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Motivation
• Imprecise information on the web
• Partial Information
• Contradictions
• Imprecise queries
Interpreting the ‘~’
• For the actors name we can use edit distance, frequency similarity measures…
• For the films rating we can use user preferences, analysis of previous queries,…
• But how to combine them?• And how to assign a score for a tuple w.r.t. the entire
query?
Probabilistic DB
• Each tuple has a probability of appearing in the DB
• Assume tuple independence• Distribution over all possible DB instances• Possible Worlds Semantics
Semantics
• A query is evaluated on every possible world• Note that for each concrete world, the query
may have several answers• In this case, sum, for each answer, the
probabilities of the worlds in which it appeared in the set of answers
• Example
Solution attempt
• Obtain a query plan• Compute intermediate results along with
probabilities• A plan in our (first) example: First compute
the join, then project on D
Wrong!
• The tuples in the original DB were independent
• The tuples in the intermediate DB are not!
• Thus the multiplication (for the projection) is incorrect.
The problem is hard
• Theorem: Answering a query over a general probabilistic DB is #P-hard (Data Complexity)
• #P-hard is the “equivalent” of NP-hard for functional problems
• E.g. #SAT - given a Boolean formula, compute how many satisfying assignments it has.
• Likely not to have a polynomial solution
Other plans
• Some query plans are OK• These are plans that preserve independencies• Let us represent the query as a logical formula• Tuples that support the answer ‘p’ satisfy: (s1 or s2) and t1
Plans and formulas
• The query was P((s1 or s2) and t1)• First join, then project corresponds to P((s1 and t1) or (s2 and t1)).
This conversion is fine in classic DBBut (s1 and t1), (s2 and t1) are not independent
events!
Safe Plan
• A plan that preserves independencies is called safe
• In our example: first project s over b, only then join with t
• = first compute the ‘OR’, then the ‘AND’
Probabilistic Events
• Atomic events tuples in the original DB
• Complex Events – boolean combination of events tuples in intermediate DBs
• Translate a query plan to a complex event
Safe Plans
• A relational algebra expression has multiple equivalent expressions
• Each corresponds to a concrete execution plan.
• Some of these plans may correspond to correct or incorrect probabilistic computations
• Let us try to detect what makes a plan safe.s
Approximation
• Most common is called Monte-carlo approximation
• Originally by Karp, improved in [suciu07]
• Guarantees convergence
• The error is greater than e with a probability of less than d after (4*n / e^2)* ln(2/d)
Functional Dependencies (FDs)
• A functional dependency {A1,…An} -> B holds for a relation R if the values of the A1,…An decide the value of B
Safe plans using FDs
• Selections and joins (over conjunctive queries) are always safe (but may cause unsafe successions..)
• Projection of a1,…,ak over the result obtained from q is safe if for every R, there is an FD a1,...,ak -> Head(q)
Where Head(q) are the attributes in the result of q
Intuition• Projection over a1,…,an OR over all tuples that have the
same values of {a1,…,an}
• To be independent, each atomic event must be sufficient to distinguish tuples that are ORed (otherwise it appears in more than one)
• I.e it uniquely determines the other atomic events appearing in the tuple
• Hence the FD (valid only in combination with a1,…,an)
Safe Plan algorithm
• Top-Down• Push all safe projections late in the plan• When you can’t, split the query q into two
sub-queries q1 and q2 such that their join is q (when possible)
• If stuck, the query is unsafe
(Union of )Conjunctive Queriesby example
• T(x):- R(x,y),S(y,30)• T(x):- P(x,y)
• In relational algebra?– Multiple Possible translations– Correspond to different ordering of operations– Each option is called a “query plan”
More notations• Head(q) is the set of head variables in q,
FreeVar(q) is the set of free variables (i.e. non-head variables) in q
• R.Key is the set of variables in the key position of the relation R
• R.NonKey is the set of variables in the non-key positions of the relation R,
• R.Pred is the predicate that q applies to R. For x in FreeVar(q), denote qx a new query whose body is identical with q and where
Head(qx) = Head(q) U {x}.
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