The Productive use of Failure to Generate Witnesses from Divergent Proof Attempts for Coinduction

Dennis, Louise Abigail; Bundy, Alan; Green, Ian

Abstract

Coinduction is a proof rule. It is the dual of induction. It allows reasoning about non--well--founded structures such as lazy lists or streams and is of particular use for reasoning about equivalences. A central difficulty in the automation of coinductive proof is the choice of a relation (called a bisimulation).

We present an automation of coinductive theorem proving. This automation is based on the idea of proof planning. Proof planning constructs the higher level steps in a proof, using knowledge of the general structure of a family of proofs and exploiting this knowledge to control the proof search. Part of proof planning involves the use of failure information to modify the plan by the use of a proof critic which exploits the information gained from the failed proof attempt.

Our approach to the problem was to develop a strategy that makes an initial simple guess at a bisimulation and then uses generalisation techniques, motivated by a critic, to refine this guess, so that a larger class of coinductive problems can be automatically verified.

The implementation of this strategy has focused on the use of coinduction to prove the equivalence of programs in a small lazy functional language which is similar to Haskell.

We have developed a proof plan for coinduction and a critic associated with this proof plan. These have been implemented in CoClam, an extended version of Clam with encouraging results. The planner has been successfully tested on a number of theorems.

Citation

Dennis, L. A., Bundy, A., & Green, I. (2000). The Productive use of Failure to Generate Witnesses from Divergent Proof Attempts for Coinduction. Annals of Mathematics and Artificial Intelligence, 29(1-4),

• Journal Article Type: Article
• Publication Date: Jan 1, 2000
• Deposit Date: Dec 8, 2005
• Publicly Available Date: Oct 9, 2007
• Journal: Annals of Mathematics and Artificial Intelligence
• Print ISSN: 1012-2443
• Publisher: Springer Verlag
• Peer Reviewed: Peer Reviewed
• Volume: 29
• Issue: 1-4
• Public URL: https://nottingham-repository.worktribe.com/output/1023659