1 \chapter[chap:future]{Future work}
3 During the development of the prototype, it has become clear that Haskell is
4 not the perfect language for this work. There are two main problems:
6 \item Haskell is not expressive enough. Haskell is still quite new in the area
7 of dependent typing, something we use extensively. Also, Haskell has some
8 special syntax to write monadic composition and arrow composition, which is
9 well suited to normal Haskell programs. However, for our hardware
10 descriptions, we could use some similar, but other, syntactic sugar (as we
12 \item Haskell is too expressive. There are some things that Haskell can
13 express, but we can never properly translate. There are certain types (both
14 primitive types and certain type constructions like recursive types) we can
15 never translate, as well as certain forms of recursion.
18 It might be good to reevaluate the choice of language for Cλash, perhaps there
19 are other languages which are better suited to describe hardware, now that
20 we have learned a lot about it.
22 Perhaps Haskell (or some other language) can be extended by using a
23 preprocessor. There has been some (point of proof) work on a the Strathclyde
24 Haskell Enhancement (\small{SHE}) preprocessor for type-level programming,
25 perhaps we can extend that, or create something similar for hardware-specific
28 It is not unlikely that the most flexible way
29 forward would be to define a completely new language with exactly the needed
30 features. This is of course an enormous effort, which should not be taken
33 \section{Correctness proofs of the normalization system}
34 As stated in \in{section}[sec:normalization:properties], there are a
35 number of properties we would like to see verified about the
36 normalization system. In particular, the \emph{termination} and
37 \emph{completeness} of the system would be a good candidate for future
38 research. Specifying formal semantics for the Core language in
39 order to verify the \emph{soundness} of the system would be an even more
42 \section{Improved notation for hierarchical state}
43 The hierarchical state model requires quite some boilerplate code for unpacking
44 and distributing the input state and collecting and repacking the output
47 \in{Example}[ex:NestedState] shows a simple composition of two stateful
48 functions, \hs{funca} and \hs{funcb}. The state annotation using the
49 \hs{State} newtype has been left out, for clarity and because the proposed
50 approach below cannot handle that (yet).
52 \startbuffer[NestedState]
53 type FooState = ( AState, BState )
54 foo :: Word -> FooState -> (FooState, Word)
58 (sa', outa) = funca in sa
59 (sb', outb) = funcb outa sb
62 \placeexample[here][ex:NestedState]{Simple function composing two stateful
63 functions.}{\typebufferhs{NestedState}}
65 Since state handling always follows strict rules, there is really no other way
66 in which this state handling can be done (you can of course move some things
67 around from the where clause to the pattern or vice versa, but it would still
68 be effectively the same thing). This means it makes extra sense to hide this
69 boilerplate away. This would incur no flexibility cost at all, since there are
70 no other ways that would work.
72 One particular notation in Haskell that seems promising, is the \hs{do}
73 notation. This is meant to simplify Monad notation by hiding away some
74 details. It allows one to write a list of expressions, which are composed
75 using the monadic \emph{bind} operator, written in Haskell as \hs{>>}. For
84 will be desugared into:
87 (somefunc a) >> (otherfunc b)
90 The main reason to have the monadic notation, is to be able to wrap
91 results of functions in a datatype (the \emph{monad}) that can contain
92 extra information, and hide extra behavior in the binding operators.
94 The \hs{>>=} operator allows extracting the actual result of a function
95 and passing it to another function. The following snippet should
104 will be desugared into:
107 (somefunc a) >>= (\x -> otherfunc x)
110 The \hs{\x -> ...} notation creates a lambda abstraction in Haskell,
111 that binds the \hs{x} variable. Here, the \hs{>>=} operator is supposed
112 to extract whatever result somefunc has and pass it to the lambda
113 expression created. This will probably not make the monadic notation
114 completely clear to a reader without prior experience with Haskell, but
115 it should serve to understand the following discussion.
117 The monadic notation could perhaps be used to compose a number of
118 stateful functions into another stateful computation. Perhaps this could
119 move all the boilerplate code into the \hs{>>} and \hs{>>=} operators.
120 Because the boilerplate is still there (it has not magically disappeared,
121 just moved into these functions), the compiler should still be able to compile
122 these descriptions without any special magic (though perhaps it should
123 always inline the binding operators to reveal the flow of values).
125 This highlights an important aspect of using a functional language for our
126 descriptions: we can use the language itself to provide abstractions of common
127 patterns, making our code smaller and easier to read.
129 \subsection{Breaking out of the Monad}
130 However, simply using the monad notation is not as easy as it sounds. The main
131 problem is that the Monad type class poses a number of limitations on the
132 bind operator \hs{>>}. Most importantly, it has the following type signature:
135 (>>) :: (Monad m) => m a -> m b -> m b
138 This means that any expression in our composition must use the same Monad
139 instance as its type, only the "return" value can be different between
142 Ideally, we would like the \hs{>>} operator to have a type like the following
144 type Stateful s r = s -> (s, r)
145 (>>) :: Stateful s1 r1 -> Stateful s2 r2 -> Stateful (s1, s2) r2
148 What we see here, is that when we compose two stateful functions (that
149 have already been applied to their inputs, leaving just the state
150 argument to be applied to), the result is again a stateful function
151 whose state is composed of the two \emph{substates}. The return value is
152 simply the return value of the second function, discarding the first (to
153 preserve that result, the \hs{>>=} operator can be used).
155 There is a trick we can apply to change the signature of the \hs{>>} operator.
156 \small{GHC} does not require the bind operators to be part of the \hs{Monad}
157 type class, as long as it can use them to translate the do notation. This
158 means we can define our own \hs{>>} and \hs{>>=} operators, outside of the
159 \hs{Monad} type class. This does conflict with the existing methods of the
160 \hs{Monad} type class, so we should prevent \small{GHC} from loading those (and
161 all of the Prelude) by passing \type{-XNoImplicitPrelude} to \type{ghc}. This
162 is slightly inconvenient, but since we hardly using anything from the prelude,
163 this is not a big problem. We might even be able to replace some of the
164 Prelude with hardware-translatable versions by doing this.
166 The binding operators can now be defined exactly as they are needed. For
167 completeness, the \hs{return} function is also defined. It is not called
168 by \small{GHC} implicitly, but can be called explicitly by a hardware
169 description. \in{Example}[ex:StateMonad] shows these definitions.
171 \startbuffer[StateMonad]
172 (>>) :: Stateful s1 r1 -> Stateful s2 r2 -> Stateful (s1, s2) r2
173 f1 >> f2 = f1 >>= \_ -> f2
175 (>>=) :: Stateful s1 r1 -> (r1 -> Stateful s2 r2) -> Stateful (s1, s2) r2
176 f1 >>= f2 = \(s1, s2) -> let (s1', r1) = f1 s1
180 return :: r -> Stateful () r
181 return r = \s -> (s, r)
183 \placeexample[here][ex:StateMonad]{Binding operators to compose stateful computations}
184 {\typebufferhs{StateMonad}}
186 These definitions closely resemble the boilerplate of unpacking state,
187 passing it to two functions and repacking the new state. With these
188 definitions, we could have written \in{example}[ex:NestedState] a lot
189 shorter, see \in{example}[ex:DoState]. In this example the type signature of
190 foo is the same (though it is now written using the \hs{Stateful} type
191 synonym, it is still completely equivalent to the original: \hs{foo :: Word ->
192 FooState -> (FooState, Word)}.
194 Note that the \hs{FooState} type has changed (so indirectly the type of
195 \hs{foo} as well). Since the state composition by the \hs{>>} works on two
196 stateful functions at a time, the final state consists of nested two-tuples.
197 The final \hs{()} in the state originates from the fact that the \hs{return}
198 function has no real state, but is part of the composition. We could have left
199 out the return expression (and the \hs{outb <-} part) to make \hs{foo}'s return
200 value equal to \hs{funcb}'s, but this approach makes it clearer what is
203 \startbuffer[DoState]
204 type FooState = ( AState, (BState, ()) )
205 foo :: Word -> Stateful FooState Word
211 \placeexample[][ex:DoState]{Simple function composing two stateful
212 functions, using do notation.}
213 {\typebufferhs{DoState}}
215 An important implication of this approach is that the order of writing
216 function applications affects the state type. Fortunately, this problem can be
217 localized by consistently using type synonyms for state types (see
218 \in{section}[sec:prototype:substatesynonyms]), which should prevent
219 changes in other function's source when a function changes.
221 A less obvious implications of this approach is that the scope of variables
222 produced by each of these expressions (using the \hs{<-} syntax) is limited to
223 the expressions that come after it. This prevents values from flowing between
224 two functions (components) in two directions. For most Monad instances, this
225 is a requirement, but here it could have been different.
227 \subsection{Alternative syntax}
228 Because of these typing issues, misusing Haskell's do notation is probably not
229 the best solution here. However, it does show that using fairly simple
230 abstractions, we could hide a lot of the boilerplate code. Extending
231 \small{GHC} with some new syntax sugar similar to the do notation might be a
235 Another abstraction mechanism offered by Haskell are arrows. Arrows are
236 a generalization of monads \cite[hughes98], for which \GHC\ also supports
237 some syntax sugar \cite[paterson01]. Their use for hiding away state
238 boilerplate is not directly evident, but since arrows are a complex
239 concept further investigation is appropriate.
241 \section[sec:future:pipelining]{Improved notation or abstraction for pipelining}
242 Since pipelining is a very common optimization for hardware systems, it should
243 be easy to specify a pipelined system. Since it introduces quite some registers
244 into an otherwise regular combinational system, we might look for some way to
245 abstract away some of the boilerplate for pipelining.
247 Something similar to the state boilerplate removal above might be appropriate:
248 Abstract away some of the boilerplate code using combinators, then hide away
249 the combinators in special syntax. The combinators will be slightly different,
250 since there is a (typing) distinction between a pipeline stage and a pipeline
251 consisting of multiple stages. Also, it seems necessary to treat either the
252 first or the last pipeline stage differently, to prevent an off-by-one error
253 in the amount of registers (which is similar to the extra \hs{()} state type
254 in \in{example}[ex:DoState], which is harmless there, but would be a problem
255 if it introduced an extra, useless, pipeline stage).
257 This problem is slightly more complex than the problem we have seen before. One
258 significant difference is that each variable that crosses a stage boundary
259 needs a register. However, when a variable crosses multiple stage boundaries,
260 it must be stored for a longer period and should receive multiple registers.
261 Since we cannot find out from the combinator code where the result of the
262 combined values is used (at least not without using Template Haskell to
263 inspect the \small{AST}), there seems to be no easy way to find how much
264 registers are needed.
266 There seem to be two obvious ways of handling this problem:
269 \item Limit the scoping of each variable produced by a stage to the next
270 stage only. This means that any variables that are to be used in subsequent
271 stages should be passed on explicitly, which should allocate the required
274 This produces cumbersome code, where there is still a lot of explicitness
275 (though this could be hidden in syntax sugar).
276 \todo{The next sentence is unclear}
277 \item Scope each variable over every subsequent pipeline stage and allocate
278 the maximum number of registers that \emph{could} be needed. This means we
279 will allocate registers that are never used, but those could be optimized
280 away later. This does mean we need some way to introduce a variable number
281 of variables (depending on the total number of stages), assign the output of
282 a different register to each (\eg., a different part of the state) and scope
283 a different one of them over each the subsequent stages.
285 This also means that when adding a stage to an existing pipeline will change
286 the state type of each of the subsequent pipeline stages, and the state type
287 of the added stage depends on the number of subsequent stages.
289 Properly describing this will probably also require quite explicit syntax,
290 meaning this is not feasible without some special syntax.
293 Some other interesting issues include pipeline stages which are already
294 stateful, mixing pipelined with normal computation, etc.
297 The main problems of recursion have been described in
298 \in{section}[sec:recursion]. In the current implementation, recursion is
299 therefore not possible, instead we rely on a number of implicitly list-recursive
302 Since recursion is a very important and central concept in functional
303 programming, it would very much improve the flexibility and elegance of our
304 hardware descriptions if we could support (full) recursion.
306 For this, there are two main problems to solve:
309 \item For list recursion, how to make a description type check? It is quite
310 possible that improvements in the \small{GHC} type-checker will make this
311 possible, though it will still stay a challenge. Further advances in
312 dependent typing support for Haskell will probably help here as well.
314 \todo{Reference Christiaan and other type-level work
315 (http://personal.cis.strath.ac.uk/conor/pub/she/)}
316 \item For all recursion, there is the obvious challenge of deciding when
317 recursion is finished. For list recursion, this might be easier (Since the
318 base case of the recursion influences the type signatures). For general
319 recursion, this requires a complete set of simplification and evaluation
320 transformations to prevent infinite expansion. The main challenge here is how
321 to make this set complete, or at least define the constraints on possible
322 recursion that guarantee it will work.
324 \todo{Reference Christian for loop unrolling?}
327 \section{Multiple clock domains and asynchronicity}
328 Cλash currently only supports synchronous systems with a single clock domain.
329 In a lot of real-world systems, both of these limitations pose problems.
331 There might be asynchronous events to which a system wants to respond. The
332 most obvious asynchronous event is of course a reset signal. Though a reset
333 signal can be synchronous, that is less flexible (and a hassle to describe in
334 Cλash, currently). Since every function in Cλash describes the behavior on
335 each cycle boundary, we really cannot fit in asynchronous behavior easily.
337 Due to the same reason, multiple clock domains cannot be easily supported. There is
338 currently no way for the compiler to know in which clock domain a function
339 should operate and since the clock signal is never explicit, there is also no
340 way to express circuits that synchronize various clock domains.
342 A possible way to express more complex timing behavior would be to make
343 functions more generic event handlers, where the system generates a stream of
344 events (Like \quote{clock up}, \quote{clock down}, \quote{input A changed},
345 \quote{reset}, etc.). When working with multiple clock domains, each domain
346 could get its own clock events.
348 \startbuffer[AsyncDesc]
349 data Event = ClockUp | Reset | ...
351 type MainState = State Word
354 initstate :: MainState
357 main :: Word -> Event -> MainState -> (MainState, Word)
358 main inp event (State acc) = (State acc', acc')
361 -- On a reset signal, reset the accumulator and output
363 -- On a normal clock cycle, accumulate the result of func
364 ClockUp -> acc + (func inp event)
365 -- On any other event, leave state and output unchanged
368 -- func is some combinational expression
369 func :: Word -> Event -> Word
370 func inp _ = inp * 2 + 3
373 \placeexample[][ex:AsyncDesc]{Hardware description using asynchronous events.}
374 {\typebufferhs{AsyncDesc}}
378 \in{Example}[ex:AsyncDesc] shows a simple example of this event-based
379 approach. In this example we see that every function takes an input of
380 type \hs{Event}. The function \hs{main} that takes the output of
381 \hs{func} and accumulates it on every clock cycle. On a reset signal,
382 the accumulator is reset. The function \hs{func} is just a combinational
383 function, with no synchronous elements. We can see this because there
384 is no state and the event input is completely ignored. If the compiler
385 is smart enough, we could even leave the event input out for functions
386 that do not use it, either because they are completely combinational (like
387 in this example), or because they rely on the the caller to select the
390 This structure is similar to the event handling structure used to perform I/O
391 in languages like Amanda. \todo{ref} There is a top level case expression that
392 decides what to do depending on the current input event.
394 A slightly more complex example is show in
395 \in{example}[ex:MulticlockDesc]. It shows a system with two clock
396 domains. There is no real integration between the clock domains in this
397 example (there is one input and one output for each clock domain), but
398 it does show how multiple clocks can be distinguished.
400 \startbuffer[MulticlockDesc]
401 data Event = ClockUpA | ClockUpB | ...
403 type MainState = State (SubAState, SubBState)
406 initstate :: MainState
407 initstate = State (initsubastate, initsubbstate)
409 main :: Word -> Word -> Event -> MainState -> (MainState, Word, Word)
410 main inpa inpb event (State (sa, sb)) = (State (sa', sb'), outa, outb)
412 -- Only update the substates if the corresponding clock has an up
420 -- Always call suba and subb, so we can always have a value for our output
422 (sa'', outa) = suba inpa sa
423 (sb'', outb) = subb inpb sb
425 type SubAState = State ...
426 suba :: Word -> SubAState -> (SubAState, Word)
429 type SubBState = State ...
430 subb :: Word -> SubAState -> (SubAState, Word)
434 \placeexample[][ex:MulticlockDesc]{Hardware description with multiple clock domains through events.}
435 {\typebufferhs{MulticlockDesc}}
437 Note that in \in{example}[ex:MulticlockDesc] the \hs{suba} and \hs{subb}
438 functions are \emph{always} called, to get at their combinational
439 outputs. The new state is calculated as well, but only saved when the
440 right clock has an up transition.
442 As you can see, there is some code duplication in the case expression that
443 selects the right clock. One of the advantages of an explicit approach like
444 this, is that some of this duplication can be extracted away into helper
445 functions. For example, we could imagine a \hs{select_clock} function, which
446 takes a stateful function that is not aware of events, and changes it into a
447 function that only updates its state on a specific (clock) event. Such a
448 function is shown in \in{example}[ex:SelectClock].
450 \startbuffer[SelectClock]
451 select_clock :: Event
452 -> (input -> State s -> (State s, output))
453 -> (input -> State s -> Event -> (State s, output))
454 select_clock clock func inp state event = (state', out)
456 state' = if clock == event then state'' else state
457 (state'', out) = func inp state
459 main :: Word -> Word -> Event -> MainState -> (MainState, Word, Word)
460 main inpa inpb event (State (sa, sb)) = (State (sa', sb'), outa, outb)
462 (sa'', outa) = (select_clock ClockUpA suba) inpa sa event
463 (sb'', outb) = (select_clock ClockUpB subb) inpb sb event
465 \placeexample[][ex:SelectClock]{A function to filter clock events.}
466 {\typebufferhs{SelectClock}}
468 As you can see, this can greatly reduce the length of the main function, while
469 increasing the readability. As you might also have noted, the select\_clock
470 function takes any stateful function from the current Cλash prototype and
471 turns it into an event-aware function!
473 Going along this route for more complex timing behavior seems promising,
474 especially since it seems possible to express very advanced timing behaviors,
475 while still allowing simple functions without any extra overhead when complex
476 behavior is not needed.
478 The main cost of this approach will probably be extra complexity in the
479 compiler: the paths (state) data can take become very non-trivial, and it
480 is probably hard to properly analyze these paths and produce the
481 intended \VHDL\ description.
483 \section{Multiple cycle descriptions}
484 In the current Cλash prototype, every description is a single-cycle
485 description. In other words, every function describes behavior for each
486 separate cycle and any recursion (or folds, maps, etc.) is expanded in space.
488 Sometimes, you might want to have a complex description that could possibly
489 take multiple cycles. Some examples include:
492 \item Some kind of map or fold over a list that could be expanded in time
493 instead of space. This would result in a function that describes n cycles
494 instead of just one, where n is the length of the list.
495 \item A large combinational expressions that would introduce a very long
496 combinational path and thus limit clock frequency. Such an expression could
497 be broken up into multiple stages, which effectively results in a pipelined
498 system (see also \in{section}[sec:future:pipelining]) with a known delay.
499 There should probably be some way for the developer to specify the cycle
500 division of the expression, since automatically deciding on such a division
501 is too complex and contains too many trade-offs, at least initially.
502 \item Unbounded recursion. It is not possible to expand unbounded (or even
503 recursion with a depth that is not known at compile time) in space, since
504 there is no way to tell how much hardware to create (it might even be
507 When expanding infinite recursion over time, each step of the recursion can
508 simply become a single clock cycle. When expanding bounded but unknown
509 recursion, we probably need to add an extra data valid output bit or
513 Apart from translating each of these multiple cycle descriptions into a per-cycle
514 description, we also need to somehow match the type signature of the multiple
515 cycle description to the type signature of the single cycle description that
516 the rest of the system expects (assuming that the rest of the system is
517 described in the \quote{normal} per-cycle manner). For example, an infinitely
518 recursive expression typically has the return type \lam{[a]}, while the rest
519 of the system would expect just \lam{a} (since the recursive expression
520 generates just a single element each cycle).
522 Naively, this matching could be done using a (built-in) function with a
523 signature like \lam{[a] -> a}, which also serves as an indicator to the
524 compiler that some expanding over time is required. However, this poses a
525 problem for simulation: how will our Haskell implementation of this magical
526 built-in function know which element of the input list to return. This
527 obviously depends on the current cycle number, but there is no way for this
528 function to know the current cycle without breaking all kinds of safety and
529 purity properties. Making this function have some state input and output could
530 help, though this state is not present in the actual hardware (or perhaps
531 there is some state, if there are value passed from one recursion step to the
532 next, but how can the type of that state be determined by the type-checker?).
534 It seems that this is the most pressing problem for multi-cycle descriptions:
535 How to interface with the rest of the system, without sacrificing safety and
536 simulation capabilities?
538 \section{Higher order values in state}
539 Another interesting idea is putting a higher-order value inside a function's
540 state value. Since we can use higher-order values anywhere, why not in the
543 \startbuffer[HigherAccum]
544 -- The accumulator function that takes a word and returns a new accumulator
545 -- and the result so far. This is the function we want to put inside the
547 type Acc = Word -> (Acc, Word)
548 acc = \a -> (\b -> acc ( a + b ), a )
550 main :: Word -> State Acc -> (State Acc, Word)
551 main a s = (State s', out)
552 where (s', out) = s a
554 \placeexample[][ex:HigherAccum]{An accumulator using a higher-order state.}
555 {\typebufferhs{HigherAccum}}
557 As a (contrived) example, consider the accumulator in
558 \in{example}[ex:HigherAccum]. This code uses a function as its state,
559 which implicitly contains the value accumulated so far. This is a
560 fairly trivial example, that is more easy to write with a simple
561 \hs{Word} state value, but for more complex descriptions this style
562 might pay off. Note that in a way we are using the \emph{continuation
563 passing style} of writing functions, where we pass a sort of
564 \emph{continuation} from one cycle to the next.
566 Our normalization process completely removes higher-order values inside a
567 function by moving applications and higher-order values around so that every
568 higher-order value will eventually be full applied. However, if we would put a
569 higher-order value inside our state, the path from the higher-order value
570 definition to its application runs through the state, which is external to the
571 function. A higher-order value defined in one cycle is not applied until a
572 later cycle. Our normalization process only works within the function, so it
573 cannot remove this use of a higher-order value.
575 However, we cannot leave a higher-order value in our state value, since it is
576 impossible to generate a register containing a higher-order value, we simply
577 cannot translate a function type to a hardware type. To solve this, we must
578 replace the higher-order value inside our state with some other value
579 representing these higher-order values.
581 On obvious approach here is to use an algebraic datatype where each
582 constructor represents one possible higher-order value that can end up in the
583 state and each constructor has an argument for each free variable of the
584 higher-order value replaced. This allows us to completely reconstruct the
585 higher-order value at the spot where it is applied, and thus the higher-order
588 This approach is commonly known as the \quote{Reynolds approach to
589 defunctionalization}, first described by J.C. Reynolds \cite[reynolds98]\ and
590 seems to apply well to this situation. One note here is that Reynolds'
591 approach puts all the higher-order values in a single datatype. For a typed
592 language, we will at least have to have a single datatype for each function
593 type, since we cannot mix them. It would be even better to split these
594 data-types a bit further, so that such a datatype will never hold a constructor
595 that is never used for a particular state variable. This separation is
596 probably a non-trivial problem, though.
598 \section{Don't care values}
599 A powerful value in \VHDL\ is the \emph{don't care} value, given as
600 \type{'-'}. This value tells the compiler that you do not really care about
601 which value is assigned to a signal, allowing the compiler to make some
602 optimizations. Since choice in hardware is often implemented using
603 a collection of logic gates instead of multiplexers only, synthesizers can
604 often reduce the amount of hardware needed by smartly choosing values for
605 these don't care cases.
607 There is not really anything comparable with don't care values in normal
608 programming languages. The closest thing is an undefined or uninitialized
609 value, though those are usually associated with error conditions.
611 It would be useful if Cλash also has some way to specify don't care values.
612 When looking at the Haskell typing system, there are really two ways to do
616 \item Add a special don't care value to every datatype. This includes the
617 obvious \hs{Bit} type, but will also need to include every user defined
618 type. An exception can be made for vectors and product types, since those
619 can simply use the don't care values of the types they contain.
621 This would also require some kind of \quote{Don't careable} type class
622 that allows each type to specify what its don't care value is. The
623 compiler must then recognize this constant and replace it with don't care
624 values in the final \VHDL\ code.
626 This is of course a very intrusive solution. Every type must become member
627 of this type class, and there is now some member in every type that is a
628 special don't care value. Guaranteeing the obvious don't care semantics
629 also becomes harder, since every pattern match or case expressions must now
630 also take care of the don't care value (this might actually be an
631 advantage, since it forces designers to specify how to handle don't care
632 for different operations).
634 \item Use the special \hs{undefined}, or \emph{bottom} value present in
635 Haskell. This is a type that is member of all types automatically, without
636 any explicit declarations.
638 Any expression that requires evaluation of an undefined value
639 automatically becomes undefined itself (or rather, there is some exception
640 mechanism). Since Haskell is lazy, this means that whenever it tries to
641 evaluate undefined, it is apparently required for determining the output
642 of the system. This property is useful, since typically a don't care
643 output is used when some output is not valid and should not be read. If
644 it is in fact not read, it should never be evaluated and simulation should
647 In practice, this works less ideal. In particular, pattern matching is not
648 always smart enough to deal with undefined. Consider the following
649 definition of an \hs{and} operator:
652 and :: Bit -> Bit -> Bit
658 When using the \hs{and} operation on an undefined (don't care) and a Low
659 value should always return a Low value. Its value does not depend on the
660 value chosen for the don't care value. However, though when applying the
661 above and function to \hs{Low} and \hs{undefined} results in exactly that
662 behavior, the result is \hs{undefined} when the arguments are swapped.
663 This is because the first pattern forces the first argument to be
664 evaluated. If it is \hs{undefined}, evaluation is halted and an exception
665 is show, which is not what is intended.
668 These options should be explored further to see if they provide feasible
669 methods for describing don't care conditions. Possibly there are completely
670 other methods which work better.
672 % vim: set sw=2 sts=2 expandtab: