Grammatical Framework (GF) is a special-purpose functional language for defining grammars. It uses a Logical Framework (LF) for a description of abstract syntax, and adds to this a notation for defining concrete syntax. GF grammars themselves are purely declarative, but can be used both for linearizing syntax trees and parsing strings. GF can describe both formal and natural languages. The key notion of this description is a grammatical object, which is not just a string, but a record that contains all information on inflection and inherent grammatical features such as number and gender in natural languages, or precedence in formal languages. Grammatical objects have a type system, which helps to eliminate run-time errors in language processing. In the same way as an LF, GF uses...

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Abstract. This paper reports on refinements and extensions to the MathLang framework that add substantial support for natural language text. We show how the extended framework supports multiple views of mathematical texts, including natural language views using the exact text that the mathematician wants to use. Thus, MathLang now supports the ability to capture the essential mathematical structure of mathematics written using natural language text. We show examples of how arbitrary mathematical text can be encoded in MathLang without needing to change any of the words or symbols of the texts or their order. In particular, we show the encoding of a theorem and its proof that has been used by Wiedijk for comparing many theorem prover representations of mathematics, namely the irrationality of √ 2 (originally due to Pythagoras). We encode a 1960 version by Hardy and Wright, and a more recent version by Barendregt. 1 On the way to a mathematical vernacular for computers Mathematicians now use computer software for a variety of tasks: typing mathematical texts, performing calculation, analyzing theories, verifying proofs. Software tools like

. We present `Plastic', an implementation of LF with Coercive Subtyping, and focus on its implementation of Universes. LF is a variant of Martin-Lof's logical framework, with explicitly typed -abstractions. We outline the system of LF with its extensions of inductive types and coercions. Plastic is the first implementation of this extended system; we discuss motivations and basic architecture, and give examples of its use. LF is used to specify type theories. The theory UTT includes a hierarchy of universes which is specified in Tarski style. We outline the theory of these universes and explain how they are implemented in Plastic. Of particular interest is the relationship between universes and inductive types, and the relationship between universes and coercive subtyping. We claim that the combination of Tarski-style universes together with coercive subtyping provides an ideal formulation of universes which is both semantically clear and practical to use. Keywords: type theory, un...

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 meta-theory, 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...

) 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, Martin-Lof'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 ...

. In the context of Plastic, a proof assistant for a variant of Martin-Lof'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 non-linear pattern matching; we propose this match can be ignored in well-typed 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 Martin-Lof's Logical Framework LF which has explicitly typed -abstractions, and a schema for inductive types within this LF which is...

In this paper we present the Durham Mathematical Vernacular (MV) project, discuss the general design of a prototype to support experimentation with issues of MV, explain current work on the prototype -- specifically in the type theory basis of the work, and end with a brief discussion of methodology and future directions. The current work concerns an implementation of Luo's typed logical framework LF, and making it more flexible with respect to the demands of implementing MV -- in particular, meta-variables, multiple contexts, subtyping, and automation. This part of the project may be of particular interest to the general theorem proving community. We will demonstrate a prototype at the conference. 1 Introduction: Defining a Mathematical Vernacular The long term aim of the project is to develop theory and techniques with which the complementary strengths of NLP (Natural Language Processing) and CAFR (Computer-Assisted Formal Reasoning) can be combined to support computer-assisted reas...

. This paper concerns techniques for providing a convenient syntax for object languages implemented via a type-theoretic Logical Framework, and reports on work in progress. We first motivate the need for a type-theoretic logical framework. Firstly, we take the logical framework seriously as a meta-language for implementing object languages (including object type theories). Another reason is the goal of building domain-specific reasoning tools which are implemented using type theory technology but do not require great expertise in type theory to use productively. We then present several examples of bi-directional translations between an encoding in the framework language and a more convenient syntax. The paper ends by discussing several techniques for implementing the translations and properties that we may require for the translation. Coercive subtyping is shown to help in the translation. 1

This paper presents the Durham Mathematical Vernacular (MV) project. The project aims to develop the technology for interactive tools based on proof checking with type theory, which allow mathematicians to develop mathematics in their usual vernacular. Its sub-goals are to develop type theory technology to support MV and to develop the corresponding NL technology. Mathematical language has many significant differences from everyday language, hence a different approach is required to automatically process such language. We discuss some important differences and how they affect implementation. A key requirement is for correctness. Another key feature is that the user defineshis own terminology. We then discuss work in progress, namely the issue of semantic well-formedness in mathematical descriptions, and a prototype being developed to experiment with aspects of the project. 1 Introduction: Defining a Mathematical Vernacular The long term aim of this project is to develop theory and tec...