## Types, Subtypes, and ASL+ (1995)

Venue: | In Recent Trends in Data Type Specification, Lecture Notes in Computer Science 906 |

Citations: | 1 - 0 self |

### BibTeX

@INPROCEEDINGS{Aspinall95types,subtypes,,

author = {David Aspinall},

title = {Types, Subtypes, and ASL+},

booktitle = {In Recent Trends in Data Type Specification, Lecture Notes in Computer Science 906},

year = {1995},

pages = {116--131},

publisher = {Springer-Verlag}

}

### OpenURL

### Abstract

. ASL+ is a formalism for specification and programming in-the-large, based on an arbitrary institution. It has rules for proving the satisfaction and refinement of specifications, which can be seen as a type theory with subtyping, including contravariant refinement for \Pi-abstracted specifications and a notion of stratified equality for higher-order objects. We describe the syntax of the language and a partial equivalence relation semantics. This style of semantics is familiar from subtyping calculi, but a novelty here is the use of a hierarchy of typed domains instead of a single untyped domain. We introduce the formal system for proving satisfaction and refinement and describe how it is linked to proof systems of the underlying programming and specification languages. 1 Introduction There is a simple correspondence between the worlds of type theory and algebraic specification: elementhood, M : A () satisfaction, M 2 Mod(A) subtyping, A A 0 () refinement, A 0 /A The element...

### Citations

751 | Rewrite systems
- Dershowitz, Jouannaud
- 1992
(Show Context)
Citation Context ...e : \Sigma ! \Sigma 0 . If we use the institution of equational logic EQ, then signatures and algebras are the usual algebraic notions and sentences are equations; an orthogonal term rewriting system =-=[9]-=- is a suitable notion of program. PBOs might include R ~ \Phi with arity (ffl; \Sigma) for a set of (oriented) \Sigma-equations ~ \Phi that define an orthogonal term rewriting system, and + \Sigma 1 ;... |

475 |
Institutions: Abstract Model Theory for Specification and Programming
- Goguen, Burstall
- 1992
(Show Context)
Citation Context ...fi-conversion. These aspects of a stratified equality (stratified over a subtyping relation) are well known in subtyping type-systems. 4 2 The Formalism ASL+ ASL+ is based on an arbitrary institution =-=[12]-=-, together with an associated programming language and specification language. This means we are given a collection of signatures \Sigma 2 Sign, and for each signature a collection of algebras Alg(\Si... |

116 | Inheritance as implicit coercion
- Breazu-Tannen, Coquand, et al.
- 1991
(Show Context)
Citation Context ...Sigma 2s\Sigma 1 by virtue of the reduct functor. This would allow many renamings to be avoided at the level of syntax; in the semantics, we would have to reconstruct them. The subtyping semantics in =-=[4]-=- uses a similar technique. 20 As well as subtyping parallels, there are some specification languages related to ASL+. Cengarle and Wirsing [7] describe a language with higherorder parameterisation, in... |

102 | Bounded quantification is undecidable
- Pierce
- 1994
(Show Context)
Citation Context ...in the system, so that showing satisfaction in ASL+ is reduced to showing underlying consequences in a canonical way. Unfortunately, similar things can be tricky to show for related subtyping systems =-=[8, 14]-=-, and the added complication of dependent \Pi-types makes things harder still. Also, the concise presentation of the rules shown here is somewhat provisional; although it is nice to describe, it may p... |

99 | Simple consequence relations
- Avron
- 1991
(Show Context)
Citation Context ... S (or T ) is built using only SBOs and variables from a Sign-sorted set V spec . We abstract away from the details of the underlying proof systems and assume that there are two consequence relations =-=[3]-=- which describe the properties of interest: \Delta j\Gamma\Gamma sat V P : S basic satisfaction; \Delta j\Gamma\Gamma ref V SsT basic refinement. In each case, V is the set of variables (from V prog a... |

63 | A semantic basis for Quest
- Cardelli, Longo
- 1991
(Show Context)
Citation Context ...lain these choices before presenting the definitions. Partial equivalences. The use of partial equivalence relation models for interpreting subtyping and polymorphic type-systems is well-known (e.g., =-=[6]-=-). Recall that a partial equivalence relation R is a set jRj (the carrier) together with a symmetric and transitive relation \GammaR\Gamma ` jRj \Theta jRj. The reason why we use PERs is that they mod... |

58 | Structural subtyping and the notion of power type
- Cardelli
- 1988
(Show Context)
Citation Context ... of the ASL specification language. The rules of their formal system are remarkably similar to those of a subtyping system with power types, designed by Cardelli as a programming language type-system =-=[5]-=-. Cardelli's system has terms of the form Power (A), which stands for the type of all subtypes of A. He uses the elementhood A : Power (A 0 ) to stand for the subtyping AsA 0 . In ASL+, there is a cor... |

46 | Subtyping with Singleton Types
- Aspinall
- 1995
(Show Context)
Citation Context ...for some B We have explained the "subtyping" abbreviation already. A relation of typedequality \Gamma ` M = N : A is included in the calculus via singleton types; this has been investigated =-=elsewhere [1]-=-. The judgement \Gamma ` A asserts that A is a denoting type. More generally, a denoting term M is one for which there exists an A such that \Gamma ` M : A. Thus the rules in Table 3 can also be used ... |

38 |
and Andrzej Tarlecki. Toward formal development of programs from algebraic specifications
- Sannella
- 1988
(Show Context)
Citation Context ...ation a model-theoretic semantics is paramount, and we search for sound proof-theoretic counterparts afterward. An example of this is a formal system introduced by Sannella, Soko/lowski, and Tarlecki =-=[15]-=- for proving satisfaction in ASL+, an extended version of the ASL specification language. The rules of their formal system are remarkably similar to those of a subtyping system with power types, desig... |

24 | and Andrzej Tarlecki. Specifications in an arbitrary institution - Sannella - 1988 |

2 |
Verification in ASL and related Specification Languages
- Farres-Cassals
- 1992
(Show Context)
Citation Context ...s and S stands for a basic \Sigma 0 -specification. Examples of proof rules which generate a consequence relation for j\Gamma\Gamma ref on the institution independent operators of ASL can be found in =-=[11]-=-. Some assumptions must be made of j\Gamma\Gamma sat and j\Gamma\Gamma ref , the most important of which is that they are sound with respect to the semantics of basic programs and specifications. For ... |

2 |
Structuring specifications inthe -large and in-the-small: Higher-order functions, dependent types and inheritance
- Krieg-Bruckner, Sannella
- 1991
(Show Context)
Citation Context ...meterisation on signatures. Drossopoulou and Paterson [10] have explored similar type theory connections when designing a language for writing parameterised programs and their interfaces. SPECTRAL in =-=[13]-=- is a design for a rich language that combines an institution-level language that has subtyping, polymorphism and dependent types with a module-level language similarly endowed for specifications and ... |

2 | Parametricity in algebraic specifications: a case study
- Sokolowski
- 1990
(Show Context)
Citation Context ...s5 Conclusions ASL+ provides a unified framework for programming and specification inthe -large which seems powerful. Some larger and more detailed examples than those shown here have been studied in =-=[17, 15, 2]-=- but work is still needed to appreciate the practical merits of the different mechanisms of parameterisation. Real specification languages built on ASL+ might not use all of the parameterisation possi... |

1 |
Type systems for algebraic specification in-the-large. Forthcoming
- Aspinall
- 1996
(Show Context)
Citation Context ...ics. In Section 3 we describe the semantics based on partial equivalence relations. Section 4 provides only an outline of the formal system for proving satisfaction; a fuller treatment will appear in =-=[2]-=-. Section 5 concludes. 1.1 Overview of the ASL+ framework ASL+ is proposed as a kernel language for writing programs and specifications in-the-large; it extends the -calculus parameterisation mechanis... |

1 |
A calculus of parameterization for algebraic specifications
- Cengarle, Wirsing
- 1994
(Show Context)
Citation Context ...ould have to reconstruct them. The subtyping semantics in [4] uses a similar technique. 20 As well as subtyping parallels, there are some specification languages related to ASL+. Cengarle and Wirsing =-=[7]-=- describe a language with higherorder parameterisation, including parameterisation on signatures. Drossopoulou and Paterson [10] have explored similar type theory connections when designing a language... |

1 |
Higher order module parameterization
- Drossopoulou, Paterson
- 1993
(Show Context)
Citation Context ...e are some specification languages related to ASL+. Cengarle and Wirsing [7] describe a language with higherorder parameterisation, including parameterisation on signatures. Drossopoulou and Paterson =-=[10]-=- have explored similar type theory connections when designing a language for writing parameterised programs and their interfaces. SPECTRAL in [13] is a design for a rich language that combines an inst... |