Dissertation - Michael Becker

(147)C1 winner1 C1 winner2 C2 C3a. winner1 ≻ loser1 W Lb. winner2 ≻ loser2 L Wc. winner3 ≻ loser3 L Wd. winner4 ≻ loser4 W LCloning C1 left a Support that is still inconsistent, as the columns for C2 and C3 stillhave both W’s and L’s in them. Looking again for the constraint that has the fewest nonemptycells in its column, C3 is chosen for cloning, since it has fewer non-empty cellsthan C2. Once C3 is cloned, C3 winner3 is installed, winner3’s winner-loser pair is removedfrom the Support, and this allows C2 to be installed. The winner-loser pairs of winner2 andwinner4 are removed, leaving the Support empty, which in turn lets the remaining C1 winner2and C3 winner4 be installed, leading to the grammar in (148).(148) C1 winner1 ≫ C3 winner3 ≫ C2 ≫ C1 winner2 , C3 winner4The two trends are successfully captured by the clones of C1 and C3, with C2 servingas a pivot for both.Cloning C3 first would have resulted in almost exactly the samegrammar, just with C3 winner3 ≫ C1 winner1 . Since C1 and C3 don’t interact directly, theirrelative ranking doesn’t matter.4.2.4 Interim summary: Choosing the least populated columnTo summarize so far: The minimal situation where cloning constraints is a useful toolfor resolving inconsistencies involves two constraints, each with both W’s and L’s in theircolumns. When the W’s and L’s that the two constraints assign are exactly opposite, as in153