  The classical approach of solvability using group theory is well known and
one original motivation is to solve polynomials by radicals. Radicals are
square, cube, square root, cube root etc of the original coefficients for the
polynomial. A polynomial is solvable by radicals if the permutation group is
solvable. This is exact solvability via group theory. With modern computers, we
might need to relax our definition of exact solvability and move towards
practical solvability. We will address seismic imaging as an example of
practical solvability by semigroup theory. The difference between semigroup and
group is that the semigroup operators do not have to be invertible as in group
operators. Using the metaphor of complex decomposition, we will decompose an
operator into simple part and complex part. The simple part of the operator is
solvable by numerical methods. The complex part of the operator is
interpretable but not numerically solvable. It is sometimes called the
evanescent energy in geophysics.
