  We explore the possibility for an evaluation of non-leptonic $\Delta S=1$ $K$
decay amplitudes through the calculation of $K\to\pi$ matrix elements using
domain-wall QCD. The relation between the physical $K\to\pi\pi$ matrix elements
and $K\to\pi$ matrix elements deduced from chiral perturbation theory is
recapitulated. Quenched numerical simulations are performed on an $16^3\times
32\times 16$ lattice at a lattice spacing $a^{-1}\approx 2$GeV for both the
standard plaquette gauge action and a renormalization-group improved gauge
action, and reasonable signals for $K\to\pi$ matrix elements are obtained.
Preliminary results are reported on the $K\to\pi\pi$ matrix elements, and
results from two actions are compared.
