The lambda-calculus studies the dynamics of computation, but the complexity of one computation path cannot be forecast, neither in time nor in space, and there is no direct control on the execution environment.

One approach for the former is to gain more control over duplication and deletion of terms, by introducing sharings of common subterms inside a lambda-term, similarly to optimal reduction graphs. This has been done by Gundersen, Heijltjes and Parigot, via the atomic lambda-calculus, a calculus extending the lambda-calculus with explicit sharings, and allowing duplications on individual constructors. It enjoys full-laziness, avoiding unnecessary duplications of constant expressions. 

For the latter, an extension of the lambda-calculus linking classical formulas to control flow constructs via Curry-Howard correspondence, the lambda-mu-calculus, was developed by Parigot. New operators in the lambda-mu-calculus correspond to continuations, representing the remaining work in a program. 

Can we combine sharing with control operators and thus obtain another extension of the lambda-calculus satisfying full-laziness and featuring control operators? A first challenge is to combine explicit sharings or substitutions with mu-reduction and substitution, and we introduce a right-associative application as in Saurin-Gaboardi's lambda-mu-calculus with streams to overcome this challenge. 
Another difficulty is to adapt the type system for lambda-mu with multiple conclusions, distinguishing one main conclusion (with a meta-disjunction connective), to a type system in which each rule can be applied to atoms (which corresponds to duplications on individual term constructors), and where no distinction is made between object and meta-level. I will present how to combine these two type systems in the simplest possible way.

I will present the atomic lambda-mu-calculus, which aims to naturally extend the two calculi presented above, giving the simplest type system combining and preserving their properties.