Add sitenote about arguments vs. inputs.

[matthijs/master-project/report.git] / Chapters / HardwareDescription.tex

diff --git a/Chapters/HardwareDescription.tex b/Chapters/HardwareDescription.tex
index ada3817f4c94e61cf03810b72d24f379ed026789..ff4b0c0b3143c78983f6464ad52d95e090b5488f 100644 (file)
--- a/Chapters/HardwareDescription.tex
+++ b/Chapters/HardwareDescription.tex
@@ -1,8 +1,9 @@
 \chapter[chap:description]{Hardware description}
 \chapter[chap:description]{Hardware description}

-  This chapter will provide an overview of the hardware description language
-  that was created and the issues that have arisen in the process. It will
-  focus on the issues of the language, not the implementation. The prototype
-  implementation will be discussed in \in{chapter}[chap:prototype].
+  In this chapter an overview will be provided of the hardware
+  description language that was created and the issues that have arisen
+  in the process. The focus will be on the issues of the language, not
+  the implementation. The prototype implementation will be discussed in
+  \in{chapter}[chap:prototype].

 
   \todo{Shortshort introduction to Cλash (Bit, Word, and, not, etc.)}
 
 
   \todo{Shortshort introduction to Cλash (Bit, Word, and, not, etc.)}
 


@@ -15,25 +16,64 @@
   hardware whenever possible, to make working with Cλash as intuitive as
   possible.
    
   hardware whenever possible, to make working with Cλash as intuitive as
   possible.
    

+  \placeintermezzo{}{
+    \defref{top level binder}
+    \defref{top level function}
+    \startframedtext[width=8cm,background=box,frame=no]
+    \startalignment[center]
+      {\tfa Top level binders and functions}
+    \stopalignment
+    \blank[medium]
+      A top level binder is any binder (variable) that is declared in
+      the \quote{global} scope of a Haskell program (as opposed to a
+      binder that is bound inside a function.
+
+      In Haskell, there is no sharp distinction between a variable and a
+      function: A function is just a variable (binder) with a function
+      type. This means that a top level function is just any top level
+      binder with a function type.
+
+      As an example, consider the following Haskell snippet:
+
+      \starthaskell
+        foo :: Int -> Int
+        foo x = inc x
+          where
+            inc = \y -> y + 1
+      \stophaskell
+
+      Here, \hs{foo} is a top level binder, whereas \hs{inc} is a
+      function (since it is bound to a lambda extraction, indicated by
+      the backslash) but is not a top level binder or function.  Since
+      the type of \hs{foo} is a function type, namely \hs{Int -> Int},
+      it is also a top level function.
+    \stopframedtext
+  }

   In this chapter we describe how to interpret a Haskell program from a
   hardware perspective. We provide a description of each Haskell language
   element that needs translation, to provide a clear picture of what is
   supported and how.
 
   In this chapter we describe how to interpret a Haskell program from a
   hardware perspective. We provide a description of each Haskell language
   element that needs translation, to provide a clear picture of what is
   supported and how.
 

+

   \section[sec:description:application]{Function application}
   \section[sec:description:application]{Function application}

-  \todo{Sidenote: Inputs vs arguments}

   The basic syntactic elements of a functional program are functions and
   The basic syntactic elements of a functional program are functions and

-  function application. These have a single obvious \small{VHDL} translation: Each
-  function becomes a hardware component, where each argument is an input port
-  and the result value is the (single) output port. This output port can have
-  a complex type (such as a tuple), so having just a single output port does
-  not pose a limitation.
+  function application. These have a single obvious \small{VHDL}
+  translation: Each top level function becomes a hardware component, where each
+  argument is an input port and the result value is the (single) output
+  port. This output port can have a complex type (such as a tuple), so
+  having just a single output port does not pose a limitation.

 
   Each function application in turn becomes component instantiation. Here, the
   result of each argument expression is assigned to a signal, which is mapped
   to the corresponding input port. The output port of the function is also
   mapped to a signal, which is used as the result of the application.
 
 
   Each function application in turn becomes component instantiation. Here, the
   result of each argument expression is assigned to a signal, which is mapped
   to the corresponding input port. The output port of the function is also
   mapped to a signal, which is used as the result of the application.
 

+  Since every top level function generates its own component, the
+  hierarchy of of function calls is reflected in the final \VHDL\ output
+  as well, creating a hierarchical \VHDL\ description of the hardware.
+  This separation in different components makes the resulting \VHDL\
+  output easier to read and debug.
+

   \in{Example}[ex:And3] shows a simple program using only function
   application and the corresponding architecture.
 
   \in{Example}[ex:And3] shows a simple program using only function
   application and the corresponding architecture.
 


@@ -44,7 +84,6 @@ and3 :: Bit -> Bit -> Bit -> Bit
 and3 a b c = and (and a b) c
 \stopbuffer
 
 and3 a b c = and (and a b) c
 \stopbuffer
 

-\todo{Mirror this image vertically}

   \startuseMPgraphic{And3}
     save a, b, c, anda, andb, out;
 
   \startuseMPgraphic{And3}
     save a, b, c, anda, andb, out;
 


@@ -82,9 +121,6 @@ and3 a b c = and (and a b) c
       {\boxedgraphic{And3}}{The architecture described by the Haskell description.}
     \stopcombination
 
       {\boxedgraphic{And3}}{The architecture described by the Haskell description.}
     \stopcombination
 

-  \todo{This section should also mention hierarchy, top level functions and
-  subcircuits}
-

   \section{Choice}
     Although describing components and connections allows us to describe a lot of
     hardware designs already, there is an obvious thing missing: Choice. We
   \section{Choice}
     Although describing components and connections allows us to describe a lot of
     hardware designs already, there is an obvious thing missing: Choice. We


@@ -99,8 +135,7 @@ and3 a b c = and (and a b) c
     the condition is false.
 
     This \hs{if} function would then essentially describe a multiplexer and
     the condition is false.
 
     This \hs{if} function would then essentially describe a multiplexer and

-    allows us to describe any architecture that uses multiplexers. \fxnote{Are
-    there other mechanisms of choice in hardware?}
+    allows us to describe any architecture that uses multiplexers.

 
     However, to be able to describe our hardware in a more convenient way, we
     also want to translate Haskell's choice mechanisms. The easiest of these
 
     However, to be able to describe our hardware in a more convenient way, we
     also want to translate Haskell's choice mechanisms. The easiest of these


@@ -109,7 +144,26 @@ and3 a b c = and (and a b) c
     simply be translated to a conditional assignment, where the conditions use
     equality comparisons against the constructors in the \hs{case} expressions.
 
     simply be translated to a conditional assignment, where the conditions use
     equality comparisons against the constructors in the \hs{case} expressions.
 

-    \todo{Assignment vs multiplexers}
+    \placeintermezzo{}{
+      \defref{substitution notation}
+      \startframedtext[width=8cm,background=box,frame=no]
+      \startalignment[center]
+        {\tfa Arguments / results vs. inputs / outputs}
+      \stopalignment
+      \blank[medium]
+        Due to the translation chosen for function application, there is a
+        very strong relation between arguments, results, inputs and outputs.
+        For clarity, the former two will always refer to the arguments and
+        results in the functional description (either Haskell or Core). The
+        latter two will refer to input and output ports in the generated
+        \VHDL.
+
+        Even though these concepts seem to be nearly identical, when stateful
+        functions are introduces we will see arguments and results that will
+        not get translated into input and output ports, making this
+        distinction more important.
+      \stopframedtext
+    }

 
     In \in{example}[ex:CaseInv] a simple \hs{case} expression is shown,
     scrutinizing a boolean value. The corresponding architecture has a
 
     In \in{example}[ex:CaseInv] a simple \hs{case} expression is shown,
     scrutinizing a boolean value. The corresponding architecture has a


@@ -120,14 +174,12 @@ and3 a b c = and (and a b) c
     parallel, regardless of which output would be chosen eventually).
     
     If we would translate a Boolean to a bit value, we could of course remove
     parallel, regardless of which output would be chosen eventually).
     
     If we would translate a Boolean to a bit value, we could of course remove

-    the comparator and directly feed 'in' into the multiplex (or even use an
+    the comparator and directly feed 'in' into the multiplexer (or even use an

     inverter instead of a multiplexer).  However, we will try to make a
     general translation, which works for all possible \hs{case} expressions.
     inverter instead of a multiplexer).  However, we will try to make a
     general translation, which works for all possible \hs{case} expressions.

-    Optimizations such as these are left for the \VHDL synthesizer, which
+    Optimizations such as these are left for the \VHDL\ synthesizer, which

     handles them very well.
 
     handles them very well.
 

-    \todo{Be more explicit about >2 alternatives}
-

     \startbuffer[CaseInv]
     inv :: Bool -> Bool
     inv x = case x of
     \startbuffer[CaseInv]
     inv :: Bool -> Bool
     inv x = case x of


@@ -198,30 +250,36 @@ and3 a b c = and (and a b) c
     nested) multiplexers. These multiplexers are driven by comparators and
     other logic, that check each pattern in turn.
 
     nested) multiplexers. These multiplexers are driven by comparators and
     other logic, that check each pattern in turn.
 

+    In these examples we have seen only binary case expressions and pattern
+    matches (\ie, with two alternatives). In practice, case expressions can
+    choose between more than two values, resulting in a number of nested
+    multiplexers.
+

   \section{Types}
   \section{Types}

-    We've seen the two most basic functional concepts translated to hardware:
-    Function application and choice. Before we look further into less obvious
-    concepts like higher order expressions and polymorphism, we will have a
-    look at the types of the values we use in our descriptions.
-
-    When working with values in our descriptions, we'll also need to provide
-    some way to translate the values used to hardware equivalents. In
-    particular, this means having to come up with a hardware equivalent for
-    every \emph{type} used in our program.
-
-    Since most functional languages have a lot of standard types that are
-    hard to translate (integers without a fixed size, lists without a static
-    length, etc.), we will start out by defining a number of \quote{builtin}
-    types ourselves. These types are builtin in the sense that our compiler
-    will have a fixed VHDL type for these. User defined types, on the other
-    hand, will have their hardware type derived directly from their Haskell
-    declaration automatically, according to the rules we sketch here.
+    Translation of two most basic functional concepts has been
+    discussed: Function application and choice. Before looking further
+    into less obvious concepts like higher-order expressions and
+    polymorphism, the possible types that can be used in hardware
+    descriptions will be discussed.
+
+    Some way is needed to translate every values used to its hardware
+    equivalents. In particular, this means a hardware equivalent for
+    every \emph{type} used in a hardware description is needed
+
+    Since most functional languages have a lot of standard types that
+    are hard to translate (integers without a fixed size, lists without
+    a static length, etc.), a number of \quote{built-in} types will be
+    defined first. These types are built-in in the sense that our
+    compiler will have a fixed VHDL type for these. User defined types,
+    on the other hand, will have their hardware type derived directly
+    from their Haskell declaration automatically, according to the rules
+    sketched here.

 
     \todo{Introduce Haskell type syntax (type constructors, type application,
     :: operator?)}
 
 
     \todo{Introduce Haskell type syntax (type constructors, type application,
     :: operator?)}
 

-    \subsection{Builtin types}
-      The language currently supports the following builtin types. Of these,
+    \subsection{Built-in types}
+      The language currently supports the following built-in types. Of these,

       only the \hs{Bool} type is supported by Haskell out of the box (the
       others are defined by the Cλash package, so they are user-defined types
       from Haskell's point of view).
       only the \hs{Bool} type is supported by Haskell out of the box (the
       others are defined by the Cλash package, so they are user-defined types
       from Haskell's point of view).


@@ -236,7 +294,7 @@ and3 a b c = and (and a b) c
         \todo{Sidenote bit vs stdlogic}
       \stopdesc
       \startdesc{\hs{Bool}}
         \todo{Sidenote bit vs stdlogic}
       \stopdesc
       \startdesc{\hs{Bool}}

-        This is the only builtin Haskell type supported and is translated
+        This is the only built-in Haskell type supported and is translated

         exactly like the Bit type (where a value of \hs{True} corresponds to a
         value of \hs{High}). Supporting the Bool type is particularly
         useful to support \hs{if ... then ... else ...} expressions, which
         exactly like the Bit type (where a value of \hs{True} corresponds to a
         value of \hs{High}). Supporting the Bool type is particularly
         useful to support \hs{if ... then ... else ...} expressions, which


@@ -303,28 +361,30 @@ and3 a b c = and (and a b) c
 
         Here, a type synonym \hs{RegisterIndex} is defined that is equal to
         the \hs{RangedWord} type constructor applied to the type \hs{D7}. In
 
         Here, a type synonym \hs{RegisterIndex} is defined that is equal to
         the \hs{RangedWord} type constructor applied to the type \hs{D7}. In

-        other words, this defines an unsigned word with values from 0 to 7
-        (inclusive). This word can be be used to index the 8 element vector
-        \hs{RegisterState} above.
+        other words, this defines an unsigned word with values from
+        {\definedfont[Serif*normalnum]0 to 7} (inclusive). This word can be be used to index the
+        8 element vector \hs{RegisterState} above.

 
         This type is translated to the \type{unsigned} \small{VHDL} type.
       \stopdesc
 
         This type is translated to the \type{unsigned} \small{VHDL} type.
       \stopdesc

-      \fxnote{There should be a reference to Christiaan's work here.}
+
+      The integer and vector built-in types are discussed in more detail
+      in \cite[baaij09].

 
     \subsection{User-defined types}
       There are three ways to define new types in Haskell: Algebraic
       datatypes with the \hs{data} keyword, type synonyms with the \hs{type}
 
     \subsection{User-defined types}
       There are three ways to define new types in Haskell: Algebraic
       datatypes with the \hs{data} keyword, type synonyms with the \hs{type}

-      keyword and type renamings with the \hs{newtype} keyword. \GHC
+      keyword and type renamings with the \hs{newtype} keyword. \GHC\

       offers a few more advanced ways to introduce types (type families,
       existential typing, \small{GADT}s, etc.) which are not standard
       Haskell.  These will be left outside the scope of this research.
 
       Only an algebraic datatype declaration actually introduces a
       offers a few more advanced ways to introduce types (type families,
       existential typing, \small{GADT}s, etc.) which are not standard
       Haskell.  These will be left outside the scope of this research.
 
       Only an algebraic datatype declaration actually introduces a

-      completely new type, for which we provide the \VHDL translation
+      completely new type, for which we provide the \VHDL\ translation

       below. Type synonyms and renamings only define new names for
       existing types (where synonyms are completely interchangeable and
       below. Type synonyms and renamings only define new names for
       existing types (where synonyms are completely interchangeable and

-      renamings need explicit conversion). Therefore, these don't need
-      any particular \VHDL translation, a synonym or renamed type will
+      renamings need explicit conversion). Therefore, these do not need
+      any particular \VHDL\ translation, a synonym or renamed type will

       just use the same representation as the original type. The
       distinction between a renaming and a synonym does no longer matter
       in hardware and can be disregarded in the generated \VHDL.
       just use the same representation as the original type. The
       distinction between a renaming and a synonym does no longer matter
       in hardware and can be disregarded in the generated \VHDL.


@@ -335,14 +395,14 @@ and3 a b c = and (and a b) c
         A product type is an algebraic datatype with a single constructor with
         two or more fields, denoted in practice like (a,b), (a,b,c), etc. This
         is essentially a way to pack a few values together in a record-like
         A product type is an algebraic datatype with a single constructor with
         two or more fields, denoted in practice like (a,b), (a,b,c), etc. This
         is essentially a way to pack a few values together in a record-like

-        structure. In fact, the builtin tuple types are just algebraic product
+        structure. In fact, the built-in tuple types are just algebraic product

         types (and are thus supported in exactly the same way).
 
         The \quote{product} in its name refers to the collection of values belonging
         types (and are thus supported in exactly the same way).
 
         The \quote{product} in its name refers to the collection of values belonging

-        to this type. The collection for a product type is the cartesian
+        to this type. The collection for a product type is the Cartesian

         product of the collections for the types of its fields.
 
         product of the collections for the types of its fields.
 

-        These types are translated to \VHDL record types, with one field for
+        These types are translated to \VHDL\ record types, with one field for

         every field in the constructor. This translation applies to all single
         constructor algebraic datatypes, including those with just one
         field (which are technically not a product, but generate a VHDL
         every field in the constructor. This translation applies to all single
         constructor algebraic datatypes, including those with just one
         field (which are technically not a product, but generate a VHDL


@@ -357,7 +417,7 @@ and3 a b c = and (and a b) c
         Note that Haskell's \hs{Bool} type is also defined as an
         enumeration type, but we have a fixed translation for that.
         
         Note that Haskell's \hs{Bool} type is also defined as an
         enumeration type, but we have a fixed translation for that.
         

-        These types are translated to \VHDL enumerations, with one value for
+        These types are translated to \VHDL\ enumerations, with one value for

         each constructor. This allows references to these constructors to be
         translated to the corresponding enumeration value.
       \stopdesc
         each constructor. This allows references to these constructors to be
         translated to the corresponding enumeration value.
       \stopdesc


@@ -365,15 +425,15 @@ and3 a b c = and (and a b) c
         A sum type is an algebraic datatype with multiple constructors, where
         the constructors have one or more fields. Technically, a type with
         more than one field per constructor is a sum of products type, but
         A sum type is an algebraic datatype with multiple constructors, where
         the constructors have one or more fields. Technically, a type with
         more than one field per constructor is a sum of products type, but

-        for our purposes this distinction does not really make a difference,
-        so we'll leave it out.
+        for our purposes this distinction does not really make a
+        difference, so this distinction is note made.

 
         The \quote{sum} in its name refers again to the collection of values
         belonging to this type. The collection for a sum type is the
         union of the the collections for each of the constructors.
 
         Sum types are currently not supported by the prototype, since there is
 
         The \quote{sum} in its name refers again to the collection of values
         belonging to this type. The collection for a sum type is the
         union of the the collections for each of the constructors.
 
         Sum types are currently not supported by the prototype, since there is

-        no obvious \VHDL alternative. They can easily be emulated, however, as
+        no obvious \VHDL\ alternative. They can easily be emulated, however, as

         we will see from an example:
 
         \starthaskell
         we will see from an example:
 
         \starthaskell


@@ -396,17 +456,17 @@ and3 a b c = and (and a b) c
         last one if \hs{B}), all the other ones have no useful value.
 
         An obvious problem with this naive approach is the space usage: The
         last one if \hs{B}), all the other ones have no useful value.
 
         An obvious problem with this naive approach is the space usage: The

-        example above generates a fairly big \VHDL type. Since we can be
+        example above generates a fairly big \VHDL\ type. Since we can be

         sure that the two \hs{Word}s in the \hs{Sum} type will never be valid
         at the same time, this is a waste of space.
 
         sure that the two \hs{Word}s in the \hs{Sum} type will never be valid
         at the same time, this is a waste of space.
 

-        Obviously, we could do some duplication detection here to reuse a
+        Obviously, duplication detection could be used to reuse a

         particular field for another constructor, but this would only
         particular field for another constructor, but this would only

-        partially solve the problem. If I would, for example, have an array of
-        8 bits and an 8 bit unsiged word, these are different types and could
-        not be shared. However, in the final hardware, both of these types
-        would simply be 8 bit connections, so we have a 100\% size increase by
-        not sharing these.
+        partially solve the problem. If two fields would be, for
+        example, an array of 8 bits and an 8 bit unsiged word, these are
+        different types and could not be shared. However, in the final
+        hardware, both of these types would simply be 8 bit connections,
+        so we have a 100\% size increase by not sharing these.

       \stopdesc
 
       Another interesting case is that of recursive types. In Haskell, an
       \stopdesc
 
       Another interesting case is that of recursive types. In Haskell, an


@@ -423,33 +483,35 @@ and3 a b c = and (and a b) c
       immediately shows the problem with recursive types: What hardware type
       to allocate here? 
       
       immediately shows the problem with recursive types: What hardware type
       to allocate here? 
       

-      If we would use the naive approach for sum types we described above, we
-      would create a record where the first field is an enumeration to
-      distinguish \hs{Empty} from \hs{Cons}. Furthermore, we would add two
-      more fields: One with the (\VHDL equivalent of) type \hs{t} (assuming we
-      actually know what type this is at compile time, this should not be a
-      problem) and a second one with type \hs{List t}.  The latter one is of
-      course a problem: This is exactly the type we were trying to translate
-      in the first place. 
+      If the naive approach for sum types described above would be used,
+      a record would be created where the first field is an enumeration
+      to distinguish \hs{Empty} from \hs{Cons}. Furthermore, two more
+      fields would be added: One with the (\VHDL\ equivalent of) type
+      \hs{t} (assuming this type is actually known at compile time, this
+      should not be a problem) and a second one with type \hs{List t}.
+      The latter one is of course a problem: This is exactly the type
+      that was to be translated in the first place.

       
       

-      Our \VHDL type will thus become infinitely deep. In other words, there
-      is no way to statically determine how long (deep) the list will be
-      (it could even be infinite).
+      The resulting \VHDL\ type will thus become infinitely deep. In
+      other words, there is no way to statically determine how long
+      (deep) the list will be (it could even be infinite).

       
       

-      In general, we can say we can never properly translate recursive types:
-      All recursive types have a potentially infinite value (even though in
+      In general, recursive types can never be properly translated: All
+      recursive types have a potentially infinite value (even though in

       practice they will have a bounded value, there is no way for the
       compiler to automatically determine an upper bound on its size).
 
   \subsection{Partial application}
       practice they will have a bounded value, there is no way for the
       compiler to automatically determine an upper bound on its size).
 
   \subsection{Partial application}

-  Now we've seen how to to translate application, choice and types, we will
-  get to a more complex concept: Partial applications. A \emph{partial
-  application} is any application whose (return) type is (again) a function
-  type.
+  Now the translation of application, choice and types has been
+  discussed, a more complex concept can be considered: Partial
+  applications. A \emph{partial application} is any application whose
+  (return) type is (again) a function type.

 
 

-  From this, we can see that the translation rules for full application do not
-  apply to a partial application. \in{Example}[ex:Quadruple] shows an example
-  use of partial application and the corresponding architecture.
+  From this, it should be clear that the translation rules for full
+  application does not apply to a partial application: There are not
+  enough values for all the input ports in the resulting \VHDL.
+  \in{Example}[ex:Quadruple] shows an example use of partial application
+  and the corresponding architecture.

 
 \startbuffer[Quadruple]
 -- | Multiply the input word by four.
 
 \startbuffer[Quadruple]
 -- | Multiply the input word by four.


@@ -496,18 +558,20 @@ quadruple n = mul (mul n)
   Here, the definition of mul is a partial function application: It applies
   the function \hs{(*) :: Word -> Word -> Word} to the value \hs{2 :: Word},
   resulting in the expression \hs{(*) 2 :: Word -> Word}. Since this resulting
   Here, the definition of mul is a partial function application: It applies
   the function \hs{(*) :: Word -> Word -> Word} to the value \hs{2 :: Word},
   resulting in the expression \hs{(*) 2 :: Word -> Word}. Since this resulting

-  expression is again a function, we can't generate hardware for it directly.
-  This is because the hardware to generate for \hs{mul} depends completely on
-  where and how it is used. In this example, it is even used twice.
-
-  However, it is clear that the above hardware description actually describes
-  valid hardware. In general, we can see that any partial applied function
-  must eventually become completely applied, at which point we can generate
-  hardware for it using the rules for function application given in
-  \in{section}[sec:description:application]. It might mean that a partial
-  application is passed around quite a bit (even beyond function boundaries),
-  but eventually, the partial application will become completely applied.
-  \todo{Provide a step-by-step example of how this works}
+  expression is again a function, hardware cannot be generated for it
+  directly.  This is because the hardware to generate for \hs{mul}
+  depends completely on where and how it is used. In this example, it is
+  even used twice.
+
+  However, it is clear that the above hardware description actually
+  describes valid hardware. In general, any partial applied function
+  must eventually become completely applied, at which point hardware for
+  it can be generated using the rules for function application given in
+  \in{section}[sec:description:application]. It might mean that a
+  partial application is passed around quite a bit (even beyond function
+  boundaries), but eventually, the partial application will become
+  completely applied. An example of this principe is given in
+  \in{section}[sec:normalization:defunctionalization].

 
   \section{Costless specialization}
     Each (complete) function application in our description generates a
 
   \section{Costless specialization}
     Each (complete) function application in our description generates a


@@ -572,14 +636,14 @@ quadruple n = mul (mul n)
 
       Specialization is useful for our hardware descriptions for functions
       that contain arguments that cannot be translated to hardware directly
 
       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
+      (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
       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
+    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 \hs{case}
     expression returning lambda's, for example).
     variables (e.g., a binder referring to a top level function), lambda
     expressions and more complex expressions with a function type (a \hs{case}
     expression returning lambda's, for example).


@@ -590,7 +654,7 @@ quadruple n = mul (mul n)
     become complete applications, applied lambda expressions disappear by
     applying β-reduction, etc.
 
     become complete applications, applied lambda expressions disappear by
     applying β-reduction, etc.
 

-    So any higher order value will be \quote{pushed down} towards its
+    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.
     
     application just like partial applications. Whenever a function boundary
     needs to be crossed, the called function can be specialized.
     


@@ -600,7 +664,7 @@ quadruple n = mul (mul n)
     In Haskell, values can be \emph{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 two element types. Haskell
     In Haskell, values can be \emph{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 two element types. Haskell

-    typeclasses allow a function to work on a specific set of types, but the
+    type classes allow a function to work on a specific set of types, but the

     general idea is the same. The opposite of this is a \emph{monomorphic}
     value, which has a single, fixed, type.
 
     general idea is the same. The opposite of this is a \emph{monomorphic}
     value, which has a single, fixed, type.
 


@@ -611,19 +675,20 @@ quadruple n = mul (mul n)
 %    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 
 
 %    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
+    When generating hardware, polymorphism cannot be easily translated. How

     many wires 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)?
     many wires 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)?

-    Note that Cλash currently does not allow user-defined typeclasses,
-    but does partly support some of the builtin typeclasses (like \hs{Num}).
+    Note that Cλash currently does not allow user-defined type classes,
+    but does partly support some of the built-in type classes (like \hs{Num}).

 
     Fortunately, we can again use the principle of specialization: Since every
     function application generates a separate piece of hardware, we can know
 
     Fortunately, we can again use the principle of specialization: Since every
     function application generates a separate piece 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).
+    the types of all arguments exactly. Provided that existential typing
+    (which is a \GHC\ extension) is not used 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).

 
     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
 
     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


@@ -631,8 +696,9 @@ quadruple n = mul (mul n)
     specialized to work just for these specific types, removing the
     polymorphism from the applied functions as well.
 
     specialized to work just for these specific types, removing the
     polymorphism from the applied functions as well.
 

-    \defref{top level function}Our top level function must not have a polymorphic type (otherwise we
-    wouldn't know the hardware interface to our top level function).
+    \defref{entry function}The entry function must not have a
+    polymorphic type (otherwise no hardware interface could be generated
+    for the entry function).

 
     By induction, this means that all functions that are (indirectly) called
     by our top level function (meaning all functions that are translated in
 
     By induction, this means that all functions that are (indirectly) called
     by our top level function (meaning all functions that are translated in


@@ -640,7 +706,7 @@ quadruple n = mul (mul n)
 
   \section{State}
     A very important concept in hardware designs is \emph{state}. In a
 
   \section{State}
     A very important concept in hardware designs is \emph{state}. In a

-    stateless (or, \emph{combinatoric}) design, every output is  directly and solely dependent on the
+    stateless (or, \emph{combinational}) design, every output is  directly and solely dependent on the

     inputs. In a stateful design, the outputs can depend on the history of
     inputs, or the \emph{state}. State is usually stored in \emph{registers},
     which retain their value during a clockcycle, and are typically updated at
     inputs. In a stateful design, the outputs can depend on the history of
     inputs, or the \emph{state}. State is usually stored in \emph{registers},
     which retain their value during a clockcycle, and are typically updated at


@@ -650,9 +716,8 @@ quadruple n = mul (mul n)
 
     \todo{Sidenote? Registers can contain any (complex) type}
   
 
     \todo{Sidenote? Registers can contain any (complex) type}
   

-    To make our hardware description language useful to describe more than
-    simple combinatoric designs, we'll need to be able to describe state in
-    some way.
+    To make Cλash useful to describe more than simple combinational
+    designs, it needs to be able to describe state in some way.

 
     \subsection{Approaches to state}
       In Haskell, functions are always pure (except when using unsafe
 
     \subsection{Approaches to state}
       In Haskell, functions are always pure (except when using unsafe


@@ -661,22 +726,48 @@ quadruple n = mul (mul n)
       evaluate a given function with given inputs, it will always provide the
       same output.
 
       evaluate a given function with given inputs, it will always provide the
       same output.
 

-      \todo{Define pure}
+      \placeintermezzo{}{
+        \defref{purity}
+        \startframedtext[width=8cm,background=box,frame=no]
+        \startalignment[center]
+          {\tfa Purity}
+        \stopalignment
+        \blank[medium]
+
+        A function is said to be pure if it satisfies two conditions:
+
+        \startitemize[KR]
+        \item When a pure function is called with the same arguments twice, it should
+        return the same value in both cases.
+        \item When a pure function is called, it should have not
+        observable side-effects.
+        \stopitemize
+
+        Purity is an important property in functional languages, since
+        it enables all kinds of mathematical reasoning and
+        optimizattions with pure functions, that are not guaranteed to
+        be correct for impure functions.
+
+        An example of a pure function is the square root function
+        \hs{sqrt}. An example of an impure function is the \hs{today}
+        function that returns the current date (which of course cannot
+        exist like this in Haskell).
+        \stopframedtext
+      }

 
 

-      This is a perfect match for a combinatoric circuit, where the output
+      This is a perfect match for a combinational circuit, where the output

       also solely depends on the inputs. However, when state is involved, this
       also solely depends on the inputs. However, when state is involved, this

-      no longer holds. Since we're in charge of our own language (or at least
-      let's pretend we aren't using Haskell and we are), we could remove this
-      purity constraint and allow a function to return different values
-      depending on the cycle in which it is evaluated (or rather, the current
-      state). However, this means that all kinds of interesting properties of
-      our functional language get lost, and all kinds of transformations and
-      optimizations might no longer be meaning preserving.
-
-      Provided that we want to keep the function pure, the current state has
-      to be present in the function's arguments in some way. There seem to be
-      two obvious ways to do this: Adding the current state as an argument, or
-      including the full history of each argument.
+      no longer holds. Of course this purity constraint cannot just be
+      removed from Haskell. But even when designing a completely new (hardware
+      description) language, it does not seem to be a good idea to
+      remove this purity. This would that all kinds of interesting properties of
+      the functional language get lost, and all kinds of transformations
+      and optimizations are no longer be meaning preserving.
+
+      So our functions must remain pure, meaning the current state has
+      to be present in the function's arguments in some way. There seem
+      to be two obvious ways to do this: Adding the current state as an
+      argument, or including the full history of each argument.

 
       \subsubsection{Stream arguments and results}
         Including the entire history of each input (\eg, the value of that
 
       \subsubsection{Stream arguments and results}
         Including the entire history of each input (\eg, the value of that


@@ -688,7 +779,7 @@ quadruple n = mul (mul n)
         An obvious downside of this solution is that on each cycle, all the
         previous cycles must be resimulated to obtain the current state. To do
         this, it might be needed to have a recursive helper function as well,
         An obvious downside of this solution is that on each cycle, all the
         previous cycles must be resimulated to obtain the current state. To do
         this, it might be needed to have a recursive helper function as well,

-        wich might be hard to be properly analyzed by the compiler.
+        which might be hard to be properly analyzed by the compiler.

 
         A slight variation on this approach is one taken by some of the other
         functional \small{HDL}s in the field: \todo{References to Lava,
 
         A slight variation on this approach is one taken by some of the other
         functional \small{HDL}s in the field: \todo{References to Lava,


@@ -708,6 +799,13 @@ quadruple n = mul (mul n)
         well (note that changing a single-element operation to a stream
         operation can done with \hs{map}, \hs{zipwith}, etc.).
 
         well (note that changing a single-element operation to a stream
         operation can done with \hs{map}, \hs{zipwith}, etc.).
 

+        This also means that primitive operations from an existing
+        language such as Haskell cannot be used directly (including some
+        syntax constructs, like \hs{case} expressions and \hs{if}
+        expressions).  This mkes this approach well suited for use in
+        \small{EDSL}s, since those already impose these same
+        limitations. \refdef{EDSL}
+

         Note that the concept of \emph{state} is no more than having some way
         to communicate a value from one cycle to the next. By introducing a
         \hs{delay} function, we can do exactly that: Delay (each value in) a
         Note that the concept of \emph{state} is no more than having some way
         to communicate a value from one cycle to the next. By introducing a
         \hs{delay} function, we can do exactly that: Delay (each value in) a


@@ -765,13 +863,9 @@ acc in = out
         definition of out), but is essentially easy to interpret. There is a
         single call to delay, resulting in a circuit with a single register,
         whose input is connected to \hs{out} (which is the output of the
         definition of out), but is essentially easy to interpret. There is a
         single call to delay, resulting in a circuit with a single register,
         whose input is connected to \hs{out} (which is the output of the

-        adder), and it's output is the expression \hs{delay out 0} (which is
+        adder), and its output is the expression \hs{delay out 0} (which is

         connected to one of the adder inputs).
 
         connected to one of the adder inputs).
 

-        This notation has a number of downsides, amongst which are limited
-        readability and ambiguity in the interpretation. \todo{Reference
-        Christiaan, who has done further investigation}
-        

       \subsubsection{Explicit state arguments and results}
         A more explicit way to model state, is to simply add an extra argument
         containing the current state value. This allows an output to depend on
       \subsubsection{Explicit state arguments and results}
         A more explicit way to model state, is to simply add an extra argument
         containing the current state value. This allows an output to depend on


@@ -780,9 +874,9 @@ acc in = out
         the current state is now an argument.
 
         In Haskell, this would look like
         the current state is now an argument.
 
         In Haskell, this would look like

-        \in{example}[ex:ExplicitAcc]\footnote[notfinalsyntax]{This example is not in the final
-  Cλash syntax}. \todo{Referencing notfinalsyntax from Introduction.tex doesn't
-  work}
+        \in{example}[ex:ExplicitAcc]\footnote[notfinalsyntax]{This
+        example is not in the final Cλash syntax}. \todo{Referencing
+        notfinalsyntax from Introduction.tex doesn't work}

 
 \startbuffer[ExplicitAcc]
 -- input -> current state -> (new state, output)
 
 \startbuffer[ExplicitAcc]
 -- input -> current state -> (new state, output)


@@ -804,14 +898,18 @@ acc in s = (s', out)
         variables are used by a function can be completely determined from its
         type signature (as opposed to the stream approach, where a function
         looks the same from the outside, regardless of what state variables it
         variables are used by a function can be completely determined from its
         type signature (as opposed to the stream approach, where a function
         looks the same from the outside, regardless of what state variables it

-        uses or whether it's stateful at all).
-
-        This approach is the one chosen for Cλash and will be examined more
-        closely below.
+        uses or whether it is stateful at all).
+        
+        This approach to state has been one of the initial drives behind
+        this research. Unlike a stream based approach it is well suited
+        to completely use existing code and language features (like
+        \hs{if} and \hs{case} expressions) because it operates on normal
+        values. Because of these reasons, this is the approach chosen
+        for Cλash. It will be examined more closely below.

 
     \subsection{Explicit state specification}
 
     \subsection{Explicit state specification}

-      We've seen the concept of explicit state in a simple example below, but
-      what are the implications of this approach?
+      The concept of explicit state has been introduced with some
+      examples above, but what are the implications of this approach?

 
       \subsubsection{Substates}
         Since a function's state is reflected directly in its type signature,
 
       \subsubsection{Substates}
         Since a function's state is reflected directly in its type signature,


@@ -889,7 +987,7 @@ acc in s = (s', out)
         \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 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
+        \item explicitly annotate state arguments and results in the input

         description.
         \stopitemize
 
         description.
         \stopitemize
 


@@ -912,11 +1010,11 @@ acc in s = (s', out)
         result is stateful. This means that the annotation lives
         \quote{outside} of the function, it is completely invisible when
         looking at the function body.
         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, \ie give stateful
+        \item Use some kind of annotation on the type level, \ie\ give stateful

         arguments and stateful (parts 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
         arguments and stateful (parts 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
+        tell if it is 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
         the translation process a lot easier, since less analysis of the
         program flow might be required.
       \stopitemize


@@ -928,7 +1026,7 @@ acc in s = (s', out)
   \todo{Note about conditions on state variables and checking them}
 
   \section[sec:recursion]{Recursion}
   \todo{Note about conditions on state variables and checking them}
 
   \section[sec:recursion]{Recursion}

-  An import concept in functional languages is recursion. In it's most basic
+  An important concept in functional languages is recursion. In its most basic

   form, recursion is a definition that is described in terms of itself. A
   recursive function is thus a function that uses itself in its body. This
   usually requires multiple evaluations of this function, with changing
   form, recursion is a definition that is described in terms of itself. A
   recursive function is thus a function that uses itself in its body. This
   usually requires multiple evaluations of this function, with changing


@@ -943,7 +1041,7 @@ acc in s = (s', out)
   problem for generating hardware.
 
   This is expected for functions that describe infinite recursion. In that
   problem for generating hardware.
 
   This is expected for functions that describe infinite recursion. In that

-  case, we can't generate hardware that shows correct behaviour in a single
+  case, we cannot generate hardware that shows correct behaviour in a single

   cycle (at best, we could generate hardware that needs an infinite number of
   cycles to complete).
   
   cycle (at best, we could generate hardware that needs an infinite number of
   cycles to complete).
   


@@ -999,8 +1097,8 @@ acc in s = (s', out)
   \stopitemize
 
   As you can see, using a simple \hs{case} expression  at value level causes
   \stopitemize
 
   As you can see, using a simple \hs{case} expression  at value level causes

-  the type checker to always typecheck both alternatives, which can't be
-  done! The typechecker is unable to distinguish the two case
+  the type checker to always typecheck both alternatives, which cannot be
+  done. The typechecker is unable to distinguish the two case

   alternatives (this is partly possible using \small{GADT}s, but that
   approach faced other problems \todo{ref christiaan?}).
 
   alternatives (this is partly possible using \small{GADT}s, but that
   approach faced other problems \todo{ref christiaan?}).
 


@@ -1009,7 +1107,7 @@ acc in s = (s', out)
   implementations of the sum function, based on the type of the
   argument, this sounds like the perfect problem to solve with a type
   class. However, this approach has its own problems (not the least of
   implementations of the sum function, based on the type of the
   argument, this sounds like the perfect problem to solve with a type
   class. However, this approach has its own problems (not the least of

-  them that you need to define a new typeclass for every recursive
+  them that you need to define a new type class for every recursive

   function you want to define).
 
   \todo{This should reference Christiaan}
   function you want to define).
 
   \todo{This should reference Christiaan}


@@ -1020,17 +1118,19 @@ acc in s = (s', out)
   counter could be expressed, but only translated to hardware for a fixed
   number of iterations. Also, this would require extensive support for compile
   time simplification (constant propagation) and compile time evaluation
   counter could be expressed, but only translated to hardware for a fixed
   number of iterations. Also, this would require extensive support for compile
   time simplification (constant propagation) and compile time evaluation

-  (evaluation of constant comparisons), to ensure non-termination. Even then, it
-  is hard to really guarantee termination, since the user (or GHC desugarer)
-  might use some obscure notation that results in a corner case of the
-  simplifier that is not caught and thus non-termination.
+  (evaluation of constant comparisons), to ensure non-termination.
+  Supporting general recursion will probably require strict conditions
+  on the input descriptions. Even then, it will be hard (if not
+  impossible) to really guarantee termination, since the user (or \GHC\
+  desugarer) might use some obscure notation that results in a corner
+  case of the simplifier that is not caught and thus non-termination.

 
   Evaluating all possible (and non-possible) ways to add recursion to
   our descriptions, it seems better to limit the scope of this research
   to exclude recursion. This allows for focusing on other interesting
 
   Evaluating all possible (and non-possible) ways to add recursion to
   our descriptions, it seems better to limit the scope of this research
   to exclude recursion. This allows for focusing on other interesting

-  areas instead. By including (builtin) support for a number of higher
-  order functions like \hs{map} and \hs{fold}, we can still express most
-  of the things we would use (list) recursion for.
+  areas instead. By including (built-in) support for a number of
+  higher-order functions like \hs{map} and \hs{fold}, we can still
+  express most of the things we would use (list) recursion for.

   
 
 % vim: set sw=2 sts=2 expandtab:
   
 
 % vim: set sw=2 sts=2 expandtab: