Local-as-View Mediators Priya Gangaraju(Class Id:203)

Local-as-View Mediators. Global predicates defined are not views of the source data. Global predicates describe the tuples that the source is able to produce. For each source: – expressions are defined – expressions involve the global predicates Queries are answered at the mediator by discovering all possible ways to construct the query using the views provided by the source.

Motivation for LAV Mediators Sometimes the the relationship between what the mediator should provide and what the sources provide is more subtle. For example, consider the predicate Par(c, p) meaning that p is a parent of c which represents the set of all child parent facts that could ever exist. The sources will provide information about whatever child-parent facts they know.

Motivation(contd..) There can be sources which may provide child- grandparent facts but not child- parent facts at all. This source can never be used to answer the child-parent query under GAV mediators. LAV mediators allow to say that a certain source provides grand parent facts. They help discover how and when to use that source in a given query.

Terminology for LAV Mediation. single Datalog rules: The queries at the mediator and the queries that describe the source Conjunctive query : A query that is a single Datalog rule. They can also define views. Global predicates: they are used as the sub- goals of mediator queries.

Contd.. Head: Each head have a unique view Predicate : The name of a view. Body: Consist of global predicates and is associated with a particular source. It is assumed that each view can be constructed with an all-free adornment.

Example.. Consider global predicate Par(c, p) meaning that p is a parent of c. One source produces parent facts. Its view is defined by the conjunctive query- V 1 (c, p)  Par(c, p) Another source produces some grand parents facts. Then its conjunctive query will be – V 2 (c, g)  Par(c, p) AND Par(p, g)

Example contd.. The query at the mediator will ask for great- grand parent facts that can be obtained from the sources. The mediator query is – Q(w, z)  Par(w, x) AND Par(x, y) AND Par(y, z) One solution can be using the parent predicate(V 1 ) directly three times. Q(w, z)  V 1 (w, x) AND V 1 (x, y) AND V 1 (y, z)

Example contd.. Another solution can be to use V 1 (parent facts) and V 2 (grandparent facts). Q(w, z)  V 1 (w, x) AND V 2 (x, z) Or Q(w, z)  V 2 (w, y) AND V 1 (y, z)

Expanding Solutions. Consider a query Q, a solution S that has a body whose subgoals are views and each view V is defined by a conjunctive query with that view as the head. The body of V’s conjunctive query can be substituted for a subgoal in S that uses the predicate V to have a body consisting of only global predicates.

Expansion Algorithm A solution S has a subgoal V(a 1, a 2,…an) where a i ’s can be any variables or constants. The view V can be of the form V(b 1, b 2,….b n )  B Where B represents the entire body. V(a 1, a 2, … a n ) can be replaced in solution S by a version of body B that has all the subgoals of B with variables possibly altered.

Expansion Algorithm contd..  The rules for altering the variables of B are: 1.First identify the local variables B, variables that appear in the body but not in the head. 2.If there are any local variables of B that appear in B or in S, replace each one by a distinct new variable that appears nowhere in the rule for V or in S. 3.In the body B, replace each b i by a i for i = 1,2…n.

Example. Consider the view definitions, V 1 (c, p)  Par(c, p) V 2 (c, g)  Par(c, p) AND Par(p, g) One of the proposed solutions S is Q(w, z)  V 1 (w, x) AND V 2 (x, z) The first subgoal with predicate V 1 in the solution can be expanded as Par(w, x) as there are no local variables.

Example Contd. The V2 subgoal has a local variable p which doesn’t appear in S nor it has been used as a local variable in another substitution. So p can be left as it is. Only x and z are to be substituted for variables c and g. The Solution S now will be Q(w, z)  Par(w, x) AND Par(x, p) AND Par(p,z)

Containment of Conjunctive Queries  A containment mapping from Q to E is a function т from the variables of Q to the variables and constants of E, such that: 1.If x is the ith argument of the head of Q, then т(x) is the ith argument of the head of E. 2.Add to т the rule that т(c)=c for any constant c. If P(x 1,x 2,… x n ) is a subgoal of Q, then P(т(x 1 ), т(x 2 ),… т(x n )) is a subgoal of E.

Example. Consider two Conjunctive queries: Q 1 : H(x, y)  A(x, z) and B(z, y) Q 2 : H(a, b)  A(a, c) AND B(d, b) AND A(a, d) When we apply the substitution, Т(x) = a, Т(y) = b, Т(z) = d, the head of Q 1 becomes H(a, b) which is the head of Q 2. So,there is a containment mapping from Q 1 to Q 2.

Example contd.. The first subgoal of Q 1 becomes A(a, d) which is the third subgoal of Q 2. The second subgoal of Q 1 becomes the second subgoal of Q 2. There is also a containment mapping from Q 2 to Q 1 so the two conjunctive queries are equivalent.

Why the Containment-Mapping Test Works Suppose there is a containment mapping т from Q 1 to Q 2. When Q 2 is applied to the database, we look for substitutions σ for all the variables of Q 2. The substitution for the head becomes a tuple t that is returned by Q 2. If we compose т and then σ, we have a mapping from the variables of Q 1 to tuples of the database that produces the same tuple t for the head of Q 1.

Finding Solutions to a Mediator Query There can be infinite number of solutions built from the views using any number of subgoals and variables. LMSS Theorem can limit the search which states that o If a query Q has n subgoals, then any answer produced by any solution is also produced by a solution that has at most n subgoals. If the conjunctive query that defines a view V has in its body a predicate P that doesn’t appear in the body of the mediator query, then we need not consider any solution that uses V.

Example. Recall the query Q1: Q(w, z)  Par(w, x) AND Par(x, y) AND Par(y, z) This query has three subgoals, so we don’t have to look at solutions with more than three subgoals.

Why the LMSS Theorem Holds Suppose we have a query Q with n subgoals and there is a solution S with more than n subgoals. The expansion E of S must be contained in Query Q, which means that there is a containment mapping from Q to E. We remove from S all subgoals whose expansion was not the target of one of Q’s subgoals under the containment mapping.

Contd.. We would have a new conjunctive query S’ with at most n subgoals. If E’ is the expansion of S’ then, E’ is a subset of Q. S is a subset of S’ as there is an identity mapping. Thus S need not be among the solutions to query Q.