Generalized realizability for extensions of arithmetic language

Let $L$ be an extension of the language of arithmetic, $V$ be a class of number-theoretical functions. A notion of the $V$-realizability for $L$-formulas is defined in such a way that indexes of functions in $V$ are used for interpreting the implication and the universal quantifier. It is proved that the semantics for $L$ based on the $V$-realizability coincides with the classic semantics iff $V$ contains all $L$-definable functions.