## 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] .