This special issue collects selected articles from the Third International Workshop on Linearity (LINEARITY 2014), which was held in Vienna, on July 13th, 2014. The workshop was a one-day satellite event of FLoC 2014, the sixth Federated Logic Conference, which was held as part of the 2014 Vienna Summer of Logic.

We investigate cut elimination in multi-focused sequent calculi and the impact on the cut elimination proof of design choices in such calculi. The particular design we advocate is illustrated by a multi-focused calculus for full linear logic using an explicitly polarised syntax and incremental focus handling, for which we provide a syntactic cut elimination procedure. We discuss the effect of cut elimination on the structure of proofs, leading to a conceptually simple proof exploiting the strong structure of multi-focused proofs.

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 a classical propositional multiplicative subexponential logic (MSEL) with one unrestricted and two linear subexponentials can encode the halting problem for two register Minsky machines, and is hence undecidable. We then show how the additive connectives can be directly simulated by giving an encoding of propositional multiplicative additive linear logic (MALL) in an MSEL with one unrestricted and four linear subexponentials.

The resource λ-calculus is a variation of the λ-calculus where arguments are superpositions of terms and must be linearly used; hence, it is a model for linear and non-deterministic programming languages. Moreover, it is the target language of the Taylor–Ehrhard–Regnier expansion of λ-terms, a linearisation of the λ-calculus which develops ordinary terms into infinite series of resource terms. In a strictly typed restriction of the resource λ-calculus, we study the notion of path persistence, and define a remarkably simple geometry of resource interaction (GoRI) that characterises it. In addition, GoRI is invariant under reduction and counts addends in normal forms. We also analyse expansion on paths in ordinary terms, showing that reduction commutes with expansion and, consequently, that persistence can be transferred back and forth between a path and its expansion. Lastly, we also provide an expanded counterpart of the execution formula, which computes paths as series of objects of GoRI; thus, exchanging determinism and conciseness for linearity and simplicity.

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 reflecting the LPC structure. The paper's meta-theoretic results include admissibility theorems for the cut and duality rules, and a translation of the LPC logic into category theory. The work also presents several concrete instances of the LPC model.