Risch Algorithm - Decidability

Decidability

The Risch algorithm applied to general elementary functions is not an algorithm but a semi-algorithm because it needs to check, as a part of its operation, if certain expressions are equivalent to zero (constant problem), in particular in the constant field. For expressions that involve only functions commonly taken to be elementary it is not known whether an algorithm performing such a check exists or not (current computer algebra systems use heuristics); moreover, if one adds the absolute value function to the list of elementary functions, it is known that no such algorithm exists; see Richardson's theorem.

Note that this issue also arises in the polynomial division algorithm; this algorithm will fail if it cannot correctly determine whether coefficients vanish identically. Virtually every non-trivial algorithm relating to polynomials uses the polynomial division algorithm, the Risch algorithm included. If the constant field is computable, i.e., for elements not dependent on x, the problem of zero-equivalence is decidable, then the Risch algorithm is a complete algorithm. Examples of computable constant fields are and, i.e., rational numbers and rational functions in y with rational number coefficients, respectively, where y is an indeterminate that does not depend on x.

This is also an issue in the Gaussian elimination matrix algorithm (or any algorithm that can compute the nullspace of a matrix), which is also necessary for many parts of the Risch algorithm. Gaussian elimination will produce incorrect results if it cannot correctly determine if a pivot is identically zero).