  We describe {\it ab initio} calculations of doubly strange, $S=-2$, $s$-shell
hypernuclei ($^4_{\Lambda\Lambda}$H, $^5_{\Lambda\Lambda}$H,
  $^5_{\Lambda\Lambda}$He and $^6_{\Lambda\Lambda}$He) as a first attempt to
explore the few-body problem of the {\it full}-coupled channel scheme for these
systems. The wave function includes $\Lambda\Lambda$, $\Lambda\Sigma$,
  $N\Xi$ and $\Sigma\Sigma$ channels. Minnesota $NN$, D2$^\prime$ $YN$, and
simulated $YY$ potentials based on the Nijmegen hard-core model, are used.
  Bound state solutions of these systems are obtained. We find that a set of
phenomenological $B_8B_8$ interactions among the octet baryons in $S=0, -1$ and
-2 sectors, which is consistent with all of the available experimental binding
energies of $S=0, -1$ and -2 $s$-shell (hyper-)nuclei, can predict a particle
stable bound state of $^4_{\Lambda\Lambda}$H.
  For $^5_{\Lambda\Lambda}$H and $^5_{\Lambda\Lambda}$He, $\Lambda N-\Sigma N$
and $\Xi N-\Lambda\Sigma$ potentials enhance the net $\Lambda\Lambda-N\Xi$
coupling, and a large $\Xi$ probability is obtained even for a weaker
$\Lambda\Lambda-N\Xi$ potential.
