O-Plan is an AI planner based on previous experience with the Nonlin planner and its derivatives. Nonlin and other similar planning systems had limited control architectures and were only partially successful at limiting their search spaces. O-Plan is a design and implementation of a more flexible system aimed at supporting planning research and development, opening up new planning methods and supporting strong search control heuristics. O-Plan takes an engineering approach to the construction of an efficient domain independent planning system which includes a mixture of AI and numerical techniques from Operations Research. The main contributions of the work are centred around the control of search within the OPlan planning framework, and this paper outlines the search control heuristics employed within the planner. These involve the use of condition typing, time and resource constraints and domain constraints to allow knowledge about an application domain to be used to prune the searc...

We propose the use of explicit proof plans to guide the search for a proof in automatic theorem proving. By representing proof plans as the specifications of LCF-like tactics, [Gordon et al 79], and by recording these specifications in a sorted meta-logic, we are able to reason about the conjectures to be proved and the methods available to prove them. In this way we can build proof plans of wide generality, formally account for and predict their successes and failures, apply them flexibly, recover from their failures, and learn them from example proofs. We illustrate this technique by building a proof plan based on a simple subset of the implicit proof plan embedded in the Boyer-Moore theorem prover, [Boyer & Moore 79]. Keywords Proof plans, inductive proofs, theorem proving, automatic programming, formal methods, planning. Acknowledgements I am grateful for many long conversations with other members of the mathematical reasoning group, from which many of the ideas in this paper e...

We describe a program for finding closed form solutions to finite 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 difference 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 i...

We propose using proof plans to implement expression normalizers in automatic theorem proving. We outline some general-purpose proof plans and show how these can be combined in various ways to yield some standard normalizers. We claim that using proof plans facilitates the flexible application of these normalizers so that they can interact with the theorem prover in which they are embedded. We intend to extend this technique to decision procedures. 1 Introduction In [Boyer & Moore 88], Boyer and Moore investigate a case study in the use of decision procedures in automated deduction, namely a decision procedure for linear arithmetic. Their conclusions were as follows. ffl Such decision procedures have a vital role to play in reducing the combinatorial explosion, but ffl they cannot be treated as black boxes. In practice, few sub-goals exactly fit the requirements of a decision procedure, but many almost do. In these cases it is necessary to augment the decision procedure with ad...

The transformation of constructive program synthesis proofs is discussed and compared with the more traditional approaches to program transformation. An example system for adapting programs to special situations by transforming constructive synthesis proofs has been reconstructed and is compared with the original implementation [Goad 80]. A brief account of more general proof transformation applications is also presented. The overall moral is that constructiveexistence proofs contain more information over and above that required for simple execution and that this can be exploited by a proof transformation system.

We describe proof planning: a technique for both describing the hierarchical structure of proofs and then using this structure to guide proof attempts. When such a proof attempt fails, these failures can be analyzed and a patch formulated and applied. We also describe rippling: a powerful proof method used in proof planning. We pose and answer a number of common questions about proof planning and rippling.

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