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

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