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On Subtyping in Type Theories with Canonical Objects

Georgiana Elena Lungu, Zhaohui Luo, Michael Wagner
2018 Types for Proofs and Programs  
How should one introduce subtyping into type theories with canonical objects such as Martin-Löf's type theory?  ...  In this paper, we introduce a type system with signatures where coercive subtyping relations can be specified, and argue that this provides a suitable subtyping mechanism for type theories with canonical  ...  :14 On Subtyping in Type Theories with Canonical Objects Γ Σ [[x:K]x(k 0 )/x]K = [k 0 /x]K ):[k 0 /x]K Theorem 13 (Conservativity).  ... 
doi:10.4230/lipics.types.2016.13 dblp:conf/types/LunguL16 fatcat:cgf77i6v4bd7nen3up5kjuu5nq

Type-theoretical semantics with coercive subtyping

Zhaohui Luo
2010 Semantics and Linguistic Theory  
expressed in a type theory with coercive subtyping.  ...  The theory of coercive subtyping adequately extends the modern type theories with a notion of subtyping and, as shown in this paper, plays a very useful role in making type theories more expressive for  ...  The notion of canonical object is so important that modern type theories are sometimes called type theories with canonical objects.  ... 
doi:10.3765/salt.v20i0.2580 fatcat:bwc4jufqfrhkhnhfw5dnnj6cgq

Type-theoretical semantics with coercive subtyping

Zhaohui Luo
2015 Semantics and Linguistic Theory  
expressed in a type theory with coercive subtyping.  ...  The theory of coercive subtyping adequately extends the modern type theories with a notion of subtyping and, as shown in this paper, plays a very useful role in making type theories more expressive for  ...  The notion of canonical object is so important that modern type theories are sometimes called type theories with canonical objects.  ... 
doi:10.3765/salt.v0i20.2580 fatcat:khdxuecxmffi5ke22bwn5chf7q

Coercive subtyping: Theory and implementation

Z. Luo, S. Soloviev, T. Xue
2013 Information and Computation  
We also compare coercive subtyping with the 'ordinary' notion of subtyping -subsumptive subtyping, and show that the former is adequate for type theories with canonical objects while the latter is not.  ...  In this paper, we give a new and adequate formulation of T [C], the system that extends a type theory T with coercive subtyping based on a set C of basic subtyping judgements, and show that coercive subtyping  ...  Subtyping for Type Theories with Canonical Objects Coercive subtyping is not the first notion one would consider when introducing subtyping into a type theory.  ... 
doi:10.1016/j.ic.2012.10.020 fatcat:lmuvh3rwqjeubdygevnuezqigm

Formal semantics in modern type theories with coercive subtyping

Zhaohui Luo
2012 Linguistics and Philosophy  
In particular, we show how to formally introduce dot-types in a type theory with coercive subtyping and study some type-theoretic constructs that provide useful representational tools for reference transfers  ...  The theory of coercive subtyping adequately extends the modern type theories and, as shown in this paper, plays a very useful role in making type theories more expressive for formal semantics.  ...  Acknowledgments Thanks to the anonymous referees for very useful comments and to the members of the Type Theory and Applications research group at RHUL for helpful discussions.  ... 
doi:10.1007/s10988-013-9126-4 fatcat:4xrofgt6mzd7rideataxdyzyue

Coercive subtyping in type theory [chapter]

Zhaohui Luo
1997 Lecture Notes in Computer Science  
The use of coercive subtyping in formal development and in reasoning about subsets of objects is discussed in the context of computerassisted formal reasoning. 1 type theories with subtyping.  ...  We propose and study coercive subtyping, a formal extension with subtyping of dependent type theories such as Martin-L of's type theory NPS90] and the type theory UTT Luo94].  ...  Communication and discussion with Anthony Bailey on his implementation of coercions in Lego has been very useful.  ... 
doi:10.1007/3-540-63172-0_45 fatcat:f2ygwdrulrbjnfzxvahqz2yg2a

Transitivity in coercive subtyping

Zhaohui Luo, Yong Luo
2005 Information and Computation  
Coercive subtyping is a general approach to abbreviation and subtyping in dependent type theories with inductive types.  ...  admissible in an intensional type theory.  ...  To do so, one would have to ask questions like: Would the canonical objects of a subtype also be canonical objects of a supertype?  ... 
doi:10.1016/j.ic.2004.10.008 fatcat:pmcxqle6svachi4ij7rpkgqycm

Coercive subtyping

Z Luo
1999 Journal of Logic and Computation  
We propose and study coercive subtyping, a formal extension with subtyping of dependent type theories such as Martin-L of's type theory 38] and the type theory UTT 30].  ...  This provides a conceptually simple and uniform framework to understand subtyping and inheritance relations in type theories with sophisticated type structures such as inductive types and universes.  ...  This work is partly supported by the UK EPSRC grant on `Subtyping, Inheritance and Reuse' (GR/K79130).  ... 
doi:10.1093/logcom/9.1.105 fatcat:dv2xyk5nzbf3dk5od6wuwsdovi

Formal Semantics in Modern Type Theories: Is It Model-Theoretic, Proof-Theoretic, or Both? [chapter]

Zhaohui Luo
2014 Lecture Notes in Computer Science  
 Definition Any closed object of an inductive type is computationally equal to a canonical object of that type.  ...   Claim: Formal semantics in Modern Type Theories is both model-theoretic and proof-theoretic.  Modern type theories  Many types of entities -"many-sorted"   So, in any hole that requires an object  ... 
doi:10.1007/978-3-662-43742-1_14 fatcat:f3kpyzn6fbefbfjfds24y4xnri

Dependent Coercions1 1This work is partly supported by the UK EPSRC grant on "Subtyping, Inheritance and Reuse" (GR/K79130)

Zhaohui Luo, Sergei Soloviev
1999 Electronical Notes in Theoretical Computer Science  
A dependent coercion introduces a subtyping relation between a type and a family of types in that an object of the type is mapped into one of the types in the family.  ...  It extends our earlier work on coercive subtyping into a uniform framework which increases the expressive p o wer with new applications.  ...  It is not clear (if possible) how the traditional approach t o s u b t yping can be applied to type theory with inductive t ypes in accordance with the view that types consist of canonical objects.  ... 
doi:10.1016/s1571-0661(05)80314-7 fatcat:b2zi4m5wkrdixdw7qmqq7vumqu

Coercive subtyping in lambda-free logical frameworks

Robin Adams
2009 Proceedings of the Fourth International Workshop on Logical Frameworks and Meta-Languages Theory and Practice - LFMTP '09  
We develop the metatheory of the resulting framework, including providing some general conditions under which typecasting in an object theory with coercive subtyping is decidable.  ...  We show how coercive subtyping may be added to a lambda-free logical framework, by constructing the logical framework TF<, an extension of the lambda-free logical framework TF with coercive subtyping.  ...  As we described in the Introduction, TF is designed so that, when an object theory has been specified, the objects of TF are in one-to-one correspondence with the terms and types of the object theory.  ... 
doi:10.1145/1577824.1577830 dblp:conf/lfmtp/Adams09 fatcat:xjncvoon4fhxbek5iifeil33dy

A Vector Model for Type-Theoretical Semantics

Konstantin Sokolov
2016 Proceedings of the 1st Workshop on Representation Learning for NLP  
Vector models of distributional semantics can be viewed as a geometric interpretation of a fragment of dependent type theory.  ...  word meanings in the compositional setting.  ...  Computation The notion of canonical object is central to the dependent type theory.  ... 
doi:10.18653/v1/w16-1627 dblp:conf/rep4nlp/Sokolov16 fatcat:2jfcecoghfg3vdjxcrxmpxcyga

Formal Verification of a Combination Decision Procedure [chapter]

Jonathan Ford, Natarajan Shankar
2002 Lecture Notes in Computer Science  
Shostak, in 1984, published a decision procedure for the combination of canonizable and solvable theories.  ...  The formalization itself posed significant challenges and the verification revealed some gaps in the informal argument.  ...  The type thry func is the predicate subtype corresponding to the predicate thry func. The type of equalities is defined as a record type with fields lhs and rhs.  ... 
doi:10.1007/3-540-45620-1_29 fatcat:zcwzcgxssbg2hfjzdm623tl6bu

Reflexive toolbox for regular expression matching

Vladimir Komendantsky
2012 Proceedings of the sixth workshop on Programming languages meets program verification - PLPV '12  
A Coq formalisation of a canonical non-deterministic finite automaton construction on a regular expression is presented.  ...  A method that we develop in the paper, called shadowing, allows for a smooth program extraction from decision procedures whatever the complexity of the dependently typed proofs.  ...  Acknowledgments I am grateful to the Automata Group of the University of Porto and especially to Nelma Moreira for the kind invitation and interesting discussions on the subject.  ... 
doi:10.1145/2103776.2103784 dblp:conf/plpv/Komendantsky12 fatcat:3rvy5sdllff55ddcpl2y7fli7u

Conceptual structures for modeling in CIM [chapter]

Michel Wermelinger, Alex Bejan
1993 Lecture Notes in Computer Science  
In this paper we develop the foundations for the utilization of conceptual structures (CS) in combination with EXPRESS and STEP Application Protocols in the eld of Computer Integrated Manufacturing (CIM  ...  based on logic to be used for enterprise information modeling and integration.  ...  A canon is a tuple hI; T; ::; Bi where I is a set of individual markers, T is a type hierarchy, :: is a conformity relation relating labels in T with markers in I, and B is a nite set of graphs with all  ... 
doi:10.1007/3-540-56979-0_19 fatcat:zjm6nqt4bja4bkgc534sc7gdim
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