(Ex.1 in Ch9 of the book KRR) In this chapter, we considered the semanticsof a description logic language that includes concept-forming operators such as FILLS andEXISTS but no role-forming operators. In this question, we extend the language with newconcept-forming operators and role-forming operators.(a) Present a formal semantics in the style of Section 9.3.1 for the following concept-formingoperators: [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 ther1’s. So[ALL[COMPOSE :Parent:BrotherInLaw]Rich]would mean something all of whose uncles are rich (where an uncle is a brother-in-lawof 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 ofthe former. (Ex.1 in Ch9 of the book KRR) In this chapter, we considered the semanticsof a description logic language that includes concept-forming operators such as FILLS andEXISTS but no role-forming operators. In this question, we extend the language with newconcept-forming operators and role-forming operators.(a) Present a formal semantics in the style of Section 9.3.1 for the following concept-formingoperators:[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 ... Tn-1 rn ] Role composition. The rn's of the rn-1's... of theri's. So[ALL [COMPOSE :Parent: Brother In Law] Rich]would mean something all of whose uncles are rich (where an uncle is a brother-in-lawof 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 ofthe former.

Introduction: In this question, we are required to extend the formal semantics of a description…

Answered: (Ex. 1 in Ch9 of the book KRR) In this…

Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

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(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 ... Tn-1 rn ] Role composition. The rn's of the rn-1's... of the
ri's. So
[ALL [COMPOSE :Parent: Brother In Law] 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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