## Coercive subtyping (1998)

Citations: | 46 - 21 self |

### BibTeX

@MISC{Luo98coercivesubtyping,

author = {Zhaohui Luo},

title = {Coercive subtyping},

year = {1998}

}

### Years of Citing Articles

### OpenURL

### Abstract

### Citations

921 | The generative lexicon
- Pustejovsky
- 1995
(Show Context)
Citation Context ...ch are subtypes of the type e of all entities, and interpret the adjective ‘heavy’ and CN ‘book’ as follows: [[heavy]] : (Phy → t) → (Phy → t) [[book]] : Phy • Info → t where the dot-type Phy • Info (=-=Pustejovsky 1995-=-) is a type of entities with both physical and informational aspects (and a book has both). However, this has a problem: 1 The idea of using coercive subtyping in linguistic semantics was considered i... |

847 |
A formulation of the simple theory of types
- Church
- 1940
(Show Context)
Citation Context ...eference transfers and multiple word meanings in formal lexical semantics. Keywords Formal semantics · Type theory · Coercive subtyping · Lexical semantics 1 Introduction Church’s simple type theory (=-=Church 1940-=-), as employed in Montague’s semantics (Montague 1974), has traditionally served as a logical language for formal semantics. Powerful alternatives, arguably more advantageous ones, may be offered by t... |

758 | On understanding types, data abstraction, and polymorphism
- Cardelli, Wegner
- 1985
(Show Context)
Citation Context ...8]. Subtyping has also been studied extensively by many researchers in the context of typed functional programming, inspired by the notion of inheritance as found in object-oriented programming (cf., =-=[13, 12]-=-). Semantic studies on subtyping in type systems (without dependent types) such as the second-order lambda calculus (ie, the system F) include the use of the PER model [9] and the coercion-based appro... |

696 | A Framework for Defining Logics
- Harper, Honsel, et al.
- 1993
(Show Context)
Citation Context ...s a typed version of Martin-Lof's logical framework (see Chapter 19 of [38] for a presentation of the latter). We should also point out that LF is different from the Edinburgh Logical Framework (ELF) =-=[22]-=-. 2 The presentation of LF and discussions on how it should be used in specifying type theories can be found in Chapter 9 of [30]. The inference rules of LF are given in Figure 1, which include genera... |

518 | Lambda calculi with types
- Barendregt
- 1992
(Show Context)
Citation Context ...ing in the meta-level logical framework, rather than in some object-level type systems is important and beneficial. 2 2 The so-called object-level type systems include, for example, Pure Type Systems =-=[5]-=- such as the Calculus of Constructions (CC) [16]. See [6] for an attempt to introduce coercions into PTS. 7 3 Coercive subtyping In this section, we first introduce the basic ideas and give informal m... |

471 |
The calculus of constructions
- Coquand, Huet
- 1988
(Show Context)
Citation Context ... than in some object-level type systems is important and beneficial. 2 2 The so-called object-level type systems include, for example, Pure Type Systems [5] such as the Calculus of Constructions (CC) =-=[16]-=-. See [6] for an attempt to introduce coercions into PTS. 7 3 Coercive subtyping In this section, we first introduce the basic ideas and give informal meaning explanations of the judgements in the ext... |

441 |
The formulae-as-types notion of construction
- Howard
- 1980
(Show Context)
Citation Context ...ssed in Luo (2010, 2011b) and the current paper. 2.1.2 Embedded logic A modern type theory has an embedded logic (or internal logic) based on the propositions-as-types principle (Curry and Feys 1958; =-=Howard 1980-=-). For example, in Martin-Löf’s predicative type theory, the logical proposition A & B corresponds to the product type A × B (a special case of �-type—see below) and a pair of a proof of A and a proof... |

342 |
Intuitionistic type theory
- Martin-Löf
- 1984
(Show Context)
Citation Context ...s: type theory, subtyping, coercion, formal reasoning, logical framework. 1 Introduction A type in type theory is often intuitively thought of as a set. For example, types in Martin-Lof's type theory =-=[36, 38]-=- can be considered as inductively defined sets. A fundamental difference between type theory and set theory is that in the former we do not have a notion of subtype that corresponds to the notion of s... |

264 |
Temporal ontology and temporal reference
- Moens, Steedman
- 1988
(Show Context)
Citation Context ...Ts, when equipped with them, become more powerful in practical applications to formal semantics. In linguistic semantics, various notions of coercion have been proposed and studied (see, for example, =-=Moens and Steedman 1988-=-). Obvious future work includes that to study to what extent the theory of coercive subtyping covers the linguistic notions of type-shifting and type coercion (see, for example, Partee and Rooth 1983;... |

263 |
Foundations of Constructive Mathematics
- Beeson
- 1985
(Show Context)
Citation Context ... : (x:K)K 0 \Gamma; x:K ` f(x) = g(x) : K 0 \Gamma ` f = g : (x:K)K 0 1 One may want to consider a more philosophical argument of whether an operation should be considered as typed. See, for example, =-=[7]-=- for some relevant discussions. 3 Contexts and assumptions hi valid \Gamma ` K kind x 62 FV (\Gamma) \Gamma; x:K valid \Gamma; x:K; \Gamma 0 valid \Gamma; x:K; \Gamma 0 ` x : K General equality rules ... |

261 |
Programming in Martin-Löf’s Type Theory: An Introduction
- Nordstrom, Petersson, et al.
- 1990
(Show Context)
Citation Context ...ce University of Durham Email: Zhaohui.Luo@durham.ac.uk Abstract We propose and study coercive subtyping, a formal extension with subtyping of dependent type theories such as Martin-Lof's type theory =-=[38]-=- and the type theory UTT [30]. In this approach, subtyping with specified implicit coercions is treated as a feature at the level of the logical framework; in particular, the meaning of an object bein... |

175 |
Noun phrase interpretation and type-shifting principles
- Partee
- 1987
(Show Context)
Citation Context .... Obvious future work includes that to study to what extent the theory of coercive subtyping covers the linguistic notions of type-shifting and type coercion (see, for example, Partee and Rooth 1983; =-=Partee 1986-=-; Pustejovsky 2011). Some of the well-known type-shifting principles can be captured by means of coercive subtyping in a straightforward way. For example, consider the type-shifting law for NPs that l... |

154 |
Generalized conjunction and type ambiguity
- Partee, Rooth
- 1983
(Show Context)
Citation Context ...oens and Steedman 1988). Obvious future work includes that to study to what extent the theory of coercive subtyping covers the linguistic notions of type-shifting and type coercion (see, for example, =-=Partee and Rooth 1983-=-; Partee 1986; Pustejovsky 2011). Some of the well-known type-shifting principles can be captured by means of coercive subtyping in a straightforward way. For example, consider the type-shifting law f... |

147 |
LEGO Proof Development System: User's Manual
- Luo, Pollack
- 1992
(Show Context)
Citation Context ...at corresponds to the notion of subset in the latter. The lack of useful subtyping mechanisms in dependent type theories with inductive types [17, 20, 30] and the associated proof development systems =-=[35, 14, 19, 34]-=- is one of the obstacles in their applications to large-scale formal development. Although subtyping is conceptually natural and pragmatically important, it has not been clear how useful and suitable ... |

141 | Intuitionistic Type Theory. Bibliopolis - Martin-Löf - 1984 |

125 | On the meanings of the logical constants and the justifications of the logical laws - Martin-Löf - 1996 |

109 |
Irlductively defined types
- Coquand, Paulin
- 1990
(Show Context)
Citation Context ...s that in the former we do not have a notion of subtype that corresponds to the notion of subset in the latter. The lack of useful subtyping mechanisms in dependent type theories with inductive types =-=[17, 20, 30]-=- and the associated proof development systems [35, 14, 19, 34] is one of the obstacles in their applications to large-scale formal development. Although subtyping is conceptually natural and pragmatic... |

108 |
Computation and Reasoning, A Type Theory for Computer Science
- Luo
- 1994
(Show Context)
Citation Context ...: Zhaohui.Luo@durham.ac.uk Abstract We propose and study coercive subtyping, a formal extension with subtyping of dependent type theories such as Martin-Lof's type theory [38] and the type theory UTT =-=[30]-=-. In this approach, subtyping with specified implicit coercions is treated as a feature at the level of the logical framework; in particular, the meaning of an object being in a supertype is given by ... |

104 |
Type-Theoretical Grammar
- Ranta
- 1995
(Show Context)
Citation Context ...eas of developing formal semantics in Martin-Löf’s type theory have been studied in Ranta (1994). Ranta himself may not regard his work as studying logical semantics (see, for example, the preface of =-=Ranta 1994-=-). However, if one looks at it from a technical (and non-philosophical) point of view, the work has studied the basics of formal semantics in a modern type theory and made many valuable proposals. 123... |

98 | Martin-Löf Type Theory
- Nordström, Petersson, et al.
- 2000
(Show Context)
Citation Context ...tionally served as a logical language for formal semantics. Powerful alternatives, arguably more advantageous ones, may be offered by the modern type theories (MTTs) such as Martin-Löf’s type theory (=-=Nordström et al. 1990-=-; Z. Luo (B) Department of Computer Science, Royal Holloway, University of London, Egham, Surrey TW20 0EX, UK e-mail: zhaohui.luo@hotmail.co.uk 123Author's personal copy 492 Z. Luo Martin-Löf 1984) a... |

94 |
References and generality
- Geach
- 1968
(Show Context)
Citation Context ...the current paper may be seen as a further development of those initial ideas. 2 The idea of interpreting CNs as types is further investigated in Luo (2012) where, in particular, Geach’s observation (=-=Geach 1962-=-) that CNs have criteria of identity is studied to justify this idea and, furthermore, it is pointed out that proof irrelevance should be adopted in a modern type theory in order for CNs to be interpr... |

93 |
The ALF proof editor and its proof engine
- Magnusson, Nordström
- 1994
(Show Context)
Citation Context ...at corresponds to the notion of subset in the latter. The lack of useful subtyping mechanisms in dependent type theories with inductive types [17, 20, 30] and the associated proof development systems =-=[35, 14, 19, 34]-=- is one of the obstacles in their applications to large-scale formal development. Although subtyping is conceptually natural and pragmatically important, it has not been clear how useful and suitable ... |

87 |
A modest model of records, inheritance, and bounded quantification
- Bruce, Longo
(Show Context)
Citation Context ...ented programming (cf., [13, 12]). Semantic studies on subtyping in type systems (without dependent types) such as the second-order lambda calculus (ie, the system F) include the use of the PER model =-=[9]-=- and the coercion-based approach [8]. A more recent logical study of subtyping in system F can be found in [24]. There is not much work on subtyping in dependent type systems in the context of proof d... |

84 | An Extended Calculus of Constructions
- Luo
- 1990
(Show Context)
Citation Context ...since they do not take into the account (the forms of) the canonical objects in the subtypes. For instance, subtyping between two \Sigma-types as found in the Extended Calculus of Constructions (ECC) =-=[25, 26]-=- is not quite compatible with the general elimination rules as found in Martin-Lof's type theory and UTT. A simple combination would lead to a system for which the subject reduction property fails to ... |

76 | Inductive sets and families in Martin-Löfs Type Theory and their set-theoretic semantics: An inversion principle for Martin-Löfs type theory
- Dybjer
- 1991
(Show Context)
Citation Context ...s that in the former we do not have a notion of subtype that corresponds to the notion of subset in the latter. The lack of useful subtyping mechanisms in dependent type theories with inductive types =-=[17, 20, 30]-=- and the associated proof development systems [35, 14, 19, 34] is one of the obstacles in their applications to large-scale formal development. Although subtyping is conceptually natural and pragmatic... |

74 |
Transfers of meaning
- Nunberg
- 1995
(Show Context)
Citation Context ...sages are only meaningful in such special situations or local contexts in which, for instance, the meanings of some words change. Example 3 (reference transfer) Consider the following utterance (cf., =-=Nunberg 1995-=-): (20) The ham sandwich shouts. Assuming that the act of shouting requires that the argument be human (ie, [[shout]] : [[human]] → Prop), it is obvious that sentence (20) is not well-formed, unless i... |

73 |
Formal Philosophy
- Montague
- 1974
(Show Context)
Citation Context ...rmal lexical semantics. Keywords Formal semantics · Type theory · Coercive subtyping · Lexical semantics 1 Introduction Church’s simple type theory (Church 1940), as employed in Montague’s semantics (=-=Montague 1974-=-), has traditionally served as a logical language for formal semantics. Powerful alternatives, arguably more advantageous ones, may be offered by the modern type theories (MTTs) such as Martin-Löf’s t... |

68 | Subtyping dependent types
- Aspinall, Compagnoni
- 1996
(Show Context)
Citation Context ... subkinding in our terminology) in Martin-Lof's logical framework [43], Pfenning's work on refinement types [39], and Aspinall and Compagnoni's work on the decidability of Edinburgh LF with subtyping =-=[2]-=- are among the recent research development. It is clear that coercive subtyping is strongly motivated by the need in proof development, where inductive types are essential. Traditional approaches base... |

68 | The theory of LEGO: A proof checker for the extended calculus of constructions
- Pollack
- 1994
(Show Context)
Citation Context ..., UTT has been applied to verification of functional programs [10, 11], imperative programs [42] and concurrent programs [44], specification and data refinement [29], and formalisation of mathematics =-=[40]-=-. 2.3 Definitional equality and computational equality We use LF seriously as a meta-level language (see Section 9.1.2 of [30] for a discussion). Along the same line, we make a distinction between the... |

52 |
A Typed Operational Semantics for Type Theory
- Goguen
- 1994
(Show Context)
Citation Context ...ductive data types (and inductive families, not covered above), and predicative type universes. It has nice meta-theoretic properties such as ChurchRosser, Subject Reduction, and Strong Normalisation =-=[21]-=-. Implemented in the Lego proof development system, UTT has been applied to verification of functional programs [10, 11], imperative programs [42] and concurrent programs [44], specification and data ... |

50 |
The Coq proof assistant user's guide. Version 5.8
- Dowek, Felty, et al.
- 1993
(Show Context)
Citation Context ...at corresponds to the notion of subset in the latter. The lack of useful subtyping mechanisms in dependent type theories with inductive types [17, 20, 30] and the associated proof development systems =-=[35, 14, 19, 34]-=- is one of the obstacles in their applications to large-scale formal development. Although subtyping is conceptually natural and pragmatically important, it has not been clear how useful and suitable ... |

42 | Refinement types for logical frameworks
- Pfenning
- 1993
(Show Context)
Citation Context ... type theory. Betarte and Tasistro's work on record types and subtyping (or record kinds and subkinding in our terminology) in Martin-Lof's logical framework [43], Pfenning's work on refinement types =-=[39]-=-, and Aspinall and Compagnoni's work on the decidability of Edinburgh LF with subtyping [2] are among the recent research development. It is clear that coercive subtyping is strongly motivated by the ... |

40 | Typing algorithm in type theory with inheritance
- Saibi
- 1997
(Show Context)
Citation Context ...ystems such as Lego and solves the above problem. 4.4 Interpreting practical forms of coercion Anthony Bailey has implemented various interesting and useful coercion mechanisms in Lego [3]. (Also see =-=[41]-=- for a related development in the Coq system.) Bailey has given interesting examples to illustrate the use of implicit coercion to make proof development easier (more readable etc.). We briefly discus... |

36 |
extended calculus of constructions
- ECC
- 1989
(Show Context)
Citation Context ...since they do not take into the account (the forms of) the canonical objects in the subtypes. For instance, subtyping between two \Sigma-types as found in the Extended Calculus of Constructions (ECC) =-=[25, 26]-=- is not quite compatible with the general elimination rules as found in Martin-Lof's type theory and UTT. A simple combination would lead to a system for which the subject reduction property fails to ... |

34 |
Inheritance and explicit coercion
- Breazu-Tannen, Coquand, et al.
- 1991
(Show Context)
Citation Context ...finitional equality. This gives a proof-theoretic (and direct meaning-theoretic) treatment of subtyping as implicit coercions, as compared with its possible model-theoretic semantic counterpart (cf., =-=[8]-=-). ffl Subtyping is treated as an extension of the underlying logical framework---the meta-language in which type theories are formulated. Making essential use of a typed logical framework LF [30], it... |

30 |
The machine-checked literate formalisation of algebra in type theory
- Bailey
- 1998
(Show Context)
Citation Context ...finite sequence" = "finite 2 sequence" and, for A : set, "A is finite" = "A is finite 1 ". Bailey has further developed this idea in his PhD thesis and applied it to=-= formal development of mathematics [4]-=-. It also has applications, for example, in lexical analysis of natural languages [33]. 5 Conclusions, related work, and further research The central idea of coercive subtyping is to introduce coerciv... |

29 |
Typechecking dependent types and subtypes
- Cardelli
- 1988
(Show Context)
Citation Context ...8]. Subtyping has also been studied extensively by many researchers in the context of typed functional programming, inspired by the notion of inheritance as found in object-oriented programming (cf., =-=[13, 12]-=-). Semantic studies on subtyping in type systems (without dependent types) such as the second-order lambda calculus (ie, the system F) include the use of the PER model [9] and the coercion-based appro... |

28 | Program specification and data refinement in type theory
- Luo
- 1993
(Show Context)
Citation Context ...ed in the Lego proof development system, UTT has been applied to verification of functional programs [10, 11], imperative programs [42] and concurrent programs [44], specification and data refinement =-=[29]-=-, and formalisation of mathematics [40]. 2.3 Definitional equality and computational equality We use LF seriously as a meta-level language (see Section 9.1.2 of [30] for a discussion). Along the same ... |

26 | The philosophical basis of intuitionistic logic - Dummett - 1973 |

25 | Coercive subtyping in type theory
- Luo
- 1996
(Show Context)
Citation Context ...nfinite reduction sequences. Remark 7 Note that the rule () makes an essential use of the judgement form k :: K; in other words, it cannot be reasonably stated with only the basic judgement forms. In =-=[32]-=-, where coercive subtyping was first introduced, k :: K is taken as a basic judgement form. The presentation of coercive subtyping in this paper makes it clearer from another angle that the () rule is... |

24 | A higher-order calculus and theory abstraction
- Luo
- 1991
(Show Context)
Citation Context ...pplications such as specification and data refinement (with refinement maps between specifications [29] as coercions), development of mathematical theories in proof development (with theory morphisms =-=[27]-=- as coercions [3]), and library structuring for proof reuse [31]. Subtyping is in general a subtle issue partly because, in the presence of (arbitrary) subtyping, a judgement of the form k :: K is a s... |

19 | Auxiliary variables and recursive procedures
- Schreiber
- 1997
(Show Context)
Citation Context ...chRosser, Subject Reduction, and Strong Normalisation [21]. Implemented in the Lego proof development system, UTT has been applied to verification of functional programs [10, 11], imperative programs =-=[42]-=- and concurrent programs [44], specification and data refinement [29], and formalisation of mathematics [40]. 2.3 Definitional equality and computational equality We use LF seriously as a meta-level l... |

17 |
Deliverables: a categorical approach to program development in type theory
- Burstall, McKinna
- 1992
(Show Context)
Citation Context ...oretic properties such as ChurchRosser, Subject Reduction, and Strong Normalisation [21]. Implemented in the Lego proof development system, UTT has been applied to verification of functional programs =-=[10, 11]-=-, imperative programs [42] and concurrent programs [44], specification and data refinement [29], and formalisation of mathematics [40]. 2.3 Definitional equality and computational equality We use LF s... |

17 |
Analytic and Synthetic Judgements in Type Theory", Kant and Contemporary
- MARTIN-LÖF
- 1994
(Show Context)
Citation Context ...r proof reuse [31]. Subtyping is in general a subtle issue partly because, in the presence of (arbitrary) subtyping, a judgement of the form k :: K is a synthetic judgement in the sense of Martin-Lof =-=[37]-=-; that is, the judgement form is essentially existential and hence in general undecidable, unless the formulation of the system has certain restrictions. Coercive subtyping offers one such restriction... |

16 | Programming in Martin-L"of's Type Theory. An Introduction - Nordstrom, Petersson, et al. - 1990 |

15 |
Lexical Meaning in context – a web of words
- Asher
- 2011
(Show Context)
Citation Context ...ected. We show that modern type theories, together with the theory of coercive subtyping, may offer a powerful language in which interesting lexical phenomena such as copredication (Pustejovsky 1995; =-=Asher 2011-=-) can be properly interpreted. This paper also studies how the dot-types (or sometimes called dot-objects), as proposed by Pustejovsky (1995) in studying logical polysemy and co-composition in lexical... |

15 | Z.: An implementation of LF with coercive subtyping and universes
- Callaghan, Luo
- 2001
(Show Context)
Citation Context ... been studied and implemented in many proof assistants such as Coq (Coq Development Team 2007; Saïbi 1997), Lego (Luo and Pollack 1992; Bailey 1999), Matita (Matita proof assistant 2008) and Plastic (=-=Callaghan and Luo 2001-=-), and used effectively in interactive theorem proving. In this paper, coercive subtyping is applied to linguistic semantics. We shall now introduce informally the basics of coercive subtyping and exp... |

15 | Coercion completion and conservativity in coercive subtyping
- Soloviev, Luo
- 2000
(Show Context)
Citation Context ...s with coercive subtyping and, fortunately, it meets the requirement. In fact, the coercive subtyping extension is not only consistent but conservative as long as the employed coercions are coherent (=-=Soloviev and Luo 2002-=-; Luo et al. 2012). Informally, coherence means that 7 It is not difficult to see this if one is familiar with how the induction principles of inductive types are formulated in a modern type theory. A... |

14 | A logic of subtyping
- Longo, Milsted, et al.
- 1995
(Show Context)
Citation Context ...h as the second-order lambda calculus (ie, the system F) include the use of the PER model [9] and the coercion-based approach [8]. A more recent logical study of subtyping in system F can be found in =-=[24]-=-. There is not much work on subtyping in dependent type systems in the context of proof development systems based on type theory. Betarte and Tasistro's work on record types and subtyping (or record k... |

14 |
A unifying theory of dependent types: the schematic approach
- Luo
- 1992
(Show Context)
Citation Context ...ve subtyping is considered in Section 4. Related work and further research topics are discussed in the Conclusion. 2 The logical framework LF and formulation of type theories The logical framework LF =-=[28, 30]-=- is a typed version of Martin-Lof's logical framework (see Chapter 19 of [38] for a presentation of the latter). We should also point out that LF is different from the Edinburgh Logical Framework (ELF... |