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Containee problem

For queries Q1 and Q2, we say Q1 is containedin Q2, denoted Q1 C Q2, iff Q1(D) C Q2(D) for every database D.

The container problern for a fixed Query Qo is the following decision problem:
Given a query Q, decide whether Qo C Q.

The containee probletn for a fixed guery Qo is the following decision problem:
Given a query Q, decide whether Q C Qo.

Formally prove or disprove the following statements:

(a) For every conjunctive query Q0, there is a polynomial-time algorithm to decide the container problem for Q0 and for given conjunctive queries Q.

(b) For every conjunctive query Q0, there is a polynomial-time algorithm to decide the container problem for Q0 and for given conjunctive queries Q that can be obtained from Qo by adding some atoms.

(c) For every conjunctive euery Qo, there is a polynomial-time algorithm to decide the containee problem for Q0 and for grven conjunctive queries  Q.

(d) For every flrst-order Query Q0, there is an algorithm to decide the containee problem for Qo and for given first-order queries Q.

To prove a statement, sketch an algorithm, along with an argument why it is polynomial, if possible. To disprove it, provide an M-hardness or undecidability proof.

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