  In this paper, we introduce a foundation for computable model theory of
rational Pavelka logic (an extension of {\L}ukasiewicz logic) and continuous
logic, and prove effective versions of some theorems in model theory. We show
how to reduce continuous logic to rational Pavelka logic. We also define
notions of computability and decidability of a model for logics with
computable, but uncountable, set of truth values; show that provability degree
of a formula w.r.t. a linear theory is computable, and use this to carry out an
effective Henkin construction. Therefore, for any effectively given consistent
linear theory in continuous logic, we effectively produce its decidable model.
This is the best possible, since we show that the computable model theory of
continuous logic is an extension of computable model theory of classical logic.
We conclude with noting that the unique separable model of a separably
categorical and computably axiomatizable theory (such as that of a probability
space or an $L^p$ Banach lattice) is decidable.
