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

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.

Inclusion, Participation and the Emergence of British Chinese Websites

Previous work has drawn attention to the relative absence of British Chinese voices in public culture. No one is more aware of this invisibility than British-born Chinese people themselves. Since 2000 the emergence of Internet discussion sites produced by British Chinese young people has provided an important forum for many of them to grapple with questions concerning their identities, experiences and status in Britain. In this paper we explore the ways in which Internet usage by British-born Chinese people has facilitated forms of self-expression, collective identity production and social and political action. This examination of British Chinese websites raises important questions about inclusion and exclusion, citizenship, participation and the development of a sense of belonging in Britain, issues which are usually overlooked in relation to a group which appears to be well integrated and successful in higher education.