## Stretching First Order Equational Logic: Proofs with Partiality, Subtypes and Retracts (1998)

Venue: | Proceedings, Workshop on First Order Theorem Proving |

Citations: | 4 - 2 self |

### BibTeX

@INPROCEEDINGS{Goguen98stretchingfirst,

author = {Joseph A. Goguen},

title = {Stretching First Order Equational Logic: Proofs with Partiality, Subtypes and Retracts},

booktitle = {Proceedings, Workshop on First Order Theorem Proving},

year = {1998},

pages = {pages}

}

### OpenURL

### Abstract

It is widely recognized that equational logic is simple, (relatively) decidable, and (relatively) easily mechanized. But it is also widely thought that equational logic has limited applicability because it cannot handle subtypes or partial functions. We show that a modest stretch of equational logic effectively handles these features. Space limits preclude a full theoretical treatment, so we often sketch, motivate and exemplify.

### Citations

281 | An initial algebra approach to the specification, correctness, and implementation of abstract data types - Goguen, Thatcher, et al. - 1978 |

213 | Order-sorted algebra I: equational deduction for multiple inheritance, overloading, exceptions and partial operations
- Goguen, Meseguer
- 1989
(Show Context)
Citation Context ... conditional equations were also important; see [10] for technical and historical details. Section 2 quickly reviews many sorted EL, and Section 3 covers the next important extension, order sorted EL =-=[11]-=-, including an inductive proof for a typical partial function. A final section discusses a further extension to hidden EL. 2 Many Subsorted Equational Logic Overloaded many sorted EL (MSEL) permits ov... |

187 |
On the structure of abstract algebras
- Birkhoff
- 1935
(Show Context)
Citation Context ... order EL does not seem sufficiently expressive. This paper describes an extension that greatly expands its expressiveness and applicability, at little cost to its advantages. EL was untyped at birth =-=[3]-=-, but later extended to many sorts in various ways, of which [1] was perhaps first and [5] notationally simplest; extensions to overloaded function symbols and conditional equations were also importan... |

176 | General logics - Meseguer |

155 | Membership algebra as a logical framework for equational specification - Meseguer |

117 | A hidden agenda
- Goguen, Malcolm
- 2000
(Show Context)
Citation Context ...res of current interest in computer science, including concurrency, nondeterminism, distribution and inheritance. This adventure is still at an early stage, but some promising results can be found in =-=[9]-=- with further references. Many related adventures are under way, including a dual adventure called coalgebra [16], and the new CafeOBJ industrial strength OBJ implementation, which directly implements... |

109 |
Algebraic Semantics of Imperative Programs
- Goguen, Malcolm
- 1996
(Show Context)
Citation Context ... It says we can regard the universally variables as new constants instead. 79 2.1 Basic OBJ Notation OBJ gives a notation which has been implemented to permit proving things about such specifications =-=[8, 13]-=-; this is sufficiently powerful that it can be used as a programming language for small applications, and as a prototyping language for larger applications. OBJ modules to be interpreted loosely, i.e.... |

96 | An approach to object semantics based on terminal co-algebras - Reichel - 1995 |

65 | Completeness of many-sorted equational logic
- Goguen, Meseguer
- 1985
(Show Context)
Citation Context ...ater extended to many sorts in various ways, of which [1] was perhaps first and [5] notationally simplest; extensions to overloaded function symbols and conditional equations were also important; see =-=[10]-=- for technical and historical details. Section 2 quickly reviews many sorted EL, and Section 3 covers the next important extension, order sorted EL [11], including an inductive proof for a typical par... |

52 | Moving between logical systems - Tarlecki - 1996 |

45 |
Order-sorted algebra solves the constructor-selector, multiple representation and coercion problems
- Goguen, Meseguer
- 1987
(Show Context)
Citation Context ...rigorous theory for many kinds of partial function, and that reasoning about such functions can be mechanized in OBJ3 [13]. This section sketches overloaded order sorted EL (OSEL), including retracts =-=[11, 6, 15]-=- and their implementation and use in OBJ3, with examples. Things are much the same as for MSEL, modulo some technicalities; therefore we shall be rather informal. An OSEL signature adds an ordering re... |

44 |
JosÃ© Meseguer, Kokichi Futatsugi, and Jean-Pierre Jouannaud. Applications of Algebraic Specifications Using OBJ, chapter Introducing OBJ
- Goguen, Winkler
- 1993
(Show Context)
Citation Context ... It says we can regard the universally variables as new constants instead. 79 2.1 Basic OBJ Notation OBJ gives a notation which has been implemented to permit proving things about such specifications =-=[8, 13]-=-; this is sufficiently powerful that it can be used as a programming language for small applications, and as a prototyping language for larger applications. OBJ modules to be interpreted loosely, i.e.... |

43 |
A characterization of computable data types by means of a finite, equational specification method
- Bergstra, Tucker
- 1980
(Show Context)
Citation Context ...e pun, but a significant fact about software engineering. Another sign we are on the right track is that any computable algebra has an equational specification, as first proved by Bergstra and Tucker =-=[2]-=-; moreover, this specification tends to be simple and intuitive in practice. (M is reachable iff the unique \Sigma-homomorphism T \Sigma !M is surjective.) Theorem 3: (Computability) Given a reachable... |

33 |
Operational semantics of order-sorted algebra
- Goguen, Jouannaud, et al.
- 1985
(Show Context)
Citation Context ...t signatures do not seem to occur in natural examples. The OBJ3 notation and implementation handle OSEL much the same as MSEL, modulo some subtle points about order sorted rewriting that we omit here =-=[7, 14]-=-. OSEL can specify not just computable functions, but also semi-computable functions. 3.1 Retracts Untyped logics are too permissive, allowing many expressions that make no sense. But strongly typed l... |

32 | Higher-Order Functions Considered Unnecessry for Higher-Order Programming - Goguen - 1990 |

31 |
and R*azvan Diaconescu. An Oxford survey of order sorted algebra
- Goguen
- 1994
(Show Context)
Citation Context ...rigorous theory for many kinds of partial function, and that reasoning about such functions can be mechanized in OBJ3 [13]. This section sketches overloaded order sorted EL (OSEL), including retracts =-=[11, 6, 15]-=- and their implementation and use in OBJ3, with examples. Things are much the same as for MSEL, modulo some technicalities; therefore we shall be rather informal. An OSEL signature adds an ordering re... |

28 |
Operational semantics of OBJ3
- Kirchner, Kirchner, et al.
- 1988
(Show Context)
Citation Context ...t signatures do not seem to occur in natural examples. The OBJ3 notation and implementation handle OSEL much the same as MSEL, modulo some subtle points about order sorted rewriting that we omit here =-=[7, 14]-=-. OSEL can specify not just computable functions, but also semi-computable functions. 3.1 Retracts Untyped logics are too permissive, allowing many expressions that make no sense. But strongly typed l... |

26 | Formality and Informality in Requirements Engineering - Goguen - 1996 |

24 |
Structures alg'ebriques dans les cat'egories. Cahiers de Topologie et G'eometrie Diff'erentiel
- Benabou
- 1968
(Show Context)
Citation Context ...ribes an extension that greatly expands its expressiveness and applicability, at little cost to its advantages. EL was untyped at birth [3], but later extended to many sorts in various ways, of which =-=[1]-=- was perhaps first and [5] notationally simplest; extensions to overloaded function symbols and conditional equations were also important; see [10] for technical and historical details. Section 2 quic... |

20 |
Semantics of computation
- Goguen
- 1975
(Show Context)
Citation Context ...eatly expands its expressiveness and applicability, at little cost to its advantages. EL was untyped at birth [3], but later extended to many sorts in various ways, of which [1] was perhaps first and =-=[5]-=- notationally simplest; extensions to overloaded function symbols and conditional equations were also important; see [10] for technical and historical details. Section 2 quickly reviews many sorted EL... |

20 | Parameterized programming and software architecture, in - Goguen - 1996 |

19 | From total equational to partial first order - Cerioli, Mossakowski, et al. - 1998 |

13 | An overview of Cafe specification environment - Futatsugi, Nakagawa - 1997 |

13 | Partially ordered sorts in algebraic specifications - Gogolla - 1984 |

11 | Proving and rewriting - Goguen - 1990 |

10 | Occasioned practises in the work of implementing development methodlogies, in - Sharrock - 1994 |

9 | Pushouts of order-sorted algebraic specifications - Haxthausen, Nickl - 1996 |

6 | Partial algebras, subsorting, and dependent types: Prerequesites for error handling in algebraic specifications - Poign'e - 1987 |

3 | A Model Theoretic Oriented Appraoch to Partial Algebras. Akademie-Verlag - Burmeister - 1986 |

3 | Parameterization for order-sorted algebraic specification - Poign'e - 1990 |

2 | Structural Induction on Partial Algebras. Akademie-Verlag - Reichel - 1984 |

1 | Uber partiell goerdnete Sortenmengen und deren Anwendung zur Fehlerbehandlung in Abstrakten Datentypen - Gogolla - 1983 |

1 | Theorem Proving and Algebra. MIT, to appear. [18] Joseph Goguen and Rod Burstall. Institutions: Abstract model theory for specification and programming - Goguen - 1992 |

1 | A hidden agenda. To appear - Goguen, Malcolm - 1997 |

1 | 30] Jos'e Meseguer. General logics - Macmillan - 1967 |

1 | Error handling for parametrized data types - Poign'e - 1984 |