Definitions by Rewriting in the Calculus of Constructions

The main novelty of this paper is to consider an extension of the Calculus of Constructions where predicates can be defined with a general form of rewrite rules. We prove the strong normalization of the reduction relation generated by the beta-rule and the user-defined rules under some general syntactic conditions including confluence. As examples, we show that two important systems satisfy these conditions: a sub-system of the Calculus of Inductive Constructions which is the basis of the proof assistant Coq, and the Natural Deduction Modulo a large class of equational theories.Comment: Best student paper (Kleene Award

Elaborating Inductive Definitions

We present an elaboration of inductive definitions down to a universe of datatypes. The universe of datatypes is an internal presentation of strictly positive families within type theory. By elaborating an inductive definition -- a syntactic artifact -- to its code -- its semantics -- we obtain an internalized account of inductives inside the type theory itself: we claim that reasoning about inductive definitions could be carried in the type theory, not in the meta-theory as it is usually the case. Besides, we give a formal specification of that elaboration process. It is therefore amenable to formal reasoning too. We prove the soundness of our translation and hint at its correctness with respect to Coq's Inductive definitions. The practical benefits of this approach are numerous. For the type theorist, this is a small step toward bootstrapping, ie. implementing the inductive fragment in the type theory itself. For the programmer, this means better support for generic programming: we shall present a lightweight deriving mechanism, entirely definable by the programmer and therefore not requiring any extension to the type theory.Comment: 32 pages, technical repor

Strong normalisation for applied lambda calculi

We consider the untyped lambda calculus with constructors and recursively defined constants. We construct a domain-theoretic model such that any term not denoting bottom is strongly normalising provided all its `stratified approximations' are. From this we derive a general normalisation theorem for applied typed lambda-calculi: If all constants have a total value, then all typeable terms are strongly normalising. We apply this result to extensions of G\"odel's system T and system F extended by various forms of bar recursion for which strong normalisation was hitherto unknown.Comment: 14 pages, paper acceptet at electronic journal LMC

Inductive types in the Calculus of Algebraic Constructions

In a previous work, we proved that an important part of the Calculus of Inductive Constructions (CIC), the basis of the Coq proof assistant, can be seen as a Calculus of Algebraic Constructions (CAC), an extension of the Calculus of Constructions with functions and predicates defined by higher-order rewrite rules. In this paper, we prove that almost all CIC can be seen as a CAC, and that it can be further extended with non-strictly positive types and inductive-recursive types together with non-free constructors and pattern-matching on defined symbols.Comment: Journal version of TLCA'0

Developing an ontological sandbox : investigating multi-level modelling’s possible Metaphysical Structures

One of the central concerns of the multi-level modelling (MLM) community is the hierarchy of classifications that appear in conceptual models; what these are, how they are linked and how they should be organised into levels and modelled. Though there has been significant work done in this area, we believe that it could be enhanced by introducing a systematic way to investigate the ontological nature and requirements that underlie the frameworks and tools proposed by the community to support MLM (such as Orthogonal Classification Architecture and Melanee). In this paper, we introduce a key component for the investigation and understanding of the ontological requirements, an ontological sandbox. This is a conceptual framework for investigating and comparing multiple variations of possible ontologies – without having to commit to any of them – isolated from a full commitment to any foundational ontology. We discuss the sandbox framework as well as walking through an example of how it can be used to investigate a simple ontology. The example, despite its simplicity, illustrates how the constructional approach can help to expose and explain the metaphysical structures used in ontologies, and so reveal the underlying nature of MLM levelling

JWalk: a tool for lazy, systematic testing of java classes by design introspection and user interaction

Popular software testing tools, such as JUnit, allow frequent retesting of modified code; yet the manually created test scripts are often seriously incomplete. A unit-testing tool called JWalk has therefore been developed to address the need for systematic unit testing within the context of agile methods. The tool operates directly on the compiled code for Java classes and uses a new lazy method for inducing the changing design of a class on the fly. This is achieved partly through introspection, using Java’s reflection capability, and partly through interaction with the user, constructing and saving test oracles on the fly. Predictive rules reduce the number of oracle values that must be confirmed by the tester. Without human intervention, JWalk performs bounded exhaustive exploration of the class’s method protocols and may be directed to explore the space of algebraic constructions, or the intended design state-space of the tested class. With some human interaction, JWalk performs up to the equivalent of fully automated state-based testing, from a specification that was acquired incrementally

Inductive and Coinductive Components of Corecursive Functions in Coq

In Constructive Type Theory, recursive and corecursive definitions are subject to syntactic restrictions which guarantee termination for recursive functions and productivity for corecursive functions. However, many terminating and productive functions do not pass the syntactic tests. Bove proposed in her thesis an elegant reformulation of the method of accessibility predicates that widens the range of terminative recursive functions formalisable in Constructive Type Theory. In this paper, we pursue the same goal for productive corecursive functions. Notably, our method of formalisation of coinductive definitions of productive functions in Coq requires not only the use of ad-hoc predicates, but also a systematic algorithm that separates the inductive and coinductive parts of functions.Comment: Dans Coalgebraic Methods in Computer Science (2008

Impredicative Encodings of (Higher) Inductive Types

Postulating an impredicative universe in dependent type theory allows System F style encodings of finitary inductive types, but these fail to satisfy the relevant {\eta}-equalities and consequently do not admit dependent eliminators. To recover {\eta} and dependent elimination, we present a method to construct refinements of these impredicative encodings, using ideas from homotopy type theory. We then extend our method to construct impredicative encodings of some higher inductive types, such as 1-truncation and the unit circle S1