  An S-restricted composition of a positive integer n is an ordered partition
of n where each summand is drawn from a given subset S of positive integers.
There are various problems regarding such compositions which have received
attention in recent years. This paper is an attempt at finding a closed- form
formula for the number of S-restricted compositions of n. To do so, we reduce
the problem to finding solutions to corresponding so-called interpreters which
are linear homogeneous recurrence relations with constant coefficients. Then,
we reduce interpreters to Diophantine equations. Such equations are not in
general solvable. Thus, we restrict our attention to those S-restricted
composition problems whose interpreters have a small number of coefficients,
thereby leading to solvable Diophantine equations. The formalism developed is
then used to study the integer sequences related to some well-known cases of
the S-restricted composition problem.
