## Section: New Results

### Compact Proof Certificates By Bounded Contractions

Participant : Kaustuv Chaudhuri.

An important engineering question in the ProofCert project is that
of communicating, manipulating, and storing formal proof
certificates. A fully detailed proof certificate, especially one
generated by proof search, can be very large. Using such proofs
would require a high bandwidth interface between the proof producer
and consumer, which limits the scalability of the *ensemble of
proving systems* approach. It is therefore natural to ask if there
are more compact formats for proof certificates. The ideal format
would have a tunable level of detail, so that the size of the
certificates can be tailored to the application domain.

Suppose the proof consumer is equipped with some proof search
capabilities. What then needs to be transmitted to the consumer to
guarantee that it can check a proof within desired complexity
bounds? It turns out that there is a systematic and general answer
to this problem: use *focusing* and record only the
“decision” rules of focusing in the proof certificate. From a high
level perspective, this answer is equivalent to designing a proof
system where the contraction rules are carefully bounded.

A proposal along these lines was published at the CPP 2012
conference [21] . In fact, this paper solves a
harder than necessary problem by building proof certificates for
linear logic, where unconstrained proof search has very high
complexity even in the propositional fragment. The proposed solution
is a spectrum of certificates that trades off the size of the
certificate for the complexity of checking the certificate. At one
end we have a very compact certificate that essentially amounts to a
maximum depth of the proof, but reconstructing a proof with only a
depth bound tends to be infeasible as the search space grows
super-exponentially with the depth. Certificates at other end of the
spectrum contain information about all the contractions in the
proof; these certificates can be checked deterministically, in time
proportional to the size of the certificate. Moreover, there is a
simple abstraction mechanism between different levels of detail in
this spectrum that allows for a *proof elaborator* to alter the
level of detail in the certificate.