  Gravitational waves from neutron star binary inspirals contain information
about the equation of state of supranuclear matter. In the absence of
definitive experimental evidence that determines the correct equation of state,
a number of diverse models that give the pressure in a neutron star as function
of its density have been proposed. These models differ not only in the
approximations and techniques they use to solve the many-body Schr\"odinger
equation, but also in the neutron star composition they assume. We study
whether gravitational wave observations of neutron star binaries in
quasicircular inspirals will allow us to distinguish between equations of state
of differing internal composition, thereby providing important information
about the properties of extremely high density matter. We carry out a Bayesian
model selection analysis, and find that second generation gravitational wave
detectors can heavily constrain equations of state that contain only quark
matter, but hybrid stars containing both normal and quark matter are harder to
distinguish from normal matter stars. A gravitational wave detection with a
signal-to-noise ratio of 30 and masses around $1.4M_{\odot}$ could either
detect or rule out strange quark stars with a 20 to 1 confidence. The presence
of kaon condensates or hyperons in neutron star inner cores cannot be easily
confirmed. For example, for the equations of state studied in this paper, even
a gravitational wave signal with a signal-to-noise ratio as high as 60 would
not allow us to claim a detection of kaon condensates or hyperons with
confidence greater than 5 to 1. On the other hand, if kaon condensates and
hyperons do not form in neutron stars, a gravitational wave signal with similar
signal-to-noise ratio would be able to constrain their existence with an 11 to
1 confidence for high-mass systems.
