Affine Lambda Calculus Research Articles

On the Elementary Affine Lambda-Calculus with and Without Fixed Points

The elementary affine lambda-calculus was introduced as a polyvalent setting for implicit computational complexity, allowing for characterizations of polynomial time and hyperexponential time predicates. But these results rely on type fixpoints (a.k.a. recursive types), and it was unknown whether this feature of the type system was really necessary. We give a positive answer by showing that without type fixpoints, we get a characterization of regular languages instead of polynomial time. The proof uses the semantic evaluation method. We also propose an aesthetic improvement on the characterization of the function classes FP and k-FEXPTIME in the presence of recursive types.

Algebras and coalgebras in the light affine Lambda calculus

Algebra and coalgebra are widely used to model data types in functional programming languages and proof assistants. Their use permits to better structure the computations and also to enhance the expressivity of a language or of a proof system. Interestingly, parametric polymorphism à la System F provides a way to encode algebras and coalgebras in strongly normalizing languages without losing the good logical properties of the calculus. Even if these encodings are sometimes unsatisfying because they provide only limited forms of algebras and coalgebras, they give insights on the expressivity of System F in terms of functions that we can program in it. With the goal of contributing to a better understanding of the expressivity of Implicit Computational Complexity systems, we study the problem of defining algebras and coalgebras in the Light Affine Lambda Calculus, a system characterizing the complexity class FPTIME. This system limits the computational complexity of programs but it also limits the ways we can use parametric polymorphism, and in general the way we can write our programs. We show here that while the restrictions imposed by the Light Affine Lambda Calculus pose some issues to the standard System F encodings, they still permit to encode some form of algebra and coalgebra. Using the algebra encoding one can define in the Light Affine Lambda Calculus the traditional inductive types. Unfortunately, the corresponding coalgebra encoding permits only a very limited form of coinductive data types. To extend this class we study an extension of the Light Affine Lambda Calculus by distributive laws for the modality §. This extension has been discussed but not studied before.

A Graph Model for Imperative Computation

Scott's graph model is a lambda-algebra based on the observation that continuous endofunctions on the lattice of sets of natural numbers can be represented via their graphs. A graph is a relation mapping finite sets of input values to output values. We consider a similar model based on relations whose input values are finite sequences rather than sets. This alteration means that we are taking into account the order in which observations are made. This new notion of graph gives rise to a model of affine lambda-calculus that admits an interpretation of imperative constructs including variable assignment, dereferencing and allocation. Extending this untyped model, we construct a category that provides a model of typed higher-order imperative computation with an affine type system. An appropriate language of this kind is Reynolds's Syntactic Control of Interference. Our model turns out to be fully abstract for this language. At a concrete level, it is the same as Reddy's object spaces model, which was the first "state-free" model of a higher-order imperative programming language and an important precursor of games models. The graph model can therefore be seen as a universal domain for Reddy's model.

Light affine lambda calculus and polynomial time strong normalization

Light Linear Logic (LLL) and Intuitionistic Light Affine Logic (ILAL) are logics that capture polynomial time computation. It is known that every polynomial time function can be represented by a proof of these logics via the proofs-as-programs correspondence. Furthermore, there is a reduction strategy which normalizes a given proof in polynomial time. Given the latter polynomial time “weak” normalization theorem, it is natural to ask whether a “strong” form of polynomial time normalization theorem holds or not. In this paper, we introduce an untyped term calculus, called Light Affine Lambda Calculus (λLA), which corresponds to ILAL. λLA is a bi-modal λ-calculus with certain constraints, endowed with very simple reduction rules. The main property of LALC is the polynomial time strong normalization: any reduction strategy normalizes a given λLA term in a polynomial number of reduction steps, and indeed in polynomial time. Since proofs of ILAL are structurally representable by terms of λLA, we conclude that the same holds for ILAL.

Intersection Types for Light Affine Lambda Calculus

Light Affine Lambda Calculus is a term calculus for polynomial time computation ([K. Terui. Light affine lambda calculus and polytime strong normalization. In Proceedings of LICS'01, pages 209–220, 2001]). Some of the terms of Light Affine Lambda Calculus must however be regarded as errors. Intuitionistic Light Affine Logic (ILAL) types only terms without errors, but not all of them. We introduce two type assignment systems with intersection types : in the first one, typable pseudo-terms are exactly the terms without errors ; in the second one, they are exactly those that reduce to normal terms without errors.

Preface: Volume 69

This volume contains the proceedings of the 9th Conference on Category Theory and Computer Science (CTCS'02), which was held at the University of Ottawa from August 15-17, 2002. The purpose of this conference series is the advancement of the foundations of computing using the tools of category theory. Indeed, category theory provides one of the key tools in the analysis of the interaction between logic and the theory of computation. The extent to which category theory has influenced these areas can be seen from the following list of topics, which are typical of the interests of this conference: • coalgebras and computing • concurrent and distributed systems • constructive mathematics • declarative programming and term rewriting • domain theory and topology • foundations of computer security • linear logic • modal and temporal logics • models of computation • program logics, data refinement, and specification • programming language semantics • type theory The list is by no means exhaustive. The vitality of the field is well displayed by the extremely high quality and the diversity of the 18 papers in this volume. For the first time in the history of CTCS, this year's conference was preceded by a “Graduate Student Preconference”, which took place from August 12-14 and which offered introductory courses in areas of importance to the conference. The response to this preconference was extraordinary, with more than 50 students and interested others taking part. This certainly suggests that the field of research of this conference will be strong and active for many years to come. The preconference offered courses in: • Introductory category theory (Susan Niefield) • Categorical logic (Philip Scott) • Concurrency theory (Peter Selinger) • Coalgebraic methods (Jiri Adamek) • Game theory (Robin Cockett) • Linear logic (Rick Blute) To hold this preconference, we received substantial funding from the Centre de Recherches Mathematiques (CRM). We thank them and Jacques Hurtubise, the Director of CRM, for their kind support. Without the help of many individuals in many different capacities, this conference would not have been possible. The editors would especially like to thank the following people: • The organizing committee: ◦ E. Moggi, Chair, (Genova) ◦ S. Abramsky (Oxford) ◦ P. Dybjer (Chalmers) ◦ B. Jay (Sydney) ◦ A. Pitts (Cambridge) • Local organizers: ◦ Rick Blute ◦ Philip Scott • The programme committee: ◦ Rick Blute, Chair (Ottawa) ◦ Robin Cockett (Calgary) ◦ Thierry Coquand (Chalmers) ◦ Andrea Corradini (Pisa) ◦ Thomas Ehrhard (Luminy) ◦ Ryu Hasegawa (Tokyo) ◦ Martin Hofmann (Munich) ◦ Bart Jacobs (Nijmegen) ◦ Michael Johnson (Macquarie) ◦ Dusko Pavlovic (Kestrel Institute) ◦ Alex Simpson (Edinburgh) • Lecturers in the student preconference: ◦ Susan Niefield (Union) ◦ Philip Scott (Ottawa) ◦ Robin Cockett (Calgary) ◦ Rick Blute (Ottawa) ◦ Jiri Adamek (Braunschweig) ◦ Peter Selinger (Ottawa)