## Section: New Results

### Separating Functional Computation from Relations

Participants : Ulysse Gérard, Dale Miller.

The logical foundation of arithmetic generally starts with a quantificational logic over relations. Of course, one often wishes to have a formal treatment of functions within this setting. Both Hilbert and Church added choice operators (such as the epsilon operator) to logic in order to coerce relations that happen to encode functions into actual functions. Others have extended the term language with confluent term rewriting in order to encode functional computation as rewriting to a normal form (e.g., the Dedukti proof checking project [46]) It is possible to take a different approach that does not extend the underlying logic with either choice principles or with an equality theory. Instead, we use the familiar two-phase construction of focused proofs and capture functional computation entirely within one of these phases. As a result, computation of functions can remain purely relational even when it is computing functions. This result, which appearred in [22], could be used to add to the Abella theorem prover a primitive method for doing deterministic computations.