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A type theory is presented that combines (intuitionistic) linear types with type dependency, thus properly generalising both intuitionistic dependent type theory and full linear logic. A syntax and complete categorical semantics are…
We reconstruct some of the development in Richard Bird's [2008] paper Zippy Tabulations of Recursive Functions, using dependent types and string diagrams rather than mere simple types. This paper serves as an intuitive introduction to and…
We prove normalization for MTT, a general multimodal dependent type theory capable of expressing modal type theories for guarded recursion, internalized parametricity, and various other prototypical modal situations. We prove that deciding…
Recently we presented a concise survey of the formulation of the induction and coinduction principles, and some concepts related to them, in programming languages type theory and four other mathematical disciplines. The presentation in type…
We introduce a new logic that combines Adjoint Logic with Graded Necessity Modalities. This results in a very expressive system capable of controlling when and how structural rules are used. We give a sequent calculus, natural deduction,…
In the pure Calculus of Constructions (CC) one can define data types and function over these, and there is a powerful higher order logic to reason over these functions and data types. This is due to the combination of impredicativity and…
We present an extension of Martin-L\"of Type Theory that contains a tiny object; a type for which there is a right adjoint to the formation of function types as well as the expected left adjoint. We demonstrate the practicality of this type…
The modal logic S4 can be used via a Curry-Howard style correspondence to obtain a lambda-calculus. Modal (boxed) types are intuitively interpreted as `closed syntax of the calculus'. This lambda-calculus is called modal type theory ---…
We consider a simple modal logic whose non-modal part has conjunction and disjunction as connectives and whose modalities come in adjoint pairs, but are not in general closure operators. Despite absence of negation and implication, and of…
We present a prescriptive type system with parametric polymorphism and subtyping for constraint logic programs. The aim of this type system is to detect programming errors statically. It introduces a type discipline for constraint logic…
The expression problem describes a fundamental tradeoff between two types of extensibility: extending a type with new operations, such as by pattern matching on an algebraic data type in functional programming, and extending a type with new…
The Dependent Object Type (DOT) calculus was designed to put Scala on a sound basis, but while DOT relies on structural subtyping, Scala is a fundamentally class-based language. This impedance mismatch means that a proof of DOT soundness by…
Using dependent type theory to formalise the syntax of dependent type theory is a very active topic of study and goes under the name of "type theory eating itself" or "type theory in type theory." Most approaches are at least loosely based…
Higher inductive types (HITs) in Homotopy Type Theory (HoTT) allow the definition of datatypes which have constructors for equalities over the defined type. HITs generalise quotient types and allow to define types which are not sets in the…
We present a doctrinal approach to category theory, obtained by abstracting from the indexed inclusions (via discrete fibrations and opfibrations) of the left and of the right actions of X in Cat in categories over X. Namely, a "weak…
Guided by Tarksi's fixpoint theorem in order theory, we show how to derive monotone recursive types with constant-time roll and unroll operations within Cedille, an impredicative, constructive, and logically consistent pure typed lambda…
In standard quantum theory, time is not an observable. It enters as a parameter in the Schr\"odinger equation, but there is no measurement operator associated to it. Nevertheless, one may take an operational viewpoint and regard time as the…
Proof assistants and programming languages based on type theories usually come in two flavours: one is based on the standard natural deduction presentation of type theory and involves eliminators, while the other provides a syntax in…
Recently, in order to mix algebraic and logic styles of specification in a uniform framework, the notion of a logic labelled transition system (Logic LTS or LLTS for short) has been introduced and explored. A variety of constructors over…
We introduce a notion of compatibility between constraint encoding and compositional structure. Phrased in the language of category theory, it is given by a "composable constraint encoding". We show that every composable constraint encoding…