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.

New ethnicities online: reflexive racialisation and the internet

In this article we analyse the emergence of Internet activity addressing the experiences of young people in two British communities: South Asian and Chinese.We focus on two web sites: www.barficulture.com and www.britishbornchinese.org.uk, drawing on interviews with site editors, content analysis of the discussion forums, and E-mail exchanges with site users. Our analysis of these two web sites shows how collective identities still matter, being redefined rather than erased by online interaction. We understand the site content through the notion of reflexive racialisation. We use this term to modify the stress given to individualisation in accounts of reflexive modernisation. In addition we question the allocation of racialised meaning from above implied by the concept of racialisation. Internet discussion forums can act as witnesses to social inequalities and through sharing experiences of racism and marginalisation, an oppositional social perspective may develop. The online exchanges have had offline consequences: social gatherings, charitable donations and campaigns against adverse media representations. These web sites have begun to change the terms of engagement between these ethnic groups and the wider society,and they have considerable potential to develop new forms of social action.