Satisfiable Formula (Satisfiability Modulo Theory)

Definition

Satisfiable Formula (Satisfiability Modulo Theory)

Let $I$ be a $\Sigma$-interpretation and let $\phi$ be a $\Sigma$-formula with $\mathrm{FV}(\phi )=X$. The formula $\phi$ is satisfiable in $I$ if some $X$-variant $I^{\prime }$ of $I$ makes it true:

$$\exists I^{\prime }{\equiv }_{X}I:I^{\prime }(\phi )=1.$$

The formula is satisfiable if it is satisfiable in at least one $\Sigma$-interpretation.