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 L03 FS: Unification


          Unification

o  Two major operations needed on feature structures
   -  Merging contents of two feature structures
   -  Rejecting incompatible structures.

   Unification provides both.

o  Unification of two simple feature structures (F1 U F2 = F3)
   +  Ex. Failure to unify:

    [CAT      v]    U    [CAT     n]
    [NUMBER  sg]         [PERSON  3]
      where U stands for unification

    Above fails to unify, as the value of attribute CAT is
    different in both.

o  Unification of two simple feature structures (F1 U F2 = F3)
   +  Ex. Successful unification:

    [CAT      n]    U    [CAT     n]  =  [CAT      n ]
    [NUMBER  sg]         [PERSON  3]     [NUMBER   sg]
                                         [PERSON   3 ]
      where U stands for unification

  *  Construct F3  as follows:
      - Take union of attributes in F1 and F2
         * They are made the attributes of F3

      - For an attribute A that occurs in both F1, F2
         * Values of attribute A in F1 and F2 should be equal
           (Note:  Values are atomic in simple FSs)
         * If not equal, reject. 
            (Cannot combine incompatible info.)
         * If equal, value of A in F3 copied from F1 or F2
            + Ex. Value of CAT in F3

     -  For an attribute A that occurs in one of F1, F2 
        but not in both
         * Value of A is copied from F1 or F2, into F3
           (as value of A in F3).

     + Ex.  Value of NUMBER and PERSON are copied in output 
            F3 from F1, F2, respectively.
     + Ex.:
         [NUMBER sg]  U  [NUMBER Pl]
           Unification fails
     + Ex.:
         [NUMBER sg]  U  [PERSON  3] =  [NUMBER sg]
                                        [PERSON  3]
o  Use of FS with lexicon and grammar

    + FS associated with 'child' in lexicon
        [CAT       n]
        [NUMBER   sg]
        [PERSON    3]
 
    + FS associated with 'a' in lexicon
        [SPEC  indef ]
        [NUMBER  sg  ]
        [PERSON   3  ]
 
    + Example CF rule with constraint 
        +  NP ->  det n
        +  Constraint:
             (FS of NP) = (FS of det)  U  (FS of n)

        +  Resulting FS (of NP)

                   [CAT     n  ]
                   [SPEC  indef]
                   [NUMBER  sg ]
                   [PERSON  3  ]

o  Unification of Complex feature strucutres (F1 U F2 = F3)
    * Do same as unification of simple FSs except 
       - (When an attribute A has values in F1 and F2) AND
          (Values of A in F1 (or F2) is a feature structure)
      Do the following in such a case:
       - If value of A is feature structure in one, atomic
         in the other:
          * Reject. (F1 and F2 incompatible)
       - If value of A is FS in both:
          * Unify value of A in F1 and value of A2 in F2
             - If reject, then return reject
             - If accept, then resulting FS is made value
               of A in F3
                  A(F3)  =  A(F1)  U   A(F2)
    + Ex.

      [ CAT     n           ]      [ CAT     n           ]
      [ AGRMNT  [NUMBER sg] ]  U   [ AGRMNT  [PERSON  3] ]    

             [ CAT        n             ]
          =  [ AGRMNT      [NUMBER  sg] ]
             [             [PERSON   3] ]

         
     + Ex.

       [ CAT     n            ]      [ CAT    n           ]
       [ AGRMNT  [NUMBER sg]  ]  U   [ AGRMNT [NUMBER pl] ]    

        Reject. Incompatible structures. Because in trying to
        unify structures associated with attribute AGRMNT:
          [NUMBER sg] and
          [NUMBER pl]
        The values of NUMBER attribute are incompatible.

o  Subsumption relation between two FS
        F1  =<  F2

    %% To define relation and expand explanation.

o  Summary:  
    - FS allows partial information to be kept
    - Unification allows us to integrate information from
      two different sources (two different FSs).
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