## Section: New Results

### Computation with meta-variables

Participant : Stéphane Lengrand.

Meta-variables are central in proof search mechanisms to represent incomplete proofs and incomplete objects. They are used in almost all implementations of proof-related software, yet their meta-theory remains less explored than that of complete proofs and objects such as in the -calculus.

In 2009, Stéphane Lengrand and Jamie Murdoch Gabbay have published in [13] a first proposal for a computational model taking these features into account.

This proposal, extending the calculus with a particular kind
of meta-variables originating from nominal logic, is more sparing than
previous approaches like *Higher-Order Abstract Syntax* , which
explicitly represents *all potential* dependencies between
incomplete objects (this leads to computational inefficiencies as
potential dependencies that are not effectively used still incur a
computational cost).

Lengrand and Gabbay's proposal is only a first step, as it does not have a neat theory of normal forms (i.e. output values). A more complete version of such a -calculus, with incomplete objects and arbitrary binding dependencies but also with better normalization properties, has been in development since.