Expressive Power: The Finite Case

Abstract

As we have seen in the previous chapter, usual relational databases are just a special case of the constraint model; indeed, a tuple $\vec a = ({a_1},...,{a_n})$ can be represented as a constraint ${c_{\vec a}}({x_1},...,{x_n}) \equiv {x_1} = {a_1} \wedge ... \wedge {x_n} = {a_n}$ in free variables x ₁,..., x _n . A relation $R = \left\{ {\vec a,...,{{\vec a}_m}} \right\}$ is then represented by a formula ${c_R}({x_1},...,{x_n}) \equiv {c_{\vec a}} \vee \cdots \vee {c_{{{\vec a}_m}}}$ stating that the interpretation of a tuple (x ₁ ,..., x _n) must be among the ${\vec a_i}S.$ Consequently, constraint query languages, over a structure $M = \left\langle {U,\Omega } \right\rangle$, can be considered as query languages over ordinary relational databases whose elements come from the set u. For example, FO + Lin and FO + Poly, the relational calculus with linear and polynomial constraints, can be considered as query languages over ordinary relational databases that store numbers.