  We show how to solve initial-boundary value problems for integrable nonlinear
differential-difference equations on a finite set of integers. The method we
employ is the discrete analogue of the unified transform (Fokas method). The
implementation of this method to the Ablowitz-Ladik system yields the solution
in terms of the unique solution of a matrix Riemann-Hilbert problem, which has
a jump matrix with explicit $(n,t)$-dependence involving certain functions
referred to as spectral functions. Some of these functions are defined in terms
of the initial value, while the remaining spectral functions are defined in
terms of two sets of boundary values. These spectral functions are not
independent but satisfy an algebraic relation called global relation. We
analyze the global relation to characterize the unknown boundary values in
terms of the given initial and boundary values. We also discuss the
linearizable boundary conditions.
