Conjunctive Queries

Cores
Cores
endomorphism h: {Y b}
B = { p(X,Y), p(X,b), p(a,b), p(U,c), p(U,V), q(a,c,d) }
B = { p(X,Y), p(X,b), p(a,b), p(U,c), p(U,V), q(a,c,d) }
h(B) =
{ p(X,b), p(a,b), p(U,c), p(U,V), q(a,c,d) }
h(B) =
{ p(X,b), p(a,b), p(U,c), p(U,V), q(a,c,d) }
{ p(a,b), p(U,c), q(a,c,d) }
f(B)=g(h(B))=gh(B)=
{ p(a,b), p(U,c), q(a,c,d) }
h(B) can be further reduced by endomorphism g: {X a, V c}
endomorphism f: {X a, Yb, V c}
45
46
Cores
Cores
B = { p(X,Y), p(X,b), p(a,b), p(U,c), p(U,V), q(a,c,d) }
B = { p(X,Y), p(X,b), p(a,b), p(U,c), p(U,V), q(a,c,d) }
h(B) =
{ p(X,b), p(a,b), p(U,c), p(U,V), q(a,c,d) }
h(B) =
{ p(X,b), p(a,b), p(U,c), p(U,V), q(a,c,d) }
f(B)=g(h(B))=gh(B)= { p(a,b), p(U,c), q(a,c,d) }
no refinement by endomorphisms possible !
endomorphism f: {X a, Yb, V c}
f(B)=g(h(B))=gh(B)=
{ p(a,b), p(U,c), q(a,c,d) }
Core(I)
unique up to variable-renaming!
47
endomorphism f: {X a, Yb, V c}
48