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CSE 233 Spring, 2012
Problem Set
Due on Tuesday, May 15
This is an individual assignment and the usual criteria for academic integrity apply. Please typeset your answer (latex recommended). More difficult problems are marked with ?.
- (3 points) Prove the following inference rule (R has attributes ABC and S has attributes DEF ):
If R[AB] ⊆ S[DE], R[AC] ⊆ S[DF ] and S satisfies D → E then R[ABC] ⊆ S[DEF ]
- (9 points) Let CQ=^ denote the set of conjunctive queries with equality (i.e. CQs allowing atoms x = c and x = y where x, y are variables and c is a constant). Let σ be a relational schema. We define the following dependency on instances over σ, called query inclusion dependency. For q 1 , q 2 ∈ CQ= over schema σ, an instance I over σ satisfies q 1 ⊆ q 2 iff q 1 (I) ⊆ q 2 (I). The implication problem for CQ=^ query inclusions is to determine, given a finite set Σ of CQ=^ query inclusion dependencies and an additional CQ=^ query inclusion dependency q 1 ⊆ q 2 over the same schema σ, whether Σ |= q 1 ⊆ q 2 , meaning that every instance over σ satisfying all inclusions in Σ also satisfies q 1 ⊆ q 2.
(i) (4 points) Show that for every relation schema R and FD f over R, there exist CQ=^ queries q, q′^ such that for every instance I over R, I |= f iff q(I) ⊆ q′(I).
(ii) (3 points) Show that (i) does not hold if equality is not allowed.
(iii) (2 points) Show that the implication problem for CQ=^ query inclusion dependencies is undecidable.
(iv) (brownie point) Do you think the implication problem for CQ query inclusion dependencies (without equality) is decidable? (stream of consciousness intuition is ok here, as long as you don’t say something terribly wrong).
- (?) (9 points) Let R be a relation schema of arity 2. Let q be a CQ (no equality) over R. Since R is binary, the body of q can be viewed as a directed graph among its variables and constants. The undirected body U (q) of q is obtained from the body of q by making all edges undirected. We say that q is acyclic if U (q) is acyclic. For example, (P 1) below is acyclic, while (P 2) is cyclic:
(P 1) : answer(x) ← R(x, y), R(y, x), R(z, y), R(y, w) (P 2) : answer(x) ← R(x, y), R(y, x), R(z, y), R(y, w), R(w, z)
(i) (7 points) Show that the combined recognition complexity^1 for evalua- tion of acyclic CQs over R is in PTIME (recall that this is NP-complete in general, by reduction from graph 3-colorability).
(ii) (2 points) Show that testing whether^2 q 1 ⊆ q 2 is in PTIME for CQs q 1 , q 2 over σ where q 2 is acyclic.
- (6 points) (Automorphisms)
(i) (4 points) Show that CALC queries are invariant under automor- phisms. In other words, if ϕ(¯x) is a CALC query without constants over schema σ, then for every instance I over σ and one-to-one map- ping f : dom → dom, if f (I) = I then f (ϕ(I)) = ϕ(I). Hint: Assume without loss of generality that ϕ uses only ∧, ∃, ¬, and use structural induction on the formula.
(ii) (2 points) Let σ consist of a binary relation R. Using (i), show that there is no CALC query without constants which on input
R a b b c c d d a
produces as answer (^1) Combined complexity: both the query and the instance are part of the input to the algorithm. (^2) Here q 1 ⊆ q 2 is usual inclusion of CQs, meaning that q 1 (I) ⊆ q 2 (I) for every instance I.
