The use of proof plans to sum series

Abstract

We describe a program for nding closed form solutions to nite sums. The program was built to test the applicability of the proof planning search control technique in a domain of mathematics outwith induction. This experiment was successful. The series summing program extends previous work in this area and was built in a short time just by providing new series summing methods to our existing inductive theorem proving system CLAM.One surprising discovery was the usefulness of the ripple tactic in summing series. Rippling is the key tactic for controlling inductive proofs, and was previously thought to be specialised to such proofs. However, it turns out to be the key sub-tactic used by all the main tactics for summing series. The only change required was that it had to be supplemented by a di erence matching algorithm to set up some initial meta-level annotations to guide the rippling process. In inductive proofs these annotations are provided by the application of mathematical induction. This evidence suggests that rippling, supplemented by di erence matching, will nd wide application in controlling mathematical proofs.

Adventures in Mathematical Reasoning

“…To explore the use of proof planning in general, and rippling in particular outside of inductive proof, I chanced on the domain of summing series [18]. Inductive proofs can be used to verify identities about finite sums.…”

The Use of Embeddings to Provide a Clean Separation of Term and Annotation for Higher Order Rippling

“…It is a heuristic that was originally tailored for use with induction but has also been used for summing series [29], limit theorems [30], proofs in logical frameworks [24] and equational theories [15]. It guides the application of rewriting by a process of meta-level annotation of the object-level terms.…”

Artificial Intelligence and Creativity

“…It is possible to give a formal account of this intuitive concept in terms of a Knuth-Bendix term order (see Walsh, Nunes and Bundy (1992) for more details). In our example, V (which is instantiated to xi+ 1 ) is considered a simpler expression than U (which is instantiated to (i + 1) .x i ).…”