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--- sav07_lecture_18 [2007/05/28 22:22]
vasu.singh
+++ sav07_lecture_18 [2007/05/29 00:37]
vasu.singh
@@ Line 27: / Line 27: @@
 This suggests that the two programs above are not equivalent for the purpose of reasoning
 in concurrently executing programs.
@@ Line 42: / Line 47: @@
      c ::= [r]  |  c[]c  |  c;c  |  c*  |  c||c  |  atomic(c)
-We express the semantics of c as the set of all possible sequences of relations of c.
+We express the semantics of c, $[[c]] \subseteq 2^{R^*}$ as the set of all possible sequences of relations of c.
 $[[c]] =  { r_1^1 r_2^1 \ldots r_{n1}^1, r_1^2 r_2^2 \ldots r_{n2}^2, \ldots, r_1^l r_2^2 \ldots r_{nk}^k}$.
@@ Line 61: / Line 66: @@
 $[[ C^* ]] = \cup_{n \ge 0} [[C^n]]$ where $[[C^0]] = \emptyset$ and $[[C^{n+1}]] = [[ C^n ; C]]$.
-$[[ $atomic$(C) ]] = $eval$([[C]])$ where eval is a function $eval: R* \rightarrow R$ such that
+$[[ $atomic$(C) ]] = $eval$([[C]])$ where eval is a function $eval: 2^{R^*} \rightarrow R$ such that
 eval$(\{r_1^1 \ldots r_{n1}^1, r_1^2 \ldots r_{n2}^2, \ldots, r_1^k \ldots r_{nk}^k \}) = (r_1^1 \circ r_1^2 \ldots \circ r_{n1}^1) \cup \ldots \cup (r_1^k \circ
 r_2^k \ldots \circ r_{nk}^k)$.
+An example
+  { b = x + y }
+  while (*) do
+    if (*) then
+      x++;
+      y--;
+    else
+      y++;
+      x--;
+    endif
+  end
+  ||
+  while (*) do
+    if (*) then
+      atomic{
+        x++;
+        y--;
+      }
+    else
+      atomic{
+        y++;
+        x--;
+      }
+    endif
+  end
+  { b = x + y }
 ==== Global reachability invariants ====