As mentioned in Section 2, polarity filtering (Gardent and Kow, 2005) eliminates from the search space those sets of rules which cover the input but cannot possibly lead to a valid derivation either because a substitution node cannot be filled or because a root node fails to have a matching substitution site 4 While (Gardent and Kow, 2005) applies polarity filtering to the initial search space (the set of rules selected by all literals in the input), we apply polarity filtering to each local tree while going up the input tree.
this filtering will weed out all combinations of mother rules and completed immediate daughter rules which cannot possibly yield a complete tree either because some daughter rule cannot be used or because some requirement of the mother rule cannot be satisfied.