Section: New Results

Foundations and applications of explicit substitutions

Participant : Beniamino Accattoli.

Starting from the study of Linear Logic proof nets, a new approach to explicit substitutions for ł-calculus has recently been introduced by Accattoli and D. Kesner [31] . This approach has been systematically explored by Accattoli and his co-authors.

The rewriting theory of these new explicit substitutions at a distance has been studied in [11] and [16] . In [11] Accattoli and Kesner study the preservation of $\lambda$-calculus strong normalization (PSN) when explicit substitutions are extended with permutative axioms allowing to swap constructors in the term, generalizing considerably the already difficult case of PSN with composition of substitutions. In [16] Accattoli developed an abstract technique for proving factorizations theorems for generic explicit substitution calculi. The factorization theorem for $\lambda$-calculus says that any reduction can be re-organized as an head reduction followed by a non-head reduction.

In [16] it is shown how to prove this theorem in an uniform way for many explicit subsitutions calculi. The technique emerged as a generalization of the proofs for explicit substitutions at a distance, which are simpler than usual explicit substitutions and thus lead to cleaner and more compact arguments, easier to generalize.

Applications of explicit substitutions at a distance have been studied in [19] , [18] , [20] . In [19] Accattoli and Dal Lago show that the length of the head reduction in calculi at a distance is a measure of time complexity. More precisely, they show that such a quantity is polynomially related (in both directions) to the cost of evaluating with Turing Machines. This result is an important step forward towards the solution of the long-standing open problem of finding a time cost model for ł-calculus.

In [20] Accattoli and Paolini apply substitutions at a distance in a call-by-value setting. They show that in this new framework there is a natural characterization of solvability, an important notion related to denotational semantics and the representation of partial recursive functions. In [26] (a work presented to a workshop and currently submitted to the post-proceedings of the workshop) Accattoli shows the tight relations between the framework in [20] and linear logic proof nets, providing a new characterization of the proof nets representing the call-by-value $\lambda$-calculus.

Finally, in [18] Accattoli and Kesner introduce a calculus generalizing many different extensions of $\lambda$-calculus with permutations, appeared in various contexts (studies about call-by-value, postponing of reductions, monadic languages, etc) and prove confluence and preservation of strong normalization, exploiting and extending their own results in [11] .