X-Git-Url: https://git.stderr.nl/gitweb?p=matthijs%2Fmaster-project%2Freport.git;a=blobdiff_plain;f=Chapters%2FNormalization.tex;h=d564d231405a00ab7f31622ef525b15f987c2218;hp=e609cb3d2874655ea38297c8c75637368b1fa2d7;hb=d92aa4307ca45f07c6ae50056e08ffc874839756;hpb=4521ef7f14ae52781329c17250f67d25d9a67dca diff --git a/Chapters/Normalization.tex b/Chapters/Normalization.tex index e609cb3..d564d23 100644 --- a/Chapters/Normalization.tex +++ b/Chapters/Normalization.tex @@ -1,809 +1,2476 @@ -\chapter{Normalization} - -% A helper to print a single example in the half the page width. The example -% text should be in a buffer whose name is given in an argument. -% -% The align=right option really does left-alignment, but without the program -% will end up on a single line. The strut=no option prevents a bunch of empty -% space at the start of the frame. -\define[1]\example{ - \framed[offset=1mm,align=right,strut=no]{ - \setuptyping[option=LAM,style=sans,before=,after=] - \typebuffer[#1] - \setuptyping[option=none,style=\tttf] +\chapter[chap:normalization]{Normalization} + % A helper to print a single example in the half the page width. The example + % text should be in a buffer whose name is given in an argument. + % + % The align=right option really does left-alignment, but without the program + % will end up on a single line. The strut=no option prevents a bunch of empty + % space at the start of the frame. + \define[1]\example{ + \framed[offset=1mm,align=right,strut=no,background=box,frame=off]{ + \setuptyping[option=LAM,style=sans,before=,after=,strip=auto] + \typebuffer[#1] + \setuptyping[option=none,style=\tttf,strip=auto] + } } -} - - -% A transformation example -\definefloat[example][examples] -\setupcaption[example][location=top] % Put captions on top - -\define[3]\transexample{ - \placeexample[here]{#1} - \startcombination[2*1] - {\example{#2}}{Original program} - {\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 -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 -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. - -\section{Goal} -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. - -This {\em normal form} is again a Core program, but with a very specific -structure. A function in normal form has nested lambda's at the top, which -produce a let expression. This let expression binds every function application -in the function and produces a simple identifier. Every bound value in -the let expression is either a simple function application or a case -expression to extract a single element from a tuple returned by a -function. - -An example of a program in canonical form would be: - -\startlambda - -- All arguments are an inital lambda - λa.λd.λsp. - -- There are nested let expressions at top level - let - -- Unpack the state by coercion - s = sp :: (Word, Word) - -- Extract both registers from the state - r1 = case s of (fst, snd) -> fst - r2 = case s of (fst, snd) -> snd - -- Calling some other user-defined function. - d' = foo d - -- Conditional connections - out = case a of - High -> r1 - Low -> r2 - r1' = case a of - High -> d - Low -> r1 - r2' = case a of - High -> r2 - Low -> d - -- Packing a tuple - s' = (,) r1' r2' - -- Packing the state by coercion - sp' = s' :: State (Word, Word) - -- Pack our return value - res = (,) sp' out - in - -- The actual result - res -\stoplambda - -\startlambda -\italic{normal} = \italic{lambda} -\italic{lambda} = λvar.\italic{lambda} (representable(typeof(var))) - | \italic{toplet} -\italic{toplet} = let \italic{binding} in \italic{toplet} - | letrec [\italic{binding}] in \italic{toplet} - | var (representable(typeof(var)), fvar(var)) -\italic{binding} = var = \italic{rhs} (representable(typeof(rhs))) - -- State packing and unpacking by coercion - | var0 = var1 :: State ty (fvar(var1)) - | var0 = var1 :: ty (var0 :: State ty) (fvar(var1)) -\italic{rhs} = userapp - | builtinapp - -- Extractor case - | case var of C a0 ... an -> ai (fvar(var)) - -- Selector case - | case var of (fvar(var)) - DEFAULT -> var0 (fvar(var0)) - C w0 ... wn -> resvar (\forall{}i, wi \neq resvar, fvar(resvar)) -\italic{userapp} = \italic{userfunc} - | \italic{userapp} {userarg} -\italic{userfunc} = var (tvar(var)) -\italic{userarg} = var (fvar(var)) -\italic{builtinapp} = \italic{builtinfunc} - | \italic{builtinapp} \italic{builtinarg} -\italic{builtinfunc} = var (bvar(var)) -\italic{builtinarg} = \italic{coreexpr} -\stoplambda - --- TODO: Define tvar, fvar, typeof, representable --- TODO: Limit builtinarg further - --- TODO: There can still be other casts around (which the code can handle, -e.g., ignore), which still need to be documented here. - --- TODO: Note about the selector case. It just supports Bit and Bool -currently, perhaps it should be generalized in the normal form? - -When looking at such a program from a hardware perspective, the top level -lambda's define the input ports. The value produced by the 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} statement) or call a builtin function -(\eg \lam{add} or \lam{sub}). For these, a hardcoded VHDL translation is -available. - -\subsection{Normal definition} -Formally, the normal form is a core program obeying the following -constraints. TODO: Update this section, this is probably not completely -accurate or relevant anymore. - -\startitemize[R,inmargin] -%\item All top level binds must have the form $\expr{\bind{fun}{lamexpr}}$. -%$fun$ is an identifier that will be bound as a global identifier. -%\item A $lamexpr$ has the form $\expr{\lam{arg}{lamexpr}}$ or -%$\expr{letexpr}$. $arg$ is an identifier which will be bound as an $argument$. -%\item[letexpr] A $letexpr$ has the form $\expr{\letexpr{letbinds}{retexpr}}$. -%\item $letbinds$ is a list with elements of the form -%$\expr{\bind{res}{appexpr}}$ or $\expr{\bind{res}{builtinexpr}}$, where $res$ is -%an identifier that will be bound as local identifier. The type of the bound -%value must be a $hardware\;type$. -%\item[builtinexpr] A $builtinexpr$ is an expression that can be mapped to an -%equivalent VHDL expression. Since there are many supported forms for this, -%these are defined in a separate table. -%\item An $appexpr$ has the form $\expr{fun}$ or $\expr{\app{appexpr}{x}}$, -%where $fun$ is a global identifier and $x$ is a local identifier. -%\item[retexpr] A $retexpr$ has the form $\expr{x}$ or $\expr{tupexpr}$, where $x$ is a local identifier that is bound as an $argument$ or $result$. A $retexpr$ must -%be of a $hardware\;type$. -%\item A $tupexpr$ has the form $\expr{con}$ or $\expr{\app{tupexpr}{x}}$, -%where $con$ is a tuple constructor ({\em e.g.} $(,)$ or $(,,,)$) and $x$ is -%a local identifier. -%\item A $hardware\;type$ is a type that can be directly translated to -%hardware. This includes the types $Bit$, $SizedWord$, tuples containing -%elements of $hardware\;type$s, and will include others. This explicitely -%excludes function types. -\stopitemize - -TODO: Say something about uniqueness of identifiers - -\subsection{Builtin expressions} -A $builtinexpr$, as defined at \in[builtinexpr] can have any of the following forms. - -\startitemize[m,inmargin] -%\item -%$tuple\_extract=\expr{\case{t}{\alt{\app{con}{x_0\;x_1\;..\;x_n}}{x_i}}}$, -%where $t$ can be any local identifier, $con$ is a tuple constructor ({\em -%e.g.} $(,)$ or $(,,,)$), $x_0$ to $x_n$ can be any identifier, and $x_i$ can -%be any of $x_0$ to $x_n$. A case expression must have a $hardware\;type$. -%\item TODO: Many more! -\stopitemize - -\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 -expressions can be applied anymore, the program is in normal form. 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. - -\starttrans -E \lam{E :: * -> *} --------------- \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} - -\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 + + \define[4]\transexample{ + \placeexample[here][ex:trans:#1]{#2} + \startcombination[2*1] + {\example{#3}}{Original program} + {\example{#4}}{Transformed program} + \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 + subsequently translate into \small{VHDL} easily. This normal form is needed because + the full core language is more expressive than \small{VHDL} in some + areas (higher-order expressions, limited polymorphism using type + 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 + \VHDL, while fully preserving the semantics of the program. We refer + to this form as the \emph{normal form} of the program. The formal + definition of this normal form is quite simple: + + \placedefinition{}{\startboxed A program is in \emph{normal form} if none of the + 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 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 + cannot generate any signals that can have multiple types. All types must be + completely known to generate hardware. + + \item All \emph{higher-order} constructions must be removed. We cannot + generate a hardware signal that contains a function, so all values, + arguments and return values used must be first order. + + \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} + generation easy, a separate binder must be bound to ever application or + other expression. + \stopitemize + + \todo{Intermezzo: functions vs plain values} + + A very simple example of a program in normal form is given in + \in{example}[ex:MulSum]. As you can see, all arguments to the function (which + will become input ports in the generated \VHDL) are at the outer level. + This means that the body of the inner lambda abstraction is never a + function, but always a plain value. + + As the body of the inner lambda abstraction, we see a single (recursive) + let expression, that binds two variables (\lam{mul} and \lam{sum}). These + variables will be signals in the generated \VHDL, bound to the output port + of the \lam{*} and \lam{+} components. + + The final line (the \quote{return value} of the function) selects the + \lam{sum} signal to be the output port of the function. This \quote{return + value} can always only be a variable reference, never a more complex + expression. + + \todo{Add generated VHDL} + + \startbuffer[MulSum] + alu :: Bit -> Word -> Word -> Word + alu = λa.λb.λc. + let + mul = (*) a b + sum = (+) mul c + in + sum + \stopbuffer + + \startuseMPgraphic{MulSum} + save a, b, c, mul, add, sum; + + % I/O ports + newCircle.a(btex $a$ etex) "framed(false)"; + newCircle.b(btex $b$ etex) "framed(false)"; + newCircle.c(btex $c$ etex) "framed(false)"; + newCircle.sum(btex $sum$ etex) "framed(false)"; + + % Components + newCircle.mul(btex * etex); + newCircle.add(btex + etex); + + a.c - b.c = (0cm, 2cm); + b.c - c.c = (0cm, 2cm); + add.c = c.c + (2cm, 0cm); + mul.c = midpoint(a.c, b.c) + (2cm, 0cm); + sum.c = add.c + (2cm, 0cm); + c.c = origin; + + % Draw objects and lines + drawObj(a, b, c, mul, add, sum); + + ncarc(a)(mul) "arcangle(15)"; + ncarc(b)(mul) "arcangle(-15)"; + ncline(c)(add); + ncline(mul)(add); + ncline(add)(sum); + \stopuseMPgraphic + + \placeexample[here][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 + + \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 + bit. The main structure is similar to \in{example}[ex:MulSum], but this + time the \lam{res} variable is bound to a case expression. This case + 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 + alu = λopcode.λa.λb. + let + res1 = (+) a b + res2 = (-) a b + res = case opcode of + Low -> res1 + High -> res2 + in + res + \stopbuffer + + \startuseMPgraphic{AddSubAlu} + save opcode, a, b, add, sub, mux, res; + + % I/O ports + newCircle.opcode(btex $opcode$ etex) "framed(false)"; + newCircle.a(btex $a$ etex) "framed(false)"; + newCircle.b(btex $b$ etex) "framed(false)"; + newCircle.res(btex $res$ etex) "framed(false)"; + % Components + newCircle.add(btex + etex); + newCircle.sub(btex - etex); + newMux.mux; + + opcode.c - a.c = (0cm, 2cm); + add.c - a.c = (4cm, 0cm); + sub.c - b.c = (4cm, 0cm); + a.c - b.c = (0cm, 3cm); + mux.c = midpoint(add.c, sub.c) + (1.5cm, 0cm); + res.c - mux.c = (1.5cm, 0cm); + b.c = origin; + + % Draw objects and lines + drawObj(opcode, a, b, res, add, sub, mux); + + ncline(a)(add) "posA(e)"; + ncline(b)(sub) "posA(e)"; + nccurve(a)(sub) "posA(e)", "angleA(0)"; + nccurve(b)(add) "posA(e)", "angleA(0)"; + nccurve(add)(mux) "posB(inpa)", "angleB(0)"; + nccurve(sub)(mux) "posB(inpb)", "angleB(0)"; + nccurve(opcode)(mux) "posB(n)", "angleA(0)", "angleB(-90)"; + ncline(mux)(res) "posA(out)"; + \stopuseMPgraphic + + \placeexample[here][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 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 + examples, there is a direct match between the normal form structure, + the generated VHDL and the architecture shown in the images. + + \startbuffer[NormalComplete] + regbank :: Bit + -> Word + -> State (Word, Word) + -> (State (Word, Word), Word) + + -- All arguments are an inital lambda (address, data, packed state) + regbank = λa.λd.λsp. + -- There are nested let expressions at top level + let + -- Unpack the state by coercion (\eg, cast from + -- State (Word, Word) to (Word, Word)) + s = sp ▶ (Word, Word) + -- Extract both registers from the state + r1 = case s of (a, b) -> a + r2 = case s of (a, b) -> b + -- Calling some other user-defined function. + d' = foo d + -- Conditional connections + out = case a of + High -> r1 + Low -> r2 + r1' = case a of + High -> d' + Low -> r1 + r2' = case a of + High -> r2 + Low -> d' + -- Packing a tuple + s' = (,) r1' r2' + -- pack the state by coercion (\eg, cast from + -- (Word, Word) to State (Word, Word)) + sp' = s' ▶ State (Word, Word) + -- Pack our return value + res = (,) sp' out + in + -- The actual result + res + \stopbuffer + + \startuseMPgraphic{NormalComplete} + save a, d, r, foo, muxr, muxout, out; + + % I/O ports + newCircle.a(btex \lam{a} etex) "framed(false)"; + newCircle.d(btex \lam{d} etex) "framed(false)"; + newCircle.out(btex \lam{out} etex) "framed(false)"; + % Components + %newCircle.add(btex + etex); + newBox.foo(btex \lam{foo} etex); + newReg.r1(btex $\lam{r1}$ etex) "dx(4mm)", "dy(6mm)"; + newReg.r2(btex $\lam{r2}$ etex) "dx(4mm)", "dy(6mm)", "reflect(true)"; + newMux.muxr1; + % Reflect over the vertical axis + reflectObj(muxr1)((0,0), (0,1)); + newMux.muxr2; + newMux.muxout; + rotateObj(muxout)(-90); + + d.c = foo.c + (0cm, 1.5cm); + a.c = (xpart r2.c + 2cm, ypart d.c - 0.5cm); + foo.c = midpoint(muxr1.c, muxr2.c) + (0cm, 2cm); + muxr1.c = r1.c + (0cm, 2cm); + muxr2.c = r2.c + (0cm, 2cm); + r2.c = r1.c + (4cm, 0cm); + r1.c = origin; + muxout.c = midpoint(r1.c, r2.c) - (0cm, 2cm); + out.c = muxout.c - (0cm, 1.5cm); + + % % Draw objects and lines + drawObj(a, d, foo, r1, r2, muxr1, muxr2, muxout, out); + + ncline(d)(foo); + nccurve(foo)(muxr1) "angleA(-90)", "posB(inpa)", "angleB(180)"; + nccurve(foo)(muxr2) "angleA(-90)", "posB(inpb)", "angleB(0)"; + nccurve(muxr1)(r1) "posA(out)", "angleA(180)", "posB(d)", "angleB(0)"; + nccurve(r1)(muxr1) "posA(out)", "angleA(0)", "posB(inpb)", "angleB(180)"; + nccurve(muxr2)(r2) "posA(out)", "angleA(0)", "posB(d)", "angleB(180)"; + nccurve(r2)(muxr2) "posA(out)", "angleA(180)", "posB(inpa)", "angleB(0)"; + nccurve(r1)(muxout) "posA(out)", "angleA(0)", "posB(inpb)", "angleB(-90)"; + nccurve(r2)(muxout) "posA(out)", "angleA(180)", "posB(inpa)", "angleB(-90)"; + % Connect port a + nccurve(a)(muxout) "angleA(-90)", "angleB(180)", "posB(sel)"; + nccurve(a)(muxr1) "angleA(180)", "angleB(-90)", "posB(sel)"; + nccurve(a)(muxr2) "angleA(180)", "angleB(-90)", "posB(sel)"; + ncline(muxout)(out) "posA(out)"; + \stopuseMPgraphic + + \todo{Don't split registers in this image?} + \placeexample[here][ex:NormalComplete]{Simple architecture consisting of an adder and a + subtractor.} + \startcombination[2*1] + {\typebufferlam{NormalComplete}}{Core description in normal form.} + {\boxedgraphic{NormalComplete}}{The architecture described by the normal form.} + \stopcombination + + + + \subsection[sec:normalization:intendednormalform]{Intended normal form definition} + Now we have some intuition for the normal form, we can describe how we want + the normal form to look like in a slightly more formal manner. The following + EBNF-like description captures most of the intended structure (and + generates a subset of \GHC's core format). + + There are two things missing: Cast expressions are sometimes + allowed by the prototype, but not specified here and the below + definition allows uses of state that cannot be translated to \VHDL\ + 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}. + + \todo{Fix indentation} + \startbuffer[IntendedNormal] + \italic{normal} := \italic{lambda} + \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)) + -- State packing and unpacking by coercion + | 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)) + -- Selector case + | 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)) + \italic{userapp} := \italic{userfunc} + | \italic{userapp} {userarg} + \italic{userfunc} := var (gvar(var)) + \italic{userarg} := var (lvar(var)) + \italic{builtinapp} := \italic{builtinfunc} + | \italic{builtinapp} \italic{builtinarg} + \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 + + \placedefinition[][def:IntendedNormal]{Definition of the intended nnormal 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 define the input ports. Lambda + abstractions cannot appear anywhere else. The variable reference + in the body of the recursive let expression is the output port. + Most function applications bound by the let expression define a + component instantiation, where the input and output ports are + mapped to local signals or arguments. Some of the others use a + built-in construction (\eg\ the \lam{case} expression) or call a + built-in function (\eg\ \lam{+} or \lam{map}). For these, a + hardcoded \small{VHDL} translation is available. + + \section[sec:normalization:transformation]{Transformation notation} + To be able to concisely present transformations, we use a specific format + for them. It is a simple format, similar to one used in logic reasoning. + + Such a transformation description looks like the following. + + \starttrans + + ~ + + -------------------------- + + ~ + + \stoptrans + + This format describes a transformation that applies to \lam{} and transforms it into \lam{}, 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{} 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 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 + (references to) binders. + + For example, the pattern \lam{a + B} will match the expression + \lam{v + (2 * w)} (binding \lam{a} to \lam{v} and \lam{B} to + \lam{(2 * w)}), but not \lam{(2 * w) + v}. + \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, commonly to prevent a transformation from + causing a loop with itself or another transformation. + + Only if these conditions are \emph{all} satisfied, the 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, + the 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{} is completely replaced by 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 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 + binder placeholder that has no value bound to it yet will be bound to (and + replaced with) a fresh binder. + \stopdesc + + To understand this notation better, the step by step application of + the η-abstraction transformation to a simple \small{ALU} will be + shown. Consider η-abstraction, which is a common transformation from + labmda calculus, described using above notation as follows: + + \starttrans + 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. + \stoptrans + + η-abstraction 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). + + 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 + alu = λopcode. case opcode of + Low -> (+) + High -> (-) + \stoplambda + + 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. + + 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. + + \startlambda + λopcode. case opcode of + Low -> (+) + High -> (-) + \stoplambda + + 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}). + + 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. + + The next expression we could apply this transformation to, is the body of + the lambda abstraction: + + \startlambda + case opcode of + Low -> (+) + High -> (-) + \stoplambda + + 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. + + 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 us use the binder + \lam{a}. This results in the following replacement expression: + + \startlambda + λa.(case opcode of + Low -> (+) + High -> (-)) a + \stoplambda + + 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 + lambda abstraction: + + \startlambda + (case opcode of + Low -> (+) + High -> (-)) a + \stoplambda + + Here, the transformation does apply, binding \lam{E} to the entire + 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 + Low -> (+) + High -> (-)) a b + \stoplambda + + 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 + (case opcode of Low -> (+); High -> (-)) a b + \stoplambda + + 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: + + \startlambda + (case opcode of Low -> (+); High -> (-)) a + \stoplambda + + Obviously, the transformation does not apply here, since it occurs in + function position (which makes the second condition false). In the same + way the transformation does not apply to both components of this + expression (\lam{case opcode of Low -> (+); High -> (-)} and \lam{a}), so + we will skip to the components of the case expression: The scrutinee and + both alternatives. Since the opcode is not a function, it does not apply + here. + + The first alternative is \lam{(+)}. This expression has a function type + (the operator still needs two arguments). It does not occur in function + position of an application and it is not a lambda expression, so the + transformation applies. + + We look at the \lam{} pattern, which is \lam{E}. + This means we bind \lam{E} to \lam{(+)}. We then replace the expression + with the \lam{}, replacing all occurences of + \lam{E} with \lam{(+)}. In the \lam{}, 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}). + + The complete function then becomes: + \startlambda + (case opcode of Low -> λa1.(+) a1; High -> (-)) a + \stoplambda + + Now the transformation no longer applies to the complete first alternative + (since it is a lambda expression). It does not apply to the addition + operator again, since it is now in function position in an application. It + does, however, apply to the application of the addition operator, since + that is neither a lambda expression nor does it occur in function + position. This means after one more application of the transformation, the + function becomes: + + \startlambda + (case opcode of Low -> λa1.λb1.(+) a1 b1; High -> (-)) a + \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: + + \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 + + \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. + + In particular, we define no particular order of transformations. Since + transformation order should not influence the resulting normal form, + this leaves the implementation free to choose any application order that + results in an efficient implementation. Unfortunately this is not + entirely true for the current set of transformations. See + \in{section}[sec:normalization:non-determinism] for a discussion of this + problem. + + When applying a single transformation, we try to apply it to every (sub)expression + in a function, not just the top level function body. This allows us to + keep the transformation descriptions concise and powerful. + + \subsection{Definitions} + 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 runtime representable notably include (but are not limited to): Types, + dictionaries, functions. + \defref{representable} + + A \emph{built-in function} is a function supplied by the Cλash framework, whose + implementation is not valid Cλash. The implementation is of course valid + Haskell, for simulation, but it is not expressable in Cλash. + \defref{built-in function} \defref{user-defined function} + + For these functions, Cλash has a \emph{built-in hardware translation}, so calls + to these functions can still be translated. These are functions like + \lam{map}, \lam{hwor} and \lam{length}. + + A \emph{user-defined} function is a function for which we do have a Cλash + implementation available. + + \subsubsection[sec:normalization:predicates]{Predicates} + Here, we define a number of predicates 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 complement of \emph{gvar}; it is true when \emph{expr} + references a local variable, false when it references a global variable. + + \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 + expression: + + \startlambda + (λa.λb.λc. a * b * c) x c + \stoplambda + + By applying β-reduction (see \in{section}[sec:normalization:beta]) once, + we can simplify this expression to: + + \startlambda + (λb.λc. x * b * c) c + \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. + 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: + + \startlambda + λc. x * c * c + \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, + 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} + 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 + does not use \small{GHC}'s subsitution code). Also, the following points + can be observed. + + \startitemize + \item Deshadowing does not guarantee overall uniqueness. For example, the + following (slightly contrived) expression shows the identifier \lam{x} bound in + two seperate places (and to different values), even though no shadowing + occurs. + + \startlambda + (let x = 1 in x) + (let x = 2 in x) + \stoplambda + + \item In our normal form (and the resulting \small{VHDL}), all binders + (signals) within the same function (entity) will end up in the same + scope. To allow this, all binders within the same function should be + 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 + cause any problems either. The only way to cause conflicts is thus to + duplicate an existing subexpression. + \stopitemize + + Given the above, our prototype maintains a unique binder invariant. This + means 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. + + \todo{Define fresh binders and unique supplies} + + \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 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. + \item Whenever (a 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 + + \section{Transform passes} + In this section we describe the actual transforms. + + Each transformation will be described informally first, explaining + the need for and goal of the transformation. Then, we will formally define + the transformation using the syntax introduced in + \in{section}[sec:normalization:transformation]. + + \subsection{General cleanup} + These transformations are general cleanup transformations, that aim to + make expressions simpler. These transformations usually clean up the + mess left behind by other transformations or clean up expressions to + expose new transformation opportunities for other transformations. + + Most of these transformations are standard optimizations in other + compilers as well. However, in our compiler, most of these are not just + optimizations, but they are required to get our program into intended + normal form. + + \placeintermezzo{}{ + \defref{substitution notation} + \startframedtext[width=8cm,background=box,frame=no] + \startalignment[center] + {\tfa Substitution notation} + \stopalignment + \blank[medium] + + In some of the transformations in this chapter, we need to perform + substitution on an expression. Substitution means replacing every + occurence of some expression (usually a variable reference) with + another expression. + + There have been a lot of different notations used in literature for + specifying substitution. The notation that will be used in this report + is the following: + + \startlambda + E[A=>B] + \stoplambda + + This means expression \lam{E} with all occurences of \lam{A} replaced + with \lam{B}. + \stopframedtext + } + + \subsubsection[sec:normalization:beta]{β-reduction} + β-reduction is a well known transformation from lambda calculus, where it is + the main reduction step. It reduces applications of lambda 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. + + 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 + expression applied to is a type. + + \starttrans + (λx.E) M + ----------------- + E[x=>M] + \stoptrans + + % And an example + \startbuffer[from] + (λa. 2 * a) (2 * b) + \stopbuffer + + \startbuffer[to] + 2 * (2 * b) + \stopbuffer + + \transexample{beta}{β-reduction}{from}{to} + + \startbuffer[from] + (λt.λa::t. a) @Int + \stopbuffer + + \startbuffer[to] + (λa::Int. a) + \stopbuffer + + \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 + (which usually occurs when unused let binding removal removes the last + binding from it). + + Note that there is no need to define this transformation for + non-recursive lets, since they always contain exactly one binding. + + \starttrans + letrec in M + -------------- + M + \stoptrans + + % 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\ + a = b). + + This transformation is not needed to get an expression into intended + normal form (since these bindings are part of the intended normal + form), but makes the resulting \small{VHDL} a lot shorter. + + \refdef{substitution notation} + \starttrans + letrec + a0 = E0 + \vdots + ai = b + \vdots + an = En + in + M + ----------------------------- \lam{b} is a variable reference + letrec \lam{ai} ≠ \lam{b} + a0 = E0 [ai=>b] + \vdots + ai-1 = Ei-1 [ai=>b] + ai+1 = Ei+1 [ai=>b] + \vdots + an = En [ai=>b] + in + M[ai=>b] + \stoptrans + + \todo{example} + + \subsubsection{Cast propagation / simplification} + This transform pushes casts down into the expression as far as + possible. This transformation has been added to make a few + 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]. + + \starttrans + (let binds in E) ▶ T + ------------------------- + let binds in (E ▶ T) + \stoptrans + + \starttrans + (case S of + p0 -> E0 + \vdots + pn -> En + ) ▶ T + ------------------------- + case S of + p0 -> E0 ▶ T + \vdots + pn -> En ▶ T + \stoptrans + + \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 contain just a (partial) function appliation 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 components + that do not add any real structure, but do hide away operations and + cause extra clutter). + + This transform takes any top level binding generated by \GHC, + whose normalized form contains only a single let binding. + + \starttrans + x = λa0 ... λan.let y = E in y + ~ + x + -------------------------------------- \lam{x} is generated by the compiler + λa0 ... λan.let y = E in y + \stoptrans + + \startbuffer[from] + (+) :: Word -> Word -> Word + (+) = GHC.Num.(+) @Word \$dNum + ~ + (+) a b + \stopbuffer + \startbuffer[to] + GHC.Num.(+) @ Alu.Word \$dNum a b + \stopbuffer + + \transexample{toplevelinline}{Top level binding inlining}{from}{to} + + \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]. + + Without this transformation, there would be a \lam{(+)} entity + 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 to use non-alphanumerics when using extended + identifiers, but it seems that none of the tooling likes + extended identifiers in filenames, so it effectively does not + work.}, so the entity would be called \quote{w7aA7f} or + something similarly meaningless and autogenerated). + + \subsection{Program structure} + These transformations are aimed at normalizing the overall structure + into the intended form. This means ensuring there is a lambda abstraction + at the top for every argument (input port or current state), putting all + of the other value definitions in let bindings and making the final + return value a simple variable reference. + + \subsubsection[sec:normalization:eta]{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 non-representable binding inlining, all function-typed + expressions should be lambda abstractions or global identifiers. + + \starttrans + 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. + \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{eta}{η-abstraction}{from}{to} + + \subsubsection[sec:normalization:appprop]{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 + opportunities for other transformations (like β-reduction and + specialization). + + Since all binders in our expression are unique (see + \in{section}[sec:normalization:uniq]), there is no risk that we will + introduce unintended shadowing by moving an expression into a lower + scope. Also, since only move expression into smaller scopes (down into + our expression), there is no risk of moving a variable reference out + of the scope in which it is defined. + + \starttrans + (letrec binds in E) M + ------------------------ + letrec binds in E M + \stoptrans + + % And an example + \startbuffer[from] + ( letrec + val = 1 + in + add val + ) 3 + \stopbuffer + + \startbuffer[to] + letrec + val = 1 + in + add val 3 + \stopbuffer + + \transexample{appproplet}{Application propagation for a let expression}{from}{to} + + \starttrans + (case x of + p0 -> E0 + \vdots + pn -> En) M + ----------------- + case x of + p0 -> E0 M + \vdots + pn -> En M + \stoptrans + + % And an example + \startbuffer[from] + ( 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} - -\subsection{Let flattening} -This transform turns two nested lets (\lam{let x = (let ... in ...) in -...}) into a single let. - -\subsection{Empty let removal} - -\subsection{Simple let binding removal} -This transforms inlines simple let bindings (\eg a = b). - -\subsection{Unused let binding removal} - -\subsection{Non-representable binding inlining} -This transform inlines let bindings of a funtion type. TODO: This should -be generelized to anything that is non representable at runtime, or -something like that. - -\subsection{Scrutinee simplification} -This transform ensures that the scrutinee of a case expression is always -a simple variable reference. - -\subsection{Case simplification} - -\subsection{Case removal} -This transform removes any case statements with a single alternative and -only wild binders. - -\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. - \item Make all arguments of builtin functions either: - \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 with 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. + ) 1 + \stopbuffer + + \startbuffer[to] + case x of + True -> id 1 + False -> neg 1 + \stopbuffer + + \transexample{apppropcase}{Application propagation for a case expression}{from}{to} + + \subsubsection[sec:normalization:letrecurse]{Let recursification} + This transformation makes all non-recursive lets recursive. In the + end, we want a single recursive let in our normalized program, so all + non-recursive lets can be converted. This also makes other + transformations simpler: They only need to be specified for recursive + let expressions (and simply will not apply to non-recursive let + expressions until this transformation has been applied). + + \starttrans + let + a = E + in + M + ------------------------------------------ + letrec + a = E + in + M + \stoptrans + + \subsubsection{Let flattening} + This transformation puts nested lets in the same scope, by lifting the + binding(s) of the inner let into the outer let. Eventually, this will + cause all let bindings to appear in the same scope. + + This transformation only applies to recursive lets, since all + non-recursive lets will be made recursive (see + \in{section}[sec:normalization:letrecurse]). + + Since we are joining two scopes together, there is no risk of moving a + variable reference out of the scope where it is defined. + + \starttrans + letrec + a0 = E0 + \vdots + ai = (letrec bindings in M) + \vdots + an = En + in + N + ------------------------------------------ + letrec + a0 = E0 + \vdots + ai = M + \vdots + an = En + bindings + in + N + \stoptrans + + \startbuffer[from] + letrec + a = 1 + b = letrec + x = a + y = c + in + x + y + c = 2 + in + b + \stopbuffer + \startbuffer[to] + letrec + a = 1 + b = x + y + c = 2 + x = a + y = c + in + b + \stopbuffer + + \transexample{letflat}{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. + + This transformation only applies to the entire body of a + function instead of any subexpression in a function. This is + achieved by the contexts, like \lam{x = E}, though this is + strictly not correct (you could read this as "if there is any + function \lam{x} that binds \lam{E}, any \lam{E} can be + transformed, while we only mean the \lam{E} that is bound by + \lam{x}). + + Note that the return value is not simplified if its not + representable. Otherwise, this would cause a direct loop with + the inlining of unrepresentable bindings. If the return value is + not representable because it has a function type, η-abstraction + should make sure that this transformation will eventually apply. + If the value is not representable for other reasons, the + function result itself is not representable, meaning this + function is not translatable anyway. + + \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 + ~ \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 + \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 + \stoptrans + + \startbuffer[from] + x = add 1 2 + \stopbuffer + + \startbuffer[to] + x = letrec x = add 1 2 in x + \stopbuffer + + \transexample{retvalsimpl}{Return value simplification}{from}{to} - 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. -\stopitemize - -When looking at the arguments of a builtin function, we can divide them -into categories: - -\startitemize - \item Arguments with a runtime representable type. + \todo{More examples} + + \subsection[sec:normalization:argsimpl]{Representable arguments simplification} + This section contains just a single transformation that deals with + representable arguments in applications. Non-representable arguments are + handled by the transformations in + \in{section}[sec:normalization:nonrep]. + + 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 + 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. + + 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. + + \refdef{global variable} + Note that references to \emph{global variables} (like a top level + function without arguments, but also an argumentless dataconstructors + like \lam{True}) are also simplified. Only local variables generate + signals in the resulting architecture. Even though argumentless + dataconstructors generate constants in generated \VHDL\ code and could be + mapped to an input port directly, they are still simplified to make the + normal form more regular. + + \refdef{representable} + \starttrans + M N + -------------------- \lam{N} is representable + letrec x = N in M x \lam{N} is not a local variable reference + \stoptrans + \refdef{local variable} + + \startbuffer[from] + add (add a 1) 1 + \stopbuffer + + \startbuffer[to] + letrec x = add a 1 in add x 1 + \stopbuffer + + \transexample{argsimpl}{Argument simplification}{from}{to} + + \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 built-in function + can receive. + + \startitemize[KR] + \item A representable local variable reference. This is the most + 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 built-in functions) is turned into a local variable + reference. + \item (A partial application of) a top level function (either built-in on + user-defined). The function extraction transformation described in + this section takes care of turning every functiontyped argument into + (a partial application of) a top level function. + \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 built-in function will be either + transformed into one of the above categories, or end up in this + categorie. In any case, the result is in normal form. + \stopitemize + + As noted, the argument simplification will handle any representable + arguments to a built-in function. The following transformation is needed + to handle non-representable arguments with a function type, all other + non-representable arguments do not need any special handling. + + \subsubsection[sec:normalization:funextract]{Function extraction} + This transform deals with function-typed arguments to built-in + functions. + 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 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 + 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 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 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 + x = λf0 ... λfn.N + \stoptrans + + \startbuffer[from] + addList = λb.λxs.map (λa . add a b) xs + \stopbuffer + + \startbuffer[to] + addList = λb.λxs.map (f b) xs + ~ + f = λb.λa.add a b + \stopbuffer + + \transexample{funextract}{Function extraction}{from}{to} + + Note that the function \lam{f} will still need normalization after + this. + + \subsection{Case normalisation} + The transformations in this section ensure that case statements end up + in normal form. + + \subsubsection{Scrutinee simplification} + This transform ensures that the scrutinee of a case expression is always + a simple variable reference. + + \starttrans + case E of + alts + ----------------- \lam{E} is not a local variable reference + letrec x = E in + case x of + alts + \stoptrans + + \startbuffer[from] + case (foo a) of + True -> a + False -> b + \stopbuffer + + \startbuffer[to] + letrec x = foo a in + case x of + True -> a + False -> b + \stopbuffer + + \transexample{letflat}{Case normalisation}{from}{to} + + + \placeintermezzo{}{ + \defref{wild binders} + \startframedtext[width=7cm,background=box,frame=no] + \startalignment[center] + {\tfa Wild binders} + \stopalignment + \blank[medium] + In a functional expression, a \emph{wild binder} refers to any + binder that is never referenced. This means that even though it + will be bound to a particular value, that value is never used. + + The Haskell syntax offers the underscore as a wild binder that + cannot even be referenced (It can be seen as introducing a new, + anonymous, binder everytime it is used). + + In these transformations, the term wild binder will sometimes be + used to indicate that a binder must not be referenced. + \stopframedtext + } + + \subsubsection{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: + + \startitemize + \item An extractor case with a single alternative that picks a 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 + + 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 defintion 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. - 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 with a function type. + \starttrans + case E of + C0 v0,0 ... v0,m -> E0 + \vdots + Cn vn,0 ... vn,m -> En + --------------------------------------------------- \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 x0,0 .. x0,m -> x0,m + \vdots + vn,m = case E of Cn xn,0 .. xn,m -> xn,m + y0 = E0 + \vdots + yn = En + in + case E of + C0 w0,0 ... w0,m -> y0 + \vdots + Cn wn,0 ... wn,m -> yn + \stoptrans + + Note that this transformation applies to case expressions with any + scrutinee. If the scrutinee is a complex expression, this might + result in duplication of work (hardware). An extra condition to + 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 + True -> add b 1 + False -> add b 2 + \stopbuffer + + \startbuffer[to] + letrec + x0 = add b 1 + x1 = add b 2 + in + case a of + True -> x0 + False -> x1 + \stopbuffer + + \transexample{selcasesimpl}{Selector case simplification}{from}{to} + + \startbuffer[from] + case a of + (,) b c -> add b c + \stopbuffer + \startbuffer[to] + letrec + 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{excasesimpl}{Extractor case simplification}{from}{to} + + \refdef{selector case} + In \in{example}[ex:trans:excasesimpl] the case expression is expanded + into multiple case expressions, including a pretty useless expression + (that is neither a selector or extractor case). This case can be + removed by the Case removal transformation in + \in{section}[sec:transformation:caseremoval]. + + \subsubsection[sec:transformation:caseremoval]{Case removal} + This transform removes any case expression with a single alternative and + only wild binders.\refdef{wild binders} + + These "useless" case expressions 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{caserem}{Case removal}{from}{to} + + \subsection[sec:normalization:nonrep]{Removing unrepresentable values} + The transformations in this section are aimed at making all the + 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 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], + 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 + the particular type applied to it. The same goes for dictionary + arguments. To remove polymorphism from let bound values, we + simply inline the let bindings that have a polymorphic type, + which should (eventually) make sure that the polymorphic + expression is applied to a type and/or dictionary, which can + then be removed by β-reduction (\in{section}[sec:normalization:beta]). + + Since both type and dictionary arguments are not representable, + \refdef{representable} + the non-representable argument specialization and + non-representable let binding inlining transformations below + take care of exactly this. + + There is one case where polymorphism cannot be completely + removed: Built-in functions are still allowed to be polymorphic + (Since we have no function body that we could properly + specialize). However, the code that generates \VHDL\ for built-in + functions knows how to handle this, so this is not a problem. + + \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. - 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. + For example, the following expression is a higher-order expression + that is not a lambda expression itself: - 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. + \refdef{id function} + \startlambda + case x of + High -> id + Low -> λx.x + \stoplambda + + The reference to the \lam{id} function shows that we can introduce a + 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. - 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 extraction} -This transform deals with arguments to functions that -are of a runtime representable type. - -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. - -%\transform{Argument extract} -%{ -%\lam{Y} is of a hardware representable type -% -%\lam{Y} is not a variable referene -% -%\conclusion -% -%\trans{X Y}{let z = Y in X z} -%} - -\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. - -%\transform{Function extraction} -%{ -%\lam{X} is a (partial application of) a builtin function -% -%\lam{Y} is not an application -% -%\lam{Y} is not a variable reference -% -%\conclusion -% -%\lam{f0 ... fm} = free local vars of \lam{Y} -% -%\lam{y} is a new global variable -% -%\lam{y = λf0 ... fn.Y} -% -%\trans{X Y}{X (y f0 ... fn)} -%} - -\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. - -TODO: Move these definitions somewhere sensible. - -Definition: A global variable is any variable that is bound at the -top level of a program. A local variable is any other variable. - -Definition: A 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. - -Definition: A 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. - -\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' = λy0 ... yi-1 f0 ... fm yi+1 ... yn . \lam{f0 ... fm} = free local vars of \lam{Y_i} - E y0 ... yi-1 Yi yi+1 ... yn -~ -x' y0 ... yi-1 f0 ... fm Yi+1 ... Yn -\stoptrans - -\subsection{Cast propagation / simplification} -This transform pushes casts down into the expression as far as possible. - -\subsection{Return value simplification} -Currently implemented using lambda simplification, let simplification, and -top simplification. Should change. - -\subsection{Example sequence} - -This section lists an example expression, with a sequence of transforms -applied to it. The exact transforms given here probably don't exactly -match the transforms given above anymore, but perhaps this can clarify -the big picture a bit. - -TODO: Update or remove this section. - -\startlambda - λx. - let s = foo x - in - case s of - (a, b) -> - case a of - High -> add - Low -> let - op' = case b of - High -> sub - Low -> λc.λd.c - in - λc.λd.op' d c -\stoplambda - -After top-level η-abstraction: - -\startlambda - λx.λc.λd. - (let s = foo x - in - case s of - (a, b) -> - case a of - High -> add - Low -> let - op' = case b of - High -> sub - Low -> λc.λd.c - in - λc.λd.op' d c - ) c d -\stoplambda - -After (extended) β-reduction: - -\startlambda - λx.λc.λd. - let s = foo x - in - case s of - (a, b) -> - case a of - High -> add c d - Low -> let - op' = case b of - High -> sub - Low -> λc.λd.c - in - op' d c -\stoplambda - -After return value extraction: - -\startlambda - λx.λc.λd. - let s = foo x - r = case s of - (a, b) -> - case a of - High -> add c d - Low -> let - op' = case b of - High -> sub - Low -> λc.λd.c - in - op' d c - in - r -\stoplambda - -Scrutinee simplification does not apply. - -After case binder wildening: - -\startlambda - λx.λc.λd. - let s = foo x - a = case s of (a, _) -> a - b = case s of (_, b) -> b - r = case s of (_, _) -> - case a of - High -> add c d - Low -> let op' = case b of - High -> sub - Low -> λc.λd.c - in - op' d c - in - r -\stoplambda - -After case value simplification - -\startlambda - λx.λc.λd. - let s = foo x - a = case s of (a, _) -> a - b = case s of (_, b) -> b - r = case s of (_, _) -> r' - rh = add c d - rl = let rll = λc.λd.c - op' = case b of - High -> sub - Low -> rll - in - op' d c - r' = case a of - High -> rh - Low -> rl - in - r -\stoplambda - -After let flattening: - -\startlambda - λx.λc.λd. - let s = foo x - a = case s of (a, _) -> a - b = case s of (_, b) -> b - r = case s of (_, _) -> r' - rh = add c d - rl = op' d c - rll = λc.λd.c - op' = case b of - High -> sub - Low -> rll - r' = case a of - High -> rh - Low -> rl - in - r -\stoplambda - -After function inlining: - -\startlambda - λx.λc.λd. - let s = foo x - a = case s of (a, _) -> a - b = case s of (_, b) -> b - r = case s of (_, _) -> r' - rh = add c d - rl = (case b of - High -> sub - Low -> λc.λd.c) d c - r' = case a of - High -> rh - Low -> rl - in - r -\stoplambda - -After (extended) β-reduction again: - -\startlambda - λx.λc.λd. - let s = foo x - a = case s of (a, _) -> a - b = case s of (_, b) -> b - r = case s of (_, _) -> r' - rh = add c d - rl = case b of - High -> sub d c - Low -> d - r' = case a of - High -> rh - Low -> rl - in - r -\stoplambda - -After case value simplification again: - -\startlambda - λx.λc.λd. - let s = foo x - a = case s of (a, _) -> a - b = case s of (_, b) -> b - r = case s of (_, _) -> r' - rh = add c d - rlh = sub d c - rl = case b of - High -> rlh - Low -> d - r' = case a of - High -> rh - Low -> rl - in - r -\stoplambda - -After case removal: - -\startlambda - λx.λc.λd. - let s = foo x - a = case s of (a, _) -> a - b = case s of (_, b) -> b - r = r' - rh = add c d - rlh = sub d c - rl = case b of - High -> rlh - Low -> d - r' = case a of - High -> rh - Low -> rl - in - r -\stoplambda - -After let bind removal: - -\startlambda - λx.λc.λd. - let s = foo x - a = case s of (a, _) -> a - b = case s of (_, b) -> b - rh = add c d - rlh = sub d c - rl = case b of - High -> rlh - Low -> d - r' = case a of - High -> rh - Low -> rl - in - r' -\stoplambda - -Application simplification is not applicable. + 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 + 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: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 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 + allowed, since it is inside a complete application. + \stopitemize + + 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 + the example, no polymorphism is shown. In reality, at least map would + be polymorphic. + + \startlambda + λy.let double = λx. x + x in + case y of + Low -> map double + 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 + 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} 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 have already seen are used. The η-abstraction + 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 + becomes a bit bigger: + + \startlambda + λy.λq.(let double = λx. x + x in + case y of + Low -> map double + High -> λz. z + ) q + \stoplambda + + η-abstraction 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 + example, the \lam{q} and \lam{r} variable will be propagated into the + let expression and then into the case expression: + + \startlambda + λy.λq.let double = λx. x + x in + case y of + Low -> map double q + High -> (λz. z) q + \stoplambda + + 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: + + \startlambda + λy.λq.let double = λx. x + x in + case y of + Low -> map double q + High -> q + \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 + 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 + non-representable binding inlining transformation below. When applying + it to our example, we get the following: + + \startlambda + λy.λq.case y of + Low -> map (λx. x + x) q + High -> q + \stoplambda + + 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 + and put the lambda abstraction inside. This is done by the function + extraction transformation from + \in{section}[sec:normalization:funextract]. + + \startlambda + λy.λq.case y of + Low -> map func q + High -> q + \stoplambda + + This also introduces a new function, that we have called \lam{func}: + + \startlambda + func = λx. x + x + \stoplambda + + Note that this does not actually remove the lambda, but now it is a + lambda at the highest level of a function, which is allowed in the + 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 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 + expression shows an example: + + \startlambda + twice :: (Word -> Word) -> Word -> Word + twice = λf.λa.f (f a) + + main = λa.app (λx. x + x) a + \stoplambda + + 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 have made the function monomorphic for clarity, even though + this function would be a lot more useful if it was polymorphic. The + function \lam{main} uses \lam{twice} to apply a lambda epression twice. + + When faced with a user defined function, a body is available for that + function. This means we could create a specialized version of the + function that only works for this particular higher-order argument + (\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: + + \startlambda + twice' :: Word -> Word + twice' = λb.(λf.λa.f (f a)) (λx. x + x) b + + 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 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 + twice' = λb.(b + b) + (b + b) + \stoplambda + + This example also shows that the resulting normal form might not be as + efficient as we might hope it to be (it is calculating \lam{(b + b)} + twice). This is discussed in more detail in + \in{section}[sec:normalization:duplicatework]. + + \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 + 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 + literals for all three integer types supported (\hs{SizedWord}, + \hs{SizedInt} and \hs{RangedWord}). This is implemented using + Haskell's \hs{Num} type class, which offers a \hs{fromInteger} method + that converts any \hs{Integer} to the Cλash datatypes. + + 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. + + \starthaskell + 1 :: SizedWord D32 + \stophaskell + + This Haskell code results in the following Core expression: + + \startlambda + fromInteger @(SizedWord D32) \$dNum (smallInteger 10) + \stoplambda + + The literal 10 will have the type \lam{GHC.Prim.Int\#}, which is + converted into an \lam{Integer} by \lam{smallInteger}. Finally, the + \lam{fromInteger} function will finally convert this into a + \lam{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 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. + For example, the following expression is not in the intended normal + form, since one of the let bindings has an unrepresentable type + (\lam{Integer}): + + \startlambda + let l = smallInteger 10 in fromInteger @(SizedWord D32) \$dNum l + \stoplambda + + By inlining these let-bindings, we can ensure that unrepresentable + literals bound by a let binding end up in an application of the + 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} + 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 + with a non-representable type, for obvious reasons), we have no choice + 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 + unrepresentable literals in an expression. + + \refdef{substitution notation} + \starttrans + letrec + a0 = E0 + \vdots + ai = Ei + \vdots + an = En + in + M + -------------------------- \lam{Ei} has a non-representable type. + letrec + a0 = E0 [ai=>Ei] \vdots + ai-1 = Ei-1 [ai=>Ei] + ai+1 = Ei+1 [ai=>Ei] + \vdots + an = En [ai=>Ei] + in + M[ai=>Ei] + \stoptrans + + \startbuffer[from] + letrec + a = smallInteger 10 + inc = λb -> add b 1 + inc' = add 1 + x = fromInteger a + in + inc (inc' x) + \stopbuffer + + \startbuffer[to] + letrec + x = fromInteger (smallInteger 10) + in + (λb -> add b 1) (add 1 x) + \stopbuffer + + \transexample{nonrepinline}{Nonrepresentable 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 + \emph{specialized} version of the function that only works for one + particular value of that argument (in other words, the argument can be + removed). + + Specialization means to create a specialized version of the called + function, with one argument already filled in. As a simple example, in + the following program (this is not actual Core, since it directly uses + a literal with the unrepresentable type \lam{GHC.Prim.Int\#}). + + \startlambda + f = λa.λb.a + b + inc = λa.f a 1 + \stoplambda + + We could specialize the function \lam{f} against the literal argument + 1, with the following result: + + \startlambda + f' = λa.a + 1 + inc = λa.f' a + \stoplambda + + In some way, this transformation is similar to β-reduction, but it + operates across function boundaries. It is also similar to + non-representable let binding inlining above, since it sort of + \quote{inlines} an expression into a called function. + + Special care must be taken when the argument has any free variables. + If this is the case, the original argument should not be removed + completely, but replaced by all the free variables of the expression. + In this way, the original expression can still be evaluated inside the + new function. + + 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 or β-reduction transformation to replace such a + variable with an expression we can propagate again. + + \starttrans + x = E + ~ + 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} + ~ \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 + \stoptrans + + This is a bit of a complex transformation. It transforms an + application of the function \lam{x}, where one of the arguments + (\lam{Y_i}) is not representable. A new + function \lam{x'} is created that wraps the body of the old function. + The body of the new function becomes a number of nested lambda + abstractions, one for each of the original arguments that are left + unchanged. + + The ith argument is replaced with the free variables of + \lam{Y_i}. Note that we reuse the same binders as those used in + \lam{Y_i}, since we can then just use \lam{Y_i} inside the new + function body and have all of the variables it uses be in scope. + + 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. + + 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. + + \todo{Examples. Perhaps reference the previous sections} + + \section{Unsolved problems} + The above system of transformations has been implemented in the prototype + and seems to work well to compile simple and more complex examples of + hardware descriptions. \todo{Ref christiaan?} However, this normalization + system has not seen enough review and work to be complete and work for + every Core expression that is supplied to it. A number of problems + have already been identified and are discussed in this section. + + \subsection[sec:normalization:duplicatework]{Work duplication} + A possible problem of β-reduction is that it could duplicate work. + When the expression applied is not a simple variable reference, but + requires calculation and the binder the lambda abstraction binds to + is used more than once, more hardware might be generated than strictly + needed. + + As an example, consider the expression: + + \startlambda + (λx. x + x) (a * b) + \stoplambda + + When applying β-reduction to this expression, we get: + + \startlambda + (a * b) + (a * b) + \stoplambda + + which of course calculates \lam{(a * b)} twice. + + 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 + given here for future reference: + + \starttrans + (λx.E) M + ----------------- + letrec x = M in E + \stoptrans + + 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 + the argument happens to be a variable reference, then simple let + binding removal (\in{section}[sec:normalization:simplelet]) will + remove it, making the result identical to that of the original + β-reduction transformation. + + When also applying argument simplification to the above example, we + get the following expression: + + \startlambda + let y = (a * b) + z = (a * b) + in y + z + \stoplambda + + 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: + + \startlambda + let y = (a * b) in y + y + \stoplambda + + \subsection[sec:normalization:non-determinism]{Non-determinism} + As an example, again consider the following expression: + + \startlambda + (λx. x + x) (a * b) + \stoplambda + + We can apply both β-reduction (\in{section}[sec:normalization:beta]) + as well as argument simplification + (\in{section}[sec:normalization:argsimpl]) to this expression. + + When applying argument simplification first and then β-reduction, we + get the following expression: + + \startlambda + let y = (a * b) in y + y + \stoplambda + + When applying β-reduction first and then argument simplification, we + get the following expression: + + \startlambda + let y = (a * b) + z = (a * b) + in y + z + \stoplambda + + As you can see, this is a different expression. This means that the + order of expressions, does in fact change the resulting normal form, + which is something that we would like to avoid. In this particular + case one of the alternatives is even clearly more efficient, so we + would of course like the more efficient form to be the normal form. + + 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. + + \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 + transformations will probably need updating to handle them in all + cases. + + \subsection[sec:normalization:stateproblems]{Normalization of stateful descriptions} + Currently, the intended normal form definition\refdef{intended + 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 + (\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). + + 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 + 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 + 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 + extended as well to be more specific about how state handling + should look like in normal form. + \in{Section}[sec:prototype:statelimits] already contains a + tight description of the limitations on the use of state + variables, which could be adapted into the intended normal form. + + \section[sec:normalization:properties]{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. This typically happens when one + transformation produces a result that is transformed back to the original + by another transformation, or when one or more transformations keep + 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 + 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 + + 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 (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 also the termination property. The soundness + property probably holds, since it is easier to manually verify (each + transformation can be reviewed separately). + + Even though no complete proofs have been made, some ideas for + possible proof strategies are shown below. + + \subsection{Graph representation} + 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 + system). 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 lambda calculus + system with β and η reduction (solid lines) and expansion (dotted lines).} + \boxedgraphic{TransformGraph} + + 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 + 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 + 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-expansion and η-expansion (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 expressions to which + no transformation applies anymore. We call this set of nodes the \emph{normal + set}. The set of nodes containing expressions in intended normal + form \refdef{intended normal form} is called the \emph{intended + normal set}. + + From such a graph, we can derive some properties easily: + \startitemize[KR] + \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 + the normal set are \emph{not} in the intended normal form) is not + strictly required. In other words, our normal set must be a + subset of the intended normal form, but they do not need to be + the same set. + form. + \item A system is deterministic if all paths starting at a particular + 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 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.) + + If we would consider the system with both expansion and reduction, there + would no longer be termination either, 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 node in the normal set, it must obviously be + \emph{deterministic} as well. + + \subsection{Termination} + In general, proving termination of an arbitrary program is a very + hard problem. \todo{Ref about arbitrary termination} Fortunately, + we only have to prove termination for our specific transformation + system. + + A common approach for these kinds of proofs is to associate a + measure with each possible expression in our system. If we can + show that each transformation strictly decreases this measure + (\ie, the expression transformed to has a lower measure than the + expression transformed from). \todo{ref about measure-based + termination proofs / analysis} + + A good measure for a system consisting of just β-reduction would + be the number of lambda expressions in the expression. Since every + application of β-reduction removes a lambda abstraction (and there + is always a bounded number of lambda abstractions in every + expression) we can easily see that a transformation system with + just β-reduction will always terminate. + + For our complete system, this measure would be fairly complex + (probably the sum of a lot of things). Since the (conditions on) + our transformations are pretty complex, we would need to include + both simple things like the number of let expressions as well as + more complex things like the number of case expressions that are + not yet in normal form. + + No real attempt has been made at finding a suitable measure for + our system yet. + + \subsection{Soundness} + Soundness is a property that can be proven for each transformation + separately. Since our system only runs separate transformations + sequentially, if each of our transformations leaves the + \emph{meaning} of the expression unchanged, then the entire system + will of course leave the meaning unchanged and is thus + \emph{sound}. + + 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. 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 + 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. + + 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 + 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 + the actual expression is different). + + \subsection{Completeness} + Proving completeness is probably not hard, but it could be a lot + of work. We have seen above that to prove completeness, we must + show that the normal set of our graph representation is a subset + of the intended normal set. + + However, it is hard to systematically generate or reason about the + normal set, since it is defined as any nodes to which no + transformation applies. To determine this set, each transformation + must be considered and when a transformation is added, the entire + set should be re-evaluated. This means it is hard to show that + 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} + + 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 + 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 + that applies to it (since that means it is not in normal form + either and since $A \subseteq C \Leftrightarrow \forall x (x \in A + \rightarrow x \in C)$). + + 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 + 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 + is immediately clear which expressions these are and adding + (or modifying) transformations to fix this should be relatively + easy. + + As observed above, applying this approach is a lot of work, since + we need to check every (set of) transformation(s) separately. + + \todo{Perhaps do a few steps of the proofs as proof-of-concept} + +% vim: set sw=2 sts=2 expandtab: