X-Git-Url: https://git.stderr.nl/gitweb?p=matthijs%2Fmaster-project%2Freport.git;a=blobdiff_plain;f=Chapters%2FHardwareDescription.tex;h=9e4061c9a4d059ee7fd8182c9d1e04565ecf94df;hp=2f4545d469eb1d4f46bb2a794fa9c793339b7755;hb=a472d8d94908d8f466a4cafe4d55c4c9410161d8;hpb=af95dbcdd9ed866f9ae9100dfc82e3bc051ed54c diff --git a/Chapters/HardwareDescription.tex b/Chapters/HardwareDescription.tex index 2f4545d..9e4061c 100644 --- a/Chapters/HardwareDescription.tex +++ b/Chapters/HardwareDescription.tex @@ -76,6 +76,281 @@ and3 a b c = and (and a b) c TODO: Define top level function and subfunctions/circuits. + \section{Choice} + Since describing components and connections allows us to describe a lot of + hardware designs already, there is an obvious thing missing: Choice. We + need some way to be able to choose between values based on another value. + In Haskell, choice is achieved by case expressions, if expressions or + pattern matching (all of which can be reduced to case expressions). + + An obvious way to add choice to our language without having to recognize + any of Haskell's syntax, would be to add a primivite \quote{\hs{if}} + function. This function would take three arguments: The condition, the + value to return when the condition is true and the value to return when + the condition is false. + + This \hs{if} function would then essentially describe a multiplexer and + allows us to describe any architecture that uses multiplexers. (TODO: Are + there other mechanisms of choice in hardware?) + + However, to be able to describe our hardware in a more convenient way, we + also need to translate Haskell's choice mechanisms. The easiest of these + are of course case expressions (and if expressions, which can be very + directly translated to case expressions). A case expression can in turn + simply be translated to a conditional assignment, where the conditions use + equality comparisons against the constructors in the case expressions. + + In \in{example}[ex:CaseInv] a simple case statement is shown, + scrutinizing a boolean value. If we would translate a Boolean to a bit + value, we could of course remove the comparator and directly feed 'in' + into the multiplex (or even use an inverter instead of a multiplexer). + However, we will try to make a general translation, which works for all + possible case statements. Optimizations such as these are left for the + \VHDL synthesizer, which handles them very well. + + \startbuffer[CaseInv] + inv :: Bool -> Bool + inv in = case in of + True -> False + False -> True + \stopbuffer + + \startuseMPgraphic{CaseInv} + save in, truecmp, falseout, trueout, out, cmp, mux; + + % I/O ports + newCircle.in(btex $in$ etex) "framed(false)"; + newCircle.out(btex $out$ etex) "framed(false)"; + % Constants + newBox.truecmp(btex $True$ etex) "framed(false)"; + newBox.trueout(btex $True$ etex) "framed(false)"; + newBox.falseout(btex $False$ etex) "framed(false)"; + + % Components + newCircle.cmp(btex $==$ etex); + newMux.mux; + + in.c = origin; + cmp.c = in.c + (3cm, 0cm); + truecmp.c = cmp.c + (-1cm, 1cm); + mux.sel = cmp.e + (1cm, -1cm); + falseout.c = mux.inpa - (2cm, 0cm); + trueout.c = mux.inpb - (2cm, 0cm); + out.c = mux.out + (2cm, 0cm); + + % Draw objects and lines + drawObj(in, out, truecmp, trueout, falseout, cmp, mux); + + ncline(in)(cmp); + ncarc(truecmp)(cmp); + nccurve(cmp.e)(mux.sel) "angleA(0)", "angleB(-90)"; + ncline(falseout)(mux) "posB(inpa)"; + ncline(trueout)(mux) "posB(inpb)"; + ncline(mux)(out) "posA(out)"; + \stopuseMPgraphic + + \placeexample[here][ex:CaseInv]{Simple inverter.} + \startcombination[2*1] + {\typebufferhs{CaseInv}}{Haskell description using a Case expression.} + {\boxedgraphic{CaseInv}}{The architecture described by the Haskell description.} + \stopcombination + + Slightly more complex (but very powerful) forms of choice are pattern + matches. A function can be defined in multiple clauses, where each clause + specifies a pattern. When the arguments match the pattern, the + corresponding clause will be used. + + \startbuffer[PatternInv] + inv :: Bool -> Bool + inv True = False + inv False = True + \stopbuffer + + \placeexample[here][ex:PatternInv]{Simple inverter using pattern matching.} + {\typebufferhs{CaseInv}} + + The architecture described in \in{example}[ex:PatternInv] is of course the + same one as the one in \in{example}[ex:CaseInv]. The general interpration + of pattern matching is also similar to that of case expressions: Generate + hardware for each of the clause (like each of the clauses of a case + expression) and connect them to the function output through (a number of + nested) multiplexers. These multiplexers are driven by comparators and + other logic, that check each pattern in turn. + + \section{Types} + Apart from giving structure to our hardware, we'll also need to provide + some way to translate the values used to hardware equivalents. In + particular, this means having to come up with a hardware equivalent for + every \emph{type} used in our program. + + Since most functional languages have a lot of standard types that are + hard to translate (integers without a fixed size, lists without a static + length, etc.), we will start out by defining a number of \quote{builtin} + types ourselves. These types are builtin in the sense that our compiler + will have a fixed VHDL type for these. User defined types, on the other + hand, will have their hardware type derived directly from their Haskell + declaration automatically, according to the rules we sketch here. + + \subsection{Builtin types} + \startdesc{Bit} + This is the most basic type available. It is mapped directly onto + the \type{std_logic} \small{VHDL} type. Mapping this to the + \type{bit} type might make more sense (since the Haskell version + only has two values), but using \type{std_logic} is more standard + (and allowed for some experimentation with don't care values) + \stopdesc + \startdesc{Bool} + This is a builtin Haskell type, but is translated exactly like the + Bit type. Supporting the Bool type is particularly useful to support + \hs{if ... then ... else ...} expressions, which always have a + \hs{Bool} value for the condition. + + A \hs{Bool} is translated to a \type{std_logic}, just like \hs{Bit}. + \stopdesc + \startdesc{SizedWord, SizedInt} + These are types to represent integers. \hs{SizedWord} is unsigned, + while \hs{SizedInt} is signed. These types are parameterized by a + length type, so you can define an unsigned word of 32 bits wide as + follows: + + \starthaskell + type Word32 = SizedWord D32 + \stophaskell + + Here, \hs{D32} is the \emph{type level representation} of the number + 32. TODO: Introduce dependent types somewhere. + + These types are translated to the \small{VHDL} \type{unsigned} and + \type{signed} respectively. + \stopdesc + \startdesc{RangedWord} + This is another type to describe integers, but unlike the previous + two it has no specific width, but an upper bound. This means that + its range is not limited to powers of two, but can be any number. + A \hs{RangedWord} only has an upper bound, its lower bound is + implicitly zero. There is a lot of added implementation complexity + when adding a lower bound and having just an upper bound was enough + for the primary purpose of this type: Typesafely indexing vectors. + + This type is translated to the \type{unsigned} \small{VHDL} type. + \stopdesc + \startdesc{TFVec} + This is a vector type, with a fixed length. It has two type + parameters: Its length and the type of the elements contained in it. + + A fixed size vector is translated to a \small{VHDL} array type. + + The \hs{TF} in its name is a reference to the implementation used in + the prototype, which uses \emph{type families}. + \stopdesc + + TODO: Reference Christiaan / describe dependent typing + \subsection{User-defined types} + There are three ways to define new types in Haskell: Algebraic + datatypes with the \hs{data} keyword, type synonyms with the \hs{type} + keyword and type renamings with the \hs{newtype} keyword. This + expclitely excludes more advanced type creation from \GHC extensions + such as type families, existential typing, \small{GADT}s, etc. + + The first of these actually introduces a new type, for which we provide + the \VHDL translation below. The latter two only define new names for + existing types (where synonyms are completely interchangeable and + renamings need explicit conversion). Therefore, these don't need any + particular \VHDL translation, a synonym or renamed type will just use + the same representation as the equivalent type. + + For algebraic types, we can make the following distinction: + + \startdesc{Product types} + A product type is an algebraic datatype with a single constructor with + two or more fields. This is essentially a way to pack a few values + together in a record-like structure. In fact, the builtin tuple types + are just product types (and are thus supported in exactly the same + way). + + The "product" in its name refers to the collection of values belonging + to this type. The collection for a product type is the cartesic + product of the collections for the types of its fields. + + These types are translated to \VHDL record types, with one field for + every field in the constructor. This translation applies to all single + constructor algebraic datatypes, including those with no fields (unit + types) and just one field (which are technically not a product). + \stopdesc + \startdesc{Enumerated types} + An enumerated type is an algebraic datatype with multiple constructors, but + none of them have fields. This is essentially a way to get an + enum-like type containing alternatives. + + Note that Haskell's \hs{Bool} type is also defined as an + enumeration type, but we have a fixed translation for that. + + These types are translated to \VHDL enumerations, with one value for + each constructor. This allows references to these constructors to be + translated to the corresponding enumeration value. + \stopdesc + \startdesc{Sum types} + A sum type is an algebraic datatype with multiple constructors, where + the constructors have one or more fields. Technically, a type with + more than one field per constructor is a sum of products type, but + for our purposes this distinction does not really make a difference, + so we'll leave it out. + + Sum types are currently not supported by the prototype, since its + translation is not so obvious as before. The can easily be emulated, + as we will see from an example: + + \starthaskell + data Sum = A Bit Word | B Word + \stophaskell + + An obvious way to translate this would be to create an enumeration to + distinguish the constructors and then create a big record that + contains all the fields of all the constructors. This is the same + translation that would result from the following enumeration and + product type (using a tuple for clarity): + + \starthaskell + data SumC = A | B + type Sum = (SumC, Bit, Word, Word) + \stophaskell + + Here, the \hs{SumC} type effectively signals which of the fields of + the \hs{Sum} type are valid, all the other ones have no useful value. + + An obvious problem with this naive approach is the space usage: The + example above generates a fairly big \VHDL type. However, we can be + sure that the two \hs{Word}s in the \hs{Sum} type will never be valid + at the same time, so this is a waste of space. + + Obviously, we could do some duplication detection here to reuse a + particular field for another constructor, but this would only + partially solve the problem. If I would, for example, have an array of + 8 bits and a 8 bit unsiged word, these are different types and could + not be shared. However, in the final hardware, both of these types + would simply be 8 bit connections, so we have a 100\% size increase by + not sharing these! + \stopdesc + + Another interesting case is that of recursive types. In Haskell, an + algebraic datatype can be recursive: Any of its field types can be (or + contain) the type being defined. The most well-known recursive type is + probably the list type, which is defined is: + + \starthaskell + data List a = Empty | Cons a (List a) + \stophaskell + + Where \hs{Empty} is usually written as \hs{[]} and \hs{Cons} as \hs{:}. + This immediately shows the problem with recursive types: What hardware + type to allocate here? There is no way to statically determine how long + the list will be + + In general, we can say we can never properly translated recursive types: + All recursive types have a potentially infinite value (even though in + practice they will have a bounded value, there is no way for the + compiler to determine an upper bound on its size. + \subsection{Partial application} It should be obvious that we cannot generate hardware signals for all expressions we can express in Haskell. The most obvious criterium for this @@ -144,6 +419,120 @@ quadruple n = mul (mul n) function boundaries), but eventually, the partial application will become completely applied. + \section{Costless specialization} + Each (complete) function application in our description generates a + component instantiation, or a specific piece of hardware in the final + design. It is interesting to note that each application of a function + generates a \emph{separate} piece of hardware. In the final design, none + of the hardware is shared between applications, even when the applied + function is the same. + + This is distinctly different from normal program compilation: Two separate + calls to the same function share the same machine code. Having more + machine code has implications for speed (due to less efficient caching) + and memory usage. For normal compilation, it is therefore important to + keep the amount of functions limited and maximize the code sharing. + + When generating hardware, this is hardly an issue. Having more \quote{code + sharing} does reduce the amount of \small{VHDL} output (Since different + component instantiations still share the same component), but after + synthesis, the amount of hardware generated is not affected. + + In particular, if we would duplicate all functions so that there is a + duplicate for every application in the program (\eg, each function is then + only applied exactly once), there would be no increase in hardware size + whatsoever. + + TODO: Perhaps these next two sections are a bit too + implementation-oriented? + + \subsection{Specialization} + Because of this, a common optimization technique called + \emph{specialization} is as good as free for hardware generation. Given + some function that has a \emph{domain} of $D$ (\eg, the set of all + possible arguments that could be applied), we create a specialized + function with exactly the same behaviour, but with an domain of $D' + \subset D$. This subset can be derived in all sort of ways, but commonly + this is done by limiting a polymorphic argument to a single type (\eg, + removing polymorphism) or by limiting an argument to just a single value + (\eg, cross-function constant propagation, effectively removing the + argument). + + Since we limit the argument domain of the specialized function, its + definition can often be optimized further (since now more types or even + values of arguments are already know). By replacing any application of + the function that falls within the reduced domain by an application of + the specialized version, the code gets faster (but the code also gets + bigger, since we now have two versions instead of one!). If we apply + this technique often enough, we can often replace all applications of a + function by specialized versions, allowing the original function to be + removed (in some cases, this can even give a net reduction of the code + compared to the non-specialized version). + + Specialization is useful for our hardware descriptions for functions + that contain arguments that cannot be translated to hardware directly + (polymorphic or higher order arguments, for example). If we can create + specialized functions that remove the argument, or make it translatable, + we can use specialization to make the original, untranslatable, function + obsolete. + + \section{Higher order values} + What holds for partial application, can be easily generalized to any + higher order expression. This includes partial applications, plain + variables (e.g., a binder referring to a top level function), lambda + expressions and more complex expressions with a function type (a case + expression returning lambda's, for example). + + Each of these values cannot be directly represented in hardware (just like + partial applications). Also, to make them representable, they need to be + applied: function variables and partial applications will then eventually + become complete applications, applied lambda expressions disappear by + applying β-reduction, etc. + + So any higher order value will be \quote{pushed down} towards its + application just like partial applications. Whenever a function boundary + needs to be crossed, the called function can be specialized. + + TODO: This is section should be improved + + \section{Polymorphism} + In Haskell, values can be polymorphic: They can have multiple types. For + example, the function \hs{fst :: (a, b) -> a} is an example of a + polymorphic function: It works for tuples with any element types. Haskell + typeclasses allow a function to work on a specific set of types, but the + general idea is the same. + +% A type class is a collection of types for which some operations are +% defined. It is thus possible for a value to be polymorphic while having +% any number of \emph{class constraints}: The value is not defined for +% every type, but only for types in the type class. An example of this is +% the \hs{even :: (Integral a) => a -> Bool} function, which can map any +% value of a type that is member of the \hs{Integral} type class + + When generating hardware, polymorphism can't be easily translated. How + many wire will you lay down for a value that could have any type? When + type classes are involved, what hardware components will you lay down for + a class method (whose behaviour depends on the type of its arguments)? + + Fortunately, we can again use the principle of specialization: Since every + function application generates separate pieces of hardware, we can know + the types of all arguments exactly. Provided that we don't use existential + typing, all of the polymorphic types in a function must depend on the + types of the arguments (In other words, the only way to introduce a type + variable is in a lambda abstraction). Our top level function must not have + a polymorphic type (otherwise we wouldn't know the hardware interface to + our top level function). + + If a function is monomorphic, all values inside it are monomorphic as + well, so any function that is applied within the function can only be + applied to monomorphic values. The applied functions can then be + specialized to work just for these specific types, removing the + polymorphism from the applied functions as well. + + By induction, this means that all functions that are (indirectly) called + by our top level function (meaning all functions that are translated in + the final hardware) become monomorphic. + \section{State} A very important concept in hardware designs is \emph{state}. In a stateless (or, \emph{combinatoric}) design, every output is a directly and solely dependent on the @@ -416,6 +805,8 @@ acc in (State s) = (State s', out) implemented in Cλash. \in{Section}[sec:prototype:statetype] expands on the possible ways this could have been implemented. + TODO: Say something about dependent types and fixed size vectors + \section[sec:recursion]{Recursion} An import concept in functional languages is recursion. In it's most basic form, recursion is a function that is defined in terms of itself. This