Damiano Mazza

Distilling abstract machines

It is well-known that many environment-based abstract machines can be seen as strategies in lambda calculi with explicit substitutions (ES). Recently, graphical syntaxes and linear logic led to the linear substitution calculus (LSC), a new approach to ES that is halfway between small-step calculi and traditional calculi with ES. This paper studies the relationship between the LSC and environment-based abstract machines. While traditional calculi with ES simulate abstract machines, the LSC rather distills them: some transitions are simulated while others vanish, as they map to a notion of structural congruence. The distillation process unveils that abstract machines in fact implement weak linear head reduction, a notion of evaluation having a central role in the theory of linear logic. We show that such a pattern applies uniformly in call-by-name, call-by-value, and call-by-need, catching many machines in the literature. We start by distilling the KAM, the CEK, and a sketch of the ZINC, and then provide simplified versions of the SECD, the lazy KAM, and Sestofts machine. Along the way we also introduce some new machines with global environments. Moreover, we show that distillation preserves the time complexity of the executions, i.e. the LSC is a complexity-preserving abstraction of abstract machines.

Linear logic by levels and bounded time complexity

We give a new characterization of elementary and deterministic polynomial time computation in linear logic through the proofs-as-programs correspondence. Girards seminal results, concerning elementary and light linear logic, achieve this characterization by enforcing a stratification principle on proofs, using the notion of depth in proof nets. Here, we propose a more general form of stratification, based on inducing levels in proof nets by means of indices, which allows us to extend Girards systems while keeping the same complexity properties. In particular, it turns out that Girards systems can be recovered by forcing depth and level to coincide. A consequence of the higher flexibility of levels with respect to depth is the absence of boxes for handling the paragraph modality. We use this fact to propose a variant of our polytime system in which the paragraph modality is only allowed on atoms, and which may thus serve as a basis for developing lambda-calculus type assignment systems with more efficient typing algorithms than existing ones.

A Core Quantitative Coeffect Calculus

Linear logic is well known for its resource-awareness, which has inspired the design of several resource management mechanisms in programming language design. Its resource-awareness arises from the distinction between linear, single-use data and non-linear, reusable data. The latter is marked by the so-called exponential modality, which, from the categorical viewpoint, is a monoidal comonad. Monadic notions of computation are well-established mechanisms used to express effects in pure functional languages. Less well-established is the notion of comonadic computation. However, recent works have shown the usefulness of comonads to structure context dependent computations. In this work, we present a language

Linear logic and polynomial time

\ell \mathcal{R}

Multiport interaction nets and concurrency

PCF inspired by a generalized interpretation of the exponential modality. In

An Infinitary Affine Lambda-Calculus Isomorphic to the Full Lambda-Calculus

\ell \mathcal{R}

The separation theorem for differential interaction nets

PCF the exponential modality carries a label--an element of a semiring

A Strong Distillery

\mathcal{R}

A denotational semantics for the symmetric interaction combinators

--that provides additional information on how a program uses its context. This additional structure is used to express comonadic type analysis.

Simple Parsimonious Types and Logarithmic Space.

Light and Elementary Linear Logic, which form key components of the interface between logic and implicit computational complexity, were originally introduced by Girard as ‘stand-alone’ logical systems with a (somewhat awkward) sequent calculus of their own. The latter was later reformulated by Danos and Joinet as a proper subsystem of linear logic, whose proofs satisfy a certain structural condition. We extend this approach to polytime computation, finding two solutions: the first is obtained by a simple extension of Danos and Joinets condition, closely resembles Aspertis Light Affine Logic and enjoys polystep strong normalisation (the polynomial bound does not depend on the reduction strategy); the second, which needs more complex conditions, exactly corresponds to Girards Light Linear Logic.