A Metatheoretic Analysis of Subtype Universes

Subtype universes were initially introduced as an expressive mechanisation of bounded quantification extending a modern type theory. In this paper, we consider a dependent type theory equipped with coercive subtyping and a generalisation of subtype universes. We prove results regarding the metatheoretic properties of subtype universes, such as consistency and strong normalisation. We analyse the causes of undecidability in bounded quantification, and discuss how coherency impacts the metatheoretic properties of theories implementing bounded quantification. We describe the effects of certain choices of subtyping inference rules on the expressiveness of a type theory, and examine various applications in natural language semantics, programming languages, and mathematics formalisation

Is Impredicativity Implicitly Implicit?

Of all the threats to the consistency of a type system, such as side effects and recursion, impredicativity is arguably the least understood. In this paper, we try to investigate it using a kind of blackbox reverse-engineering approach to map the landscape. We look at it with a particular focus on its interaction with the notion of implicit arguments, also known as erasable arguments. More specifically, we revisit several famous type systems believed to be consistent and which do include some form of impredicativity, and show that they can be refined to equivalent systems where impredicative quantification can be marked as erasable, in a stricter sense than the kind of proof irrelevance notion used for example for Prop terms in systems like Coq. We hope these observations will lead to a better understanding of why and when impredicativity can be sound. As a first step in this direction, we discuss how these results suggest some extensions of existing systems where constraining impredicativity to erasable quantifications might help preserve consistency

Engineering formal systems in constructive type theory

This thesis presents a practical methodology for formalizing the meta-theory of formal systems with binders and coinductive relations in constructive type theory. While constructive type theory offers support for reasoning about formal systems built out of inductive definitions, support for syntax with binders and coinductive relations is lacking. We provide this support. We implement syntax with binders using well-scoped de Bruijn terms and parallel substitutions. We solve substitution lemmas automatically using the rewriting theory of the -calculus. We present the Autosubst library to automate our approach in the proof assistant Coq. Our approach to coinductive relations is based on an inductive tower construction, which is a type-theoretic form of transfinite induction. The tower construction allows us to reduce coinduction to induction. This leads to a symmetric treatment of induction and coinduction and allows us to give a novel construction of the companion of a monotone function on a complete lattice. We demonstrate our methods with a series of case studies. In particular, we present a proof of type preservation for CC!, a proof of weak and strong normalization for System F, a proof that systems of weakly guarded equations have unique solutions in CCS, and a compiler verification for a compiler from a non-deterministic language into a deterministic language. All technical results in the thesis are formalized in Coq.In dieser Dissertation beschreiben wir praktische Techniken um Formale Systeme mit Bindern und koinduktiven Relationen in Konstruktiver Typtheorie zu implementieren. Während Konstruktive Typtheorie bereits gute Unterstützung für Induktive Definition bietet, gibt es momentan kaum Unterstützung für syntaktische Systeme mit Bindern, oder koinduktiven Definitionen. Wir kodieren Syntax mit Bindern in Typtheorie mit einer de Bruijn Darstellung und zeigen alle Substitutionslemmas durch Termersetzung mit dem -Kalkül. Wir präsentieren die Autosubst Bibliothek, die unseren Ansatz im Beweisassistenten Coq implementiert. Für koinduktive Relationen verwenden wir eine induktive Turmkonstruktion, welche das typtheoretische Analog zur Transfiniten Induktion darstellt. Auf diese Art erhalten wir neue Beweisprinzipien für Koinduktion und eine neue Konstruktion von Pous’ “companion” einer monotonen Funktion auf einem vollständigen Verband. Wir validieren unsere Methoden an einer Reihe von Fallstudien. Alle technischen Ergebnisse in dieser Dissertation sind mit Coq formalisiert

On Irrelevance and Algorithmic Equality in Predicative Type Theory

Dependently typed programs contain an excessive amount of static terms which are necessary to please the type checker but irrelevant for computation. To separate static and dynamic code, several static analyses and type systems have been put forward. We consider Pfenning's type theory with irrelevant quantification which is compatible with a type-based notion of equality that respects eta-laws. We extend Pfenning's theory to universes and large eliminations and develop its meta-theory. Subject reduction, normalization and consistency are obtained by a Kripke model over the typed equality judgement. Finally, a type-directed equality algorithm is described whose completeness is proven by a second Kripke model.Comment: 36 pages, superseds the FoSSaCS 2011 paper of the first author, titled "Irrelevance in Type Theory with a Heterogeneous Equality Judgement

Towards an Implicit Calculus of Inductive Constructions. Extending the Implicit Calculus of Constructions with Union and Subset Types.

International audienceWe present extensions of Miquel's Implicit Calculus of Constructions (ICC) and Barras and Bernardo's decidable Implicit Calculus of Constructions (ICC*) with union and subset types. The purpose of these systems is to solve the problem of interaction betweeen logical and computational data. This is a work in progress and our long term goal is to add the whole inductive types to ICC and ICC* in order to define a complete framework for theorem proving

Pure subtype systems: a type theory for extensible software

This thesis presents a novel approach to type theory called “pure subtype systems”, and a core calculus called DEEP which is based on that approach. DEEP is capable of modeling a number of interesting language techniques that have been proposed in the literature, including mixin modules, virtual classes, feature-oriented programming, and partial evaluation. The design of DEEP was motivated by two well-known problems: “the expression problem”, and “the tag elimination problem.” The expression problem is concerned with the design of an interpreter that is extensible, and requires an advanced module system. The tag elimination problem is concerned with the design of an interpreter that is efficient, and requires an advanced partial evaluator. We present a solution in DEEP that solves both problems simultaneously, which has never been done before. These two problems serve as an “acid test” for advanced type theories, because they make heavy demands on the static type system. Our solution in DEEP makes use of the following capabilities. (1) Virtual types are type definitions within a module that can be extended by clients of the module. (2) Type definitions may be mutually recursive. (3) Higher-order subtyping and bounded quantification are used to represent partial information about types. (4) Dependent types and singleton types provide increased type precision. The combination of recursive types, virtual types, dependent types, higher-order subtyping, and bounded quantification is highly non-trivial. We introduce “pure subtype systems” as a way of managing this complexity. Pure subtype systems eliminate the distinction between types and objects; every term can behave as either a type or an object depending on context. A subtype relation is defined over all terms, and subtyping, rather than typing, forms the basis of the theory. We show that higher-order subtyping is strong enough to completely subsume the traditional type relation, and we provide practical algorithms for type checking and for finding minimal types. The cost of using pure subtype systems lies in the complexity of the meta-theory. Unfortunately, we are unable to establish some basic meta-theoretic properties, such as type safety and transitivity elimination, although we have made some progress towards these goals. We formulate the subtype relation as an abstract reduction system, and we show that the type theory is sound if the reduction system is confluent. We can prove that reductions are locally confluent, but a proof of global confluence remains elusive. In summary, pure subtype systems represent a new and interesting approach to type theory. This thesis describes the basic properties of pure subtype systems, and provides concrete examples of how they can be applied. The Deep calculus demonstrates that our approach has a number of real-world practical applications in areas that have proved to be quite difficult for traditional type theories to handle. However, the ultimate soundness of the technique remains an open question

Type theoretic semantics for semantic networks: an application to natural language engineering

Semantic Networks have long been recognised as an important tool for natural language processing. This research has been a formal analysis of a semantic network using constructive type theory. The particular net studied is SemNet, the internal knowledge representation for LOLITA(^1): a large scale natural language engineering system. SemNet has been designed with large scale, efficiency, integration and expressiveness in mind. It supports many different forms of plausible and valid reasoning, including: epistemic reasoning, causal reasoning and inheritance. The unified theory of types (UTT) integrates two well known type theories, Coquand-Huet's (impredicative) calculus of constructions and Martin-Lof's (predicative) type theory. The result is a strong and expressive language which has been used for formalization of mathematics, program specification and natural language. Motivated by the computational and richly expressive nature of UTT, this research has used it for formalization and semantic analysis of SemNet. Moreover, because of applications to software engineering, type checkers/proof assistants have been built. These tools are ideal for organising and managing the analysis of SemNet. The contribution of the work is twofold. First the semantic model built has led to improved and deeper understanding of SemNet. This is important as many researchers that work on different aspects of LOLITA, now have a clear and un- ambigious interpertation of the meaning of SemNet constructs. The model has also been used to show soundess of the valid reasoning and to give a reasonable semantic account of epistemic reasoning. Secondly the research contributes to NLE generally, both because it demonstrates that UTT is a useful formalization tool and that the good aspects of SemNet have been formally presented