cs345-8

# cs345-8 - Searching for Solutions Careful Analysis of...

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Unformatted text preview: Searching for Solutions Careful Analysis of Expansions The Bucket Algorithm 1 Solutions and Expansions x For every subgoal p(X,Y) of a query, each solution must have a subgoal (view) whose expansion allows a containment mapping to p(X,Y). x “Bucket” for a subgoal = set of views that “cover” the subgoal. x A solution must include > 1 view from each bucket. 2 But … x There’s much more to the story. x A careful examination of how variables from the view definitions, query, solution, and expansion relate will eventually reveal additional constraints on the structure of the solutions. 3 (Non)Distinguished Variables x A variable that appears in the head of a CQ is said to be distinguished ; otherwise nondistinguished. p(X,Y) :- q(X,Z) & r(Z,Y) Distinguished Nondistinguished 4 Local Variables of Expansions x When we expand a view subgoal of a solution, the nondistinguished variables of the view definition become local.  A local variable may not appear anywhere else in the expansion. x Variables of the solution substitute for the distinguished variables of a view definition. 5 Picture Correspond to dist­ inguished of the view v(X,Y) :- p(X,Z) & q(Z,Y) sol(U,V) :… & v(U,W) & … exp(U,V) :- … & p(U,Z1) & q(Z1,W) … Distinguished in solution; may appear elsewhere Local from view; may not appear except as shown Exposed variables Nondistinguished in solution; may appear elsewhere 6 Exposed Variables x Variables of the expansion that have substituted for distinguished variables of a view. x These are the only variables that may appear in subgoals belonging to the expansion of two different solution subgoals. 7 The Variables of the Query x A query variable is shared if it appears more than once; otherwise it is unique. x A distinguished query variable can only map to the corresponding distinguished variable of the expansion/solution. x A nondistinguished, unique variable of the query maps to any variable of the expansion. 8 Mapping Shared Variables x There are two options for shared variables: 1. Map to a local variable of one expansion. 2. Map to an exposed variable. x Only in case (2) can the query subgoals with a shared variable map to expansion subgoals that come from more than one solution subgoal. 9 Picture ­­­ Map to Exposed v(X,Y) :- p(X,Z) & q(Z,Y) sol(U,V) :- … v(U,W) … W duplicated so D can be handled u(W,V) … exp(U,V) :- p(U,Z1) & q(Z1,W) … r(W,T) … que(A,B) :- … q(C,D) … r(D,E) … Shared variable D maps to exposed variable W. We can map another occurrence of D to a copy of W that comes from another view. 10 Picture v(X,Y) :- p(X,Z) & q(Z,Y) sol(U,V) :… & v(U,W) & … exp(U,V) :- … & p(U,Z1) & q(Z1,W) … que(U,V) :- … p(U,A) & q(A,D) … All occurrences of shared variable A map to local variable Z1. 11 Buckets x To help search for solutions, we create buckets: 1. One bucket for each subgoal of the query. 2. One bucket for each shared variable in the query. 12 Buckets for Subgoals x Members of the bucket for a subgoal p(A,B) are pairs consisting of: 1. A view v. 2. A particular p ­subgoal in the body of v. x There are conditions on p(A,B) and the target subgoal p(X,Y) described on the next slide. 13 Buckets for Subgoals ­­­ (2) 1. p(A,B) must be mappable to p(X,Y). That is, if A=B, then X=Y. 2. If (say) A is a distinguished variable of the query, then X is distinguished in the view. 3. If (say) A is a shared variable, then X is distinguished in the view. x Obvious extension to > 2 arguments. 14 Buckets for Shared Variables x Members of the bucket for a shared variable A consist of: 1. A view v, and 2. A set of subgoals S of v such that there is a CM from all the query subgoals containing A to S. x In this mapping, distinguished variables of the query map to distinguished variables of the view. 15 Example v(X,Y) :- p(X,Z) & p(Z,Y) w(U,V) :- p(U,S) & p(S,T) & p(T,V) q(A,B) :- p(A,C) & p(C,D) & p(D,E) & p(E,F) & p(F,G) & p(G,B) x v = “grandparent”; w = “great­grandparent”; query q = “sixth­generation ancestors.” 16 Example ­­­ p(A,C) v(X,Y) :- p(X,Z) & p(Z,Y) w(U,V) :- p(U,S) & p(S,T) & p(T,V) q(A,B) :- p(A,C) & p(C,D) & p(D,E) & p(E,F) & p(F,G) & p(G,B) x The bucket for p(A,C) is empty.  A is distinguished; C is shared.  No view subgoal has distinguished variables in both positions. 17 Example ­­­ p(C,D) v(X,Y) :- p(X,Z) & p(Z,Y) w(U,V) :- p(U,S) & p(S,T) & p(T,V) q(A,B) :- p(A,C) & p(C,D) & p(D,E) & p(E,F) & p(F,G) & p(G,B) x The bucket for p(C,D) is empty.  Both C and D are shared.  No view subgoal has distinguished variables in both positions. x Likewise, all subgoals of q have empty buckets. 18 Example ­­­ Shared Variable C v(X,Y) :- p(X,Z) & p(Z,Y) w(U,V) :- p(U,S) & p(S,T) & p(T,V) q(A,B) :- p(A,C) & p(C,D) & p(D,E) & p(E,F) & p(F,G) & p(G,B) x The bucket for C : 1. {p(X,Z), p(Z,Y)} from v. x Important: X is distinguished (since A maps to X ). 1. {p(U,S), p(S,T)} from w. x Important: U is distinguished (since A maps to U ). 19 Shared Variable C ­­­ Continued v(X,Y) :- p(X,Z) & p(Z,Y) w(U,V) :- p(U,S) & p(S,T) & p(T,V) q(A,B) :- p(A,C) & p(C,D) & p(D,E) & p(E,F) & p(F,G) & p(G,B) x The bucket for C does not contain {p(S,T), p(T,V)} from w.  Because distinguished variable A of the query would have to map to S, which is local in the view definition. 20 Example ­­­ Shared Variable D v(X,Y) :- p(X,Z) & p(Z,Y) w(U,V) :- p(U,S) & p(S,T) & p(T,V) q(A,B) :- p(A,C) & p(C,D) & p(D,E) & p(E,F) & p(F,G) & p(G,B) x The bucket for D : 1. {p(X,Z), p(Z,Y)} from v. 2. {p(U,S), p(S,T)} and {p(S,T), p(T,V)} from w. x Either is OK, since neither C nor E is distinguished. x E, F like D ; G like A. 21 Example ­­­ Continued x Each of the six query subgoals must be covered by at least one member of a bucket. x Since the subgoals themselves have empty buckets, we must group them according to their shared variables and cover them, in groups, from the buckets for the variables. 22 Example ­­­ Continued x One possibility: use the members from v in the buckets for C, E, and G. x Since shared variables D and F map to distinguished variables of the view definition, we can use v three times in the solution, and equate the corresponding variables. 23 First Solution v(X,Y):w(U,V):q(A,B):& s(A,B):e(A,B):& p(X,Z) & p(Z,Y) p(U,S) & p(S,T) & p(T,V) p(A,C) & p(C,D) & p(D,E) p(E,F) & p(F,G) & p(G,B) v(A,J) & v(J,K) & v(K,B) p(A,Z1) & p(Z1,J) & p(J,Z2) p(Z2,K) & p(K,Z3) & p(Z3,B) 24 Example ­­­ Continued x Another possibility is to use one copy of w to cover the first three query subgoals and another copy of w to cover the last three. x The first copy covers shared variables C and D ; the second covers F and G. x Shared variable E maps to distinguished variables of w. 25 Second Solution v(X,Y) :- p(X,Z) & p(Z,Y) w(U,V) :- p(U,S) & p(S,T) & p(T,V) q(A,B) :- p(A,C) & p(C,D) & p(D,E)& p(E,F) & p(F,G) & p(G,B) s(A,B) :- w(A,J) & w(J,B) e(A,B) :- p(A,S1) & p(S1,T1) & p(T1,J) & p(J,S2) & p(S2,T2) & p(T2,B) 26 Why There Are No More Solutions x For instance, we cannot use one v subgoal v(A,J) in the solution to cover shared variable C and another v(K,L) to cover D. x v(A,J) expands to p(A,Z1) & p(Z1,J), forcing D to map to J. x But v(K,L) expands to p(K,Z2) & p(Z2,L), forcing D to map to Z2. 27 ...
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## This document was uploaded on 01/06/2012.

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