An Automated Confluence Proof for an Infinite Rewrite System Parametrized over an Integro-Differential Algebra

In our symbolic approach to boundary problems for linear ordinary differential equations we use the algebra of integro-differential operators as an algebraic analogue of differential, integral and boundary operators (Section 2). They allow to express the problem statement (differential equation and boundary conditions) as well as the solution operator (an integral operator called “Green’s operator”), and they are the basis for operations on boundary problems like solving and factoring [14, 17]. A survey of the implementation is given in [18]. The integro-differential operators are realized by a noetherian and confluent rewrite system [17]. From a ring-theoretic point of view, this rewrite system constitutes a basis for the ideal of relations among the fundamental operators, and confluence means we have a noncommutative Gröbner basis [3, 4, 2, 9]. However, since the relation ideal is infinitely generated in a polynomial ring with infinitely many indeterminates, none of the known implementations [13] is applicable. This is why the confluence proof is somewhat subtle (Section 3). The generators for the relation ideal are parametrized over a given integro-differential algebra, and the reduction of S-polynomials must incorporate the computational laws of the latter. The automated proof in [15] has achieved this in an ad-hoc manner for the special case of what was called “analytic algebras” there. In our new proof, the computational laws of integro-differential algebras are internalized by using so-called integro-differential polynomials [16] in the formation of the S-polynomials. We also refer to [19] for a detailed presentation of the new automated proof and the corresponding integro-differential structures. We use a prototype implementation of integro-differential polynomials and reduction rings, based on Theorema and available at www.theorema.org. The Theorema system was designed by B. Buchberger as an integrated environment