Ex.1 in Ch9 of the book KRR) In this chapter, we considered the semantics of a description logic language that includes concept-forming operators such as FILLS and EXISTS but no role-forming operators. In this question, we extend the language with new concept-forming operators and role-forming operators. (a) Present a formal semantics in the style of Section 9.3.1 for the following concept-forming operators: [SOME r] Role existence. Something with at least 1 r. [AT-MOST n r] Maximum role cardinality. Something with at most n r’s. (b) Do the same for the following role-forming operators: [INVERSE r] Role inverse. So the :Child role could be defined as [INVERSE :Parent]. [COMPOSE r1 ... rn−1 rn ] Role composition. The rn’s of the rn−1’s . . . of the r1’s. So [ALL[COMPOSE :Parent:BrotherInLaw]Rich] would mean something all of whose uncles are rich (where an uncle is a brother-in-law of a parent). (c) Use this semantic specification to show that for any roles r, s, and t, the concept [ALL[COMPOSE r s ][SOME t]] subsumes the concept [ALL r [AND[ALL s [EXISTS 2 t ]][ALL s [AT-MOST 2 t]]]] by showing that the extension of the latter concept is always a subset of the extension of the former.
(Ex.1 in Ch9 of the book KRR) In this chapter, we considered the semantics
of a description logic language that includes concept-forming operators such as FILLS and
EXISTS but no role-forming operators. In this question, we extend the language with new
concept-forming operators and role-forming operators.
(a) Present a formal semantics in the style of Section 9.3.1 for the following concept-forming
operators:
[SOME r] Role existence. Something with at least 1 r.
[AT-MOST n r] Maximum role cardinality. Something with at most n r’s.
(b) Do the same for the following role-forming operators:
[INVERSE r] Role inverse. So the :Child role could be defined as
[INVERSE :Parent].
[COMPOSE r1 ... rn−1 rn ] Role composition. The rn’s of the rn−1’s . . . of the
r1’s. So
[ALL[COMPOSE :Parent:BrotherInLaw]Rich]
would mean something all of whose uncles are rich (where an uncle is a brother-in-law
of a parent).
(c) Use this semantic specification to show that for any roles r, s, and t, the concept
[ALL[COMPOSE r s ][SOME t]]
subsumes the concept
[ALL r [AND[ALL s [EXISTS 2 t ]][ALL s [AT-MOST 2 t]]]]
by showing that the extension of the latter concept is always a subset of the extension of
the former.
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