X-Git-Url: https://git.stderr.nl/gitweb?p=matthijs%2Fmaster-project%2Freport.git;a=blobdiff_plain;f=Chapters%2FNormalization.tex;h=d564d231405a00ab7f31622ef525b15f987c2218;hp=61170088d4b13a23166d7a7683628629f7442351;hb=d92aa4307ca45f07c6ae50056e08ffc874839756;hpb=f276067e08b153e130b305242a4e2ef87e84960a diff --git a/Chapters/Normalization.tex b/Chapters/Normalization.tex index 6117008..d564d23 100644 --- a/Chapters/Normalization.tex +++ b/Chapters/Normalization.tex @@ -24,39 +24,40 @@ The first step in the core to \small{VHDL} translation process, is normalization. We aim to bring the core description into a simpler form, which we can subsequently translate into \small{VHDL} easily. This normal form is needed because - the full core language is more expressive than \small{VHDL} in some areas and because - core can describe expressions that do not have a direct hardware - interpretation. + the full core language is more expressive than \small{VHDL} in some + areas (higher-order expressions, limited polymorphism using type + classes, etc.) and because core can describe expressions that do not + have a direct hardware interpretation. \section{Normal form} The transformations described here have a well-defined goal: To bring the - program in a well-defined form that is directly translatable to hardware, - while fully preserving the semantics of the program. We refer to this form as - the \emph{normal form} of the program. The formal definition of this normal - form is quite simple: + program in a well-defined form that is directly translatable to + \VHDL, while fully preserving the semantics of the program. We refer + to this form as the \emph{normal form} of the program. The formal + definition of this normal form is quite simple: - \placedefinition{}{A program is in \emph{normal form} if none of the - transformations from this chapter apply.} + \placedefinition{}{\startboxed A program is in \emph{normal form} if none of the + transformations from this chapter apply.\stopboxed} Of course, this is an \quote{easy} definition of the normal form, since our program will end up in normal form automatically. The more interesting part is to see if this normal form actually has the properties we would like it to have. - But, before getting into more definitions and details about this normal form, - let's try to get a feeling for it first. The easiest way to do this is by - describing the things we want to not have in a normal form. + But, before getting into more definitions and details about this normal + form, let us try to get a feeling for it first. The easiest way to do this + is by describing the things that are unwanted in the intended normal form. \startitemize \item Any \emph{polymorphism} must be removed. When laying down hardware, we - can't generate any signals that can have multiple types. All types must be + cannot generate any signals that can have multiple types. All types must be completely known to generate hardware. - \item Any \emph{higher order} constructions must be removed. We can't + \item All \emph{higher-order} constructions must be removed. We cannot generate a hardware signal that contains a function, so all values, - arguments and returns values used must be first order. + arguments and return values used must be first order. - \item Any complex \emph{nested scopes} must be removed. In the \small{VHDL} + \item All complex \emph{nested scopes} must be removed. In the \small{VHDL} description, every signal is in a single scope. Also, full expressions are not supported everywhere (in particular port maps can only map signal names and constants, not complete expressions). To make the \small{VHDL} @@ -68,13 +69,13 @@ A very simple example of a program in normal form is given in \in{example}[ex:MulSum]. As you can see, all arguments to the function (which - will become input ports in the final hardware) are at the outer level. + 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 final hardware, bound to the output port + 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 @@ -101,7 +102,7 @@ newCircle.a(btex $a$ etex) "framed(false)"; newCircle.b(btex $b$ etex) "framed(false)"; newCircle.c(btex $c$ etex) "framed(false)"; - newCircle.sum(btex $res$ etex) "framed(false)"; + newCircle.sum(btex $sum$ etex) "framed(false)"; % Components newCircle.mul(btex * etex); @@ -131,12 +132,13 @@ {\boxedgraphic{MulSum}}{The architecture described by the normal form.} \stopcombination - The previous example described composing an architecture by calling other - functions (operators), resulting in a simple architecture with components and - connections. There is of course also some mechanism for choice in the normal - form. In a normal Core program, the \emph{case} expression can be used in a - few different ways to describe choice. In normal form, this is limited to a - very specific form. + \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 @@ -145,6 +147,7 @@ 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 @@ -199,15 +202,16 @@ {\boxedgraphic{AddSubAlu}}{The architecture described by the normal form.} \stopcombination - As a more complete example, consider \in{example}[ex:NormalComplete]. This - example contains everything that is supported in normal form, with the - exception of builtin higher order functions. The graphical version of the - architecture contains a slightly simplified version, since the state tuple - packing and unpacking have been left out. Instead, two seperate registers are - drawn. Also note that most synthesis tools will further optimize this - architecture by removing the multiplexers at the register input and - instead put some gates in front of the register's clock input, but we want - to show the architecture as close to the description as possible. + As a more complete example, consider + \in{example}[ex:NormalComplete]. This example shows everything that + is allowed in normal form, except for built-in higher-order functions + (like \lam{map}). The graphical version of the architecture contains + a slightly simplified version, since the state tuple packing and + unpacking have been left out. Instead, two separate registers are + drawn. Most synthesis tools will further optimize this architecture by + removing the multiplexers at the register input and instead use the write + enable port of the register (when it is available), but we want to show + the architecture as close to the description as possible. As you can see from the previous examples, the generation of the final architecture from the normal form is straightforward. In each of the @@ -315,58 +319,73 @@ \subsection[sec:normalization:intendednormalform]{Intended normal form definition} Now we have some intuition for the normal form, we can describe how we want the normal form to look like in a slightly more formal manner. The following - EBNF-like description completely captures the intended structure (and - generates a subset of GHC's core format). - - Some clauses have an expression listed in parentheses. These are conditions - that need to apply to the clause. + 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}. - \defref{intended normal form definition} \todo{Fix indentation} - \startlambda + \startbuffer[IntendedNormal] \italic{normal} := \italic{lambda} - \italic{lambda} := λvar.\italic{lambda} (representable(var)) + \italic{lambda} := λvar.\italic{lambda} (representable(var)) | \italic{toplet} - \italic{toplet} := letrec [\italic{binding}...] in var (representable(var)) - \italic{binding} := var = \italic{rhs} (representable(rhs)) + \italic{toplet} := letrec [\italic{binding}...] in var (representable(var)) + \italic{binding} := var = \italic{rhs} (representable(rhs)) -- State packing and unpacking by coercion - | var0 = var1 ▶ State ty (lvar(var1)) - | var0 = var1 ▶ ty (var1 :: State ty ∧ lvar(var1)) - \italic{rhs} := userapp - | builtinapp + | var0 = var1 ▶ State ty (lvar(var1)) + | var0 = var1 ▶ ty (var1 :: State ty ∧ lvar(var1)) + \italic{rhs} := \italic{userapp} + | \italic{builtinapp} -- Extractor case - | case var of C a0 ... an -> ai (lvar(var)) + | case var of C a0 ... an -> ai (lvar(var)) -- Selector case - | case var of (lvar(var)) - [ DEFAULT -> var ] (lvar(var)) + | case var of (lvar(var)) + [ DEFAULT -> var ] (lvar(var)) C0 w0,0 ... w0,n -> var0 \vdots - Cm wm,0 ... wm,n -> varm (\forall{}i \forall{}j, wi,j \neq vari, lvar(vari)) + Cm wm,0 ... wm,n -> varm (\forall{}i \forall{}j, wi,j \neq vari, lvar(vari)) \italic{userapp} := \italic{userfunc} | \italic{userapp} {userarg} - \italic{userfunc} := var (gvar(var)) - \italic{userarg} := var (lvar(var)) + \italic{userfunc} := var (gvar(var)) + \italic{userarg} := var (lvar(var)) \italic{builtinapp} := \italic{builtinfunc} | \italic{builtinapp} \italic{builtinarg} - \italic{builtinfunc} := var (bvar(var)) - \italic{builtinarg} := var (representable(var) ∧ lvar(var)) - | \italic{partapp} (partapp :: a -> b) - | \italic{coreexpr} (¬representable(coreexpr) ∧ ¬(coreexpr :: a -> b)) - \italic{partapp} := \italic{userapp} | \italic{builtinapp} - \stoplambda - - \todo{There can still be other casts around (which the code can handle, - e.g., ignore), which still need to be documented here} + \italic{built-infunc} := var (bvar(var)) + \italic{built-inarg} := var (representable(var) ∧ lvar(var)) + | \italic{partapp} (partapp :: a -> b) + | \italic{coreexpr} (¬representable(coreexpr) ∧ ¬(coreexpr :: a -> b)) + \italic{partapp} := \italic{userapp} + | \italic{builtinapp} + \stopbuffer - When looking at such a program from a hardware perspective, the top level - lambda's define the input ports. The variable reference in the body of - the recursive let expression is the output port. Most function - applications bound by the let expression define a component - instantiation, where the input and output ports are mapped to local - signals or arguments. Some of the others use a builtin construction (\eg - the \lam{case} expression) or call a builtin function (\eg \lam{+} or - \lam{map}). For these, a hardcoded \small{VHDL} translation is - available. + \placedefinition[][def:IntendedNormal]{Definition of the intended 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 @@ -379,14 +398,14 @@ ~ -------------------------- - + ~ \stoptrans - This format desribes a transformation that applies to \lam{} and transforms it into \lam{}, assuming - that all conditions apply. In this format, there are a number of placeholders + 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: @@ -394,9 +413,9 @@ 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 + the expression it matches. It is convention to use an uppercase letter (\eg\ \lam{M} or \lam{E}) to refer to any expression (including a simple variable - reference) and lowercase letters (\eg \lam{v} or \lam{b}) to refer to + reference) and lowercase letters (\eg\ \lam{v} or \lam{b}) to refer to (references to) binders. For example, the pattern \lam{a + B} will match the expression @@ -410,7 +429,7 @@ transformation applies, commonly to prevent a transformation from causing a loop with itself or another transformation. - Only if these conditions are \emph{all} true, the transformation + Only if these conditions are \emph{all} satisfied, the transformation applies. \stopdesc @@ -431,7 +450,7 @@ \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 with the \lam{}. + expression>} 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 @@ -442,8 +461,9 @@ \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 functions. + These are templates for new functions to be added to the context. + This is a way to let a transformation create new top level + functions. Each addition has the form \lam{binder = template}. As above, any placeholder in the addition is replaced with the value bound to it, and any @@ -451,7 +471,10 @@ replaced with) a fresh binder. \stopdesc - As an example, we'll look at η-abstraction: + To understand this notation better, the step by step application of + the η-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} @@ -466,12 +489,13 @@ transformation does not apply infinitely (which are not necessarily part of the conventional definition of η-abstraction). - Consider the following function, which is a fairly obvious way to specify a - simple ALU (Note that \in{example}[ex:AddSubAlu] shows the normal form of this - function). The parentheses around the \lam{+} and \lam{-} operators are - commonly used in Haskell to show that the operators are used as normal - functions, instead of \emph{infix} operators (\eg, the operators appear - before their arguments, instead of in between). + Consider the following function, in Core notation, which is a fairly obvious way to specify a + simple \small{ALU} (Note that it is not yet in normal form, but + \in{example}[ex:AddSubAlu] shows the normal form of this function). + The parentheses around the \lam{+} and \lam{-} operators are + commonly used in Haskell to show that the operators are used as + normal functions, instead of \emph{infix} operators (\eg, the + operators appear before their arguments, instead of in between). \startlambda alu :: Bit -> Word -> Word -> Word @@ -521,7 +545,7 @@ By now, the placeholder \lam{E} is bound to the entire expression. The placeholder \lam{x}, which occurs in the replacement template, is not bound - yet, so we need to generate a fresh binder for that. Let's use the binder + yet, so we need to generate a fresh binder for that. Let us use the binder \lam{a}. This results in the following replacement expression: \startlambda @@ -541,8 +565,8 @@ \stoplambda Here, the transformation does apply, binding \lam{E} to the entire - expression and \lam{x} to the fresh binder \lam{b}, resulting in the - replacement: + expression (which has type \lam{Word -> Word}) and binding \lam{x} + to the fresh binder \lam{b}, resulting in the replacement: \startlambda λb.(case opcode of @@ -550,8 +574,8 @@ 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 + 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 @@ -573,7 +597,7 @@ function position (which makes the second condition false). In the same way the transformation does not apply to both components of this expression (\lam{case opcode of Low -> (+); High -> (-)} and \lam{a}), so - we'll skip to the components of the case expression: The scrutinee and + we will skip to the components of the case expression: The scrutinee and both alternatives. Since the opcode is not a function, it does not apply here. @@ -636,49 +660,45 @@ keep the transformation descriptions concise and powerful. \subsection{Definitions} - In the following sections, we will be using a number of functions and - notations, which we will define here. - - \subsubsection{Concepts} - A \emph{global variable} is any variable (binder) that is bound at the - top level of a program, or an external module. A \emph{local variable} is any - other variable (\eg, variables local to a function, which can be bound by - lambda abstractions, let expressions and pattern matches of case - alternatives). Note that this is a slightly different notion of global versus - local than what \small{GHC} uses internally. - \defref{global variable} \defref{local variable} - - A \emph{hardware representable} (or just \emph{representable}) type or value - is (a value of) a type that we can generate a signal for in hardware. For - example, a bit, a vector of bits, a 32 bit unsigned word, etc. Values that are - not runtime representable notably include (but are not limited to): Types, - dictionaries, functions. - \defref{representable} - - A \emph{builtin function} is a function supplied by the Cλash framework, whose - implementation is not valid Cλash. The implementation is of course valid - Haskell, for simulation, but it is not expressable in Cλash. - \defref{builtin function} \defref{user-defined function} - - For these functions, Cλash has a \emph{builtin hardware translation}, so calls + 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{Predicates} + \subsubsection[sec:normalization:predicates]{Predicates} Here, we define a number of predicates that can be used below to concisely - specify conditions.\refdef{global variable} + specify conditions. \emph{gvar(expr)} is true when \emph{expr} is a variable that references a global variable. It is false when it references a local variable. - \refdef{local variable}\emph{lvar(expr)} is the complement of \emph{gvar}; it is true when \emph{expr} + \emph{lvar(expr)} is the complement of \emph{gvar}; it is true when \emph{expr} references a local variable, false when it references a global variable. - \refdef{representable}\emph{representable(expr)} or \emph{representable(var)} is true when - \emph{expr} or \emph{var} is \emph{representable}. + \emph{representable(expr)} is true when \emph{expr} is \emph{representable}. \subsection[sec:normalization:uniq]{Binder uniqueness} A common problem in transformation systems, is binder uniqueness. When not @@ -709,11 +729,17 @@ \stoplambda This is obviously not what was supposed to happen! The root of this problem is - the reuse of binders: Identical binders can be bound in different scopes, such - that only the inner one is \quote{visible} in the inner expression. In the example - above, the \lam{c} binder was bound outside of the expression and in the inner - lambda expression. Inside that lambda expression, only the inner \lam{c} is - visible. + the reuse of binders: Identical binders can be bound in different, + but overlapping scopes. Any variable reference in those + overlapping scopes then refers to the variable bound in the inner + (smallest) scope. There is not way to refer to the variable in the + outer scope. This effect is usually referred to as + \emph{shadowing}: When a binder is bound in a scope where the + binder already had a value, the inner binding is said to + \emph{shadow} the outer binding. In the example above, the \lam{c} + binder was bound outside of the expression and in the inner lambda + expression. Inside that lambda expression, only the inner \lam{c} + can be accessed. There are a number of ways to solve this. \small{GHC} has isolated this problem to their binder substitution code, which performs \emph{deshadowing} @@ -723,9 +749,9 @@ binder uniqueness problems in \small{GHC}. In our transformation system, maintaining this non-shadowing invariant is - a bit harder to do (mostly due to implementation issues, the prototype doesn't - use \small{GHC}'s subsitution code). Also, the following points can be - observed. + a bit harder to do (mostly due to implementation issues, the prototype + does not use \small{GHC}'s subsitution code). Also, the following points + can be observed. \startitemize \item Deshadowing does not guarantee overall uniqueness. For example, the @@ -821,7 +847,6 @@ \stopframedtext } - \defref{beta-reduction} \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, @@ -865,6 +890,58 @@ \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 @@ -879,11 +956,22 @@ M \stoptrans - \todo{Example} + % And an example + \startbuffer[from] + let + in + 2 + \stopbuffer + + \startbuffer[to] + 2 + \stopbuffer + + \transexample{emptylet}{Empty let removal}{from}{to} \subsubsection[sec:normalization:simplelet]{Simple let binding removal} This transformation inlines simple let bindings, that bind some - binder to some other binder instead of a more complex expression (\ie + binder to some other binder instead of a more complex expression (\ie\ a = b). This transformation is not needed to get an expression into intended @@ -914,60 +1002,48 @@ \todo{example} - \subsubsection{Unused let binding removal} - This transformation removes let bindings that are never used. - Occasionally, \GHC's desugarer introduces some unused let bindings. + \subsubsection{Cast propagation / simplification} + This transform pushes casts down into the expression as far as + possible. This transformation has been added to make a few + specific corner cases work, but it is not clear yet if this + transformation handles cast expressions completely or in the + right way. See \in{section}[sec:normalization:castproblems]. - This normalization pass should really be unneeded to get into intended normal form - (since unused bindings are not forbidden by the normal form), but in practice - the desugarer or simplifier emits some unused bindings that cannot be - normalized (e.g., calls to a \type{PatError}\todo{Check this name}). Also, - this transformation makes the resulting \small{VHDL} a lot shorter. + \starttrans + (let binds in E) ▶ T + ------------------------- + let binds in (E ▶ T) + \stoptrans - \todo{Don't use old-style numerals in transformations} \starttrans - letrec - a0 = E0 + (case S of + p0 -> E0 \vdots - ai = Ei + pn -> En + ) ▶ T + ------------------------- + case S of + p0 -> E0 ▶ T \vdots - an = En - in - M \lam{ai} does not occur free in \lam{M} - ---------------------------- \forall j, 0 ≤ j ≤ n, j ≠ i (\lam{ai} does not occur free in \lam{Ej}) - letrec - a0 = E0 - \vdots - ai-1 = Ei-1 - ai+1 = Ei+1 - \vdots - an = En - in - M + pn -> En ▶ T \stoptrans - \todo{Example} - - \subsubsection{Cast propagation / simplification} - This transform pushes casts down into the expression as far as possible. - Since its exact role and need is not clear yet, this transformation is - not yet specified. - - \todo{Cast propagation} - \subsubsection{Top level binding inlining} + \refdef{top level binding} This transform takes simple top level bindings generated by the \small{GHC} compiler. \small{GHC} sometimes generates very simple \quote{wrapper} bindings, which are bound to just a variable - reference, or a partial application to constants or other variable - references. + reference, or contain just a (partial) function 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 a bunch of - components that are really boring). + the resulting VHDL output easier to read (since it removes components + that do not add any real structure, but do hide away operations and + cause extra clutter). - This transform takes any top level binding generated by the compiler, + This transform takes any top level binding generated by \GHC, whose normalized form contains only a single let binding. \starttrans @@ -996,16 +1072,16 @@ \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 + in the \VHDL\ which would just add its inputs. This generates a lot of overhead in the \VHDL, which is particularly annoying when browsing the generated RTL schematic (especially since most non-alphanumerics, like all characters in \lam{(+)}, are not - allowed in \VHDL architecture names\footnote{Technically, it is + allowed in \VHDL\ architecture names\footnote{Technically, it is allowed to use non-alphanumerics when using extended identifiers, but it seems that none of the tooling likes - extended identifiers in filenames, so it effectively doesn't + extended identifiers in filenames, so it effectively does not work.}, so the entity would be called \quote{w7aA7f} or - something similarly unreadable and autogenerated). + something similarly meaningless and autogenerated). \subsection{Program structure} These transformations are aimed at normalizing the overall structure @@ -1022,9 +1098,8 @@ \starttrans E \lam{E :: a -> b} - -------------- \lam{E} is not the first argument of an application. + -------------- \lam{E} does not occur on a function position in an application λx.E x \lam{E} is not a lambda abstraction. - \lam{x} is a variable that does not occur free in \lam{E}. \stoptrans \startbuffer[from] @@ -1081,12 +1156,12 @@ \starttrans (case x of - p1 -> E1 + p0 -> E0 \vdots pn -> En) M ----------------- case x of - p1 -> E1 M + p0 -> E0 M \vdots pn -> En M \stoptrans @@ -1111,8 +1186,9 @@ This transformation makes all non-recursive lets recursive. In the end, we want a single recursive let in our normalized program, so all non-recursive lets can be converted. This also makes other - transformations simpler: They can simply assume all lets are - recursive. + transformations simpler: They only need to be specified for recursive + let expressions (and simply will not apply to non-recursive let + expressions until this transformation has been applied). \starttrans let @@ -1188,21 +1264,22 @@ This transformation ensures that the return value of a function is always a simple local variable reference. - Currently implemented using lambda simplification, let simplification, and - top simplification. Should change into something like the following, which - works only on the result of a function instead of any subexpression. This is - achieved by the contexts, like \lam{x = E}, though this is strictly not - correct (you could read this as "if there is any function \lam{x} that binds - \lam{E}, any \lam{E} can be transformed, while we only mean the \lam{E} that - is bound by \lam{x}. This might need some extra notes or something). - - Note that the return value is not simplified if its not representable. - Otherwise, this would cause a direct loop with the inlining of - unrepresentable bindings. If the return value is not - representable because it has a function type, η-abstraction should - make sure that this transformation will eventually apply. If the value - is not representable for other reasons, the function result itself is - not representable, meaning this function is not translatable anyway. + 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 @@ -1249,7 +1326,7 @@ This transformation ensures that all representable arguments will become references to local variables. This ensures they will become references to local signals in the resulting \small{VHDL}, which is required due to - limitations in the component instantiation code in \VHDL (one can only + limitations in the component instantiation code in \VHDL\ (one can only assign a signal or constant to an input port). By ensuring that all arguments are always simple variable references, we always have a signal available to map to the input ports. @@ -1264,7 +1341,7 @@ 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 + 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. @@ -1286,10 +1363,10 @@ \transexample{argsimpl}{Argument simplification}{from}{to} - \subsection[sec:normalization:builtins]{Builtin functions} - This section deals with (arguments to) builtin functions. In the + \subsection[sec:normalization:built-ins]{Built-in functions} + This section deals with (arguments to) built-in functions. In the intended normal form definition\refdef{intended normal form definition} - we can see that there are three sorts of arguments a builtin function + we can see that there are three sorts of arguments a built-in function can receive. \startitemize[KR] @@ -1297,9 +1374,9 @@ common argument to any function. The argument simplification transformation described in \in{section}[sec:normalization:argsimpl] makes sure that \emph{any} representable argument to \emph{any} - function (including builtin functions) is turned into a local variable + function (including built-in functions) is turned into a local variable reference. - \item (A partial application of) a top level function (either builtin on + \item (A partial application of) a top level function (either built-in on user-defined). The function extraction transformation described in this section takes care of turning every functiontyped argument into (a partial application of) a top level function. @@ -1308,25 +1385,25 @@ transformation needed. Note that this category is exactly all expressions that are not transformed by the transformations for the previous two categories. This means that \emph{any} core expression - that is used as an argument to a builtin function will be either + 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 builtin function. The following transformation is needed + arguments to a built-in function. The following transformation is needed to handle non-representable arguments with a function type, all other - non-representable arguments don't need any special handling. + non-representable arguments do not need any special handling. \subsubsection[sec:normalization:funextract]{Function extraction} - This transform deals with function-typed arguments to builtin + This transform deals with function-typed arguments to built-in functions. - Since builtin functions cannot be specialized (see + Since built-in functions cannot be specialized (see \in{section}[sec:normalization:specialize]) to remove the arguments, these arguments are extracted into a new global function instead. In other words, we create a new top level function that has exactly the extracted argument as its body. This greatly simplifies the - translation rules needed for builtin functions, since they only need + translation rules needed for built-in functions, since they only need to handle (partial applications of) top level functions. Any free variables occuring in the extracted arguments will become @@ -1334,14 +1411,14 @@ with a reference to the new function, applied to any free variables from the original argument. - This transformation is useful when applying higher order builtin functions + This transformation is useful when applying higher-order built-in functions like \hs{map} to a lambda abstraction, for example. In this case, the code that generates \small{VHDL} for \hs{map} only needs to handle top level functions and partial applications, not any other expression (such as lambda abstractions or even more complicated expressions). \starttrans - M N \lam{M} is (a partial aplication of) a builtin function. + M N \lam{M} is (a partial 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 @@ -1364,6 +1441,9 @@ 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. @@ -1373,7 +1453,7 @@ alts ----------------- \lam{E} is not a local variable reference letrec x = E in - case E of + case x of alts \stoptrans @@ -1393,51 +1473,79 @@ \transexample{letflat}{Case normalisation}{from}{to} - \subsubsection{Case simplification} - This transformation ensures that all case expressions become normal form. This - means they will become one of: + \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 single field - from a datatype, \eg \lam{case x of (a, b) -> a}. + \item An extractor case with a single alternative that picks a field + from a datatype, \eg\ \lam{case x of (a, b) -> a}. \item A selector case with multiple alternatives and only wild binders, that makes a choice between expressions based on the constructor of another - expression, \eg \lam{case x of Low -> a; High -> b}. + expression, \eg\ \lam{case x of Low -> a; High -> b}. \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. - \defref{wild binder} \starttrans case E of C0 v0,0 ... v0,m -> E0 \vdots Cn vn,0 ... vn,m -> En - --------------------------------------------------- \forall i \forall j, 0 ≤ i ≤ n, 0 ≤ i < m (\lam{wi,j} is a wild (unused) binder) - letrec - v0,0 = case E of C0 v0,0 .. v0,m -> v0,0 + --------------------------------------------------- \lam{\forall i \forall j, 0 ≤ i ≤ n, 0 ≤ i < m} (\lam{wi,j} is a wild (unused) binder) + letrec The case expression is not an extractor case + v0,0 = case E of C0 x0,0 .. x0,m -> x0,0 The case expression is not a selector case \vdots - v0,m = case E of C0 v0,0 .. v0,m -> v0,m + v0,m = case E of C0 x0,0 .. x0,m -> x0,m \vdots - vn,m = case E of Cn vn,0 .. vn,m -> vn,m - x0 = E0 + vn,m = case E of Cn xn,0 .. xn,m -> xn,m + y0 = E0 \vdots - xn = En + yn = En in case E of - C0 w0,0 ... w0,m -> x0 + C0 w0,0 ... w0,m -> y0 \vdots - Cn wn,0 ... wn,m -> xn + Cn wn,0 ... wn,m -> yn \stoptrans - \todo{Check the subscripts of this transformation} - - Note that this transformation applies to case statements with any - scrutinee. If the scrutinee is a complex expression, this might result - in duplicate hardware. An extra condition to only apply this - transformation when the scrutinee is already simple (effectively - causing this transformation to be only applied after the scrutinee - simplification transformation) might be in order. - \fxnote{This transformation specified like this is complicated and misses - conditions to prevent looping with itself. Perhaps it should be split here for - discussion?} + 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 @@ -1446,7 +1554,7 @@ \stopbuffer \startbuffer[to] - letnonrec + letrec x0 = add b 1 x1 = add b 2 in @@ -1481,10 +1589,10 @@ \in{section}[sec:transformation:caseremoval]. \subsubsection[sec:transformation:caseremoval]{Case removal} - This transform removes any case statements with a single alternative and - only wild binders. + This transform removes any case expression with a single alternative and + only wild binders.\refdef{wild binders} - These "useless" case statements are usually leftovers from case simplification + These "useless" case expressions are usually leftovers from case simplification on extractor case (see the previous example). \starttrans @@ -1510,14 +1618,14 @@ values used in our expression representable. There are two main transformations that are applied to \emph{all} unrepresentable let bindings and function arguments. These are meant to address three - different kinds of unrepresentable values: Polymorphic values, higher - order values and literals. The transformation are described generically: - They apply to all non-representable values. However, non-representable - values that don't fall into one of these three categories will be moved - around by these transformations but are unlikely to completely - disappear. They usually mean the program was not valid in the first - place, because unsupported types were used (for example, a program using - strings). + different kinds of unrepresentable values: Polymorphic values, + higher-order values and literals. The transformation are described + generically: They apply to all non-representable values. However, + non-representable values that do not fall into one of these three + categories will be moved around by these transformations but are + unlikely to completely disappear. They usually mean the program was not + valid in the first place, because unsupported types were used (for + example, a program using strings). Each of these three categories will be detailed below, followed by the actual transformations. @@ -1541,21 +1649,24 @@ take care of exactly this. There is one case where polymorphism cannot be completely - removed: Builtin functions are still allowed to be polymorphic + removed: Built-in functions are still allowed to be polymorphic (Since we have no function body that we could properly - specialize). However, the code that generates \VHDL for builtin + specialize). However, the code that generates \VHDL\ for built-in functions knows how to handle this, so this is not a problem. - \subsubsection{Defunctionalization} - These transformations remove higher order expressions from our - program, making all values first-order. + \subsubsection[sec:normalization:defunctionalization]{Defunctionalization} + These transformations remove higher-order expressions from our + program, making all values first-order. The approach used for + defunctionalization uses a combination of specialization, inlining and + some cleanup transformations, was also proposed in parallel research + by Neil Mitchell \cite[mitchell09]. Higher order values are always introduced by lambda abstractions, none of the other Core expression elements can introduce a function type. However, other expressions can \emph{have} a function type, when they have a lambda expression in their body. - For example, the following expression is a higher order expression + For example, the following expression is a higher-order expression that is not a lambda expression itself: \refdef{id function} @@ -1566,24 +1677,24 @@ \stoplambda The reference to the \lam{id} function shows that we can introduce a - higher order expression in our program without using a lambda + higher-order expression in our program without using a lambda expression directly. However, inside the definition of the \lam{id} function, we can be sure that a lambda expression is present. Looking closely at the definition of our normal form in \in{section}[sec:normalization:intendednormalform], we can see that - there are three possibilities for higher order values to appear in our + there are three possibilities for higher-order values to appear in our intended normal form: \startitemize[KR] \item[item:toplambda] Lambda abstractions can appear at the highest level of a top level function. These lambda abstractions introduce the arguments (input ports / current state) of the function. - \item[item:builtinarg] (Partial applications of) top level functions can appear as an - argument to a builtin function. + \item[item:built-inarg] (Partial applications of) top level functions can appear as an + argument to a built-in function. \item[item:completeapp] (Partial applications of) top level functions can appear in function position of an application. Since a partial application - cannot appear anywhere else (except as builtin function arguments), + cannot appear anywhere else (except as built-in function arguments), all partial applications are applied, meaning that all applications will become complete applications. However, since application of arguments happens one by one, in the expression: @@ -1594,7 +1705,7 @@ allowed, since it is inside a complete application. \stopitemize - We will take a typical function with some higher order values as an + We will take a typical function with some higher-order values as an example. The following function takes two arguments: a \lam{Bit} and a list of numbers. Depending on the first argument, each number in the list is doubled, or the list is returned unmodified. For the sake of @@ -1608,15 +1719,15 @@ High -> λz. z \stoplambda - This example shows a number of higher order values that we cannot - translate to \VHDL directly. The \lam{double} binder bound in the let + This example shows a number of higher-order values that we cannot + translate to \VHDL\ directly. The \lam{double} binder bound in the let expression has a function type, as well as both of the alternatives of the case expression. The first alternative is a partial application of - the \lam{map} builtin function, whereas the second alternative is a + the \lam{map} built-in function, whereas the second alternative is a lambda abstraction. - To reduce all higher order values to one of the above items, a number - of transformations we've already seen are used. The η-abstraction + To reduce all higher-order values to one of the above items, a number + of transformations we have already seen are used. The η-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 @@ -1645,7 +1756,7 @@ High -> (λz. z) q \stoplambda - This propagation makes higher order values become applied (in + This propagation makes higher-order values become applied (in particular both of the alternatives of the case now have a representable type). Completely applied top level functions (like the first alternative) are now no longer invalid (they fall under @@ -1661,11 +1772,11 @@ \stoplambda As you can see in our example, all of this moves applications towards - the higher order values, but misses higher order functions bound by + the higher-order values, but misses higher-order functions bound by let expressions. The applications cannot be moved towards these values (since they can be used in multiple places), so the values will have to be moved towards the applications. This is achieved by inlining all - higher order values bound by let applications, by the + higher-order values bound by let applications, by the non-representable binding inlining transformation below. When applying it to our example, we get the following: @@ -1675,9 +1786,9 @@ High -> q \stoplambda - We've nearly eliminated all unsupported higher order values from this - expressions. The one that's remaining is the first argument to the - \lam{map} function. Having higher order arguments to a builtin + We have nearly eliminated all unsupported higher-order values from this + expressions. The one that is remaining is the first argument to the + \lam{map} function. Having higher-order arguments to a built-in function like \lam{map} is allowed in the intended normal form, but only if the argument is a (partial application) of a top level function. This is easily done by introducing a new top level function @@ -1702,10 +1813,10 @@ intended normal form. There is one case that has not been discussed yet. What if the - \lam{map} function in the example above was not a builtin function + \lam{map} function in the example above was not a built-in function but a user-defined function? Then extracting the lambda expression into a new function would not be enough, since user-defined functions - can never have higher order arguments. For example, the following + can never have higher-order arguments. For example, the following expression shows an example: \startlambda @@ -1717,13 +1828,13 @@ This example shows a function \lam{twice} that takes a function as a first argument and applies that function twice to the second argument. - Again, we've made the function monomorphic for clarity, even though + Again, we have made the function monomorphic for clarity, even though this function would be a lot more useful if it was polymorphic. The function \lam{main} uses \lam{twice} to apply a lambda epression twice. When faced with a user defined function, a body is available for that function. This means we could create a specialized version of the - function that only works for this particular higher order argument + function that only works for this particular higher-order argument (\ie, we can just remove the argument and call the specialized function without the argument). This transformation is detailed below. Applying this transformation to the example gives: @@ -1735,13 +1846,13 @@ main = λa.app' a \stoplambda - The \lam{main} function is now in normal form, since the only higher - order value there is the top level lambda expression. The new - \lam{twice'} function is a bit complex, but the entire original body of - the original \lam{twice} function is wrapped in a lambda abstraction - and applied to the argument we've specialized for (\lam{λx. x + x}) - and the other arguments. This complex expression can fortunately be - effectively reduced by repeatedly applying β-reduction: + The \lam{main} function is now in normal form, since the only + higher-order value there is the top level lambda expression. The new + \lam{twice'} function is a bit complex, but the entire original body + of the original \lam{twice} function is wrapped in a lambda + abstraction and applied to the argument we have specialized for + (\lam{λx. x + x}) and the other arguments. This complex expression can + fortunately be effectively reduced by repeatedly applying β-reduction: \startlambda twice' :: Word -> Word @@ -1765,14 +1876,14 @@ representable type: Integer literals. Cλash supports using integer literals for all three integer types supported (\hs{SizedWord}, \hs{SizedInt} and \hs{RangedWord}). This is implemented using - Haskell's \hs{Num} typeclass, which offers a \hs{fromInteger} method + 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 + When \GHC\ sees integer literals, it will automatically insert calls to the \hs{fromInteger} method in the resulting Core expression. For example, the following expression in Haskell creates a 32 bit unsigned word with the value 1. The explicit type signature is needed, since - there is no context for \GHC to determine the type from otherwise. + there is no context for \GHC\ to determine the type from otherwise. \starthaskell 1 :: SizedWord D32 @@ -1791,7 +1902,7 @@ Both the \lam{GHC.Prim.Int\#} and \lam{Integer} types are not representable, and cannot be translated directly. Fortunately, there - is no need to translate them, since \lam{fromInteger} is a builtin + is no need to translate them, since \lam{fromInteger} is a built-in function that knows how to handle these values. However, this does require that the \lam{fromInteger} function is directly applied to these non-representable literal values, otherwise errors will occur. @@ -1805,7 +1916,7 @@ By inlining these let-bindings, we can ensure that unrepresentable literals bound by a let binding end up in an application of the - appropriate builtin function, where they are allowed. Since it is + appropriate built-in function, where they are allowed. Since it is possible that the application of that function is in a different function than the definition of the literal value, we will always need to specialize away any unrepresentable literals that are used as @@ -1819,7 +1930,7 @@ but to inline the binding to remove it. As we have seen in the previous sections, inlining these bindings - solves (part of) the polymorphism, higher order values and + solves (part of) the polymorphism, higher-order values and unrepresentable literals in an expression. \refdef{substitution notation} @@ -1913,10 +2024,12 @@ ~ x Y0 ... Yi ... Yn \lam{Yi} is not representable --------------------------------------------- \lam{Yi} is not a local variable reference - x' y0 ... yi-1 f0 ... fm Yi+1 ... Yn \lam{f0 ... fm} are all free local vars of \lam{Yi} + x' Y0 ... Yi-1 f0 ... fm Yi+1 ... Yn \lam{f0 ... fm} are all free local vars of \lam{Yi} ~ \lam{T0 ... Tn} are the types of \lam{Y0 ... Yn} - x' = λ(y0 :: T0) ... λ(yi-1 :: Ty-1). λf0 ... λfm. λ(yi+1 :: Ty+1) ... λ(yn :: Tn). - E y0 ... yi-1 Yi yi+1 ... yn + x' = λ(y0 :: T0) ... λ(yi-1 :: Ty-1). + λf0 ... λfm. + λ(yi+1 :: Ty+1) ... λ(yn :: Tn). + E y0 ... yi-1 Yi yi+1 ... yn \stoptrans This is a bit of a complex transformation. It transforms an @@ -1941,7 +2054,7 @@ from the types of the \emph{actual} arguments (\lam{T0 ... Tn}). This means that any polymorphism in the arguments is removed, even when the corresponding explicit type lambda is not removed - yet.\refdef{type lambda} + yet. \todo{Examples. Perhaps reference the previous sections} @@ -1985,8 +2098,8 @@ letrec x = M in E \stoptrans - This doesn't seem like much of an improvement, but it does get rid of - the lambda expression (and the associated higher order value), while + This does not seem like much of an improvement, but it does get rid of + the lambda expression (and the associated higher-order value), while at the same time introducing a new let binding. Since the result of every application or case expression must be bound by a let expression in the intended normal form anyway, this is probably not a problem. If @@ -2051,7 +2164,7 @@ transformation system as well. However, it is likely that there are other occurences of this problem. - \subsection{Casts} + \subsection[sec:normalization:castproblems]{Casts} We do not fully understand the use of cast expressions in Core, so there are probably expressions involving cast expressions that cannot be brought into intended normal form by this transformation system. @@ -2060,12 +2173,12 @@ transformations will probably need updating to handle them in all cases. - \subsection{Normalization of stateful descriptions} + \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 + 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). @@ -2079,17 +2192,18 @@ matching, causing a state input to be unpacked multiple times or be unpacked and repacked only in some of the code paths. - \todo{example?} - Without going into detail about the exact problems (of which there are probably more than have shown up so far), it seems unlikely that these problems can be solved entirely by just - improving the \VHDL state generation in the final stage. The + improving the \VHDL\ state generation in the final stage. The normalization stage seems the best place to apply the rewriting needed to support more complex stateful descriptions. This does of course mean that the intended normal form definition must be 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 @@ -2126,11 +2240,13 @@ developed in the final part of the research, leaving no more time for verifying these properties. In fact, it is likely that the current transformation system still violates some of these - properties in some cases and should be improved (or extra conditions - on the input hardware descriptions should be formulated). + properties in some cases (see + \in{section}[sec:normalization:non-determinism] and + \in{section}[sec:normalization:stateproblems]) and should be improved (or + extra conditions on the input hardware descriptions should be formulated). This is most likely the case with the completeness and determinism - properties, perhaps als the termination property. The soundness + properties, perhaps also the termination property. The soundness property probably holds, since it is easier to manually verify (each transformation can be reviewed separately). @@ -2138,12 +2254,15 @@ possible proof strategies are shown below. \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. + 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; @@ -2191,10 +2310,10 @@ 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. + 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 @@ -2205,7 +2324,7 @@ \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 + (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 @@ -2213,8 +2332,9 @@ From such a graph, we can derive some properties easily: \startitemize[KR] - \item A system will \emph{terminate} if there is no path of infinite length - in the graph (this includes cycles, but can also happen without cycles). + \item A system will \emph{terminate} if there is no walk (sequence of + edges, or transformations) of infinite length in the graph (this + includes cycles, but can also happen without cycles). \item Soundness is not easily represented in the graph. \item A system is \emph{complete} if all of the nodes in the normal set have the intended normal form. The inverse (that all of the nodes outside of @@ -2229,8 +2349,8 @@ When looking at the \in{example}[ex:TransformGraph], we see that the system terminates for both the reduction and expansion systems (but note that, for - expansion, this is only true because we've limited the possible - expressions. In comlete lambda calculus, there would be a path from + expansion, this is only true because we have limited the possible + expressions. In complete lambda calculus, there would be a path from \lam{(λx.λy. (+) x y) 1} to \lam{(λx.λy.(λz.(+) z) x y) 1} to \lam{(λx.λy.(λz.(λq.(+) q) z) x y) 1} etc.) @@ -2289,12 +2409,12 @@ To be able to formally show that each transformation properly preserves the meaning of every expression, we require an exact definition of the \emph{meaning} of every expression, so we can - compare them. Currently there seems to be no formal definition of - the meaning or semantics of \GHC's core language, only informal - descriptions are available. + compare them. A definition of the operational semantics of \GHC's Core + language is available \cite[sulzmann07], but this does not seem + sufficient for our goals (but it is a good start). It should be possible to have a single formal definition of - meaning for Core for both normal Core compilation by \GHC and for + meaning for Core for both normal Core compilation by \GHC\ and for our compilation to \VHDL. The main difference seems to be that in hardware every expression is always evaluated, while in software it is only evaluated if needed, but it should be possible to @@ -2323,7 +2443,7 @@ each node in the normal set is also in the intended normal set. Reasoning about our intended normal set is easier, since we know how to generate it from its definition. \refdef{intended normal - form definition}. + form definition} Fortunately, we can also prove the complement (which is equivalent, since $A \subseteq B \Leftrightarrow \overline{B} @@ -2343,7 +2463,7 @@ This approach is especially useful for proving completeness of our system, since if expressions exist to which none of the - transformations apply (\ie if the system is not yet complete), it + transformations apply (\ie\ if the system is not yet complete), it is immediately clear which expressions these are and adding (or modifying) transformations to fix this should be relatively easy.