Philip J. Scott

Bounded linear logic: a modular approach to polynomial-time computability

Abstract Usual typed lambda-calculi yield input/output specifications; in this paper the authors show how to extend this paradigm to complexity specifications. This is achieved by means of a restricted version of linear logic in which the use of exponential connectives is bounded in advance. This bounded linear logic naturally involves polynomials in its syntax and dynamics. It is then proved that any functional term of appropriate type actually encodes a polynomial-time algorithm and that conversely any polynomial-time function can be obtained in this way.

On the p-calculus and linear logic

Abstract We detail Abramskys “proofs-as-processes” paradigm for interpreting classical linear logic (CLL) (Girard, 1987) into a “synchronous” version of the π-calculus recently proposed by Milner (1992, 1993). The translation is given at the abstract level of proof structures. We give a detailed treatment of information flow in proof-nets and show how to mirror various evaluation strategies for proof normalization. We also give soundness and completeness results for the process-calculus translations of various fragments of CLL. The paper also gives a self-contained introduction to some of the deeper proof-theory of CLL, and its process interpretation.

Geometry of Interaction and linear combinatory algebras

We present an axiomatic framework for Girards Geometry of Interaction based on the notion of linear combinatory algebra. We give a general construction on traced monoidal categories, with certain additional structure, that is sufficient to capture the exponentials of Linear Logic, which produces such algebras (and hence also ordinary combinatory algebras). We illustrate the construction on six standard examples, representing both the ‘particle-style’ as well as the ‘wave-style’ Geometry of Interaction.

Category theory for linear logicians

This paper presents an introduction to category theory with an emphasis on those aspects relevant to the analysis of the model theory of linear logic. With this in mind, we focus on the basic definitions of category theory and categorical logic. An analysis of cartesian and cartesian closed categories and their relation to intuitionistic logic is followed by a consideration of symmetric monoidal closed, linearly distributive and ∗-autonomous categories and their relation to multiplicative linear logic. We examine nonsymmetric monoidal categories, and consider them as models of noncommutative linear logic. We introduce traced monoidal categories, and discuss their relation to the geometry of interaction. The necessary aspects of the theory of monads is introduced in order to describe the categorical modelling of the exponentials. We conclude by briefly describing the notion of full completeness, a strong form of categorical completeness, which originated in the categorical model theory of linear logic. No knowledge of category theory is assumed, but we do assume knowledge of linear logic sequent calculus and the standard models of linear logic, and modest familiarity with typed lambda calculus.

Linear Läuchli semantics

Abstract We introduce a linear analogue of Lauchlis semantics for intuitionistic logic. In fact, our result is a strengthening of Lauchlis work to the level of proofs, rather than provability. This is obtained by considering continuous actions of the additive group of integers on a category of topological vector spaces. The semantics, based on functorial polymorphism, consists of dinatural transformations which are equivariant with respect to all such actions. Such dinatural transformations are called uniform. To any sequent in Multiplicative Linear Logic (MLL), we associate a vector space of“diadditive” uniform transformations. We then show that this space is generated by denotations of cut-free proofs of the sequent in the theory MLL + MIX. Thus we obtain a full completeness theorem in the sense of Abramsky and Jagadeesan, although our result differs from theirs in the use of dinatural transformations. As corollaries, we show that these dinatural transformations compose, and obtain a conservativity result: diadditive dinatural transformations which are uniform with respect to actions of the additive group of integers are also uniform with respect to the actions of arbitrary cocommutative Hopf algebras. Finally, we discuss several possible extensions of this work to noncommutative logic. It is well known that the intuitionistic version of Lauchlis semantics is a special case of the theory of logical relations, due to Plotkin and Statman. Thus, our work can also be viewed as a first step towards developing a theory of logical relations for linear logic and concurrency.

A categorical model for the geometry of interaction

We consider the multiplicative and exponential fragment of linear logic (MELL) and give a geometry of interaction (GoI) semantics for it based on unique decomposition categories. We prove a soundness and finiteness theorem for this interpretation. We show that Girards original approach to GoI 1 via operator algebras is exactly captured in this categorical framework.

Normalization and the Yoneda embedding

We show how to solve the word problem for simply typed λβη-calculus by using a few well-known facts about categories of presheaves and the Yoneda embedding. The formal setting for these results is P-category theory, a version of ordinary category theory where each hom-set is equipped with a partial equivalence relation. The part of P-category theory we develop here is constructive and thus permits extraction of programs from proofs. It is important to stress that in our method we make no use of traditional proof-theoretic or rewriting techniques. To show the robustness of our method, we give an extended treatment for more general λ-theories in the Appendix.

Linear logic in computer science

Preface List of contributors Part I. Tutorials: 1. Category theory for linear logicians R. Blute and Ph. Scott 2. Proof nets and the x-calculus S. Guerrini 3. An overview of linear logic programming D. Miller 4. Linearity and nonlinearity in distributed computation G. Winskel Part II. Refereed Articles: 5. An axiomatic approach to structural rules for locative linear logic J. M. Andreoli 6. An introduction to uniformity in ludics C. Faggian, M. R. Fleury-Donnadieu and M. Quatrini 7. Slicing polarized addictive normalization O. Laurent and L. Toratora De Falco 8. A topological correctness criterion for muliplicative noncommutative logic P.A. Mellies Part III. Invited Articles: 9. Bicategories in algebra and linguistics J. Lambek 10. Between logic and quantic: a tract J. Y. Girard.

Normal Forms and Cut-Free Proofs as Natural Transformations

What equations can we guarantee that simple functional programs must satisfy, irrespective of their obvious defining equations? Equivalently, what non-trivial identifications must hold between lambda terms, thought-of as encoding appropriate natural deduction proofs ? We show that the usual syntax guarantees that certain naturality equations from category theory are necessarily provable. At the same time, our categorical approach addresses an equational meaning of cut-elimination and asymmetrical interpretations of cut-free proofs. This viewpoint is connected to Reynolds’ relational interpretation of parametricity ([27], [2]), and to the Kelly-Lambek-Mac Lane-Mints approach to coherence problems in category theory.

Softness of hypercoherences and MALL full completeness

Abstract We prove a full completeness theorem for multiplicative–additive linear logic (i.e. MALL ) using a double gluing construction applied to Ehrhard’s *-autonomous category of hypercoherences. This is the first non-game-theoretic full completeness theorem for this fragment. Our main result is that every dinatural transformation between definable functors arises from the denotation of a cut-free MALL proof. Our proof consists of three steps. We show: • Dinatural transformations on this category satisfy Joyal’s softness property for products and coproducts. • Softness, together with multiplicative full completeness, guarantees that every dinatural transformation corresponds to a Girard MALL proof-structure. • The proof-structure associated with any dinatural transformation is a MALL proof-net, hence a denotation of a proof. This last step involves a detailed study of cycles in additive proof-structures. The second step is a completely general result, while the third step relies on the concrete structure of a double gluing construction over hypercoherences.