Initial computability, algebraic specifications, and partial algebras.

*(English)*Zbl 0634.68001
The International Series of Monographs on Computer Science, 2. Oxford Science Publications. Oxford: Clarendon Press. (Copubl. with Akademie- Verlag, Berlin). 221 p.; £30.00 (1987).

This is the second, modified edition of the author’s book “Structural induction on partial algebras. Introduction to theory and application of partial algebras. Part II” [Math. Res. 18 (1984; Zbl 0553.08002)]. [Part I: P. Burmeister, A model theoretic oriented approach to partial algebras (Math. Res. 32) (1986; Zbl 0598.08004).] The reviewed book is concerned with two areas of science: partial algebras and computer science.

Partial algebras (more precisely: partial heterogeneous algebras) are presented here in an original way. The partiality of operations is defined equationally. Notion of hep (hierarchical equationally partial) signature is introduced, and hep-varieties and hep-quasi-varieties are considered (Chapter 2). Chapter 3 is devoted to the algebraic foundation of structural induction on partial algebras.

Classes of equationally partial algebras, defined by so-called canons, i.e. initially restricting algebraic theories, are considered in Chapter 4. It is motivated by semantic problems and functional enrichments of parameterized abstract data types.

Chapters 1 and 5 are concerned with applications of the language of partial algebras to computer science, namely: Chapter 1 gives specifications of several abstract data types and of parameterized abstract data types containing partial operations. The notion of behavioural canons is introduced in Chapter 5 and computable enrichments of such canons are studied here.

References in the book are mainly concerned with computer science (an almost complete bibliography of partial algebras is given in the above mentioned book of P. Burmeister). - An index ends the book.

Partial algebras (more precisely: partial heterogeneous algebras) are presented here in an original way. The partiality of operations is defined equationally. Notion of hep (hierarchical equationally partial) signature is introduced, and hep-varieties and hep-quasi-varieties are considered (Chapter 2). Chapter 3 is devoted to the algebraic foundation of structural induction on partial algebras.

Classes of equationally partial algebras, defined by so-called canons, i.e. initially restricting algebraic theories, are considered in Chapter 4. It is motivated by semantic problems and functional enrichments of parameterized abstract data types.

Chapters 1 and 5 are concerned with applications of the language of partial algebras to computer science, namely: Chapter 1 gives specifications of several abstract data types and of parameterized abstract data types containing partial operations. The notion of behavioural canons is introduced in Chapter 5 and computable enrichments of such canons are studied here.

References in the book are mainly concerned with computer science (an almost complete bibliography of partial algebras is given in the above mentioned book of P. Burmeister). - An index ends the book.

Reviewer: B.Wojdyło

##### MSC:

68-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to computer science |

08-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to general algebraic systems |

68W30 | Symbolic computation and algebraic computation |

08A55 | Partial algebras |

68N01 | General topics in the theory of software |

68Q55 | Semantics in the theory of computing |

68Q60 | Specification and verification (program logics, model checking, etc.) |