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  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.