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 * $Q(u,v) \leftarrow T(w,u), U(v,w), R(v,u)$   * $Q(u,v) \leftarrow T(w,u), U(v,w), R(v,u)$ 
 * For this query, [[First Order LogicFOL]] meaning is   * For this query, [[First Order LogicFOL]] meaning is 
−  * $\forall \ u, v \Big[ Q(u, v) \Rightarrow \exists \ w \ : \ T(w, u) \land U(v, w) \land R(v, u) \Big]$  +  * $\forall \ u, v \ \Big[ Q(u, v) \Rightarrow \exists \ w \ : \ T(w, u) \land U(v, w) \land R(v, u) \Big]$ 
   
 Now consider that we have a tuple $(a, b)$   Now consider that we have a tuple $(a, b)$ 
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 == Sources ==   == Sources == 
−  * Web Data Management book [http://webdam.inria.fr/Jorge]  +  * [[Web Data Management (book)]] 
   
 [[Category:Logic]]   [[Category:Logic]] 
Latest revision as of 15:39, 23 November 2015
Skolem Function
Intuition
Suppose, we have the following Conjunctive Query:
 $Q(u,v) \leftarrow T(w,u), U(v,w), R(v,u)$
 For this query, FOL meaning is
 $\forall \ u, v \ \Big[ Q(u, v) \Rightarrow \exists \ w \ : \ T(w, u) \land U(v, w) \land R(v, u) \Big]$
Now consider that we have a tuple $(a, b)$
 $(a, b)$ belongs to the data source that backs $V_1$
 so we have a fact $Q(a, b)$
 from this fact $Q(a, b)$ can infer that $R(b, a)$
 $Q(a, b) \Rightarrow R(b, a)$
 (all conjuncts have to be true for a statement to be true, so it means the last conjuncts holds true)
But we can infer other things as well
 e.g. $Q(a, b) \Rightarrow \exists \ d_1 \ : \ T(d_1, a) \land U(b, d_1)$
 where $d_1$ is some constant
 we don't know its value, but we know it exists (since it's existentially qualified) and
 it depends on constants $a$ and $b$
 so we can denote this dependency as $d_1 = f_1(a, b)$
Skolem Function
 the symbol $f_1(u, v)$ is a Skolem Function of arity 2
 $f_1(u, v)$ denotes that there exists some constant that depends on values of $u$ and $v$
 given two distinct Skolem terms, e.g. $f_1(1, 2)$ and $f_1(2, v_3)$ we never can say if they belong to the same constant or not
Sources