## Section: New Results

### A logic for reasoning about logic specifications

Participants : David Baelde, Andrew Gacek, Dale Miller.

As described in SectionÂ
3.3 , there has been a
decade-long effort to design a logical framework for reasoning about
logic specifications.
Finally in 2008 and 2009 team members have reached what appears to be
a natural culmination of this development. In particular, David
Baelde's PhD [27] and Andrew Gacek's PhD thesis
[37] provided rich analysis of how the
-quantification can be related to fixed point definitions and
their associated induction and co-induction inference rules. Baelde
has concentrated on proving focusing-style results that are critical
for proof automation and on a *minimal generic* interpretation of
the -quantifier. Gacek has concentrated on a *nominal
generic* interpretation of the -quantifier. We now
understand the difference between these logics: the nominal approach
resembles much more closely the approach developed by Pitts
[59] .

Full proofs of the important meta-theory results of the logic in Gacek's thesis have been submitted for publication [41] . Gacek has also provided an implementation of his logic within the Abella prover that he has worked on as part of his PhD thesis.

We have developed extensive examples of our this new logic: significant examples taken from the -calculus have been published in [65] and the Abella distribution contains a large number of examples.