CS345 Assignment #3

Due Monday, April 26, 1999

Problem Set

1. (20 pts.) Find the modularly stratified model for the following program and EDB: p(X) :- q(X) & r(X,Y) & NOT p(Y) q(X) :- r(X,Y) & NOT q(Y) q(X) :- a(X,Y) & a(Y,X) r(X,Y) :- a(X,Y) & b(X,Y) with EDB facts a(1,2), a(2,3), a(3,1), a(3,4), a(4,3), b(1,2), and b(2,3). Indicate the locally stratified model for each module.

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2. (20 pts.) Consider the conjunctive queries Q1 and Q2. Q1: p(U,Z) :- q(U,V) & q(X,Y) & r(Y,Z) & r(V,X) Q2: p(U,V) :- q(Y,U) & q(U,X) & r(U,V) & r(X,Y) Is Q1 contained in Q2? Is Q2 contained in Q1? Justify your answers by offering containment mappings or showing that none exists.

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3. (20 pts.) Consider the conjunctive queries Q1 and Q2. Q1: p(X) :- q(X,Y) & q(Y,Z) & q(Z,X) Q2: p(X) :- q(X,Y) & q(Y,Z) & q(Z,U) & q(U,V) Is Q1 contained in Q2? Is Q2 contained in Q1? Justify your answers by using the method of canonical databases.

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4. (20 pts.) Consider the conjunctive queries Q1 and Q2. Q1: p(X) :- q(X,Y) & q(Y,X) & r(X,U,Y) & r(Y,X,Z) & s(X,V) Q2: p(X) :- q(X,Y) & q(Y,Z) & r(X,Y,T) & r(U,V,T) Test whether Q1 is contained in Q2 by using Saraiya's containment test. Start by mapping q(X,Y) to q(X,Y). Indicate at each step what inferences are made.

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5. (20 pts.) Find all equivalences and containments among the four conjunctive queries below. Note that c is a constant in Q3. Q1: p(X,X) :- q(X,A) & q(A,Z) Q2: p(X,Y) :- q(X,A) & q(A,B) & q(B,C) & q(C,Y) Q3: p(X,Y) :- q(X,A) & q(A,c) & q(c,B) & q(B,Y) Q4: p(X,Y) :- q(X,A) & q(A,C) & q(B,C) & q(C,E) & q(C,D) & q(E,Y)