 1 + \stopbuffer + + \startbuffer[to] + case x of + True -> id 1 + False -> neg 1 + \stopbuffer + + \transexample{Application propagation for a case expression}{from}{to} + + \subsubsection{Empty let removal} + This transformation is simple: It removes recursive lets that have no bindings + (which usually occurs when let derecursification removes the last binding from + it). + + \starttrans + letrec in M + -------------- + M + \stoptrans + + \subsubsection{Simple let binding removal} + This transformation inlines simple let bindings (\eg a = b). + + This transformation is not needed to get into normal form, but makes the + resulting \small{VHDL} a lot shorter. + + \starttrans + letnonrec + a = b + in + M + ----------------- + M[b/a] + \stoptrans + + \starttrans + letrec + \vdots + a = b + \vdots + in + M + ----------------- + let + \vdots [b/a] + \vdots [b/a] + in + M[b/a] + \stoptrans + + \subsubsection{Unused let binding removal} + This transformation removes let bindings that are never used. Usually, + the desugarer introduces some unused let bindings. + + This normalization pass should really be unneeded to get into normal form + (since ununsed 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 a = E in M + ---------------------------- \lam{a} does not occur free in \lam{M} + M + \stoptrans + + \starttrans + letrec + \vdots + a = E + \vdots + in + M + ---------------------------- \lam{a} does not occur free in \lam{M} + letrec + \vdots + \vdots + in + M + \stoptrans + + \subsubsection{Cast propagation / simplification} + This transform pushes casts down into the expression as far as possible. + Since its exact role and need is not clear yet, this transformation is + not yet specified. + + \subsubsection{Compiler generated top level binding inlining} + TODO + + \section{Program structure} + + \subsubsection{η-abstraction} + This transformation makes sure that all arguments of a function-typed + expression are named, by introducing lambda expressions. When combined with + β-reduction and function inlining below, all function-typed expressions should + be lambda abstractions or global identifiers. + + \starttrans + E \lam{E :: a -> b} + -------------- \lam{E} is not the first argument of 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] + foo = λa.case a of + True -> λb.mul b b + False -> id + \stopbuffer + + \startbuffer[to] + foo = λa.λx.(case a of + True -> λb.mul b b + False -> λy.id y) x + \stopbuffer + + \transexample{η-abstraction}{from}{to} + + \subsubsection{Let derecursification} + This transformation is meant to make lets non-recursive whenever possible. + This might allow other optimizations to do their work better. TODO: Why is + this needed exactly? + + \subsubsection{Let flattening} + This transformation puts nested lets in the same scope, by lifting the + binding(s) of the inner let into a new let around the outer let. Eventually, + this will cause all let bindings to appear in the same scope (they will all be + in scope for the function return value). + + Note that this transformation does not try to be smart when faced with + recursive lets, it will just leave the lets recursive (possibly joining a + recursive and non-recursive let into a single recursive let). The let + rederecursification transformation will do this instead. + + \starttrans + letnonrec x = (let bindings in M) in N + ------------------------------------------ + let bindings in (letnonrec x = M) in N + \stoptrans + + \starttrans + letrec + \vdots + x = (let bindings in M) + \vdots + in + N + ------------------------------------------ + letrec + \vdots + bindings + x = M + \vdots + in + N + \stoptrans + + \startbuffer[from] + let + a = letrec + x = 1 + y = 2 + in + x + y + in + letrec + b = let c = 3 in a + c + d = 4 + in + d + b + \stopbuffer + \startbuffer[to] + letrec + x = 1 + y = 2 + in + let + a = x + y + in + letrec + c = 3 + b = a + c + d = 4 + in + d + b + \stopbuffer + + \transexample{Let flattening}{from}{to} + + \subsubsection{Return value simplification} + This transformation ensures that the return value of a function is always a + simple local variable reference. + + Currently implemented using lambda simplification, let simplification, and + top simplification. Should change into something like the following, which + works only on the result of a function instead of any subexpression. This is + achieved by the contexts, like \lam{x = E}, though this is strictly not + correct (you could read this as "if there is any function \lam{x} that binds + \lam{E}, any \lam{E} can be transformed, while we only mean the \lam{E} that + is bound by \lam{x}. This might need some extra notes or something). + + \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 + let x = E in x + \stoptrans + + \starttrans + x = λv0 ... λvn.E + ~ \lam{E} is representable + E \lam{E} is not a let expression + --------------------------- \lam{E} is not a local variable reference + let x = E in x + \stoptrans + + \starttrans + x = λv0 ... λvn.let ... in E + ~ \lam{E} is representable + E \lam{E} is not a local variable reference + --------------------------- + let x = E in x + \stoptrans + + \startbuffer[from] + x = add 1 2 + \stopbuffer + + \startbuffer[to] + x = let x = add 1 2 in x + \stopbuffer + + \transexample{Return value simplification}{from}{to} + + \subsection{Argument simplification} + The transforms in this section deal with simplifying application + arguments into normal form. The goal here is to: + + \startitemize + \item Make all arguments of user-defined functions (\eg, of which + we have a function body) simple variable references of a runtime + representable type. This is needed, since these applications will be turned + into component instantiations. + \item Make all arguments of builtin functions one of: + \startitemize + \item A type argument. + \item A dictionary argument. + \item A type level expression. + \item A variable reference of a runtime representable type. + \item A variable reference or partial application of a function type. + \stopitemize + \stopitemize + + When looking at the arguments of a user-defined function, we can + divide them into two categories: + \startitemize + \item Arguments of a runtime representable type (\eg bits or vectors). + + These arguments can be preserved in the program, since they can + be translated to input ports later on. However, since we can + only connect signals to input ports, these arguments must be + reduced to simple variables (for which signals will be + produced). This is taken care of by the argument extraction + transform. + \item Non-runtime representable typed arguments. + + These arguments cannot be preserved in the program, since we + cannot represent them as input or output ports in the resulting + \small{VHDL}. To remove them, we create a specialized version of the + called function with these arguments filled in. This is done by + the argument propagation transform. + + Typically, these arguments are type and dictionary arguments that are + used to make functions polymorphic. By propagating these arguments, we + are essentially doing the same which GHC does when it specializes + functions: Creating multiple variants of the same function, one for + each type for which it is used. Other common non-representable + arguments are functions, e.g. when calling a higher order function + with another function or a lambda abstraction as an argument. + + The reason for doing this is similar to the reasoning provided for + the inlining of non-representable let bindings above. In fact, this + argument propagation could be viewed as a form of cross-function + inlining. + \stopitemize + + TODO: Check the following itemization. + + When looking at the arguments of a builtin function, we can divide them + into categories: + + \startitemize + \item Arguments of a runtime representable type. + + As we have seen with user-defined functions, these arguments can + always be reduced to a simple variable reference, by the + argument extraction transform. Performing this transform for + builtin functions as well, means that the translation of builtin + functions can be limited to signal references, instead of + needing to support all possible expressions. + + \item Arguments of a function type. + + These arguments are functions passed to higher order builtins, + like \lam{map} and \lam{foldl}. Since implementing these + functions for arbitrary function-typed expressions (\eg, lambda + expressions) is rather comlex, we reduce these arguments to + (partial applications of) global functions. + + We can still support arbitrary expressions from the user code, + by creating a new global function containing that expression. + This way, we can simply replace the argument with a reference to + that new function. However, since the expression can contain any + number of free variables we also have to include partial + applications in our normal form. + + This category of arguments is handled by the function extraction + transform. + \item Other unrepresentable arguments. + + These arguments can take a few different forms: + \startdesc{Type arguments} + In the core language, type arguments can only take a single + form: A type wrapped in the Type constructor. Also, there is + nothing that can be done with type expressions, except for + applying functions to them, so we can simply leave type + arguments as they are. + \stopdesc + \startdesc{Dictionary arguments} + In the core language, dictionary arguments are used to find + operations operating on one of the type arguments (mostly for + finding class methods). Since we will not actually evaluatie + the function body for builtin functions and can generate + code for builtin functions by just looking at the type + arguments, these arguments can be ignored and left as they + are. + \stopdesc + \startdesc{Type level arguments} + Sometimes, we want to pass a value to a builtin function, but + we need to know the value at compile time. Additionally, the + value has an impact on the type of the function. This is + encoded using type-level values, where the actual value of the + argument is not important, but the type encodes some integer, + for example. Since the value is not important, the actual form + of the expression does not matter either and we can leave + these arguments as they are. + \stopdesc + \startdesc{Other arguments} + Technically, there is still a wide array of arguments that can + be passed, but does not fall into any of the above categories. + However, none of the supported builtin functions requires such + an argument. This leaves use with passing unsupported types to + a function, such as calling \lam{head} on a list of functions. + + In these cases, it would be impossible to generate hardware + for such a function call anyway, so we can ignore these + arguments. + + The only way to generate hardware for builtin functions with + arguments like these, is to expand the function call into an + equivalent core expression (\eg, expand map into a series of + function applications). But for now, we choose to simply not + support expressions like these. + \stopdesc + + From the above, we can conclude that we can simply ignore these + other unrepresentable arguments and focus on the first two + categories instead. + \stopitemize + + \subsubsection{Argument simplification} + This transform deals with arguments to functions that + are of a runtime representable type. It ensures that they will all become + references to global variables, or local signals in the resulting \small{VHDL}. + + TODO: It seems we can map an expression to a port, not only a signal. + Perhaps this makes this transformation not needed? + TODO: Say something about dataconstructors (without arguments, like True + or False), which are variable references of a runtime representable + type, but do not result in a signal. + + To reduce a complex expression to a simple variable reference, we create + a new let expression around the application, which binds the complex + expression to a new variable. The original function is then applied to + this variable. + + \starttrans + M N + -------------------- \lam{N} is of a representable type + let x = N in M x \lam{N} is not a local variable reference + \stoptrans + + \startbuffer[from] + add (add a 1) 1 + \stopbuffer + + \startbuffer[to] + let x = add a 1 in add x 1 + \stopbuffer + + \transexample{Argument extraction}{from}{to} + + \subsubsection{Function extraction} + This transform deals with function-typed arguments to builtin functions. + Since these arguments cannot be propagated, we choose to extract them + into a new global function instead. + + Any free variables occuring 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 + like \hs{map} to a lambda abstraction, for example. In this case, the code + that generates \small{VHDL} for \hs{map} only needs to handle top level functions and + partial applications, not any other expression (such as lambda abstractions or + even more complicated expressions). + + \starttrans + M N \lam{M} is a (partial aplication of a) builtin function. + --------------------- \lam{f0 ... fn} = free local variables of \lam{N} + M x f0 ... fn \lam{N :: a -> b} + ~ \lam{N} is not a (partial application of) a top level function + x = λf0 ... λfn.N + \stoptrans + + \startbuffer[from] + map (λa . add a b) xs + + map (add b) ys + \stopbuffer + + \startbuffer[to] + x0 = λb.λa.add a b + ~ + map x0 xs + + x1 = λb.add b + map x1 ys + \stopbuffer + + \transexample{Function extraction}{from}{to} + + \subsubsection{Argument propagation} + This transform deals with arguments to user-defined functions that are + not representable at runtime. This means these arguments cannot be + preserved in the final form and most be {\em propagated}. + + Propagation means to create a specialized version of the called + function, with the propagated argument already filled in. As a simple + example, in the following program: + + \startlambda + f = λa.λb.a + b + inc = λa.f a 1 + \stoplambda + + we could {\em propagate} the constant argument 1, with the following + result: + + \startlambda + f' = λa.a + 1 + inc = λa.f' a + \stoplambda + + Special care must be taken when the to-be-propagated expression has any + free variables. If this is the case, the original argument should not be + removed alltogether, but replaced by all the free variables of the + expression. In this way, the original expression can still be evaluated + inside the new function. Also, this brings us closer to our goal: All + these free variables will be simple variable references. + + To prevent us from propagating the same argument over and over, a simple + local variable reference is not propagated (since is has exactly one + free variable, itself, we would only replace that argument with itself). + + This shows that any free local variables that are not runtime representable + cannot be brought into normal form by this transform. We rely on an + inlining transformation to replace such a variable with an expression we + can propagate again. + + \starttrans + x = E + ~ + x Y0 ... Yi ... Yn \lam{Yi} is not of a runtime representable type + --------------------------------------------- \lam{Yi} is not a local variable reference + x' y0 ... yi-1 f0 ... fm Yi+1 ... Yn \lam{f0 ... fm} = free local vars of \lam{Yi} + ~ + x' = λy0 ... yi-1 f0 ... fm yi+1 ... yn . + E y0 ... yi-1 Yi yi+1 ... yn + + \stoptrans + + TODO: Example + + \subsection{Case simplification} + \subsubsection{Scrutinee simplification} + This transform ensures that the scrutinee of a case expression is always + a simple variable reference. + + \starttrans + case E of + alts + ----------------- \lam{E} is not a local variable reference + let x = E in + case E of + alts + \stoptrans + + \startbuffer[from] + case (foo a) of + True -> a + False -> b + \stopbuffer + + \startbuffer[to] + let x = foo a in + case x of + True -> a + False -> b + \stopbuffer + + \transexample{Let flattening}{from}{to} + + + \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 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}. + \stopitemize + + \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) + letnonrec + v0,0 = case x of C0 v0,0 .. v0,m -> v0,0 + \vdots + v0,m = case x of C0 v0,0 .. v0,m -> v0,m + x0 = E0 + \dots + vn,m = case x of Cn vn,0 .. vn,m -> vn,m + xn = En + in + case E of + C0 w0,0 ... w0,m -> x0 + \vdots + Cn wn,0 ... wn,m -> xn + \stoptrans + + TODO: This transformation specified like this is complicated and misses + conditions to prevent looping with itself. Perhaps we should split it here for + discussion? + + \startbuffer[from] + case a of + True -> add b 1 + False -> add b 2 + \stopbuffer + + \startbuffer[to] + letnonrec + x0 = add b 1 + x1 = add b 2 + in + case a of + True -> x0 + False -> x1 + \stopbuffer + + \transexample{Selector case simplification}{from}{to} + + \startbuffer[from] + case a of + (,) b c -> add b c + \stopbuffer + \startbuffer[to] + letnonrec + b = case a of (,) b c -> b + c = case a of (,) b c -> c + x0 = add b c + in + case a of + (,) w0 w1 -> x0 + \stopbuffer + + \transexample{Extractor case simplification}{from}{to} + + \subsubsection{Case removal} + This transform removes any case statements with a single alternative and + only wild binders. + + These "useless" case statements are usually leftovers from case simplification + on extractor case (see the previous example). + + \starttrans + case x of + C v0 ... vm -> E + ---------------------- \lam{\forall i, 0 <= i <= m} (\lam{vi} does not occur free in E) + E + \stoptrans + + \startbuffer[from] + case a of + (,) w0 w1 -> x0 + \stopbuffer + + \startbuffer[to] + x0 + \stopbuffer + + \transexample{Case removal}{from}{to} + +\subsection{Monomorphisation} + TODO: Better name for this section + + Reference type-specialization (== argument propagation) + +\subsubsection{Defunctionalization} + Reference higher-order-specialization (== argument propagation) + + \subsubsection{Non-representable binding inlining} + This transform inlines let bindings that have a non-representable type. Since + we can never generate a signal assignment for these bindings (we cannot + declare a signal assignment with a non-representable type, for obvious + reasons), we have no choice but to inline the binding to remove it. + + If the binding is non-representable because it is a lambda abstraction, it is + likely that it will inlined into an application and β-reduction will remove + the lambda abstraction and turn it into a representable expression at the + inline site. The same holds for partial applications, which can be turned into + full applications by inlining. + + Other cases of non-representable bindings we see in practice are primitive + Haskell types. In most cases, these will not result in a valid normalized + output, but then the input would have been invalid to start with. There is one + exception to this: When a builtin function is applied to a non-representable + expression, things might work out in some cases. For example, when you write a + literal \hs{SizedInt} in Haskell, like \hs{1 :: SizedInt D8}, this results in + the following core: \lam{fromInteger (smallInteger 10)}, where for example + \lam{10 :: GHC.Prim.Int\#} and \lam{smallInteger 10 :: Integer} have + non-representable types. TODO: This/these paragraph(s) should probably become a + separate discussion somewhere else. + + \starttrans + letnonrec a = E in M + -------------------------- \lam{E} has a non-representable type. + M[E/a] + \stoptrans + + \starttrans + letrec + \vdots + a = E + \vdots + in + M + -------------------------- \lam{E} has a non-representable type. + letrec + \vdots [E/a] + \vdots [E/a] + in + M[E/a] + \stoptrans + + \startbuffer[from] + letrec + a = smallInteger 10 + inc = λa -> add a 1 + inc' = add 1 + x = fromInteger a + in + inc (inc' x) + \stopbuffer + + \startbuffer[to] + letrec + x = fromInteger (smallInteger 10) + in + (λa -> add a 1) (add 1 x) + \stopbuffer + + \transexample{Let flattening}{from}{to} + + +\section{Provable properties} + When looking at the system of transformations outlined above, there are a + number of questions that we can ask ourselves. The main question is of course: + \quote{Does our system work as intended?}. We can split this question into a + number of subquestions: + + \startitemize[KR] + \item[q:termination] Does our system \emph{terminate}? Since our system will + keep running as long as transformations apply, there is an obvious risk that + it will keep running indefinitely. One transformation produces a result that + is transformed back to the original by another transformation, for example. + \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 + 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 + 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 + obvious risk that different transformation orderings will result in + \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. + \stopitemize + + \subsection{Graph representation} + Before looking into how to prove these properties, we'll look at our + transformation system from a graph perspective. The nodes of the graph are + all possible Core expressions. The (directed) edges of the graph are + transformations. When a transformation α applies to an expression \lam{A} to + produce an expression \lam{B}, we add an edge from the node for \lam{A} to the + node for \lam{B}, labeled α. + + \startuseMPgraphic{TransformGraph} + save a, b, c, d; + + % Nodes + newCircle.a(btex \lam{(λx.λy. (+) x y) 1} etex); + newCircle.b(btex \lam{λy. (+) 1 y} etex); + newCircle.c(btex \lam{(λx.(+) x) 1} etex); + newCircle.d(btex \lam{(+) 1} etex); + + b.c = origin; + c.c = b.c + (4cm, 0cm); + a.c = midpoint(b.c, c.c) + (0cm, 4cm); + d.c = midpoint(b.c, c.c) - (0cm, 3cm); + + % β-conversion between a and b + ncarc.a(a)(b) "name(bred)"; + ObjLabel.a(btex $\xrightarrow[normal]{}{β}$ etex) "labpathname(bred)", "labdir(rt)"; + ncarc.b(b)(a) "name(bexp)", "linestyle(dashed withdots)"; + ObjLabel.b(btex $\xleftarrow[normal]{}{β}$ etex) "labpathname(bexp)", "labdir(lft)"; + + % η-conversion between a and c + ncarc.a(a)(c) "name(ered)"; + ObjLabel.a(btex $\xrightarrow[normal]{}{η}$ etex) "labpathname(ered)", "labdir(rt)"; + ncarc.c(c)(a) "name(eexp)", "linestyle(dashed withdots)"; + ObjLabel.c(btex $\xleftarrow[normal]{}{η}$ etex) "labpathname(eexp)", "labdir(lft)"; + + % η-conversion between b and d + ncarc.b(b)(d) "name(ered)"; + ObjLabel.b(btex $\xrightarrow[normal]{}{η}$ etex) "labpathname(ered)", "labdir(rt)"; + ncarc.d(d)(b) "name(eexp)", "linestyle(dashed withdots)"; + ObjLabel.d(btex $\xleftarrow[normal]{}{η}$ etex) "labpathname(eexp)", "labdir(lft)"; + + % β-conversion between c and d + ncarc.c(c)(d) "name(bred)"; + ObjLabel.c(btex $\xrightarrow[normal]{}{β}$ etex) "labpathname(bred)", "labdir(rt)"; + ncarc.d(d)(c) "name(bexp)", "linestyle(dashed withdots)"; + ObjLabel.d(btex $\xleftarrow[normal]{}{β}$ etex) "labpathname(bexp)", "labdir(lft)"; + + % Draw objects and lines + drawObj(a, b, c, d); + \stopuseMPgraphic + + \placeexample[right][ex:TransformGraph]{Partial graph of a labmda calculus + system with β and η reduction (solid lines) and expansion (dotted lines).} + \boxedgraphic{TransformGraph} + + Of course our graph is unbounded, since we can construct an infinite amount of + Core expressions. Also, there might potentially be multiple edges between two + given nodes (with different labels), though seems unlikely to actually happen + in our system. + + See \in{example}[ex:TransformGraph] for the graph representation of a very + simple lambda calculus that contains just the expressions \lam{(λx.λy. (+) x + y) 1}, \lam{λy. (+) 1 y}, \lam{(λx.(+) x) 1} and \lam{(+) 1}. The + transformation system consists of β-reduction and η-reduction (solid edges) or + β-reduction and η-reduction (dotted edges). + + 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}. + + From such a graph, we can derive some properties easily: + \startitemize[KR] + \item A system will \emph{terminate} if there is no path of infinite length + in the graph (this includes cycles). + \item Soundness is not easily represented in the graph. + \item A system is \emph{complete} if all of the nodes in the normal set have + the intended normal form. The inverse (that all of the nodes outside of + the normal set are \emph{not} in the intended normal form) is not + strictly required. + \item A system is deterministic if all paths from a node, which end in a node + in the normal set, end at the same node. + \stopitemize + + When looking at the \in{example}[ex:TransformGraph], we see that the system + terminates for both the reduction and expansion systems (but note that, for + expansion, this is only true because we've limited the possible expressions! + In comlete lambda calculus, there would be a path from \lam{(λx.λy. (+) x y) + 1} to \lam{(λx.λy.(λz.(+) z) x y) 1} to \lam{(λx.λy.(λz.(λq.(+) q) z) x y) 1} + etc.) + + If we would consider the system with both expansion and reduction, there would + no longer be termination, since there would be cycles all over the place. + + The reduction and expansion systems have a normal set of containing just + \lam{(+) 1} or \lam{(λx.λy. (+) x y) 1} respectively. Since all paths in + either system end up in these normal forms, both systems are \emph{complete}. + Also, since there is only one normal form, it must obviously be + \emph{deterministic} as well. + + \subsection{Termination} + Approach: Counting. + + Church-Rosser? + + \subsection{Soundness} + Needs formal definition of semantics. + Prove for each transformation seperately, implies soundness of the system. + + \subsection{Completeness} + Show that any transformation applies to every Core expression that is not + in normal form. To prove: no transformation applies => in intended form. + Show the reverse: Not in intended form => transformation applies. + + \subsection{Determinism} + How to prove this?