LINEARITY 2014 INFORMAL PROCEEDINGS: PAPERS WITH ABSTRACTS

ABSTRACT. In this paper, we introduce the notion of structure. Using the notion of structure, we study, revisit, discover and rediscover some basic properties of ludics from a very general point of view.

ABSTRACT. We study cut elimination for a multifocused variant of linear logic in the sequent calculus. The multifocused normal form of proofs yields problems that do not appear in the standard focused system, related to the constraints in grouping rule instances inside focusing phases. We show that cut elimination can be performed in a sensible way even though the proof requires specific lemmas to deal with multifocusing phases, and discuss the difficulties arising with cut elimination when considering normal norms of proofs in linear logic.

ABSTRACT. Subexponential logic is a variant of linear logic with a family of exponential connectives---called subexponentials---that are indexed and arranged in a pre-order. Each subexponential has or lacks associated structural properties of weakening and contraction. We show that classical propositional multiplicative linear logic extended with one unrestricted and two incomparable linear subexponentials can encode the halting problem for two register Minsky machines, and is hence undecidable.

ABSTRACT. Particle-style token machines are a way to interpret proofs and programs, when the latter are written following the principles of linear logic. In this paper, we show that token machines also make sense when the programs at hand are those of a simple quantum lambda-calculus. This, however, requires generalizing the concept of a token machine to one in which more than one particle travel around the term at the same time. The presence of multiple tokens is intimately related to entanglement and allows to give a simple operational semantics to the calculus, coherently with the principles of quantum computation.

ABSTRACT. Linear and substructural types are powerful tools, but adding them to standard functional programming languages often means introducing extra annotations and typing machinery. We propose a lightweight substructural type system design that recasts the structural rules of weakening and contraction as type classes; we demonstrate this design in a prototype language, Clamp.

Clamp supports polymorphic substructural types as well as an expressive system of mutable references. At the same time, it adds little additional overhead to a standard Damas–Hindley–Milner type system enriched with type classes. We have established type safety for the core model and implemented a type checker for Clamp in Haskell.

ABSTRACT. We introduce (in the multiplicative and exponential fragment of linear logic) the notion of protodifferential net. We then define a coherence relation on the set of proto-differential nets and prove the analogue of the result proven for differential lambda-terms, which does not hold for differential nets: a set of proto-differential nets is the subset of the proto-Taylor expansion of some proof-structure if and only if it is a clique.

ABSTRACT. This paper defines a new proof- and category-theoretic framework for classical linear logic that separates reasoning into one linear regime and two persistent regimes corresponding to ! and ?. The resulting linear/producer/consumer (LPC) logic puts the three classes of propositions on the same semantic footing, following Benton's linear/non-linear formulation of intuitionistic linear logic. Semantically, LPC corresponds to a system of three categories connected by adjunctions that reflect the linear/producer/consumer structure. The paper's metatheoretic results include admissibility theorems for the cut and duality rules, and a translation of the LPC logic into the category theory. The work also presents several concrete instances of the LPC model, including one based on finite vector spaces.

ABSTRACT. The inhabitation problem for intersection types is known to be undecidable. We study the problem in the case of non-idempotent intersection, in particular for a type assignment system characterizing solvable terms, and we prove decidability through a sound and complete algorithm.

ABSTRACT. The Resource λ-calculus (RC) is a variation of the λ-calculus where arguments can be superposed and must be linearly used. Hence it is a model for non-deterministic and linear programming languages, and the target language of the Taylor expansion of λ-terms. In a strictly typed restriction of RC, we study the persistency and we define a Geometry of Interaction that is invariant under reduction, characterises persistency, and counts the addends in normal forms.

ABSTRACT. Linear logic refines conventional logic by being a ``resource-conscious'' logic. We analyze the notion of resource maintained by linear logic proofs and argue that it only focuses on a ``spatial'' dimension omitting another important ``temporal'' dimension. We generalize (or further refine using even more structural rules) linear logic to maintain combined spatial and temporal resources. The resulting logic is reversible and thus allows a study of reversible computation from a proof-theoretic perspective.