Dan Connolly's tinkering lab notebook

## Existentials in ACL2 and Milawa make sense; how about level breakers?

Since my Sep 2006 visit to the ACL 2 seminar, I've been trying to get my head around existentials in ACL2. The lightbulb finally went off this week while reading Jared's Dec 2009 Milawa thesis.

3.7 Provability

Now that we have a proof checker, we can use existential quantification to
decide whether a particular formula is provable. Recall from page 61 the notion
of a witnessing (Skolem) function.
We begin by introducing a witnessing function,
logic.provable-witness, whose defining axiom is as follows.

Definition 92: logic.provable-witness
(por* (pequal* ...))

Intuitively, this axiom can be understood as: if there exists an appeal which is
a valid proof of x, then (logic.provable-witness x axioms thms atbl) is such
an appeal.

Ding! Now I get it.

This witnessing stuff is published in other ACL publications, noteably:

• Structured Theory Development for a Mechanized Logic, M. Kaufmann and J Moore, Journal of Automated Reasoning 26, no. 2 (2001), pp. 161-203.

But I can't fit those in my tiny brain.

Thanks, Jared, for explaining it at my speed!

Here's hoping I can turn this new knowledge into code that converts N3Rules to ACL2 and/or Milawa's format. N3Rules covers RDF, RDFs, and, I think, OWL2-RL and some parts of RIF. Roughly what stuff FuXi covers.

I'm somewhat hopeful that the rest of N3 is just quoting. That's the intuition that got me looking into ACL2 and Milawa again after working on some TAG stuff using N3Logic to encode ABLP logic. Last time I tried turning N3 {} terms in to lisp quote expressions was when looking at IKL as a semantic framework for N3. I didn't like the results that time; I'm not sure why I expect it to be different this time, but somehow I do...

Another question that's keeping me up at night lately: is there a way to fit level-breakers such as log:uri (or name and denotation, if not wtr from KIF) in the Milawa architecture somehow?