Some fixes to the prototype chapter.

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

diff --git a/Chapters/Normalization.tex b/Chapters/Normalization.tex
index 82305479b573f29b9b8e68a51c16c44305d17026..922322bcda1901bba315d1afb35429ec8a69d481 100644 (file)
--- a/Chapters/Normalization.tex
+++ b/Chapters/Normalization.tex
@@ -1952,7 +1952,7 @@
         twice). This is discussed in more detail in
         \in{section}[sec:normalization:duplicatework].
 
-      \subsubsection{Literals}
+      \subsubsection[sec:normalization:literals]{Literals}
         There are a limited number of literals available in Haskell and Core.
         \refdef{enumerated types} When using (enumerating) algebraic
         data-types, a literal is just a reference to the corresponding data
@@ -2559,5 +2559,24 @@
       we need to check every (set of) transformation(s) separately.
 
       \todo{Perhaps do a few steps of the proofs as proof-of-concept}
+  \subsection{Determinism}
+    A well-known technique for proving determinism in lambda calculus
+    and other reduction systems, is using the Church-Rosser property
+    \cite[church36]. A reduction system has the CR property if and only if:
+
+    \placedefinition[here]{Church-Rosser theorem}
+      {\lam{\forall A, B, C \exists D (A ->> B ∧ A ->> C => B ->> D ∧ C ->> D)}}
+
+    Here, \lam{A ->> B} means \lam{A} \emph{reduces to} \lam{B}. In
+    other words, there is a set of transformations that can transform
+    \lam{A} to \lam{B}. \lam{=>} is used to mean \emph{implies}.
+
+    For a transformation system holding the Church-Rosser property, it
+    is easy to show that it is in fact deterministic. Showing that this
+    property actually holds is a harder problem, but has been
+    done for some reduction systems in the lambda calculus
+    \cite[klop80]\ \cite[barendregt84]. Doing the same for our
+    transformation system is probably more complicated, but not
+    impossible.
 
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