## Recursive Definitions in Z (1998)

Venue: | ZUM’98: The Z Formal Specification Notation, volume 1493 of Lecture Notes in Computer Science |

Citations: | 5 - 0 self |

### BibTeX

@INPROCEEDINGS{Arthan98recursivedefinitions,

author = {R.D. Arthan},

title = {Recursive Definitions in Z},

booktitle = {ZUM’98: The Z Formal Specification Notation, volume 1493 of Lecture Notes in Computer Science},

year = {1998},

pages = {154--171},

publisher = {Springer-Verlag}

}

### OpenURL

### Abstract

This paper considers some issues in the theory and practice of defining functions over recursive data types in Z. Principles justifying such definitions are formulated. Z free types are contrasted with the free algebras of universal algebra: the notions turn out to be related but not isomorphic.

### Citations

1188 |
The Z Notation: A Reference Manual
- Spivey
- 1989
(Show Context)
Citation Context ...ut an inconsistent specification is wasted and implementation of an inconsistent specification is either impossible or trivial depending on one's point of view. The most widely used definitions of Z, =-=[8, 9]-=- do consider the consistency of some Z paragraph forms, most notably the free type paragraph. This topic is further explored in [1, 7, 11]. In this paper, we consider approaches to proving the consist... |

162 |
Naive Set Theory
- Halmos
- 1974
(Show Context)
Citation Context ... Y ffl 8 x : F FT ffl h(k(x )) = e(h(j x j)) (6) The proof of (6) is similar to the proof of definition by induction for the natural numbers that one can find in elementary texts on set theory (e.g., =-=[5]-=-). Given Y and e as in the statement of the theorem, we consider partial approximations to the desired total function h, That is to say we consider functions g : FT ! Y which satisfy g(k(x )) = e(g(j ... |

95 |
The Z Notation: A Reference Manual (second edition
- Spivey
- 1992
(Show Context)
Citation Context ...ut an inconsistent specification is wasted and implementation of an inconsistent specification is either impossible or trivial depending on one's point of view. The most widely used definitions of Z, =-=[8, 9]-=- do consider the consistency of some Z paragraph forms, most notably the free type paragraph. This topic is further explored in [1, 7, 11]. In this paper, we consider approaches to proving the consist... |

83 |
Difierence Algebra
- |Cohn
- 1965
(Show Context)
Citation Context ...section, we shall explore the analogy that gives rise to the name, and find, perhaps surprisingly, that it is not as close as one might hope. We will need some elementary ideas from universal algebra =-=[4]-=-. Universal algebra studies the features of algebraic systems such as rings, groups, fields, etc., that are independent of the fine details of the theory of rings, groups, fields, etc. Two basic conce... |

77 | Automating recursive type definitions in higher order logic
- Melham
- 1989
(Show Context)
Citation Context ... special attention to the functorial case 6 so as to exploit the additional properties it enjoys. This would let us use many of the practical techniques developed for other logical systems, e.g., see =-=[6]-=- for a treatment of recursive definitions in HOL. In particular, for functorial free type definitions, PDR actually entails all the other axioms and so may be used as a starting point for many lines o... |

15 |
On recursive free types in Z
- Smith
(Show Context)
Citation Context ...ng on one's point of view. The most widely used definitions of Z, [8, 9] do consider the consistency of some Z paragraph forms, most notably the free type paragraph. This topic is further explored in =-=[1, 7, 11]-=-. In this paper, we consider approaches to proving the consistency of a particular class of axiomatic description, namely, axiomatic descriptions that define functions on a (typically recursive) free ... |

14 | On Free Type Definitions in Z
- Arthan
- 1991
(Show Context)
Citation Context ...ng on one's point of view. The most widely used definitions of Z, [8, 9] do consider the consistency of some Z paragraph forms, most notably the free type paragraph. This topic is further explored in =-=[1, 7, 11]-=-. In this paper, we consider approaches to proving the consistency of a particular class of axiomatic description, namely, axiomatic descriptions that define functions on a (typically recursive) free ... |

5 |
Richer Types for Z
- Spivey
- 1996
(Show Context)
Citation Context ...e approaches are not isomorphic. For many purposes, the set-theoretic approach is, I would claim, both simpler and more powerful. 6 A possible line in this direction would be to adopt the approach of =-=[10]-=- which reports on an enrichment of the Z type system, extending the basic types of Z to allow finer distinctions to be made by a Z support tool, e.g., to distinguish sequences from other less structur... |

3 | Automating Recursive Type De in Higher Order Logic - Melham - 1989 |

2 |
The Consistency Theorem for Free Type Definitions in
- Spivey
- 1996
(Show Context)
Citation Context ...ng on one's point of view. The most widely used definitions of Z, [8, 9] do consider the consistency of some Z paragraph forms, most notably the free type paragraph. This topic is further explored in =-=[1, 7, 11]-=-. In this paper, we consider approaches to proving the consistency of a particular class of axiomatic description, namely, axiomatic descriptions that define functions on a (typically recursive) free ... |

1 |
Recursive Data Types in Typed Set Theory. Unpublished pre-print
- Arthan
(Show Context)
Citation Context ...m. In categorical language, these are the ones for which the operation mapping Y to E i [Y =T ] gives the objects part of a functor whose morphisms part is the operation hsdiscussed in Sect. 3.1 (see =-=[2]-=- for a more explicit account). It may well be of benefit in developing conceptual and mechanized tools for working with Z free types to pay special attention to the functorial case 6 so as to exploit ... |

1 |
Mechanizing the Z Toolkit
- Arthan
- 1997
(Show Context)
Citation Context ...free use of the Z toolkit. Techniques for informal reasoning about the toolkit are discussed in several books, e.g., [13]; progress on one approach to automated proof for the Z toolkit is reported in =-=[3]-=-. The rest of this paper is structured as follows: Sect. 2 discusses the practical issues; Sect. 3 considers general principles for definition by recursion; Sect. 4 contrasts the theory for Z with con... |

1 |
Inconsistency and Undefinedness in Z --- A Practical Guide. These proceedings
- Valentine
- 1998
(Show Context)
Citation Context ...o the consistency of f . Approaches to verifying the consistency of Z specifications are discussed in more detail in earlier work of the present author [1] and in a forthcoming paper by Sam Valentine =-=[12]-=- As a first example of a function defined on the set BINTREE, consider the following axiomatic description of a function intended to return the label at the root of a tree. labelOf : BINTREE ! Z 8 i :... |

1 | On Free Type De in Z - Arthan - 1991 |

1 | The Consistency Theorem for Free Type De - Spivey - 1996 |

1 | Inconsistency and Unde in Z | A Practical Guide. These proceedings - Valentine - 1998 |