Light affine lambda calculus and polytime strong normalization

Context semantics, linear logic, and computational complexity

“…For every n ∈ N, a !-cut at level n is fired only when any cut at level n is either a W -cut or a !-cut These two conditions induce another relation − , which is itself a reduction strategy: note that firing a cut at level n does not introduce cuts at levels strictly smaller than n, while firing a !-cut at level n only introduces cuts at level n + 1. As a consequence, − can be considered as a "level-by-level" strategy [2,19].…”

“…In particular, we here prove strong soundness theorems (any proof can be reduced in a bounded amount of time, independently on the underlying reduction strategy) for various subsystems of multiplicative linear logic by studying how restricting exponential rules reflect to W G . These proofs are simpler than similar ones from the literature [12,2,19,16], which in many cases refer to weak rather than strong soundness. The weight W G of a proof-net G will be defined from the context semantics of G following two ideas:…”

Linear Recursive Functions

“…From another perspective there have been a number of calculi, again many based on linear logic, for capturing specific complexity classes ( [6,11,14,7,18,25,8]). One of the main examples is that of bounded linear logic [14], which captures the class of polynomial time computable functions.…”

Elementary Affine Logic and the Call-by-Value Lambda Calculus

“…After their introduction, they have been shown to be relevant for optimal reduction [5,6], programming language design [4,7] and set theory [8]. However, proof languages for these logics, designed through the Curry-Howard Correspondence, are syntactically quite complex and can hardly be proposed as programming languages.…”

“…Indeed, β-reduction is more permissive than the restrictive copying discipline governing calculi directly derived from light logics. Consider, for example, the following expression in Λ LA (see [7]):…”