  We investigate Hodge-theoretic properties of Calabi-Yau complete
intersections $V$ of $r$ semi-ample divisors in $d$-dimensional toric Fano
varieties having at most Gorenstein singularities. Our main purpose is to show
that the combinatorial duality proposed by second author agrees with the
duality for Hodge numbers predicted by mirror symmetry. It is expected that the
complete verification of mirror symmetry predictions for singular Calabi-Yau
varieties $V$ of arbitrary dimension demands considerations of so called {\em
string-theoretic Hodge numbers} $h^{p,q}_{\rm st}(V)$. We restrict ourselves to
the string-theoretic Hodge numbers $h^{0,q}_{\rm st}(V)$ and $h^{1,q}_{\rm
st}(V)$ $(0 \leq q \leq d-r) which coincide with the usual Hodge numbers
$h^{0,q}(\widehat{V})$ and $h^{1,q}(\widehat{V})$ of a $MPCP$-desingularization
$\widehat{V}$ of $V$.
