{\boxedgraphic{And3}}{The architecture described by the Haskell description.}
\stopcombination
+ TODO: Define top level function and subfunctions/circuits.
+
\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
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
In Haskell, this would look like \in{example}[ex:ExplicitAcc].
\startbuffer[ExplicitAcc]
-acc :: Word -> (State Word) -> (State Word, Word)
+-- input -> current state -> (new state, output)
+acc :: Word -> Word -> (Word, Word)
acc in (State s) = (State s', out)
where
out = s + in
looks the same from the outside, regardless of what state variables it
uses (or wether it's stateful at all).
- A direct consequence of this, is that if a function calls other
- stateful functions (\eg, has subcircuits), it has to somehow know the
- current state for these called functions. The only way to do this, is
- to put these \emph{substates} inside the caller's state. This means
- that a function's state is the sum of the states of all functions it
- calls, and its own state.
-
This approach is the one chosen for Cλash and will be examined more
closely below.
\subsection{Explicit state specification}
- Note about semantic correctness of top level state.
-
- Note about automatic ``down-pushing'' of state.
-
- Note about explicit state specification as the best solution.
-
- Note about substates
-
- Note about conditions on state variables and checking them.
-
- \subsection{Explicit state implementation}
- Recording state variables at the type level.
-
- 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.
-
+ We've seen the concept of explicit state in a simple example below, but
+ what are the implications of this approach?
+
+ \subsubsection{Substates}
+ Since a function's state is reflected directly in its type signature,
+ if a function calls other stateful functions (\eg, has subcircuits) it
+ has to somehow know the current state for these called functions. The
+ only way to do this, is to put these \emph{substates} inside the
+ caller's state. This means that a function's state is the sum of the
+ states of all functions it calls, and its own state.
+
+ This also means that the type of a function (at least the "state"
+ part) is dependent on its implementation and the functions it calls.
+ This is the major downside of this approach: The separation between
+ interface and implementation is limited. However, since Cλash is not
+ very suitable for separate compilation (see
+ \in{section}[sec:prototype:separate]) this is not a big problem in
+ practice. Additionally, when using a type synonym for the state type
+ of each function, we can still provide explicit type signatures
+ while keeping the state specification for a function near its
+ definition only.
- \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.
+ \subsubsection{...}
+ We need some way to know which arguments should become input ports and
+ which argument(s?) should become the current state (\eg, be bound to
+ the register outputs). This does not hold holds not just for the top
+ level function, but also for any subfunctions. Or could we perhaps
+ deduce the statefulness of subfunctions by analyzing the flow of data
+ in the calling functions?
+
+ To explore this matter, we make an interesting observation: We get
+ completely correct behaviour when we put all state registers in the
+ top level entity (or even outside of it). All of the state arguments
+ and results on subfunctions are treated as normal input and output
+ ports. Effectively, a stateful function results in a stateless
+ hardware component that has one of its input ports connected to the
+ output of a register and one of its output ports connected to the
+ input of the same register.
+
+ TODO: Example?
+
+ Of course, even though the hardware described like this has the
+ correct behaviour, unless the layout tool does smart optimizations,
+ there will be a lot of extra wire in the design (since registers will
+ not be close to the component that uses them). Also, when working with
+ the generated \small{VHDL} code, there will be a lot of extra ports
+ just to pass one state values, which can get quite confusing.
+
+ To fix this, we can simply \quote{push} the registers down into the
+ subcircuits. When we see a register that is connected directly to a
+ subcircuit, we remove the corresponding input and output port and put
+ the register inside the subcircuit instead. This is slightly less
+ trivial when looking at the Haskell code instead of the resulting
+ circuit, but the idea is still the same.
+
+ TODO: Example?
+
+ However, when applying this technique, we might push registers down
+ too far. When you intend to store a result of a stateless subfunction
+ in the caller's state and pass the current value of that state
+ variable to that same function, the register might get pushed down too
+ far. It is impossible to distinguish this case from similar code where
+ the called function is in fact stateful. From this we can conclude
+ that we have to either:
+
+ \startitemize
+ \item accept that the generated hardware might not be exactly what we
+ intended, in some specific cases. In most cases, the hardware will be
+ what we intended.
+ \item explicitely annotate state arguments and results in the input
+ description.
+ \stopitemize
+
+ The first option causes (non-obvious) exceptions in the language
+ intepretation. Also, automatically determining where registers should
+ end up is easier to implement correctly with explicit annotations, so
+ for these reasons we will look at how this annotations could work.
+
+
+ TODO: Note about conditions on state variables and checking them.
+
+ \subsection{Explicit state annotation}
+ To make our stateful descriptions unambigious and easier to translate,
+ we need some way for the developer to describe which arguments and
+ results are intended to become stateful.
+
+ Roughly, we have two ways to achieve this:
+ \startitemize[KR]
+ \item Use some kind of annotation method or syntactic construction in
+ the language to indicate exactly which argument and (part of the)
+ result is stateful. This means that the annotation lives
+ \quote{outside} of the function, it is completely invisible when
+ looking at the function body.
+ \item Use some kind of annotation on the type level, \eg give stateful
+ arguments and (part of) results a different type. This has the
+ potential to make this annotation visible inside the function as well,
+ such that when looking at a value inside the function body you can
+ tell if it's stateful by looking at its type. This could possibly make
+ the translation process a lot easier, since less analysis of the
+ program flow might be required.
+ \stopitemize
- External init state works for hardware generation as well.
+ From these approaches, the type level \quote{annotations} have been
+ implemented in Cλash. \in{Section}[sec:prototype:statetype] expands on
+ the possible ways this could have been implemented.
- Implementation issues: state splitting, linking input to output state,
- checking usage constraints on state variables.
+ 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
Due to these complications, we leave other forms of recursion as
future work as well.
+
+ \section{Supported types}
+ TODO
projects / matthijs / master-project / report.git / blobdiff