X-Git-Url: https://git.stderr.nl/gitweb?p=matthijs%2Fmaster-project%2Freport.git;a=blobdiff_plain;f=Chapters%2FNormalization.tex;h=1e5b200c0a82922b4f1b0c0350b09d6212372980;hp=b91eb40edda218c60ccb1fc951a2da436490af10;hb=48f4c2d4c18166aac6485e1c4ce759940c320bb2;hpb=66f5758c343546733f7cda281ccff7eafedd6b73 diff --git a/Chapters/Normalization.tex b/Chapters/Normalization.tex index b91eb40..1e5b200 100644 --- a/Chapters/Normalization.tex +++ b/Chapters/Normalization.tex @@ -1,827 +1,1732 @@ -\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). -\startbuffer[from] -(case x of - p1 -> E1 - \vdots - pn -> En) M -\stopbuffer -\startbuffer[to] -case x of - p1 -> E1 M - \vdots - pn -> En M -\stopbuffer - -%\transform{Extended β-reduction} -%{ -%\conclusion -%\trans{(λx.E) M}{E[M/x]} -% -%\nextrule -%\conclusion -%\trans{(let binds in E) M}{let binds in E M} -% -%\nextrule -%\conclusion -%\transbuf{from}{to} -%} - -\startbuffer[from] -let a = (case x of - True -> id - False -> neg + + \define[3]\transexample{ + \placeexample[here]{#1} + \startcombination[2*1] + {\example{#2}}{Original program} + {\example{#3}}{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 and because + core can describe expressions that do not have a direct hardware + interpretation. + + TODO: Describe core properties not supported in \small{VHDL}, and describe how the + \small{VHDL} we want to generate should look like. + + \section{Normal form} + The transformations described here have a well-defined goal: To bring the + program in a well-defined form that is directly translatable to hardware, + while fully preserving the semantics of the program. We refer to this form as + the \emph{normal form} of the program. The formal definition of this normal + form is quite simple: + + \placedefinition{}{A program is in \emph{normal form} if none of the + transformations from this chapter apply.} + + Of course, this is an \quote{easy} definition of the normal form, since our + program will end up in normal form automatically. The more interesting part is + to see if this normal form actually has the properties we would like it to + have. + + But, before getting into more definitions and details about this normal form, + let's try to get a feeling for it first. The easiest way to do this is by + describing the things we want to not have in a normal form. + + \startitemize + \item Any \emph{polymorphism} must be removed. When laying down hardware, we + can't generate any signals that can have multiple types. All types must be + completely known to generate hardware. + + \item Any \emph{higher order} constructions must be removed. We can't + generate a hardware signal that contains a function, so all values, + arguments and returns values used must be first order. + + \item Any complex \emph{nested scopes} must be removed. In the \small{VHDL} + description, every signal is in a single scope. Also, full expressions are + not supported everywhere (in particular port maps can only map signal names, + not expressions). To make the \small{VHDL} generation easy, all values must be bound + on the \quote{top level}. + \stopitemize + + TODO: Intermezzo: functions vs plain values + + A very simple example of a program in normal form is given in + \in{example}[ex:MulSum]. As you can see, all arguments to the function (which + will become input ports in the final hardware) are at the top. This means that + the body of the final lambda abstraction is never a function, but always a + plain value. + + After the lambda abstractions, we see a single let expression, that binds two + variables (\lam{mul} and \lam{sum}). These variables will be signals in the + final hardware, bound to the output port of the \lam{*} and \lam{+} + components. + + The final line (the \quote{return value} of the function) selects the + \lam{sum} signal to be the output port of the function. This \quote{return + value} can always only be a variable reference, never a more complex + expression. + + \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 $res$ 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 an adder and a + subtractor.} + \startcombination[2*1] + {\typebufferlam{MulSum}}{Core description in normal form.} + {\boxedgraphic{MulSum}}{The architecture described by the normal form.} + \stopcombination + + The previous example described composing an architecture by calling other + functions (operators), resulting in a simple architecture with component and + connection. 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 the same as in \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}. + + \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 contains everything that is supported in normal form, with the + exception of builtin higher order functions. The graphical version of the + architecture contains a slightly simplified version, since the state tuple + packing and unpacking have been left out. Instead, two seperate registers are + drawn. Also note that most synthesis tools will further optimize this + architecture by removing the multiplexers at the register input and replace + them with some logic in the clock inputs, but we want to show the architecture + as close to the description as possible. + + \startbuffer[NormalComplete] + regbank :: Bit + -> Word + -> State (Word, Word) + -> (State (Word, Word), Word) + + -- All arguments are an inital lambda + 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 (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' + -- 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 + + \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{Intended normal form definition} + Now we have some intuition for the normal form, we can describe how we want + the normal form to look like in a slightly more formal manner. The following + EBNF-like description completely captures the intended structure (and + generates a subset of GHC's core format). + + Some clauses have an expression listed in parentheses. These are conditions + that need to apply to the clause. + + \startlambda + \italic{normal} = \italic{lambda} + \italic{lambda} = λvar.\italic{lambda} (representable(var)) + | \italic{toplet} + \italic{toplet} = letrec [\italic{binding}...] in var (representable(varvar)) + \italic{binding} = var = \italic{rhs} (representable(rhs)) + -- State packing and unpacking by coercion + | var0 = var1 :: State ty (lvar(var1)) + | var0 = var1 :: ty (var0 :: State ty) (lvar(var1)) + \italic{rhs} = userapp + | builtinapp + -- Extractor case + | case var of C a0 ... an -> ai (lvar(var)) + -- Selector case + | case var of (lvar(var)) + DEFAULT -> var0 (lvar(var0)) + C w0 ... wn -> resvar (\forall{}i, wi \neq resvar, lvar(resvar)) + \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{builtinfunc} = var (bvar(var)) + \italic{builtinarg} = \italic{coreexpr} + \stoplambda + + -- 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 \small{VHDL} translation is + available. + + \section{Transformation notation} + To be able to concisely present transformations, we use a specific format to + them. It is a simple format, similar to one used in logic reasoning. + + Such a transformation description looks like the following. + + \starttrans + + ~ + + -------------------------- + + ~ + + \stoptrans + + This format desribes a transformation that applies to \lam{original + expresssion} and transforms it into \lam{transformed expression}, assuming + that all conditions apply. In this format, there are a number of placeholders + in pointy brackets, most of which should be rather obvious in their meaning. + Nevertheless, we will more precisely specify their meaning below: + + \startdesc{} The expression pattern that will be matched + against (subexpressions of) the expression to be transformed. We call this a + pattern, because it can contain \emph{placeholders} (variables), which match + any expression or binder. Any such placeholder is said to be \emph{bound} to + the expression it matches. It is convention to use an uppercase latter (\eg + \lam{M} or \lam{E} to refer to any expression (including a simple variable + reference) and lowercase letters (\eg \lam{v} or \lam{b}) to refer to + (references to) binders. + + For example, the pattern \lam{a + B} will match the expression + \lam{v + (2 * w)} (and bind \lam{a} to \lam{v} and \lam{B} to + \lam{(2 * 2)}), but not \lam{v + (2 * w)}. + \stopdesc + + \startdesc{} + These are extra conditions on the expression that is matched. These + conditions can be used to further limit the cases in which the + transformation applies, in particular to prevent a transformation from + causing a loop with itself or another transformation. + + Only if these if these conditions are \emph{all} true, this transformation + applies. + \stopdesc + + \startdesc{} + These are a number of extra conditions on the context of the function. In + particular, these conditions can require some other top level function to be + present, whose value matches the pattern given here. The format of each of + these conditions is: \lam{binder = }. + + Typically, the binder is some placeholder bound in the \lam{}, while the pattern contains some placeholders that are used in + the \lam{transformed expression}. + + Only if a top level binder exists that matches each binder and pattern, this + transformation applies. + \stopdesc + + \startdesc{} + This is the expression template that is the result of the transformation. If, looking + at the above three items, the transformation applies, the \lam{original + expression} is completely replaced with the \lam{}. + We call this a template, because it can contain placeholders, referring to + any placeholder bound by the \lam{} or the + \lam{}. The resulting expression will have those + placeholders replaced by the values bound to them. + + Any binder (lowercase) placeholder that has no value bound to it yet will be + bound to (and replaced with) a fresh binder. + \stopdesc + + \startdesc{} + These are templates for new functions to add to the context. This is a way + to have a transformation create new top level functiosn. + + Each addition has the form \lam{binder = template}. As above, any + placeholder in the addition is replaced with the value bound to it, and any + binder placeholder that has no value bound to it yet will be bound to (and + replaced with) a fresh binder. + \stopdesc + + As an example, we'll look at η-abstraction: + + \starttrans + E \lam{E :: 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 + + Consider the following function, which is a fairly obvious way to specify a + simple ALU (Note \at{example}[ex:AddSubAlu] is the normal form of this + function): + + \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's 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 + labmda abstraction: + + \startlambda + (case opcode of + Low -> (+) + High -> (-)) a + \stoplambda + + Here, the transformation does apply, binding \lam{E} to the entire + expression and \lam{x} to the fresh binder \lam{b}, resulting in the + replacement: + + \startlambda + λb.(case opcode of + Low -> (+) + High -> (-)) a b + \stoplambda + + Again, the transformation does not apply to this lambda abstraction, so we + look at its body. For brevity, we'll put the case statement 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. In the same way the transformation does not apply to both + components of this expression (\lam{case opcode of Low -> (+); High -> (-)} + and \lam{a}), so we'll skip to the components of the case expression: The + scrutinee and both alternatives. Since the opcode is not a function, it does + not apply here, and we'll leave both alternatives as an exercise to the + reader. The final function, after all these transformations becomes: + + \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 + + In this case, the transformation does not apply anymore, though this might + not always be the case (e.g., the application of a transformation on a + subexpression might open up possibilities to apply the transformation + further up in the expression). + + \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 (see TODO + ref), this leaves the implementation free to choose any application order that + results in an efficient implementation. + + When applying a single transformation, we try to apply it to every (sub)expression + in a function, not just the top level function. This allows us to keep the + transformation descriptions concise and powerful. + + \subsection{Definitions} + In the following sections, we will be using a number of functions and + notations, which we will define here. + + TODO: Define substitution + + \subsubsection{Other concepts} + A \emph{global variable} is any variable that is bound at the + top level of a program, or an external module. A \emph{local variable} is any + other variable (\eg, variables local to a function, which can be bound by + lambda abstractions, let expressions and pattern matches of case + alternatives). Note that this is a slightly different notion of global versus + local than what \small{GHC} uses internally. + \defref{global variable} \defref{local variable} + + A \emph{hardware representable} (or just \emph{representable}) type or value + is (a value of) a type that we can generate a signal for in hardware. For + example, a bit, a vector of bits, a 32 bit unsigned word, etc. Types that are + not runtime representable notably include (but are not limited to): Types, + dictionaries, functions. + \defref{representable} + + A \emph{builtin function} is a function supplied by the Cλash framework, whose + implementation is not valid Cλash. The implementation is of course valid + Haskell, for simulation, but it is not expressable in Cλash. + \defref{builtin function} \defref{user-defined function} + + For these functions, Cλash has a \emph{builtin hardware translation}, so calls + to these functions can still be translated. These are functions like + \lam{map}, \lam{hwor} and \lam{length}. + + A \emph{user-defined} function is a function for which we do have a Cλash + implementation available. + + \subsubsection{Functions} + Here, we define a number of functions that can be used below to concisely + specify conditions. + + \refdef{global variable}\emph{gvar(expr)} is true when \emph{expr} is a variable that references a + global variable. It is false when it references a local variable. + + \refdef{local variable}\emph{lvar(expr)} is the complement of \emph{gvar}; it is true when \emph{expr} + references a local variable, false when it references a global variable. + + \refdef{representable}\emph{representable(expr)} or \emph{representable(var)} is true when + \emph{expr} or \emph{var} is \emph{representable}. + + \subsection{Binder uniqueness} + A common problem in transformation systems, is binder uniqueness. When not + considering this problem, it is easy to create transformations that mix up + bindings and cause name collisions. Take for example, the following core + expression: + + \startlambda + (λa.λb.λc. a * b * c) x c + \stoplambda + + By applying β-reduction (see below) 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 scopes, such + that only the inner one is \quote{visible} in the inner expression. In the example + above, the \lam{c} binder was bound outside of the expression and in the inner + lambda expression. Inside that lambda expression, only the inner \lam{c} is + visible. + + There are a number of ways to solve this. \small{GHC} has isolated this + problem to their binder substitution code, which performs \emph{deshadowing} + during its expression traversal. This means that any binding that shadows + another binding on a higher level is replaced by a new binder that does not + shadow any other binding. This non-shadowing invariant is enough to prevent + binder uniqueness problems in \small{GHC}. + + In our transformation system, maintaining this non-shadowing invariant is + a bit harder to do (mostly due to implementation issues, the prototype doesn't + use \small{GHC}'s subsitution code). Also, we can observe the following + points. + + \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) 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 + meanst that in any given moment during normalization, all binders \emph{within + a single function} must be unique. To achieve this, we apply the following + technique. + + 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 pose any conflict, but it does ensure that + all binders within the function are generated by the same unique supply. See + (TODO: ref fresh binder). + \item Whenever a new binder must be generated, we generate a fresh binder that + is guaranteed to be different from \emph{all binders generated so far}. This + can thus never introduce duplication and will maintain the invariant. + \item Whenever (part of) an expression is duplicated (for example when + inlining), all binders in the expression are replaced with fresh binders + (using the same method as at the start of normalization). These fresh binders + can never introduce duplication, so this will maintain the invariant. + \item Whenever we move part of an expression around within the function, there + is no need to do anything special. There is obviously no way to introduce + duplication by moving expressions around. Since we know that each of the + binders is already unique, there is no way to introduce (incorrect) shadowing + either. + \stopitemize + + \section{Transform passes} + In this section we describe the actual transforms. Here we're using + the core language in a notation that resembles lambda calculus. + + Each of these transforms is meant to be applied to every (sub)expression + in a program, for as long as it applies. Only when none of the + transformations can be applied anymore, the program is in normal form (by + definition). We hope to be able to prove that this form will obey all of the + constraints defined above, but this has yet to happen (though it seems likely + that it will). + + Each of the transforms will be described informally first, explaining + the need for and goal of the transform. Then, a formal definition is + given, using a familiar syntax from the world of logic. Each transform + is specified as a number of conditions (above the horizontal line) and a + number of conclusions (below the horizontal line). The details of using + this notation are still a bit fuzzy, so comments are welcom. + + \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 normal + form. + + \subsubsection{β-reduction} + β-reduction is a well known transformation from lambda calculus, where it is + the main reduction step. It reduces applications of labmda abstractions, + removing both the lambda abstraction and the application. + + In our transformation system, this step helps to remove unwanted lambda + abstractions (basically all but the ones at the top level). Other + transformations (application propagation, non-representable inlining) make + sure that most lambda abstractions will eventually be reducable by + β-reduction. + + \starttrans + (λx.E) M + ----------------- + E[M/x] + \stoptrans + + % And an example + \startbuffer[from] + (λa. 2 * a) (2 * b) + \stopbuffer + + \startbuffer[to] + 2 * (2 * b) + \stopbuffer + + \transexample{β-reduction}{from}{to} + + \subsubsection{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). + + \starttrans + letrec in M + -------------- + M + \stoptrans + + TODO: Example + + \subsubsection{Simple let binding removal} + This transformation inlines simple let bindings (\eg a = b). + + This transformation is not needed to get into normal form, but makes the + resulting \small{VHDL} a lot shorter. + + \starttrans + letrec + a0 = E0 + \vdots + ai = b + \vdots + an = En + in + M + ----------------------------- \lam{b} is a variable reference + letrec + a0 = E0 [b/ai] + \vdots + ai-1 = Ei-1 [b/ai] + ai+1 = Ei+1 [b/ai] + \vdots + an = En [b/ai] + in + M[b/ai] + \stoptrans + + TODO: Example + + \subsubsection{Unused let binding removal} + This transformation removes let bindings that are never used. Usually, + the desugarer introduces some unused let bindings. + + This normalization pass should really be unneeded to get into normal form + (since unused bindings are not forbidden by the normal form), but in practice + the desugarer or simplifier emits some unused bindings that cannot be + normalized (e.g., calls to a \type{PatError} (TODO: Check this name)). Also, + this transformation makes the resulting \small{VHDL} a lot shorter. + + \starttrans + letrec + a0 = E0 + \vdots + ai = Ei + \vdots + an = En + in + M \lam{a} does not occur free in \lam{M} + ---------------------------- \forall j, 0 <= j <= n, j ≠ i (\lam{a} 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 + + TODO: Example + + \subsubsection{Cast propagation / simplification} + This transform pushes casts down into the expression as far as possible. + Since its exact role and need is not clear yet, this transformation is + not yet specified. + + \subsubsection{Top level binding inlining} + This transform takes simple top level bindings generated by the + \small{GHC} compiler. \small{GHC} sometimes generates very simple + \quote{wrapper} bindings, which are bound to just a variable + reference, or a partial application to constants or other variable + references. + + Note that this transformation is completely optional. It is not + required to get any function into normal form, but it does help making + the resulting VHDL output easier to read (since it removes a bunch of + components that are really boring). + + This transform takes any top level binding generated by the compiler, + 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{Top level binding inlining}{from}{to} + + Without this transformation, the (+) function would generate an + architecture 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 + is 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 doesn't work}, so the entity would be called + \quote{w7aA7f} or something similarly unreadable 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), putting all of the other + value definitions in let bindings and making the final return value a + simple variable reference. + + \subsubsection{η-abstraction} + This transformation makes sure that all arguments of a function-typed + expression are named, by introducing lambda expressions. When combined with + β-reduction and non-representable binding inlining, all function-typed + expressions should be lambda abstractions or global identifiers. + + \starttrans + E \lam{E :: a -> b} + -------------- \lam{E} is not the first argument of an application. + λx.E x \lam{E} is not a lambda abstraction. + \lam{x} is a variable that does not occur free in \lam{E}. + \stoptrans + + \startbuffer[from] + foo = λa.case a of + True -> λb.mul b b + False -> id + \stopbuffer + + \startbuffer[to] + foo = λa.λx.(case a of + True -> λb.mul b b + False -> λy.id y) x + \stopbuffer + + \transexample{η-abstraction}{from}{to} + + \subsubsection{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). + + \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{Application propagation for a let expression}{from}{to} + + \starttrans + (case x of + p1 -> E1 + \vdots + pn -> En) M + ----------------- + case x of + p1 -> E1 M + \vdots + pn -> En M + \stoptrans + + % And an example + \startbuffer[from] + ( case x of + True -> id + False -> neg ) 1 - 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{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. - - 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. - - 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. - - These arguments are functions passed to higher order builtins, - like \lam{map} and \lam{foldl}. Since implementing these - functions for arbitrary function-typed expressions (\eg, lambda - expressions) is rather comlex, we reduce these arguments to - (partial applications of) global functions. - - We can still support arbitrary expressions from the user code, - by creating a new global function containing that expression. - This way, we can simply replace the argument with a reference to - that new function. However, since the expression can contain any - number of free variables we also have to include partial - applications in our normal form. - - This category of arguments is handled by the function extraction - transform. - \item Other unrepresentable arguments. - - These arguments can take a few different forms: - \startdesc{Type arguments} - In the core language, type arguments can only take a single - form: A type wrapped in the Type constructor. Also, there is - nothing that can be done with type expressions, except for - applying functions to them, so we can simply leave type - arguments as they are. - \stopdesc - \startdesc{Dictionary arguments} - In the core language, dictionary arguments are used to find - operations operating on one of the type arguments (mostly for - finding class methods). Since we will not actually evaluatie - the function body for builtin functions and can generate - code for builtin functions by just looking at the type - arguments, these arguments can be ignored and left as they - are. - \stopdesc - \startdesc{Type level arguments} - Sometimes, we want to pass a value to a builtin function, but - we need to know the value at compile time. Additionally, the - value has an impact on the type of the function. This is - encoded using type-level values, where the actual value of the - argument is not important, but the type encodes some integer, - for example. Since the value is not important, the actual form - of the expression does not matter either and we can leave - these arguments as they are. - \stopdesc - \startdesc{Other arguments} - Technically, there is still a wide array of arguments that can - be passed, but does not fall into any of the above categories. - However, none of the supported builtin functions requires such - an argument. This leaves use with passing unsupported types to - a function, such as calling \lam{head} on a list of functions. - - In these cases, it would be impossible to generate hardware - for such a function call anyway, so we can ignore these - arguments. - - The only way to generate hardware for builtin functions with - arguments like these, is to expand the function call into an - equivalent core expression (\eg, expand map into a series of - function applications). But for now, we choose to simply not - support expressions like these. - \stopdesc - - From the above, we can conclude that we can simply ignore these - other unrepresentable arguments and focus on the first two - categories instead. -\stopitemize - -\subsubsection{Argument 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 - -%\transform{Argument propagation} -%{ -%\lam{x} is a global variable, bound to a user-defined function -% -%\lam{x = E} -% -%\lam{Y_i} is not of a runtime representable type -% -%\lam{Y_i} is not a local variable reference -% -%\conclusion -% -%\lam{f0 ... fm} = free local vars of \lam{Y_i} -% -%\lam{x'} is a new global variable -% -%\lam{x' = λy0 ... yi-1 f0 ... fm yi+1 ... yn . E y0 ... yi-1 Yi yi+1 ... yn} -% -%\trans{x Y0 ... Yi ... Yn}{x' y0 ... yi-1 f0 ... fm Yi+1 ... Yn} -%} -% -%TODO: The above definition looks too complicated... Can we find -%something more concise? - -\subsection{Cast propagation} -This transform pushes casts down into the expression as far as possible. -\subsection{Let recursification} -This transform makes all lets recursive. -\subsection{Let simplification} -This transform makes the result value of all let expressions a simple -variable reference. -\subsection{Let flattening} -This transform turns two nested lets (\lam{let x = (let ... in ...) in -...}) into a single let. -\subsection{Simple let binding removal} -This transforms inlines simple let bindings (\eg a = b). -\subsection{Function 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 binder wildening} -This transform replaces all binders of a each case alternative with a -wild binder (\ie, one that is never referred to). This will possibly -introduce a number of new "selector" case statements, that only select -one element from an algebraic datatype and bind it to a variable. -\subsection{Case value simplification} -This transform simplifies the result value of each case alternative by -binding the value in a let expression and replacing the value by a -simple variable reference. -\subsection{Case removal} -This transform removes any case statements with a single alternative and -only wild binders. - -\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) -> + \stopbuffer + + \startbuffer[to] + case x of + True -> id 1 + False -> neg 1 + \stopbuffer + + \transexample{Application propagation for a case expression}{from}{to} + + \subsubsection{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 can simply assume all lets are + recursive. + + \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 a new let around the outer let. Eventually, + this will cause all let bindings to appear in the same scope (they will all be + in scope for the function return value). + + \starttrans + letrec + \vdots + x = (letrec bindings in M) + \vdots + in + N + ------------------------------------------ + letrec + \vdots + bindings + x = M + \vdots + in + N + \stoptrans + + \startbuffer[from] + letrec + a = letrec + x = 1 + y = 2 + in + x + y + in + a + \stopbuffer + \startbuffer[to] + letrec + x = 1 + y = 2 + a = x + y + in + a + \stopbuffer + + \transexample{Let flattening}{from}{to} + + \subsubsection{Return value simplification} + This transformation ensures that the return value of a function is always a + simple local variable reference. + + Currently implemented using lambda simplification, let simplification, and + top simplification. Should change into something like the following, which + works only on the result of a function instead of any subexpression. This is + achieved by the contexts, like \lam{x = E}, though this is strictly not + correct (you could read this as "if there is any function \lam{x} that binds + \lam{E}, any \lam{E} can be transformed, while we only mean the \lam{E} that + is bound by \lam{x}. This might need some extra notes or something). + + 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, of course. 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 representable 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{Return value simplification}{from}{to} + + \subsection{Argument simplification} + The transforms in this section deal with simplifying application + arguments into normal form. The goal here is to: + + \startitemize + \item Make all arguments of user-defined functions (\eg, of which + we have a function body) simple variable references of a runtime + representable type. This is needed, since these applications will be turned + into component instantiations. + \item Make all arguments of builtin functions one of: + \startitemize + \item A type argument. + \item A dictionary argument. + \item A type level expression. + \item A variable reference of a runtime representable type. + \item A variable reference or partial application of a function type. + \stopitemize + \stopitemize + + When looking at the arguments of a user-defined function, we can + divide them into two categories: + \startitemize + \item Arguments of a runtime representable type (\eg bits or vectors). + + These arguments can be preserved in the program, since they can + be translated to input ports later on. However, since we can + only connect signals to input ports, these arguments must be + reduced to simple variables (for which signals will be + produced). This is taken care of by the argument extraction + transform. + \item Non-runtime representable typed arguments. + + These arguments cannot be preserved in the program, since we + cannot represent them as input or output ports in the resulting + \small{VHDL}. To remove them, we create a specialized version of the + called function with these arguments filled in. This is done by + the argument propagation transform. + + Typically, these arguments are type and dictionary arguments that are + used to make functions polymorphic. By propagating these arguments, we + are essentially doing the same which GHC does when it specializes + functions: Creating multiple variants of the same function, one for + each type for which it is used. Other common non-representable + arguments are functions, e.g. when calling a higher order function + with another function or a lambda abstraction as an argument. + + The reason for doing this is similar to the reasoning provided for + the inlining of non-representable let bindings above. In fact, this + argument propagation could be viewed as a form of cross-function + inlining. + \stopitemize + + TODO: Check the following itemization. + + When looking at the arguments of a builtin function, we can divide them + into categories: + + \startitemize + \item Arguments of a runtime representable type. + + As we have seen with user-defined functions, these arguments can + always be reduced to a simple variable reference, by the + argument extraction transform. Performing this transform for + builtin functions as well, means that the translation of builtin + functions can be limited to signal references, instead of + needing to support all possible expressions. + + \item Arguments of a function type. + + These arguments are functions passed to higher order builtins, + like \lam{map} and \lam{foldl}. Since implementing these + functions for arbitrary function-typed expressions (\eg, lambda + expressions) is rather comlex, we reduce these arguments to + (partial applications of) global functions. + + We can still support arbitrary expressions from the user code, + by creating a new global function containing that expression. + This way, we can simply replace the argument with a reference to + that new function. However, since the expression can contain any + number of free variables we also have to include partial + applications in our normal form. + + This category of arguments is handled by the function extraction + transform. + \item Other unrepresentable arguments. + + These arguments can take a few different forms: + \startdesc{Type arguments} + In the core language, type arguments can only take a single + form: A type wrapped in the Type constructor. Also, there is + nothing that can be done with type expressions, except for + applying functions to them, so we can simply leave type + arguments as they are. + \stopdesc + \startdesc{Dictionary arguments} + In the core language, dictionary arguments are used to find + operations operating on one of the type arguments (mostly for + finding class methods). Since we will not actually evaluatie + the function body for builtin functions and can generate + code for builtin functions by just looking at the type + arguments, these arguments can be ignored and left as they + are. + \stopdesc + \startdesc{Type level arguments} + Sometimes, we want to pass a value to a builtin function, but + we need to know the value at compile time. Additionally, the + value has an impact on the type of the function. This is + encoded using type-level values, where the actual value of the + argument is not important, but the type encodes some integer, + for example. Since the value is not important, the actual form + of the expression does not matter either and we can leave + these arguments as they are. + \stopdesc + \startdesc{Other arguments} + Technically, there is still a wide array of arguments that can + be passed, but does not fall into any of the above categories. + However, none of the supported builtin functions requires such + an argument. This leaves use with passing unsupported types to + a function, such as calling \lam{head} on a list of functions. + + In these cases, it would be impossible to generate hardware + for such a function call anyway, so we can ignore these + arguments. + + The only way to generate hardware for builtin functions with + arguments like these, is to expand the function call into an + equivalent core expression (\eg, expand map into a series of + function applications). But for now, we choose to simply not + support expressions like these. + \stopdesc + + From the above, we can conclude that we can simply ignore these + other unrepresentable arguments and focus on the first two + categories instead. + \stopitemize + + \subsubsection{Argument simplification} + This transform deals with arguments to functions that + are of a runtime representable type. It ensures that they will all become + references to global variables, or local signals in the resulting \small{VHDL}. + + TODO: It seems we can map an expression to a port, not only a signal. + Perhaps this makes this transformation not needed? + TODO: Say something about dataconstructors (without arguments, like True + or False), which are variable references of a runtime representable + type, but do not result in a signal. + + To reduce a complex expression to a simple variable reference, we create + a new let expression around the application, which binds the complex + expression to a new variable. The original function is then applied to + this variable. + + \starttrans + M N + -------------------- \lam{N} is of a representable type + letrec x = N in M x \lam{N} is not a local variable reference + \stoptrans + + \startbuffer[from] + add (add a 1) 1 + \stopbuffer + + \startbuffer[to] + letrec x = add a 1 in add x 1 + \stopbuffer + + \transexample{Argument extraction}{from}{to} + + \subsubsection{Function extraction} + This transform deals with function-typed arguments to builtin functions. + Since these arguments cannot be propagated, we choose to extract them + into a new global function instead. + + Any free variables occuring in the extracted arguments will become + parameters to the new global function. The original argument is replaced + with a reference to the new function, applied to any free variables from + the original argument. + + This transformation is useful when applying higher order builtin functions + like \hs{map} to a lambda abstraction, for example. In this case, the code + that generates \small{VHDL} for \hs{map} only needs to handle top level functions and + partial applications, not any other expression (such as lambda abstractions or + even more complicated expressions). + + \starttrans + M N \lam{M} is a (partial aplication of a) builtin function. + --------------------- \lam{f0 ... fn} = free local variables of \lam{N} + M (x f0 ... fn) \lam{N :: a -> b} + ~ \lam{N} is not a (partial application of) a top level function + x = λf0 ... λfn.N + \stoptrans + + \startbuffer[from] + map (λa . add a b) xs + + map (add b) ys + \stopbuffer + + \startbuffer[to] + map (x0 b) xs + + map x1 ys + ~ + x0 = λb.λa.add a b + x1 = λb.add b + \stopbuffer + + \transexample{Function extraction}{from}{to} + + Note that \lam{x0} and {x1} will still need normalization after this. + + \subsubsection{Argument propagation} + This transform deals with arguments to user-defined functions that are + not representable at runtime. This means these arguments cannot be + preserved in the final form and most be {\em propagated}. + + Propagation means to create a specialized version of the called + function, with the propagated argument already filled in. As a simple + example, in the following program: + + \startlambda + f = λa.λb.a + b + inc = λa.f a 1 + \stoplambda + + We could {\em propagate} the constant argument 1, with the following + result: + + \startlambda + f' = λa.a + 1 + inc = λa.f' a + \stoplambda + + Special care must be taken when the to-be-propagated expression has any + free variables. If this is the case, the original argument should not be + removed alltogether, but replaced by all the free variables of the + expression. In this way, the original expression can still be evaluated + inside the new function. Also, this brings us closer to our goal: All + these free variables will be simple variable references. + + To prevent us from propagating the same argument over and over, a simple + local variable reference is not propagated (since is has exactly one + free variable, itself, we would only replace that argument with itself). + + This shows that any free local variables that are not runtime representable + cannot be brought into normal form by this transform. We rely on an + inlining transformation to replace such a variable with an expression we + can propagate again. + + \starttrans + x = E + ~ + x Y0 ... Yi ... Yn \lam{Yi} is not of a runtime representable type + --------------------------------------------- \lam{Yi} is not a local variable reference + x' y0 ... yi-1 f0 ... fm Yi+1 ... Yn \lam{f0 ... fm} = free local vars of \lam{Yi} + ~ + x' = λy0 ... yi-1 f0 ... fm yi+1 ... yn . + E y0 ... yi-1 Yi yi+1 ... yn + + \stoptrans + + TODO: Example + + \subsection{Case simplification} + \subsubsection{Scrutinee simplification} + This transform ensures that the scrutinee of a case expression is always + a simple variable reference. + + \starttrans + case E of + alts + ----------------- \lam{E} is not a local variable reference + letrec x = E in + case E 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{Let flattening}{from}{to} + + + \subsubsection{Case simplification} + This transformation ensures that all case expressions become normal form. This + means they will become one of: + \startitemize + \item An extractor case with a single alternative that picks a single field + from a datatype, \eg \lam{case x of (a, b) -> a}. + \item A selector case with multiple alternatives and only wild binders, that + makes a choice between expressions based on the constructor of another + expression, \eg \lam{case x of Low -> a; High -> b}. + \stopitemize + + \starttrans + case E of + C0 v0,0 ... v0,m -> E0 + \vdots + Cn vn,0 ... vn,m -> En + --------------------------------------------------- \forall i \forall j, 0 <= i <= n, 0 <= i < m (\lam{wi,j} is a wild (unused) binder) + letrec + v0,0 = case x of C0 v0,0 .. v0,m -> v0,0 + \vdots + v0,m = case x of C0 v0,0 .. v0,m -> v0,m + x0 = E0 + \dots + vn,m = case x of Cn vn,0 .. vn,m -> vn,m + xn = En + in + case E of + C0 w0,0 ... w0,m -> x0 + \vdots + Cn wn,0 ... wn,m -> xn + \stoptrans + + TODO: This transformation specified like this is complicated and misses + conditions to prevent looping with itself. Perhaps we should split it here for + discussion? + + \startbuffer[from] + case a of + True -> add b 1 + False -> add b 2 + \stopbuffer + + \startbuffer[to] + letnonrec + x0 = add b 1 + x1 = add b 2 + in case a of - 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) -> + True -> x0 + False -> x1 + \stopbuffer + + \transexample{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 - 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. + (,) w0 w1 -> x0 + \stopbuffer + + \transexample{Extractor case simplification}{from}{to} + + \subsubsection{Case removal} + This transform removes any case statements with a single alternative and + only wild binders. + + These "useless" case statements are usually leftovers from case simplification + on extractor case (see the previous example). + + \starttrans + case x of + C v0 ... vm -> E + ---------------------- \lam{\forall i, 0 <= i <= m} (\lam{vi} does not occur free in E) + E + \stoptrans + + \startbuffer[from] + case a of + (,) w0 w1 -> x0 + \stopbuffer + + \startbuffer[to] + x0 + \stopbuffer + + \transexample{Case removal}{from}{to} + + \subsection{Removing polymorphism} + Reference type-specialization (== argument propagation) + + Reference polymporphic binding inlining (== non-representable binding + inlining). + + \subsection{Defunctionalization} + These transformations remove most higher order expressions from our + program, making it completely first-order (the only exception here is for + arguments to builtin functions, since we can't specialize builtin + function. TODO: Talk more about this somewhere). + + Reference higher-order-specialization (== argument propagation) + + \subsubsection{Non-representable binding inlining} + This transform inlines let bindings that have a non-representable type. Since + we can never generate a signal assignment for these bindings (we cannot + declare a signal assignment with a non-representable type, for obvious + reasons), we have no choice but to inline the binding to remove it. + + If the binding is non-representable because it is a lambda abstraction, it is + likely that it will inlined into an application and β-reduction will remove + the lambda abstraction and turn it into a representable expression at the + inline site. The same holds for partial applications, which can be turned into + full applications by inlining. + + Other cases of non-representable bindings we see in practice are primitive + Haskell types. In most cases, these will not result in a valid normalized + output, but then the input would have been invalid to start with. There is one + exception to this: When a builtin function is applied to a non-representable + expression, things might work out in some cases. For example, when you write a + literal \hs{SizedInt} in Haskell, like \hs{1 :: SizedInt D8}, this results in + the following core: \lam{fromInteger (smallInteger 10)}, where for example + \lam{10 :: GHC.Prim.Int\#} and \lam{smallInteger 10 :: Integer} have + non-representable types. TODO: This/these paragraph(s) should probably become a + separate discussion somewhere else. + + + \starttrans + letrec + a0 = E0 + \vdots + ai = Ei + \vdots + an = En + in + M + -------------------------- \lam{Ei} has a non-representable type. + letrec + a0 = E0 [Ei/ai] + \vdots + ai-1 = Ei-1 [Ei/ai] + ai+1 = Ei+1 [Ei/ai] + \vdots + an = En [Ei/ai] + in + M[Ei/ai] + \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{None representable binding inlining}{from}{to} + + + \section{Provable properties} + When looking at the system of transformations outlined above, there are a + number of questions that we can ask ourselves. The main question is of course: + \quote{Does our system work as intended?}. We can split this question into a + number of subquestions: + + \startitemize[KR] + \item[q:termination] Does our system \emph{terminate}? Since our system will + keep running as long as transformations apply, there is an obvious risk that + it will keep running indefinitely. One transformation produces a result that + is transformed back to the original by another transformation, for example. + \item[q:soundness] Is our system \emph{sound}? Since our transformations + continuously modify the expression, there is an obvious risk that the final + normal form will not be equivalent to the original program: Its meaning could + have changed. + \item[q:completeness] Is our system \emph{complete}? Since we have a complex + system of transformations, there is an obvious risk that some expressions will + not end up in our intended normal form, because we forgot some transformation. + In other words: Does our transformation system result in our intended normal + form for all possible inputs? + \item[q:determinism] Is our system \emph{deterministic}? Since we have defined + no particular order in which the transformation should be applied, there is an + obvious risk that different transformation orderings will result in + \emph{different} normal forms. They might still both be intended normal forms + (if our system is \emph{complete}) and describe correct hardware (if our + system is \emph{sound}), so this property is less important than the previous + three: The translator would still function properly without it. + \stopitemize + + \subsection{Graph representation} + Before looking into how to prove these properties, we'll look at our + transformation system from a graph perspective. The nodes of the graph are + all possible Core expressions. The (directed) edges of the graph are + transformations. When a transformation α applies to an expression \lam{A} to + produce an expression \lam{B}, we add an edge from the node for \lam{A} to the + node for \lam{B}, labeled α. + + \startuseMPgraphic{TransformGraph} + save a, b, c, d; + + % Nodes + newCircle.a(btex \lam{(λx.λy. (+) x y) 1} etex); + newCircle.b(btex \lam{λy. (+) 1 y} etex); + newCircle.c(btex \lam{(λx.(+) x) 1} etex); + newCircle.d(btex \lam{(+) 1} etex); + + b.c = origin; + c.c = b.c + (4cm, 0cm); + a.c = midpoint(b.c, c.c) + (0cm, 4cm); + d.c = midpoint(b.c, c.c) - (0cm, 3cm); + + % β-conversion between a and b + ncarc.a(a)(b) "name(bred)"; + ObjLabel.a(btex $\xrightarrow[normal]{}{β}$ etex) "labpathname(bred)", "labdir(rt)"; + ncarc.b(b)(a) "name(bexp)", "linestyle(dashed withdots)"; + ObjLabel.b(btex $\xleftarrow[normal]{}{β}$ etex) "labpathname(bexp)", "labdir(lft)"; + + % η-conversion between a and c + ncarc.a(a)(c) "name(ered)"; + ObjLabel.a(btex $\xrightarrow[normal]{}{η}$ etex) "labpathname(ered)", "labdir(rt)"; + ncarc.c(c)(a) "name(eexp)", "linestyle(dashed withdots)"; + ObjLabel.c(btex $\xleftarrow[normal]{}{η}$ etex) "labpathname(eexp)", "labdir(lft)"; + + % η-conversion between b and d + ncarc.b(b)(d) "name(ered)"; + ObjLabel.b(btex $\xrightarrow[normal]{}{η}$ etex) "labpathname(ered)", "labdir(rt)"; + ncarc.d(d)(b) "name(eexp)", "linestyle(dashed withdots)"; + ObjLabel.d(btex $\xleftarrow[normal]{}{η}$ etex) "labpathname(eexp)", "labdir(lft)"; + + % β-conversion between c and d + ncarc.c(c)(d) "name(bred)"; + ObjLabel.c(btex $\xrightarrow[normal]{}{β}$ etex) "labpathname(bred)", "labdir(rt)"; + ncarc.d(d)(c) "name(bexp)", "linestyle(dashed withdots)"; + ObjLabel.d(btex $\xleftarrow[normal]{}{β}$ etex) "labpathname(bexp)", "labdir(lft)"; + + % Draw objects and lines + drawObj(a, b, c, d); + \stopuseMPgraphic + + \placeexample[right][ex:TransformGraph]{Partial graph of a labmda calculus + system with β and η reduction (solid lines) and expansion (dotted lines).} + \boxedgraphic{TransformGraph} + + Of course our graph is unbounded, since we can construct an infinite amount of + Core expressions. Also, there might potentially be multiple edges between two + given nodes (with different labels), though seems unlikely to actually happen + in our system. + + See \in{example}[ex:TransformGraph] for the graph representation of a very + simple lambda calculus that contains just the expressions \lam{(λx.λy. (+) x + y) 1}, \lam{λy. (+) 1 y}, \lam{(λx.(+) x) 1} and \lam{(+) 1}. The + transformation system consists of β-reduction and η-reduction (solid edges) or + β-reduction and η-reduction (dotted edges). + + TODO: Define β-reduction and η-reduction? + + Note that the normal form of such a system consists of the set of nodes + (expressions) without outgoing edges, since those are the expression to which + no transformation applies anymore. We call this set of nodes the \emph{normal + set}. + + From such a graph, we can derive some properties easily: + \startitemize[KR] + \item A system will \emph{terminate} if there is no path of infinite length + in the graph (this includes cycles). + \item Soundness is not easily represented in the graph. + \item A system is \emph{complete} if all of the nodes in the normal set have + the intended normal form. The inverse (that all of the nodes outside of + the normal set are \emph{not} in the intended normal form) is not + strictly required. + \item A system is deterministic if all paths from a node, which end in a node + in the normal set, end at the same node. + \stopitemize + + When looking at the \in{example}[ex:TransformGraph], we see that the system + terminates for both the reduction and expansion systems (but note that, for + expansion, this is only true because we've limited the possible expressions! + In comlete lambda calculus, there would be a path from \lam{(λx.λy. (+) x y) + 1} to \lam{(λx.λy.(λz.(+) z) x y) 1} to \lam{(λx.λy.(λz.(λq.(+) q) z) x y) 1} + etc.) + + If we would consider the system with both expansion and reduction, there would + no longer be termination, since there would be cycles all over the place. + + The reduction and expansion systems have a normal set of containing just + \lam{(+) 1} or \lam{(λx.λy. (+) x y) 1} respectively. Since all paths in + either system end up in these normal forms, both systems are \emph{complete}. + Also, since there is only one normal form, it must obviously be + \emph{deterministic} as well. + + \subsection{Termination} + Approach: Counting. + + Church-Rosser? + + \subsection{Soundness} + Needs formal definition of semantics. + Prove for each transformation seperately, implies soundness of the system. + + \subsection{Completeness} + Show that any transformation applies to every Core expression that is not + in normal form. To prove: no transformation applies => in intended form. + Show the reverse: Not in intended form => transformation applies. + + \subsection{Determinism} + How to prove this?