  The framework of Light Logics has been extensively studied to control the
complexity of higher-order functional programs. We propose an extension of this
framework to multithreaded programs with side effects, focusing on the case of
polynomial time. After introducing a modal \lambda-calculus with parallel
composition and regions, we prove that a realistic call-by-value evaluation
strategy can be computed in polynomial time for a class of well-formed
programs. The result relies on the simulation of call-by-value by a polynomial
shallow-first strategy which preserves the evaluation order of side effects.
Then, we provide a polynomial type system that guarantees that well-typed
programs do not go wrong. Finally, we illustrate the expressivity of the type
system by giving a programming example of concurrent iteration producing side
effects over an inductive data structure.
