X-Git-Url: https://git.stderr.nl/gitweb?p=matthijs%2Fmaster-project%2Freport.git;a=blobdiff_plain;f=Chapters%2FPrototype.tex;h=85e7fb2816ffb04aeb898b4ffebb5e36cd23fac4;hp=945e920c33a5956bb611a1eb69635124fa6b19d0;hb=dde172499703d199e396b84bb6d5c13dae8cdd8d;hpb=fc1688977fdd8ee18e027876b8d86b0c38e25540 diff --git a/Chapters/Prototype.tex b/Chapters/Prototype.tex index 945e920..85e7fb2 100644 --- a/Chapters/Prototype.tex +++ b/Chapters/Prototype.tex @@ -1,4 +1,4 @@ -\chapter{Prototype} +\chapter[chap:prototype]{Prototype} An important step in this research is the creation of a prototype compiler. Having this prototype allows us to apply the ideas from the previous chapter to actual hardware descriptions and evaluate their usefulness. Having a @@ -50,6 +50,9 @@ TODO: Was Haskell really a good choice? Perhaps say this somewhere else? + \subsection{Output language or format} + VHDL / Verilog / EDIF etc. Why VHDL? + \section{Prototype design} As stated above, we will use the Glasgow Haskell Compiler (\small{GHC}) to implement our prototype compiler. To understand the design of the @@ -149,7 +152,7 @@ newEmptyBox.inp(0,0); newBox.front(btex \small{GHC} frontend + desugarer etex); newBox.norm(btex Normalization etex); - newBox.vhdl(btex VHDL generation etex); + newBox.vhdl(btex \small{VHDL} generation etex); newEmptyBox.out(0,0); % Space the boxes evenly @@ -166,7 +169,7 @@ ObjLabel.inp(btex Haskell source etex) "labpathname(haskell)", "labdir(rt)"; ObjLabel.front(btex Core etex) "labpathname(core)", "labdir(rt)"; ObjLabel.norm(btex Normalized core etex) "labpathname(normal)", "labdir(rt)"; - ObjLabel.vhdl(btex VHDL description etex) "labpathname(vhdl)", "labdir(rt)"; + ObjLabel.vhdl(btex \small{VHDL} description etex) "labpathname(vhdl)", "labdir(rt)"; % Draw the objects (and deferred labels) drawObj (inp, front, norm, vhdl, out); @@ -185,9 +188,9 @@ order expressions, has a specific structure, etc.), but is also very close to directly describing hardware. \stopdesc - \startdesc{VHDL generation} + \startdesc{\small{VHDL} generation} The last step takes the normal formed core representation and generates - VHDL for it. Since the normal form has a specific, hardware-like + \small{VHDL} for it. Since the normal form has a specific, hardware-like structure, this final step is very straightforward. \stopdesc @@ -197,14 +200,521 @@ constructs. This step is described in a lot of detail at \in{chapter}[chap:normalization]. + \section{The Core language} + Most of the prototype deals with handling the program in the Core + language. In this section we will show what this language looks like and + how it works. - Core - description of the language (appendix?) - Implementation issues - Simplified core? + The Core language is a functional language that describes + \emph{expressions}. Every identifier used in Core is called a + \emph{binder}, since it is bound to a value somewhere. On the highest + level, a Core program is a collection of functions, each of which bind a + binder (the function name) to an expression (the function value, which has + a function type). + + The Core language itself does not prescribe any program structure, only + expression structure. In the \small{GHC} compiler, the Haskell module + structure is used for the resulting Core code as well. Since this is not + so relevant for understanding the Core language or the Normalization + process, we'll only look at the Core expression language here. + + Each Core expression consists of one of these possible expressions. + + \startdesc{Variable reference} +\startlambda +a +\stoplambda + This is a simple reference to a binder. It's written down as the + name of the binder that is being referred to, which should of course be + bound in a containing scope (including top level scope, so a reference + to a top level function is also a variable reference). Additionally, + constructors from algebraic datatypes also become variable references. + + The value of this expression is the value bound to the given binder. + + Each binder also carries around its type, but this is usually not shown + in the Core expressions. Occasionally, the type of an entire expression + or function is shown for clarity, but this is only informational. In + practice, the type of an expression is easily determined from the + structure of the expression and the types of the binders and occasional + cast expressions. This minimize the amount of bookkeeping needed to keep + the typing consistent. + \stopdesc + \startdesc{Literal} +\startlambda +10 +\stoplambda + This is a simple literal. Only primitive types are supported, like + chars, strings, ints and doubles. The types of these literals are the + \quote{primitive} versions, like \lam{Char\#} and \lam{Word\#}, not the + normal Haskell versions (but there are builtin conversion functions). + \stopdesc + \startdesc{Application} +\startlambda +func arg +\stoplambda + This is simple function application. Each application consists of two + parts: The function part and the argument part. Applications are used + for normal function \quote{calls}, but also for applying type + abstractions and data constructors. + + The value of an application is the value of the function part, with the + first argument binder bound to the argument part. + \stopdesc + \startdesc{Lambda abstraction} +\startlambda +λbndr.body +\stoplambda + This is the basic lambda abstraction, as it occurs in labmda calculus. + It consists of a binder part and a body part. A lambda abstraction + creates a function, that can be applied to an argument. + + Note that the body of a lambda abstraction extends all the way to the + end of the expression, or the closing bracket surrounding the lambda. In + other words, the lambda abstraction \quote{operator} has the lowest + priority of all. + + The value of an application is the value of the body part, with the + binder bound to the value the entire lambda abstraction is applied to. + \stopdesc + \startdesc{Non-recursive let expression} +\startlambda +let bndr = value in body +\stoplambda + A let expression allows you to bind a binder to some value, while + evaluating to some other value (where that binder is in scope). This + allows for sharing of subexpressions (you can use a binder twice) and + explicit \quote{naming} of arbitrary expressions. Note that the binder + is not in scope in the value bound to it, so it's not possible to make + recursive definitions with the normal form of the let expression (see + the recursive form below). + + Even though this let expression is an extension on the basic lambda + calculus, it is easily translated to a lambda abstraction. The let + expression above would then become: + +\startlambda +(λbndr.body) value +\stoplambda + + This notion might be useful for verifying certain properties on + transformations, since a lot of verification work has been done on + lambda calculus already. + + The value of a let expression is the value of the body part, with the + binder bound to the value. + \stopdesc + \startdesc{Recursive let expression} +\startlambda +letrec + bndr1 = value1 + \vdots + bndrn = valuen +in + body +\stoplambda + + This is the recursive version of the let expression. In \small{GHC}'s + Core implementation, non-recursive and recursive lets are not so + distinct as we present them here, but this provides a clearer overview. + + The main difference with the normal let expression is that each of the + binders is in scope in each of the values, in addition to the body. This + allows for self-recursive definitions or mutually recursive definitions. + + It should also be possible to express a recursive let using normal + lambda calculus, if we use the \emph{least fixed-point operator}, + \lam{Y}. + \stopdesc + \startdesc{Case expression} +\startlambda + case scrut of bndr + DEFAULT -> defaultbody + C0 bndr0,0 ... bndr0,m -> body0 + \vdots + Cn bndrn,0 ... bndrn,m -> bodyn +\stoplambda + +TODO: Define WHNF + + A case expression is the only way in Core to choose between values. A case + expression evaluates its scrutinee, which should have an algebraic + datatype, into weak head normal form (\small{WHNF}) and (optionally) binds + it to \lam{bndr}. It then chooses a body depending on the constructor of + its scrutinee. If none of the constructors match, the \lam{DEFAULT} + alternative is chosen. + + Since we can only match the top level constructor, there can be no overlap + in the alternatives and thus order of alternatives is not relevant (though + the \lam{DEFAULT} alternative must appear first for implementation + efficiency). + + Any arguments to the constructor in the scrutinee are bound to each of the + binders after the constructor and are in scope only in the corresponding + body. + + To support strictness, the scrutinee is always evaluated into WHNF, even + when there is only a \lam{DEFAULT} alternative. This allows a strict + function argument to be written like: + +\startlambda +function (case argument of arg + DEFAULT -> arg) +\stoplambda + + This seems to be the only use for the extra binder to which the scrutinee + is bound. When not using strictness annotations (which is rather pointless + in hardware descriptions), \small{GHC} seems to never generate any code + making use of this binder. The current prototype does not handle it + either, which probably means that code using it would break. + + Note that these case statements are less powerful than the full Haskell + case statements. In particular, they do not support complex patterns like + in Haskell. Only the constructor of an expression can be matched, complex + patterns are implemented using multiple nested case expressions. + + Case statements are also used for unpacking of algebraic datatypes, even + when there is only a single constructor. For examples, to add the elements + of a tuple, the following Core is generated: + +\startlambda +sum = λtuple.case tuple of + (,) a b -> a + b +\stoplambda + + Here, there is only a single alternative (but no \lam{DEFAULT} + alternative, since the single alternative is already exhaustive). When + it's body is evaluated, the arguments to the tuple constructor \lam{(,)} + (\eg, the elements of the tuple) are bound to \lam{a} and \lam{b}. + \stopdesc + \startdesc{Cast expression} +\startlambda +body :: targettype +\stoplambda + A cast expression allows you to change the type of an expression to an + equivalent type. Note that this is not meant to do any actual work, like + conversion of data from one format to another, or force a complete type + change. Instead, it is meant to change between different representations + of the same type, \eg switch between types that are provably equal (but + look different). + + In our hardware descriptions, we typically see casts to change between a + Haskell newtype and its contained type, since those are effectively + different representations of the same type. + + More complex are types that are proven to be equal by the typechecker, + but look different at first glance. To ensure that, once the typechecker + has proven equality, this information sticks around, explicit casts are + added. In our notation we only write the target type, but in reality a + cast expressions carries around a \emph{coercion}, which can be seen as a + proof of equality. TODO: Example + + The value of a cast is the value of its body, unchanged. The type of this + value is equal to the target type, not the type of its body. + + Note that this syntax is also used sometimes to indicate that a particular + expression has a particular type, even when no cast expression is + involved. This is then purely informational, since the only elements that + are explicitely typed in the Core language are the binder references and + cast expressions, the types of all other elements are determined at + runtime. + \stopdesc + \startdesc{Note} + + The Core language in \small{GHC} allows adding \emph{notes}, which serve + as hints to the inliner or add custom (string) annotations to a core + expression. These shouldn't be generated normally, so these are not + handled in any way in the prototype. + \stopdesc + \startdesc{Type} +\startlambda +@type +\stoplambda + It is possibly to use a Core type as a Core expression. This is done to + allow for type abstractions and applications to be handled as normal + lambda abstractions and applications above. This means that a type + expression in Core can only ever occur in the argument position of an + application, and only if the type of the function that is applied to + expects a type as the first argument. This happens for all polymorphic + functions, for example, the \lam{fst} function: + +\startlambda +fst :: \forall a. \forall b. (a, b) -> a +fst = λtup.case tup of (,) a b -> a + +fstint :: (Int, Int) -> Int +fstint = λa.λb.fst @Int @Int a b +\stoplambda + + The type of \lam{fst} has two universally quantified type variables. When + \lam{fst} is applied in \lam{fstint}, it is first applied to two types. + (which are substitued for \lam{a} and \lam{b} in the type of \lam{fst}, so + the type of \lam{fst} actual type of arguments and result can be found: + \lam{fst @Int @Int :: (Int, Int) -> Int}). + \stopdesc + + TODO: Core type system - Haskell language coverage / constraints - Recursion - Builtin types - Custom types (Sum types, product types) - Function types / higher order expressions + \section[sec:prototype:statetype]{State annotations in Haskell} + Ideal: Type synonyms, since there is no additional code overhead for + packing and unpacking. Downside: there is no explicit conversion in Core + either, so type synonyms tend to get lost in expressions (they can be + preserved in binders, but this makes implementation harder, since that + statefulness of a value must be manually tracked). + + Less ideal: Newtype. Requires explicit packing and unpacking of function + arguments. If you don't unpack substates, there is no overhead for + (un)packing substates. This will result in many nested State constructors + in a nested state type. \eg: + + \starttyping + State (State Bit, State (State Word, Bit), Word) + \stoptyping + + Alternative: Provide different newtypes for input and output state. This + makes the code even more explicit, and typechecking can find even more + errors. However, this requires defining two type synomyms for each + stateful function instead of just one. \eg: + \starttyping + type AccumStateIn = StateIn Bit + type AccumStateOut = StateOut Bit + \stoptyping + This also increases the possibility of having different input and output + states. Checking for identical input and output state types is also + harder, since each element in the state must be unpacked and compared + separately. + + Alternative: Provide a type for the entire result type of a stateful + function, not just the state part. \eg: + + \starttyping + newtype Result state result = Result (state, result) + \stoptyping + + This makes it easy to say "Any stateful function must return a + \type{Result} type, without having to sort out result from state. However, + this either requires a second type for input state (similar to + \type{StateIn} / \type{StateOut} above), or requires the compiler to + select the right argument for input state by looking at types (which works + for complex states, but when that state has the same type as an argument, + things get ambiguous) or by selecting a fixed (\eg, the last) argument, + which might be limiting. + + \subsubsection{Example} + As an example of the used approach, a simple averaging circuit, that lets + the accumulation of the inputs be done by a subcomponent. + + \starttyping + newtype State s = State s + + type AccumState = State Bit + accum :: Word -> AccumState -> (AccumState, Word) + accum i (State s) = (State (s + i), s + i) + + type AvgState = (AccumState, Word) + avg :: Word -> AvgState -> (AvgState, Word) + avg i (State s) = (State s', o) + where + (accums, count) = s + -- Pass our input through the accumulator, which outputs a sum + (accums', sum) = accum i accums + -- Increment the count (which will be our new state) + count' = count + 1 + -- Compute the average + o = sum / count' + s' = (accums', count') + \stoptyping + + And the normalized, core-like versions: + + \starttyping + accum i spacked = res + where + s = case spacked of (State s) -> s + s' = s + i + spacked' = State s' + o = s + i + res = (spacked', o) + + avg i spacked = res + where + s = case spacked of (State s) -> s + accums = case s of (accums, \_) -> accums + count = case s of (\_, count) -> count + accumres = accum i accums + accums' = case accumres of (accums', \_) -> accums' + sum = case accumres of (\_, sum) -> sum + count' = count + 1 + o = sum / count' + s' = (accums', count') + spacked' = State s' + res = (spacked', o) + \stoptyping + + + + As noted above, any component of a function's state that is a substate, + \eg passed on as the state of another function, should have no influence + on the hardware generated for the calling function. Any state-specific + \small{VHDL} for this component can be generated entirely within the called + function. So,we can completely leave out substates from any function. + + From this observation, we might think to remove the substates from a + function's states alltogether, and leave only the state components which + are actual states of the current function. While doing this would not + remove any information needed to generate \small{VHDL} from the function, it would + cause the function definition to become invalid (since we won't have any + substate to pass to the functions anymore). We could solve the syntactic + problems by passing \type{undefined} for state variables, but that would + still break the code on the semantic level (\ie, the function would no + longer be semantically equivalent to the original input). + + To keep the function definition correct until the very end of the process, + we will not deal with (sub)states until we get to the \small{VHDL} generation. + Here, we are translating from Core to \small{VHDL}, and we can simply not generate + \small{VHDL} for substates, effectively removing the substate components + alltogether. + + There are a few important points when ignore substates. + + First, we have to have some definition of "substate". Since any state + argument or return value that represents state must be of the \type{State} + type, we can simply look at its type. However, we must be careful to + ignore only {\em substates}, and not a function's own state. + + In the example above, this means we should remove \type{accums'} from + \type{s'}, but not throw away \type{s'} entirely. We should, however, + remove \type{s'} from the output port of the function, since the state + will be handled by a \small{VHDL} procedure within the function. + + When looking at substates, these can appear in two places: As part of an + argument and as part of a return value. As noted above, these substates + can only be used in very specific ways. + + \desc{State variables can appear as an argument.} When generating \small{VHDL}, we + completely ignore the argument and generate no input port for it. + + \desc{State variables can be extracted from other state variables.} When + extracting a state variable from another state variable, this always means + we're extracting a substate, which we can ignore. So, we simply generate no + \small{VHDL} for any extraction operation that has a state variable as a result. + + \desc{State variables can be passed to functions.} When passing a + state variable to a function, this always means we're passing a substate + to a subcomponent. The entire argument can simply be ingored in the + resulting port map. + + \desc{State variables can be returned from functions.} When returning a + state variable from a function (probably as a part of an algebraic + datatype), this always mean we're returning a substate from a + subcomponent. The entire state variable should be ignored in the resulting + port map. The type binder of the binder that the function call is bound + to should not include the state type either. + + \startdesc{State variables can be inserted into other variables.} When inserting + a state variable into another variable (usually by constructing that new + variable using its constructor), we can identify two cases: + + \startitemize + \item The state is inserted into another state variable. In this case, + the inserted state is a substate, and can be safely left out of the + constructed variable. + \item The state is inserted into a non-state variable. This happens when + building up the return value of a function, where you put state and + retsult variables together in an algebraic type (usually a tuple). In + this case, we should leave the state variable out as well, since we + don't want it to be included as an output port. + \stopitemize + + So, in both cases, we can simply leave out the state variable from the + resulting value. In the latter case, however, we should generate a state + proc instead, which assigns the state variable to the input state variable + at each clock tick. + \stopdesc + + \desc{State variables can appear as (part of) a function result.} When + generating \small{VHDL}, we can completely ignore any part of a function result + that has a state type. If the entire result is a state type, this will + mean the entity will not have an output port. Otherwise, the state + elements will be removed from the type of the output port. + + + Now, we know how to handle each use of a state variable separately. If we + look at the whole, we can conclude the following: + + \startitemize + \item A state unpack operation should not generate any \small{VHDL}. The binder + to which the unpacked state is bound should still be declared, this signal + will become the register and will hold the current state. + \item A state pack operation should not generate any \small{VHDL}. The binder th + which the packed state is bound should not be declared. The binder that is + packed is the signal that will hold the new state. + \item Any values of a State type should not be translated to \small{VHDL}. In + particular, State elements should be removed from tuples (and other + datatypes) and arguments with a state type should not generate ports. + \item To make the state actually work, a simple \small{VHDL} proc should be + generated. This proc updates the state at every clockcycle, by assigning + the new state to the current state. This will be recognized by synthesis + tools as a register specification. + \stopitemize + + + When applying these rules to the example program (in normal form), we will + get the following result. All the parts that don't generate any value are + crossed out, leaving some very boring assignments here and there. + + + \starthaskell + avg i --spacked-- = res + where + s = --case spacked of (State s) -> s-- + --accums = case s of (accums, \_) -> accums-- + count = case s of (--\_,-- count) -> count + accumres = accum i --accums-- + --accums' = case accumres of (accums', \_) -> accums'-- + sum = case accumres of (--\_,-- sum) -> sum + count' = count + 1 + o = sum / count' + s' = (--accums',-- count') + --spacked' = State s'-- + res = (--spacked',-- o) + \stophaskell + + When we would really leave out the crossed out parts, we get a slightly + weird program: There is a variable \type{s} which has no value, and there + is a variable \type{s'} that is never used. Together, these two will form + the state proc of the function. \type{s} contains the "current" state, + \type{s'} is assigned the "next" state. So, at the end of each clock + cycle, \type{s'} should be assigned to \type{s}. + + Note that the definition of \type{s'} is not removed, even though one + might think it as having a state type. Since the state type has a single + argument constructor \type{State}, some type that should be the resulting + state should always be explicitly packed with the State constructor, + allowing us to remove the packed version, but still generate \small{VHDL} for the + unpacked version (of course with any substates removed). + + As you can see, the definition of \type{s'} is still present, since it + does not have a state type (The State constructor. The \type{accums'} substate has been removed, + leaving us just with the state of \type{avg} itself. + \subsection{Initial state} + How to specify the initial state? Cannot be done inside a hardware + function, since the initial state is its own state argument for the first + call (unless you add an explicit, synchronous reset port). + + External init state is natural for simulation. + + External init state works for hardware generation as well. + + Implementation issues: state splitting, linking input to output state, + checking usage constraints on state variables. + + Implementation issues + \subsection[sec:prototype:separate]{Separate compilation} + - Simplified core? + \section{Haskell language coverage and constraints} + Recursion + Builtin types + Custom types (Sum types, product types) + Function types / higher order expressions