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Contextual analysis of word meanings in typetheoretical semantics
 In Logical aspects of computational linguistics (LACL’2011). LNAI 6736
, 2011
"... Abstract. Word meanings are context sensitive and may change in different situations. In this paper, we consider how contexts and the associated contextual meanings of words may be represented in typetheoretical semantics, the formal semantics based on modern type theories.Itisshown,inparticular,tha ..."
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Cited by 10 (4 self)
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Abstract. Word meanings are context sensitive and may change in different situations. In this paper, we consider how contexts and the associated contextual meanings of words may be represented in typetheoretical semantics, the formal semantics based on modern type theories.Itisshown,inparticular,thatthe framework of coercive subtyping provides various useful tools in the representation. 1
PAL+: A LambdaFree Logical Framework
, 2000
"... A lambdafree logical framework takes parameterisation and definitions as the basic notions to provide schematic mechanisms for specification of type theories and their use in practice. The framework presented here, PAL + , is a logical framework for specification and implementation of type theor ..."
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Cited by 9 (1 self)
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A lambdafree logical framework takes parameterisation and definitions as the basic notions to provide schematic mechanisms for specification of type theories and their use in practice. The framework presented here, PAL + , is a logical framework for specification and implementation of type theories, such as MartinLof's type theory or UTT. As in MartinLof's logical framework [NPS90], computational rules can be introduced and are used to give meanings to the declared constants. However, PAL + only allows one to talk about the concepts that are intuitively in the object type theories: types and their objects, and families of types and families of objects of types. In particular, in PAL + , one cannot directly represent families of families of entities, which could be done in other logical frameworks by means of lambda abstraction. PAL + is in the spirit of de Bruijn's PAL for Automath [dB80]. Compared with PAL, PAL + allows one to represent parametric concepts such as famil...
A Theory of Typed Coercions and its Applications
"... A number of important program rewriting scenarios can be recast as typedirected coercion insertion. These range from more theoretical applications such as coercive subtyping and supporting overloading in type theories, to more practical applications such as integrating static and dynamically typed ..."
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Cited by 9 (2 self)
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A number of important program rewriting scenarios can be recast as typedirected coercion insertion. These range from more theoretical applications such as coercive subtyping and supporting overloading in type theories, to more practical applications such as integrating static and dynamically typed code using gradual typing, and inlining code to enforce security policies such as access control and provenance tracking. In this paper we give a general theory of typedirected coercion insertion. We specifically explore the inherent tradeoff between expressiveness and ambiguity—the more powerful the strategy for generating coercions, the greater the possibility of several, semantically distinct rewritings for a given program. We consider increasingly powerful coercion generation strategies, work out example applications supported by the increased power (including those mentioned above), and identify the inherent ambiguity problems of each setting, along with various techniques to tame the ambiguities.
Dependent Coercions
, 1999
"... A notion of dependent coercion is introduced and studied in the context of dependent type theories. It extends our earlier work on coercive subtyping into a uniform framework which increases the expressive power with new applications. A dependent coercion introduces a subtyping relation between a ty ..."
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Cited by 8 (5 self)
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A notion of dependent coercion is introduced and studied in the context of dependent type theories. It extends our earlier work on coercive subtyping into a uniform framework which increases the expressive power with new applications. 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. We present the formal framework, discuss its metatheory, and consider applications such as its use in functional programming with dependent types. 1 Introduction Coercive subtyping, as studied in [Luo97, Luo99, JLS98], represents a new general approach to subtyping and inheritance in type theory. In particular, it provides a framework in which subtyping, inheritance, and abbreviation can be understood in dependent type theories where types are understood as consisting of canonical objects. In this paper, we extend the framework of coercive subtyping to introduce a notion of dependent coer...
Coercive subtyping and lexical semantics (Extended Abstract)
 Logical Aspects of Computational Linguistics (LACL’98
, 1998
"... ) Zhaohui Luo and Paul Callaghan Department of Computer Science, University of Durham fZhaohui.Luo, P.C.Callaghang@durham.ac.uk 1 Introduction This paper investigates the use of constructive type theory in lexical semantics. Our intention is to explore how a rich language of types with subtyping c ..."
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Cited by 8 (8 self)
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) Zhaohui Luo and Paul Callaghan Department of Computer Science, University of Durham fZhaohui.Luo, P.C.Callaghang@durham.ac.uk 1 Introduction This paper investigates the use of constructive type theory in lexical semantics. Our intention is to explore how a rich language of types with subtyping can be used to express lexical knowledge, both as an application of type theory and as an alternative to current approaches. In particular, we show that coercive subtyping [Luo97, Luo98a], provides a formal framework with useful mechanisms for lexical semantics. Coercive subtyping extends constructive type theories (eg, MartinLof's intensional type theory [NPS90] and the type theory UTT [Luo94]) with a simple abbreviational mechanism. It provides elegant and flexible means of representing inheritance and overloading. In our earlier paper on the structure of Mathematical Vernacular [LC98], coercive subtyping is used to represent the inheritance relationships between mathematical concepts and ...
Working with Mathematical Structures in Type Theory
"... Abstract. We address the problem of representing mathematical structures in a proof assistant which: 1) is based on a type theory with dependent types, telescopes and a computational version of Leibniz equality; 2) implements coercive subtyping, accepting multiple coherent paths between type familie ..."
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Cited by 8 (4 self)
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Abstract. We address the problem of representing mathematical structures in a proof assistant which: 1) is based on a type theory with dependent types, telescopes and a computational version of Leibniz equality; 2) implements coercive subtyping, accepting multiple coherent paths between type families; 3) implements a restricted form of higher order unification and type reconstruction. We show how to exploit the previous quite common features to reduce the “syntactic ” gap between pen&paper and formalised algebra. However, to reach our goal we need to propose unification and type reconstruction heuristics that are slightly different from the ones usually implemented. We have implemented them in Matita. 1
The Matita Interactive Theorem Prover
"... Abstract. Matita is an interactive theorem prover being developed by the Helm team at the University of Bologna. Its stable version 0.5.x may be downloaded at ..."
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Cited by 8 (6 self)
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Abstract. Matita is an interactive theorem prover being developed by the Helm team at the University of Bologna. Its stable version 0.5.x may be downloaded at
E.: A constructive and formal proof of Lebesgues Dominated Convergence Theorem in the interactive theorem prover Matita
 Journal of Formalized Reasoning
, 2008
"... We present a formalisation of a constructive proof of Lebesgue’s Dominated Convergence Theorem given by Sacerdoti Coen and Zoli in [CSCZ]. The proof is done in the abstract setting of ordered uniformities, also introduced by the two authors as a simplification of Weber’s lattice uniformities given i ..."
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Cited by 7 (4 self)
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We present a formalisation of a constructive proof of Lebesgue’s Dominated Convergence Theorem given by Sacerdoti Coen and Zoli in [CSCZ]. The proof is done in the abstract setting of ordered uniformities, also introduced by the two authors as a simplification of Weber’s lattice uniformities given in [Web91, Web93]. The proof is fully constructive, in the sense that it is done in Bishop’s style and, under certain assumptions, it is also fully predicative. The formalisation is done in the Calculus of (Co)Inductive Constructions using the interactive theorem prover Matita [ASTZ07]. It exploits some peculiar features of Matita and an advanced technique to represent algebraic hierarchies previously introduced by the authors in [ST07]. Moreover, we introduce a new technique to cope with duality to halve the formalisation effort.
Implementation Techniques for Inductive Types in Plastic
 Types for Proofs and Programs, volume 1956 of LNCS
, 2000
"... . In the context of Plastic, a proof assistant for a variant of MartinLof's Logical Framework LF with explicitly typed abstractions, we outline the technique used for implementing inductive types from their declarations. This form of inductive types gives rise to a problem of nonlinear patter ..."
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Cited by 4 (2 self)
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. In the context of Plastic, a proof assistant for a variant of MartinLof's Logical Framework LF with explicitly typed abstractions, we outline the technique used for implementing inductive types from their declarations. This form of inductive types gives rise to a problem of nonlinear pattern matching; we propose this match can be ignored in welltyped terms, and outline a proof of this. The paper then explains how the inductive types are realised inside the reduction mechanisms of Plastic, and briefly considers optimisations for inductive types. Key words: type theory, inductive types, LF, implementation. 1 Introduction This paper considers implementation techniques for a particular approach to inductive types in constructive type theory. The inductive types considered are those given in Chapter 9 of [15], in which Luo presents a variant of MartinLof's Logical Framework LF which has explicitly typed abstractions, and a schema for inductive types within this LF which is...
OrderSorted Inductive Types
, 1999
"... System F ! is an extension of system F ! with subtyping and bounded quantification. Ordersorted algebra is an extension of manysorted algebra with overloading and subtyping. We combine both formalisms to obtain IF ! , a higherorder typed calculus with subtyping, bounded quantification a ..."
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Cited by 4 (3 self)
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System F ! is an extension of system F ! with subtyping and bounded quantification. Ordersorted algebra is an extension of manysorted algebra with overloading and subtyping. We combine both formalisms to obtain IF ! , a higherorder typed calculus with subtyping, bounded quantification and ordersorted inductive types, i.e. data types with builtin subtyping and overloading. Moreover we show that IF ! enjoys important metatheoretic properties, including confluence, strong normalization, subject reduction and decidability of typechecking. 1 Introduction Typed functional programming languages such as Haskell and ML and typetheory based proofdevelopment systems such as Coq and Lego support the introduction of inductively defined types such as natural numbers or booleans, parameterized inductively defined types such as lists and even parameterized mutual inductively defined types such as trees and forests. In addition, those languages support the definition of functions ...