Add a new definition of the normal form.

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

diff --git a/Chapters/Normalization.tex b/Chapters/Normalization.tex
index d36556beeb775495a399f554be7da5686313efad..b91eb40edda218c60ccb1fc951a2da436490af10 100644 (file)
--- a/Chapters/Normalization.tex
+++ b/Chapters/Normalization.tex
@@ -62,29 +62,75 @@ An example of a program in canonical form would be:
 
 \startlambda
   -- All arguments are an inital lambda
-  λx.λc.λd.
-  -- There is one let expression at the top level
+  λa.λd.λsp.
+  -- There are nested let expressions at top level
   let
+    -- Unpack the state by coercion
+    s = sp :: (Word, Word)
+    -- Extract both registers from the state
+    r1 = case s of (fst, snd) -> fst
+    r2 = case s of (fst, snd) -> snd
     -- Calling some other user-defined function.
-    s = foo x
-    -- Extracting result values from a tuple
-    a = case s of (a, b) -> a
-    b = case s of (a, b) -> b
-    -- Some builtin expressions
-    rh = add c d
-    rhh = sub d c
+    d' = foo d
     -- Conditional connections
-    rl = case b of
-      High -> rhh
+    out = case a of
+      High -> r1
+      Low -> r2
+    r1' = case a of
+      High -> d
+      Low -> r1
+    r2' = case a of
+      High -> r2
       Low -> d
-    r = case a of
-      High -> rh
-      Low -> rl
+    -- Packing a tuple
+    s' = (,) r1' r2'
+    -- Packing the state by coercion
+    sp' = s' :: State (Word, Word)
+    -- Pack our return value
+    res = (,) sp' out
   in
     -- The actual result
-    r
+    res
+\stoplambda
+
+\startlambda
+\italic{normal} = \italic{lambda}
+\italic{lambda} = λvar.\italic{lambda} (representable(typeof(var)))
+                | \italic{toplet} 
+\italic{toplet} = let \italic{binding} in \italic{toplet} 
+                | letrec [\italic{binding}] in \italic{toplet}
+                | var (representable(typeof(var)), fvar(var))
+\italic{binding} = var = \italic{rhs} (representable(typeof(rhs)))
+                 -- State packing and unpacking by coercion
+                 | var0 = var1 :: State ty (fvar(var1))
+                 | var0 = var1 :: ty (var0 :: State ty) (fvar(var1))
+\italic{rhs} = userapp
+             | builtinapp
+             -- Extractor case
+             | case var of C a0 ... an -> ai (fvar(var))
+             -- Selector case
+             | case var of (fvar(var))
+                DEFAULT -> var0 (fvar(var0))
+                C w0 ... wn -> resvar (\forall{}i, wi \neq resvar, fvar(resvar))
+\italic{userapp} = \italic{userfunc}
+                 | \italic{userapp} {userarg}
+\italic{userfunc} = var (tvar(var))
+\italic{userarg} = var (fvar(var))
+\italic{builtinapp} = \italic{builtinfunc}
+                    | \italic{builtinapp} \italic{builtinarg}
+\italic{builtinfunc} = var (bvar(var))
+\italic{builtinarg} = \italic{coreexpr}
 \stoplambda
 
+-- TODO: Define tvar, fvar, typeof, representable
+-- TODO: Limit builtinarg further
+
+-- TODO: There can still be other casts around (which the code can handle,
+e.g., ignore), which still need to be documented here.
+
+-- TODO: Note about the selector case. It just supports Bit and Bool
+currently, perhaps it should be generalized in the normal form?
+
 When looking at such a program from a hardware perspective, the top level
 lambda's define the input ports. The value produced by the let expression is
 the output port. Most function applications bound by the let expression
@@ -472,8 +518,8 @@ translatable. A user-defined function is any other function.
 \starttrans
 x = E
 ~
-x Y0 ... Yi ... Yn                               \lam{Y_i} is not of a runtime representable type
----------------------------------------------    \lam{Y_i} is not a local variable reference
+x Y0 ... Yi ... Yn                               \lam{Yi} is not of a runtime representable type
+---------------------------------------------    \lam{Yi} is not a local variable reference
 x' = λy0 ... yi-1 f0 ... fm yi+1 ... yn .        \lam{f0 ... fm} = free local vars of \lam{Y_i}
       E y0 ... yi-1 Yi yi+1 ... yn   
 ~
@@ -779,4 +825,3 @@ After let bind removal:
 \stoplambda
 
 Application simplification is not applicable.
-\stoptext