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+.NH 2
+Loop detection
+.PP
+Loops are detected by using the loop construction
+algorithm of.
+.[~[
+aho compiler design
+.], section 13.1.]
+This algorithm uses \fIback edges\fR.
+A back edge is an edge from B to C in the CFG,
+whose head (C) dominates its tail (B).
+The loop associated with this back edge
+consists of C plus all nodes in the CFG
+that can reach B without going through C.
+.PP
+As an example of how the algorithm works,
+consider the piece of program of Fig. 4.1.
+First just look at the program and think for
+yourself what part of the code constitutes the loop.
+.DS
+loop
+   if cond then                       1
+      -- lots of simple
+      -- assignment
+      -- statements              2          3
+      exit; -- exit loop
+   else
+      S; -- one statement
+   end if;
+end loop;
+
+Fig. 4.1 A misleading loop
+.DE
+Although a human being may be easily deceived
+by the brackets "loop" and "end loop",
+the loop detection algorithm will correctly
+reply that only the test for "cond" and
+the single statement in the false-part
+of the if statement are part of the loop!
+The statements in the true-part only get
+executed once, so there really is no reason at all
+to say they're part of the loop too.
+The CFG contains one back edge, "3->1".
+As node 3 cannot be reached from node 2,
+the latter node is not part of the loop.
+.PP
+A source of problems with the algorithm is the fact
+that different back edges may result in
+the same loop.
+Such an ill-structured loop is
+called a \fImessy\fR loop.
+After a loop has been constructed, it is checked
+if it is really a new loop.
+.PP
+Loops can partly overlap, without one being nested
+inside the other.
+This is the case in the program of Fig. 4.2.
+.DS
+1:                              1
+   S1;
+2:
+   S2;                          2
+   if cond then
+      goto 4;
+   S3;                     3         4
+   goto 1;
+4:
+   S4;
+   goto 1;
+
+Fig. 4.2 Partly overlapping loops
+.DE
+There are two back edges "3->1" and "4->1",
+resulting in the loops {1,2,3} and {1,2,4}.
+With every basic block we associate a set of
+all loops it is part of.
+It is not sufficient just to record its
+most enclosing loop.
+.PP
+After all loops of a procedure are detected, we determine
+the nesting level of every loop.
+Finally, we find all strong and firm blocks of the loop.
+If the loop has only one back edge (i.e. it is not messy),
+the set of firm blocks consists of the
+head of this back edge and its dominators
+in the loop (including the loop entry block).
+A firm block is also strong if it is not a
+successor of a block that may exit the loop;
+a block may exit a loop if it has an (immediate) successor
+that is not part of the loop.
+For messy loops we do not determine the strong
+and firm blocks. These loops are expected
+to occur very rarely.