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**Algebraic recursion theory. Ed. by J. L. Bell.**
*(English)*
Zbl 0613.03018

Mathematics and its Applications. Statistics and Operational Research. Chichester: Ellis Horwood Limited; New York etc.: Halsted Press: a division of John Wiley & Sons. 256 p. £37.50 (1986).

The aim of this book is to do recursive function theory abstractly by means of algebraic systems satisfying certain suitable axioms.

Part A is an introduction. In part B the author defines a system called an operative space. Roughly speaking such a system is an abstraction of what one has in ordinary recursion theory with the usual pairing functions J, K, and l. The author then introduces iterative operator spaces. These involve further axioms which give the existence of certain kinds of minimal fixed points. Various classes of recursive functions are studied in this abstract setting. Among the families of examples is one dealing with fuzzy relations and one dealing with abstract programs in theoretical computer science.

Part A is an introduction. In part B the author defines a system called an operative space. Roughly speaking such a system is an abstraction of what one has in ordinary recursion theory with the usual pairing functions J, K, and l. The author then introduces iterative operator spaces. These involve further axioms which give the existence of certain kinds of minimal fixed points. Various classes of recursive functions are studied in this abstract setting. Among the families of examples is one dealing with fuzzy relations and one dealing with abstract programs in theoretical computer science.

Reviewer: H.Gonshor

### MSC:

03D75 | Abstract and axiomatic computability and recursion theory |

03-02 | Research exposition (monographs, survey articles) pertaining to mathematical logic and foundations |

03D20 | Recursive functions and relations, subrecursive hierarchies |