  Physics of the Equation of State (EoS) for proto-neutron star (PNS) concerns
properties of neutron rich matter at finite temperature over wide range of
densities. In this contribution we discuss the structure of PNS inner crust in
a relativisitc mean filed model with spherical Wigner-Setiz approximation, and
the composition of matter around neutrino-sphere in PNS in a virial expansion
of non-ideal gas composed of nucleons and nuclei. We go on to discuss several
new complete EoS for PNS and supernova, whose detailed composition is important
for the neutrino dynamics. We focus on one important distinction for various
EoS - the density dependence of symmetry energy E'sym, and its interesting
correlation with the radii of neutron star, as well as properties of neutron
distribution in neutron rich nuclei. Improved understanding of E'sym from
terrestrial experiment on neutron distribution of neutron rich nuclei,
benchmark calculations via ab initio methods, and statistical analysis on good
quality observational data will advance our knowledge on EoS.
