Type Theory with Explicit Universe Polymorphism

Electronic Proceedings in Theoretical Computer Science, 2010

We present an approach to type theory in which the typing judgments do not have explicit contexts. Instead of judgments of the shape Γ A : B, our systems just have judgments of the shape A : B. An essential feature of our systems is that already in pseudo-terms we distinguish the free from the bound variables. Specifically we give the rules of the 'Pure Type System' class of type theories in this style. We prove that the type judgments of these systems corresponds in a natural way with the judgments of the traditionally formulated Pure Type Systems. I.e., our systems have exactly the same well-typed terms as traditional presentations of type theory. Our system can be seen as a type theory in which all type judgments share an identical, infinite, typing context that has infinitely many variables for each possible type. For this reason we call our system Γ∞. This name means to suggest that our type judgment A : B should be read as Γ∞ A : B, with a fixed infinite type context called Γ∞.

development, 2007

Universe types characterise aliasing in object oriented programming languages and are used to reason modularly about programs. In this report we formalise prior work by Müller and Poetzsch-Heffter, who designed the Universe Type System for a core subset of Java. We present our work in two steps. We first give a Topological Universe Type System and show subject reduction to a small-step dynamic semantics for our language. Motivated by concerns of Modular verification, we then give an Encapsulation Universe Type System (based on the owner-as-modifier principle), prove subject reduction with respect to the former small-step semantics, and show how the type system can be used for modular verification.

2011 IEEE 26th Annual Symposium on Logic in Computer Science, 2011

We study a complex type theory, a Calculus of Inductive Constructions with a predicative hierarchy of universes and a first-order theory T built in its conversion relation. The theory T is specified abstractly, by a set of constructors, a set of defined symbols, axioms expressing that constructors are free and defined symbols completely defined, and a generic elimination principle relying on crucial properties of first-order structures satisfying the axioms. We first show that COQMTU enjoys all basic meta-theoretical properties of such calculi, confluence, subject reduction and strong normalization when restricted to weak-elimination, implying the decidability of type-checking in this case as well as consistency. The case of strong elimination is left open.

Theoria, 2008

T h i s paper presents a one-sorted version of the simple theory of types with ' E ' as its sole primitive predicate. This system will be called SST while the many-sorted version of the simple theory of types which also has ' E ' as its only primitive predicate will be called ST.2 Because SST contains a partial formalization of the type concept, its axioms do not contain schematic type indices and are more similar to the axioms of the standard set theories than those of ST and QST-Quine's one-sorted translation of ST.3 Moreover, SST is easily modified to treat negative types; and while ST and QST can be translated into SST, metatheorems two and three of Section Four show that SST is not stronger (in a technical sense) than these systems. (These results also obtain when axioms of infinity are added to all three systems.)

Programming languages often come with type systems. Some of these are simple, others are sophisticated. As a stylistic representation of types in programming languages several versions of typed lambda calculus are studied. During the last 20 years many of these systems have appeared, so there is some need of classification. Working towards a taxonomy, gives a fine-structure of the theory of constructions (Coquand and Huet 1988) in the form of a canonical cube of eight type systems ordered by inclusion. Berardi (1988) and have independently generalized the method of constructing systems in the A.-cube. Moreover, Berardi (1988 showed that the generalized type systems are flexible enough to describe many logical systems. In that way the well-known propositions-as-types interpretation obtains a nice canonical form.

The Review of Symbolic Logic, 2021

Standard Type Theory, ${\textrm {STT}}$ , tells us that $b^n(a^m)$ is well-formed iff $n=m+1$ . However, Linnebo and Rayo [23] have advocated the use of Cumulative Type Theory, $\textrm {CTT}$ , which has more relaxed type-restrictions: according to $\textrm {CTT}$ , $b^\beta (a^\alpha )$ is well-formed iff $\beta>\alpha $ . In this paper, we set ourselves against $\textrm {CTT}$ . We begin our case by arguing against Linnebo and Rayo’s claim that $\textrm {CTT}$ sheds new philosophical light on set theory. We then argue that, while $\textrm {CTT}$ ’s type-restrictions are unjustifiable, the type-restrictions imposed by ${\textrm {STT}}$ are justified by a Fregean semantics. What is more, this Fregean semantics provides us with a principled way to resist Linnebo and Rayo’s Semantic Argument for $\textrm {CTT}$ . We end by examining an alternative approach to cumulative types due to Florio and Jones [10]; we argue that their theory is best seen as a misleadingly formulated version...

2017

Type theories, from the early days of Montague Semantics (Montague 1974) to the recent work of using rich or modern type theories, have a long history of being employed as foundational languages of natural language semantics. In this introductory chapter, we will describe and discuss the development of type theories as foundational languages of mathematics, as well as their applications as foundational languages for formal semantics. In the end, a brief description of each chapter in the volume will follow.

This paper is to appear in the anthology "Reflections on the Foundations of Mathematics". It discusses some of the ways in which Martin-Löf type theory differs from set theory. The discussion concentrates on conceptual, rather than technical, differences. It revolves around four topics: sets versus types; syntax; functions ; and identity. The difference between sets and types is spelt out as the difference between unified pluralities and kinds, or sorts. A detailed comparison is then offered of the syntax of the two languages. Emphasis is put on the distinction between proposition and judgement, drawn by type theory, but not by set theory. Unlike set theory, type theory treats the notion of function as primitive. It is shown that certain inconveniences pertaining to function application that afflicts the set-theoretical account of functions are thus avoided. Finally, the distinction, drawn in type theory, between judgemental and propositional identity is discussed. It is argued that the criterion of identity for a domain cannot be formulated in terms of propositional identity. It follows that the axiom of extensionality cannot be taken as a statement of the criterion of identity for sets.

Journal of Logic, Language and Information, 2006

2003

We show how to write generic programs and proofs in Martin-Löf type theory. To this end we consider several extensions of Martin-Löf's logical framework for dependent types. Each extension has a universes of codes (signatures) for inductively defined sets with generic formation, introduction, elimination, and equality rules. These extensions are modeled on Dybjer and Setzer's finitely axiomatized theories of inductive-recursive definitions, which also have a universe of codes for sets, and generic formation, introduction, elimination, and equality rules. However, here we consider several smaller universes of interest for generic programming and universal algebra. We thus formalize one-sorted and many-sorted term algebras, as well as iterated, parameterized, generalized, and indexed inductive definitions. We also show how to extend the techniques of generic programming to these universes. Most of the definitions in the paper have been implemented using the proof assistant Alfa for dependent type theory.