\chapter[chap:future]{Future work}
+\section{New language}
+During the development of the prototype, it has become clear that Haskell is
+not the perfect language for this work. There are two main problems:
+\startitemize
+\item Haskell is not expressive enough. Haskell is still quite new in the area
+of dependent typing, something we use extensively. Also, Haskell has some
+special syntax to write monadic composition and arrow composition, which is
+well suited to normal Haskell programs. However, for our hardware
+descriptions, we could use some similar, but other, syntactic sugar (as we
+have seen above).
+\item Haskell is too expressive. There are some things that Haskell can
+express, but we can never properly translate. There are certain types (both
+primitive types and certain type constructions like recursive types) we can
+never translate, as well as certain forms of recursion.
+\stopitemize
+
+It might be good to reevaluate the choice of language for Cλash, perhaps there
+are other languages which are better suited to describe hardware, now that
+we have learned a lot about it.
+
+Perhaps Haskell (or some other language) can be extended by using a
+preprocessor. There has been some (point of proof) work on a the Strathclyde
+Haskell Enhancement (\small{SHE}) preprocessor for type-level programming,
+perhaps we can extend that, or create something similar for hardware-specific
+notation.
+
+It is not unlikely that the most flexible way
+forward would be to define a completely new language with exactly the needed
+features. This is of course an enormous effort, which should not be taken
+lightly.
+
+\section{Correctness proofs of the normalization system}
+As stated in \in{section}[sec:normalization:properties], there are a
+number of properties we would like to see verified about the
+normalization system. In particular, the \emph{termination} and
+\emph{completeness} of the system would be a good candidate for future
+research. Specifying formal semantics for the Core language in
+order to verify the \emph{soundness} of the system would be an even more
+challenging task.
+
\section{Improved notation for hierarchical state}
The hierarchical state model requires quite some boilerplate code for unpacking
and distributing the input state and collecting and repacking the output
One particular notation in Haskell that seems promising, is the \hs{do}
notation. This is meant to simplify Monad notation by hiding away some
-details. It allows one to write a list of expressions, which are composited
+details. It allows one to write a list of expressions, which are composed
using the monadic \emph{bind} operator, written in Haskell as \hs{>>}. For
example, the snippet:
The main reason to have the monadic notation, is to be able to wrap
results of functions in a datatype (the \emph{monad}) that can contain
-extra information, and hide extra behaviour in the binding operators.
+extra information, and hide extra behavior in the binding operators.
The \hs{>>=} operator allows extracting the actual result of a function
-and passing it to another function. Let's try to illustrate this from an
-example. The following snippet:
+and passing it to another function. The following snippet should
+illustrate this:
\starthaskell
do
will be desugared into:
\starthaskell
-(somefunc a) >>= (\\x -> otherfunc x)
+(somefunc a) >>= (\x -> otherfunc x)
\stophaskell
-The \hs{\\x -> ...} notation creates a lambda abstraction in Haskell,
+The \hs{\x -> ...} notation creates a lambda abstraction in Haskell,
that binds the \hs{x} variable. Here, the \hs{>>=} operator is supposed
to extract whatever result somefunc has and pass it to the lambda
expression created. This will probably not make the monadic notation
The monadic notation could perhaps be used to compose a number of
stateful functions into another stateful computation. Perhaps this could
move all the boilerplate code into the \hs{>>} and \hs{>>=} operators.
-Because the boilerplate is still there (it's not magically disappeared,
+Because the boilerplate is still there (it has not magically disappeared,
just moved into these functions), the compiler should still be able to compile
these descriptions without any special magic (though perhaps it should
always inline the binding operators to reveal the flow of values).
-This is highlights an important aspect of using a functional language for our
-descriptions: We can use the language itself to provide abstractions of common
-patterns, making our code smaller.
+This highlights an important aspect of using a functional language for our
+descriptions: we can use the language itself to provide abstractions of common
+patterns, making our code smaller and easier to read.
\subsection{Breaking out of the Monad}
However, simply using the monad notation is not as easy as it sounds. The main
(>>) :: Stateful s1 r1 -> Stateful s2 r2 -> Stateful (s1, s2) r2
\stophaskell
-What we see here, is that when we compose two stateful functions (whose inputs
-have already been applied, leaving just the state argument to be applied), the
-result is again a stateful function whose state is composed of the two
-\emph{substates}. The return value is simply the return value of the second
-function, discarding the first (to preserve that one, the \hs{>>=} operator
-can be used).
+What we see here, is that when we compose two stateful functions (that
+have already been applied to their inputs, leaving just the state
+argument to be applied to), the result is again a stateful function
+whose state is composed of the two \emph{substates}. The return value is
+simply the return value of the second function, discarding the first (to
+preserve that result, the \hs{>>=} operator can be used).
There is a trick we can apply to change the signature of the \hs{>>} operator.
\small{GHC} does not require the bind operators to be part of the \hs{Monad}
type class, as long as it can use them to translate the do notation. This
means we can define our own \hs{>>} and \hs{>>=} operators, outside of the
-\hs{Monad} typeclass. This does conflict with the existing methods of the
-\hs{Monad} typeclass, so we should prevent \small{GHC} from loading those (and
+\hs{Monad} type class. This does conflict with the existing methods of the
+\hs{Monad} type class, so we should prevent \small{GHC} from loading those (and
all of the Prelude) by passing \type{-XNoImplicitPrelude} to \type{ghc}. This
is slightly inconvenient, but since we hardly using anything from the prelude,
this is not a big problem. We might even be able to replace some of the
-Prelude with hardware-translateable versions by doing this.
+Prelude with hardware-translatable versions by doing this.
-We can now define the following binding operators. For completeness, we also
-supply the return function, which is not called by \small{GHC} implicitely,
-but can be called explicitely by a hardware description.
+The binding operators can now be defined exactly as they are needed. For
+completeness, the \hs{return} function is also defined. It is not called
+by \small{GHC} implicitly, but can be called explicitly by a hardware
+description. \in{Example}[ex:StateMonad] shows these definitions.
-\starthaskell
+\startbuffer[StateMonad]
(>>) :: Stateful s1 r1 -> Stateful s2 r2 -> Stateful (s1, s2) r2
f1 >> f2 = f1 >>= \_ -> f2
return :: r -> Stateful () r
return r = \s -> (s, r)
-\stophaskell
+\stopbuffer
+\placeexample[here][ex:StateMonad]{Binding operators to compose stateful computations}
+ {\typebufferhs{StateMonad}}
-As you can see, this closely resembles the boilerplate of unpacking state,
+These definitions closely resemble the boilerplate of unpacking state,
passing it to two functions and repacking the new state. With these
-definitions, we could have writting \in{example}[ex:NestedState] a lot
+definitions, we could have written \in{example}[ex:NestedState] a lot
shorter, see \in{example}[ex:DoState]. In this example the type signature of
foo is the same (though it is now written using the \hs{Stateful} type
synonym, it is still completely equivalent to the original: \hs{foo :: Word ->
outb <- funcb outa
return outb
\stopbuffer
-\placeexample[here][ex:DoState]{Simple function composing two stateful
+\placeexample[][ex:DoState]{Simple function composing two stateful
functions, using do notation.}
{\typebufferhs{DoState}}
An important implication of this approach is that the order of writing
function applications affects the state type. Fortunately, this problem can be
-localized by consistently using type synonyms for state types, which should
-prevent changes in other function's source when a function changes.
+localized by consistently using type synonyms for state types (see
+\in{section}[sec:prototype:substatesynonyms]), which should prevent
+changes in other function's source when a function changes.
-A less obvous implications of this approach is that the scope of variables
+A less obvious implications of this approach is that the scope of variables
produced by each of these expressions (using the \hs{<-} syntax) is limited to
the expressions that come after it. This prevents values from flowing between
two functions (components) in two directions. For most Monad instances, this
is a requirement, but here it could have been different.
-\todo{Add examples or reference for state synonyms}
-
\subsection{Alternative syntax}
Because of these typing issues, misusing Haskell's do notation is probably not
the best solution here. However, it does show that using fairly simple
\small{GHC} with some new syntax sugar similar to the do notation might be a
feasible.
+\subsection{Arrows}
+Another abstraction mechanism offered by Haskell are arrows. Arrows are
+a generalization of monads \cite[hughes98], for which \GHC\ also supports
+some syntax sugar \cite[paterson01]. Their use for hiding away state
+boilerplate is not directly evident, but since arrows are a complex
+concept further investigation is appropriate.
+
\section[sec:future:pipelining]{Improved notation or abstraction for pipelining}
Since pipelining is a very common optimization for hardware systems, it should
be easy to specify a pipelined system. Since it introduces quite some registers
-into an otherwise regular combinatoric system, we might look for some way to
+into an otherwise regular combinational system, we might look for some way to
abstract away some of the boilerplate for pipelining.
Something similar to the state boilerplate removal above might be appropriate:
in \in{example}[ex:DoState], which is harmless there, but would be a problem
if it introduced an extra, useless, pipeline stage).
-This problem is slightly more complex than the problem we've seen before. One
+This problem is slightly more complex than the problem we have seen before. One
significant difference is that each variable that crosses a stage boundary
needs a register. However, when a variable crosses multiple stage boundaries,
it must be stored for a longer period and should receive multiple registers.
-Since we can't find out from the combinator code where the result of the
+Since we cannot find out from the combinator code where the result of the
combined values is used (at least not without using Template Haskell to
inspect the \small{AST}), there seems to be no easy way to find how much
registers are needed.
\startitemize
\item Limit the scoping of each variable produced by a stage to the next
stage only. This means that any variables that are to be used in subsequent
- stages should be passed on explicitely, which should allocate the required
+ stages should be passed on explicitly, which should allocate the required
number of registers.
This produces cumbersome code, where there is still a lot of explicitness
(though this could be hidden in syntax sugar).
\todo{The next sentence is unclear}
\item Scope each variable over every subsequent pipeline stage and allocate
- the maximum amount of registers that \emph{could} be needed. This means we
+ the maximum number of registers that \emph{could} be needed. This means we
will allocate registers that are never used, but those could be optimized
away later. This does mean we need some way to introduce a variable number
of variables (depending on the total number of stages), assign the output of
This also means that when adding a stage to an existing pipeline will change
the state type of each of the subsequent pipeline stages, and the state type
- ofthe added stage depends on the number of subsequent stages.
+ of the added stage depends on the number of subsequent stages.
Properly describing this will probably also require quite explicit syntax,
meaning this is not feasible without some special syntax.
The main problems of recursion have been described in
\in{section}[sec:recursion]. In the current implementation, recursion is
therefore not possible, instead we rely on a number of implicitly list-recursive
-builtin functions.
+built-in functions.
Since recursion is a very important and central concept in functional
programming, it would very much improve the flexibility and elegance of our
\startitemize
\item For list recursion, how to make a description type check? It is quite
-possible that improvements in the \small{GHC} typechecker will make this
+possible that improvements in the \small{GHC} type-checker will make this
possible, though it will still stay a challenge. Further advances in
dependent typing support for Haskell will probably help here as well.
There might be asynchronous events to which a system wants to respond. The
most obvious asynchronous event is of course a reset signal. Though a reset
signal can be synchronous, that is less flexible (and a hassle to describe in
-Cλash, currently). Since every function in Cλash describes the behaviour on
-each cycle boundary, we really can't fit in asynchronous behaviour easily.
+Cλash, currently). Since every function in Cλash describes the behavior on
+each cycle boundary, we really cannot fit in asynchronous behavior easily.
Due to the same reason, multiple clock domains cannot be easily supported. There is
currently no way for the compiler to know in which clock domain a function
should operate and since the clock signal is never explicit, there is also no
way to express circuits that synchronize various clock domains.
-A possible way to express more complex timing behaviour would be to make
+A possible way to express more complex timing behavior would be to make
functions more generic event handlers, where the system generates a stream of
events (Like \quote{clock up}, \quote{clock down}, \quote{input A changed},
\quote{reset}, etc.). When working with multiple clock domains, each domain
could get its own clock events.
-As an example, we would have something like the following:
-
-\starthaskell
+\startbuffer[AsyncDesc]
data Event = ClockUp | Reset | ...
type MainState = State Word
-- On any other event, leave state and output unchanged
_ -> acc
--- func is some combinatoric expression
+-- func is some combinational expression
func :: Word -> Event -> Word
func inp _ = inp * 2 + 3
-\stophaskell
+\stopbuffer
+
+\placeexample[][ex:AsyncDesc]{Hardware description using asynchronous events.}
+ {\typebufferhs{AsyncDesc}}
\todo{Picture}
-In this example, we see that every function takes an input of type
-\hs{Event}. The function \hs{main} that takes the output of
-\hs{func} and accumulates it on every clock cycle. On a reset signal, the
-accumulator is reset. The function \hs{func} is just a combinatoric
-function, with no synchronous elements. We can see this because there is no
-state and the event input is completely ignored. If the compiler is smart
-enough, we could even leave the event input out for functions that don't use
-it, either because they are completely combinatoric (like in this example), or
-because they rely on the the caller to select the clock signal.
+\in{Example}[ex:AsyncDesc] shows a simple example of this event-based
+approach. In this example we see that every function takes an input of
+type \hs{Event}. The function \hs{main} that takes the output of
+\hs{func} and accumulates it on every clock cycle. On a reset signal,
+the accumulator is reset. The function \hs{func} is just a combinational
+function, with no synchronous elements. We can see this because there
+is no state and the event input is completely ignored. If the compiler
+is smart enough, we could even leave the event input out for functions
+that do not use it, either because they are completely combinational (like
+in this example), or because they rely on the the caller to select the
+clock signal.
This structure is similar to the event handling structure used to perform I/O
in languages like Amanda. \todo{ref} There is a top level case expression that
decides what to do depending on the current input event.
-A slightly more complex example that shows a system with two clock domains.
-There is no real integration between the clock domains (there is one input and
-one output for each clock domain), but it does show how multiple clocks can be
-distinguished.
+A slightly more complex example is show in
+\in{example}[ex:MulticlockDesc]. It shows a system with two clock
+domains. There is no real integration between the clock domains in this
+example (there is one input and one output for each clock domain), but
+it does show how multiple clocks can be distinguished.
-\starthaskell
+\startbuffer[MulticlockDesc]
data Event = ClockUpA | ClockUpB | ...
type MainState = State (SubAState, SubBState)
type SubBState = State ...
subb :: Word -> SubAState -> (SubAState, Word)
subb = ...
-\stophaskell
+\stopbuffer
-Note that in the above example the \hs{suba} and \hs{subb} functions are
-\emph{always} called, to get at their combinatoric outputs. The new state
-is calculated as well, but only saved when the right clock has an up
-transition.
+\placeexample[][ex:MulticlockDesc]{Hardware description with multiple clock domains through events.}
+ {\typebufferhs{MulticlockDesc}}
+
+Note that in \in{example}[ex:MulticlockDesc] the \hs{suba} and \hs{subb}
+functions are \emph{always} called, to get at their combinational
+outputs. The new state is calculated as well, but only saved when the
+right clock has an up transition.
As you can see, there is some code duplication in the case expression that
selects the right clock. One of the advantages of an explicit approach like
this, is that some of this duplication can be extracted away into helper
functions. For example, we could imagine a \hs{select_clock} function, which
takes a stateful function that is not aware of events, and changes it into a
-function that only updates its state on a specific (clock) event.
+function that only updates its state on a specific (clock) event. Such a
+function is shown in \in{example}[ex:SelectClock].
-\starthaskell
+\startbuffer[SelectClock]
select_clock :: Event
-> (input -> State s -> (State s, output))
-> (input -> State s -> Event -> (State s, output))
where
(sa'', outa) = (select_clock ClockUpA suba) inpa sa event
(sb'', outb) = (select_clock ClockUpB subb) inpb sb event
-\stophaskell
+\stopbuffer
+\placeexample[][ex:SelectClock]{A function to filter clock events.}
+ {\typebufferhs{SelectClock}}
As you can see, this can greatly reduce the length of the main function, while
increasing the readability. As you might also have noted, the select\_clock
function takes any stateful function from the current Cλash prototype and
turns it into an event-aware function!
-Going along this route for more complex timing behaviour seems promising,
-espicially since it seems possible to express very advanced timing behaviours,
+Going along this route for more complex timing behavior seems promising,
+especially since it seems possible to express very advanced timing behaviors,
while still allowing simple functions without any extra overhead when complex
-behaviour is not needed.
+behavior is not needed.
The main cost of this approach will probably be extra complexity in the
-compiler: The paths (state) data can take become very non-trivial, and it
+compiler: the paths (state) data can take become very non-trivial, and it
is probably hard to properly analyze these paths and produce the
-intended \VHDL description.
+intended \VHDL\ description.
\section{Multiple cycle descriptions}
In the current Cλash prototype, every description is a single-cycle
-description. In other words, every function describes behaviour for each
+description. In other words, every function describes behavior for each
separate cycle and any recursion (or folds, maps, etc.) is expanded in space.
Sometimes, you might want to have a complex description that could possibly
\startitemize
\item Some kind of map or fold over a list that could be expanded in time
instead of space. This would result in a function that describes n cycles
- instead of just one, where n is the lenght of the list.
- \item A large combinatoric expressions that would introduce a very long
- combinatoric path and thus limit clock frequency. Such an expression could
+ instead of just one, where n is the length of the list.
+ \item A large combinational expressions that would introduce a very long
+ combinational path and thus limit clock frequency. Such an expression could
be broken up into multiple stages, which effectively results in a pipelined
system (see also \in{section}[sec:future:pipelining]) with a known delay.
There should probably be some way for the developer to specify the cycle
division of the expression, since automatically deciding on such a division
- is too complex and contains too many tradeoffs, at least initially.
+ is too complex and contains too many trade-offs, at least initially.
\item Unbounded recursion. It is not possible to expand unbounded (or even
recursion with a depth that is not known at compile time) in space, since
there is no way to tell how much hardware to create (it might even be
infinite).
When expanding infinite recursion over time, each step of the recursion can
- simply become a single clockcycle. When expanding bounded but unknown
+ simply become a single clock cycle. When expanding bounded but unknown
recursion, we probably need to add an extra data valid output bit or
something similar.
\stopitemize
of the system would expect just \lam{a} (since the recursive expression
generates just a single element each cycle).
-Naively, this matching could be done using a (builtin) function with a
+Naively, this matching could be done using a (built-in) function with a
signature like \lam{[a] -> a}, which also serves as an indicator to the
compiler that some expanding over time is required. However, this poses a
-problem for simulation: How will our Haskell implementation of this magical
-builtin function know which element of the input list to return. This
+problem for simulation: how will our Haskell implementation of this magical
+built-in function know which element of the input list to return. This
obviously depends on the current cycle number, but there is no way for this
function to know the current cycle without breaking all kinds of safety and
purity properties. Making this function have some state input and output could
help, though this state is not present in the actual hardware (or perhaps
there is some state, if there are value passed from one recursion step to the
-next, but how can the type of that state be determined by the typechecker?).
+next, but how can the type of that state be determined by the type-checker?).
-It seems that this is the most pressing problem for multicycle descriptions:
+It seems that this is the most pressing problem for multi-cycle descriptions:
How to interface with the rest of the system, without sacrificing safety and
simulation capabilities?
\section{Higher order values in state}
-Another interesting idea is putting a higher order value inside a function's
-state value. Since we can use higher order values anywhere, why not in the
+Another interesting idea is putting a higher-order value inside a function's
+state value. Since we can use higher-order values anywhere, why not in the
state?
-As a (contrived) example, consider the following accumulator:
-
-\starthaskell
+\startbuffer[HigherAccum]
-- The accumulator function that takes a word and returns a new accumulator
-- and the result so far. This is the function we want to put inside the
-- state.
main :: Word -> State Acc -> (State Acc, Word)
main a s = (State s', out)
where (s', out) = s a
-\stophaskell
-
-This code uses a function as its state, which implicitely contains the value
-accumulated so far. This is a fairly trivial example, that is more easy to
-write with a simple \hs{Word} state value, but for more complex descriptions
-this style might pay off. Note that in a way we are using the
-\emph{continuation passing style} of writing functions, where we pass a
-sort of \emph{continuation} from one cycle to the next.
-
-Our normalization process completely removes higher order values inside a
-function by moving applications and higher order values around so that every
-higher order value will eventually be full applied. However, if we would put a
-higher order value inside our state, the path from the higher order value
+\stopbuffer
+\placeexample[][ex:HigherAccum]{An accumulator using a higher-order state.}
+ {\typebufferhs{HigherAccum}}
+
+As a (contrived) example, consider the accumulator in
+\in{example}[ex:HigherAccum]. This code uses a function as its state,
+which implicitly contains the value accumulated so far. This is a
+fairly trivial example, that is more easy to write with a simple
+\hs{Word} state value, but for more complex descriptions this style
+might pay off. Note that in a way we are using the \emph{continuation
+passing style} of writing functions, where we pass a sort of
+\emph{continuation} from one cycle to the next.
+
+Our normalization process completely removes higher-order values inside a
+function by moving applications and higher-order values around so that every
+higher-order value will eventually be full applied. However, if we would put a
+higher-order value inside our state, the path from the higher-order value
definition to its application runs through the state, which is external to the
-function. A higher order value defined in one cycle is not applied until a
+function. A higher-order value defined in one cycle is not applied until a
later cycle. Our normalization process only works within the function, so it
-cannot remove this use of a higher order value.
+cannot remove this use of a higher-order value.
-However, we cannot leave a higher order value in our state value, since it is
-impossible to generate a register containing a higher order value, we simply
-can't translate a function type to a hardware type. To solve this, we must
-replace the higher order value inside our state with some other value
-representing these higher order values.
+However, we cannot leave a higher-order value in our state value, since it is
+impossible to generate a register containing a higher-order value, we simply
+cannot translate a function type to a hardware type. To solve this, we must
+replace the higher-order value inside our state with some other value
+representing these higher-order values.
On obvious approach here is to use an algebraic datatype where each
-constructor represents one possible higher order value that can end up in the
+constructor represents one possible higher-order value that can end up in the
state and each constructor has an argument for each free variable of the
-higher order value replaced. This allows us to completely reconstruct the
-higher order value at the spot where it is applied, and thus the higher order
+higher-order value replaced. This allows us to completely reconstruct the
+higher-order value at the spot where it is applied, and thus the higher-order
value disappears.
This approach is commonly known as the \quote{Reynolds approach to
-defuntionalization}, first described by J.C. Reynolds and seems to apply well
-to this situation. One note here is that Reynolds' approach puts all the
-higher order values in a single datatype. For a typed language, we will at
-least have to have a single datatype for each function type, since we can't
-mix them. It would be even better to split these datatypes a bit further, so
-that such a datatype will never hold a constructor that is never used for a
-particular state variable. This separation is probably a non-trivial problem,
-though.
-
-\todo{Reference "Definitional interpreters for higher-order programming
-languages":
-http://portal.acm.org/citation.cfm?id=805852\&dl=GUIDE\&coll=GUIDE\&CFID=58835435\&CFTOKEN=81856623}
-
-\section{New language}
-During the development of the prototype, it has become clear that Haskell is
-not the perfect language for this work. There are two main problems:
-\startitemize
-\item Haskell is not expressive enough. Haskell is still quite new in the area
-of dependent typing, something we use extensively. Also, Haskell has some
-special syntax to write monadic composition and arrow composition, which is
-well suited to normal Haskell programs. However, for our hardware
-descriptions, we could use some similar, but other, syntactic sugar (as we
-have seen above).
-\item Haskell is too expressive. There are some things that Haskell can
-express, but we can never properly translate. There are certain types (both
-primitive types and certain type constructions like recursive types) we can
-never translate, as well as certain forms of recursion.
-\stopitemize
-
-It might be good to reevaluate the choice of language for Cλash, perhaps there
-are other languages which are better suited to describe hardware, now that
-we've learned a lot about it.
-
-Perhaps Haskell (or some other language) can be extended by using a
-preprocessor. There has been some (point of proof) work on a the Strathclyde
-Haskell Enhancement (\small{SHE}) preprocessor for type-level programming,
-perhaps we can extend that, or create something similar for hardware-specific
-notation.
-
-It is not unlikely that the most flexible way
-forward would be to define a completely new language with exactly the needed
-features. This is of course an enormous effort, which should not be taken
-lightly.
+defunctionalization}, first described by J.C. Reynolds \cite[reynolds98]\ and
+seems to apply well to this situation. One note here is that Reynolds'
+approach puts all the higher-order values in a single datatype. For a typed
+language, we will at least have to have a single datatype for each function
+type, since we cannot mix them. It would be even better to split these
+data-types a bit further, so that such a datatype will never hold a constructor
+that is never used for a particular state variable. This separation is
+probably a non-trivial problem, though.
\section{Don't care values}
- A powerful value in \VHDL is the \emph{don't care} value, given as
- \type{'-'}. This value tells the compiler that you don't really care about
+ A powerful value in \VHDL\ is the \emph{don't care} value, given as
+ \type{'-'}. This value tells the compiler that you do not really care about
which value is assigned to a signal, allowing the compiler to make some
- optimizations. Since hardware choice in hardware is often implemented using
+ optimizations. Since choice in hardware is often implemented using
a collection of logic gates instead of multiplexers only, synthesizers can
often reduce the amount of hardware needed by smartly choosing values for
these don't care cases.
type. An exception can be made for vectors and product types, since those
can simply use the don't care values of the types they contain.
- This would also require some kind of \quote{Dont careable} type class
+ This would also require some kind of \quote{Don't careable} type class
that allows each type to specify what its don't care value is. The
compiler must then recognize this constant and replace it with don't care
- values in the final \VHDL code.
+ values in the final \VHDL\ code.
This is of course a very intrusive solution. Every type must become member
- of this typeclass, and there is now some member in every type that is a
+ of this type class, and there is now some member in every type that is a
special don't care value. Guaranteeing the obvious don't care semantics
also becomes harder, since every pattern match or case expressions must now
also take care of the don't care value (this might actually be an
definition of an \hs{and} operator:
\starthaskell
- or :: Bit -> Bit -> Bit
+ and :: Bit -> Bit -> Bit
and Low _ = Low
and _ Low = Low
and High High = High
value should always return a Low value. Its value does not depend on the
value chosen for the don't care value. However, though when applying the
above and function to \hs{Low} and \hs{undefined} results in exactly that
- behviour, the result is \hs{undefined} when the arguments are swapped.
+ behavior, the result is \hs{undefined} when the arguments are swapped.
This is because the first pattern forces the first argument to be
evaluated. If it is \hs{undefined}, evaluation is halted and an exception
is show, which is not what is intended.
methods for describing don't care conditions. Possibly there are completely
other methods which work better.
-\section{Correctness proofs of the normalization system}
-As stated in \in{section}[sec:normalization:properties], there are a
-number of properties we would like to see verified about the
-normalization system. In particular, the \emph{termination} and
-\emph{completeness} of the system would be a good candidate for future
-research. Specifying formal semantics for the Core language in
-order to verify the \emph{soundness} of the system would be an even more
-challenging task.
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projects / matthijs / master-project / report.git / blobdiff