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Journals ACM SIGACT News Volume 32 Issue 2

theory is used . In addition, there are some new results . For example, it is shown that the Hilber t Nullstellensatz over any field is NP-complete . The book is well written and uses examples liberally . It is generally speaking, self-contained . In particular, no knowledge of complexity theory or traditional computability theory is necessary. However, knowledge of basic analysis and topology is necessary . One of the wonderful things about this new field is that it is so new and while much is known , much is not known . The are ample opportunities for new research . The following investigation s come to mind . 1. Explore the structure of the Turing degrees over R . Is there a Turing-complete c .e . set? Is there a Turing-intermediate c .e . set? The usual methods for answering these sort of question s are strongly based on mathematical induction and so do not apply . 2. Continuing with the above line of inquiry, does every `sufficiently low' Turing degree contai n a Julia set? 3. Computable analysis developed in the context of the field of computable real numbers . Perhaps this investigation should be re-opened

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Review of Basic proof theory: second edition by A. S. Troelstra and H. Schwichtenberg. Cambridge University Press.

Reviewer-Avigad, Jeremy
ACM SIGACT News , Volume 32 (2)
Association for Computing Machinery – Jun 1, 2001

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