Linear Type Theory Research Articles

Fibrational Modal Type Theory

This paper describes a fibrational categorical semantics for the modal necessity-only fragment of constructive modal type theory, both with and without dependent types. Constructive type theory does not usually discuss logical modalities, and modalities tend to be mostly studied within classical logic, not within type theory. But modalities should be very useful in type theory, as they are very useful in modelling theoretical computing systems. Providing constructive versions of modal logics and their associated Curry-Howard modal type theories is also a very productive program, e.g. helpful when dealing with computational effects, staged computation, and functional reactive types, for example. There seems to be renewed interest in the notion of constructive modal type theory (and in notions of linear type theory), in part because of the interest in homotopy type theory. The modal type theory presented here uses dependent types, in the style of Ritter's categorical models of the Calculus of Constructions. To build up to these, we first discuss the kinds of constructive modal type theory in the literature. Then we provide a non-dependent modal type theory, introduced in previous work, that we generalize to dependent types in the following section. Dependent type theories are usually but not always given categorical semantics in terms of fibrations. We provide semantics in terms of fibrations for both the non-dependent and the dependent type systems discussed and prove them sound and complete, thereby providing evidence that the type theory is meaningful. These fibrational models should be also applicable to the homotopy type theory setting.

Duality Theory and Categorical Universal Logic: With Emphasis on Quantum Structures

Categorical Universal Logic is a theory of monad-relativised hyperdoctrines (or fibred universal algebras), which in particular encompasses categorical forms of both first-order and higher-order quantum logics as well as classical, intuitionistic, and diverse substructural logics. Here we show there are those dual adjunctions that have inherent hyperdoctrine structures in their predicate functor parts. We systematically investigate into the categorical logics of dual adjunctions by utilising Johnstone-Dimov-Tholen's duality-theoretic framework. Our set-theoretical duality-based hyperdoctrines for quantum logic have both universal and existential quantifiers (and higher-order structures), giving rise to a universe of Takeuti-Ozawa's quantum sets via the tripos-to-topos construction by Hyland-Johnstone-Pitts. The set-theoretical hyperdoctrinal models of quantum logic, as well as all quantum hyperdoctrines with cartesian base categories, turn out to give sound and complete semantics for Faggian-Sambin's first-order quantum sequent calculus over cartesian type theory; in addition, quantum hyperdoctrines with monoidal base categories are sound and complete for the calculus over linear type theory. We finally consider how to reconcile Birkhoff-von Neumann's quantum logic and Abramsky-Coecke's categorical quantum mechanics (which is modernised quantum logic as an antithesis to the traditional one) via categorical universal logic.

Linear usage of state

We investigate the phenomenon that "every monad is a linear state monad". We do this by studying a fully-complete state-passing translation from an impure call-by-value language to a new linear type theory: the enriched call-by-value calculus. The results are not specific to store, but can be applied to any computational effect expressible using algebraic operations, even to effects that are not usually thought of as stateful. There is a bijective correspondence between generic effects in the source language and state access operations in the enriched call-by-value calculus. From the perspective of categorical models, the enriched call-by-value calculus suggests a refinement of the traditional Kleisli models of effectful call-by-value languages. The new models can be understood as enriched adjunctions.

Toposes are symmetric monoidal closed categories

Journal of Applied Mathematics and Computational Mechanics, Prace Naukowe Instytutu Matematyki i Informatyki, Politechnika Częstochowska, Scientific Research of the Institute of Mathematics and Computer Science, Czestochowa University of Technology

Higher-order functional reactive programming in bounded space

Functional reactive programming (FRP) is an elegant and successful approach to programming reactive systems declaratively. The high levels of abstraction and expressivity that make FRP attractive as a programming model do, however, often lead to programs whose resource usage is excessive and hard to predict. In this paper, we address the problem of space leaks in discrete-time functional reactive programs. We present a functional reactive programming language that statically bounds the size of the dataflow graph a reactive program creates, while still permitting use of higher-order functions and higher-type streams such as streams of streams. We achieve this with a novel linear type theory that both controls allocation and ensures that all recursive definitions are well-founded. We also give a denotational semantics for our language by combining recent work on metric spaces for the interpretation of higher-order causal functions with length-space models of space-bounded computation. The resulting category is doubly closed and hence forms a model of the logic of bunched implications.

Pattern Unification for the Lambda Calculus with Linear and Affine Types

We define the pattern fragment for higher-order unification problems in linear and affine type theory and give a deterministic unification algorithm that computes most general unifiers.

Linear type theory for asynchronous session types

AbstractSession types support a type-theoretic formulation of structured patterns of communication, so that the communication behaviour of agents in a distributed system can be verified by static typechecking. Applications include network protocols, business processes and operating system services. In this paper we define a multithreaded functional language with session types, which unifies, simplifies and extends previous work. There are four main contributions. First is an operational semantics with buffered channels, instead of the synchronous communication of previous work. Second, we prove that the session type of a channel gives an upper bound on the necessary size of the buffer. Third, session types are manipulated by means of the standard structures of a linear type theory, rather than by means of new forms of typing judgement. Fourth, a notion of subtyping, including the standard subtyping relation for session types (imported into the functional setting), and a novel form of subtyping between standard and linear function types, which allows the typechecker to handle linear types conveniently. Our new approach significantly simplifies session types in the functional setting, clarifies their essential features and provides a secure foundation for language developments such as polymorphism and object-orientation.

Parametric Domain-theoretic Models of Polymorphic Intuitionistic / Linear Lambda Calculus

We present a formalization of a version of Abadi and Plotkin's logic for parametricity for a polymorphic dual intuitionistic / linear type theory with fixed points, and show, following Plotkin's suggestions, that it can be used to define a wide collection of types, including solutions to recursive domain equations. We further define a notion of parametric LAPL-structure and prove that it provides a sound and complete class of models for the logic, and conclude that such models have solutions for a wide class of recursive domain equations. Finally, we present a concrete parametric LAPL-structure based on suitable categories of partial equivalence relations over a universal model of the untyped lambda calculus.

Synthetic Domain Theory and Models of Linear Abadi & Plotkin Logic

In a recent article [L. Birkedal, R. E. Møgelberg, and R. L. Petersen. Parametric domain-theoretic models of linear Abadi & Plotkin logic. Technical Report TR-2005-57, IT University of Copenhagen, February 2005] the first two authors and R.L. Petersen have defined a notion of parametric LAPL-structure. Such structures are parametric models of the equational theory PILLY, a polymorphic intuitionistic / linear type theory with fixed points, in which one can reason using parametricity and, for example, solve a large class of domain equations [L. Birkedal, R. E. Møgelberg, and R. L. Petersen. Parametric domain-theoretic models of linear Abadi & Plotkin logic. Technical Report TR-2005-57, IT University of Copenhagen, February 2005, L Birkedal, R. E. Møgelberg, and R. L. Petersen. Parametric domain-theoretic models of polymorphic intuitionistic / linear lambda calculus. In Proceedings of the Twenty-first Conference on the Mathematical Foundations of Programming Semantics, 2005. To appear].Based on recent work by Simpson and Rosolini [G. Rosolini and A. Simpson. Using synthetic domain theory to prove operational properties of a polymorphic programming language based on strictness. Manuscript, 2004] we construct a family of parametric LAPL-structures using synthetic domain theory and use the results of loc. cit. and results about LAPL-structures to prove operational consequences of parametricity for a strict version of the Lily programming language. In particular we can show that one can solve domain equations in the strict version of Lily up to ground contextual equivalence.

A Linear Spine Calculus

Abstract : We present the spine calculus S(approaches -o&T) as an efficient representation for the linear lambda-calculus lambda(approaches -o&T) which includes intuitionistic functions (approach) Tau linear functions (-o)Tau additive pairing (&Tau) and additive unit (T). S(approaches -o&T) enhances the representation of Church's simply typed lamda-calculus as abstract Bolum trees by enforcing extensionality and by incorporating linear constructs. This approach permits procedures such as unification to retain the efficient head access that characterizes first-order term languages without the overhead of performing n-conversions at run time. Potential applications lie in proof search(Tau) logic programming(Tau) and logical frameworks based on linear type theories. We define the spine calculus(Tau) give translations of lambda(approaches -o&T) into S(approaches -o&T) and vice-versa(Tau) prove their soundness and completeness with respect to typing and reduction(Tau) and shows that the spine calculus is strongly normalizing and admits unique canonical forms.

A type theory for memory allocation and data layout

Ordered type theory is an extension of linear type theory in which variables in the context may be neither dropped nor re-ordered. This restriction gives rise to a natural notion of adjacency . We show that a language based on ordered types can use this property to give an exact account of the layout of data in memory. The fuse constructor from ordered logic describes adjacency of values in memory, and the mobility modal describes pointers into the heap. We choose a particular allocation model based on a common implementation scheme for copying garbage collection and show how this permits us to separate out the allocation and initialization of memory locations in such a way as to account for optimizations such as the coalescing of multiple calls to the allocator.

A Linear Logical Framework

We present the linear type theory λ Π⊸&⊤ as the formal basis for LLF , a conservative extension of the logical framework LF . LLF combines the expressive power of dependent types with linear logic to permit the natural and concise representation of a whole new class of deductive systems, namely those dealing with state. As an example we encode a version of Mini-ML with mutable references including its type system and its operational semantics and describe how to take practical advantage of the representation of its computations.

Efficient resource management for linear logic proof search

The design of linear logic programming languages and theorem provers opens a number of new implementation challenges not present in more traditional logic languages such as Horn clauses ( Prolog) and hereditary Harrop formulas ( λ Prolog and Elf). Among these, the problem of efficiently managing the linear context when solving a goal is of crucial importance for the use of these systems in non-trivial applications. This paper studies this problem in the case of Lolli [10], though its results have application to other systems including those based on linear type theory. We first give a proof-theoretic presentation of the operational semantics of this language as a resolution calculus. We then present a series of resource management systems designed to eliminate the non-determinism in the distribution of linear formulas that undermines the efficiency of a direct implementation of this system.

A Type-theoretic Description of Action Calculi

Action calculi, introduced by Milner, provide a framework for investigating models of interaction. This talk will focus on the connection between action calculi and known concepts arising from type theory. The aim of this work is to isolate what is distinctive about action calculi, and to investigate the potential of action calculi as an underlying framework for many kinds of computational behaviour.The first part of the talk will introduce action calculi. In the second part, I'll give a type-theoretic account of action calculi, using the general binding operators of Aczel. I will discuss two extensions: higher-order action calculi which correspond to Moggi's commutative computational lambda-calculus, and linear action calculi which correspond to the linear type theories of Barber and Benton.This talk is based on joint work with Andrew Barber, Masahito Hasegawa and Gordon Plotkin. If time, I will also describe current work arising from the connections described above.