Re: [mizar] proof objects for mizar: already available?

[Josef Urban <josef.urban@gmail.com>]

On 10/1/11, Jesse Alama <jesse.alama@gmail.com> wrote:

> (b) Proofs objects are already available, in some sense, for mizar
>      proofs.  They do not give precisely what is wanted from the
>      concept of proof object, though:
>
>      * They are not always available.  With the current transformation
>        into a vanilla first-order format, and with current ATPs, for
>        less than half of mizar theorems from the library do we have
>        deductions.  This is clearly an important result for the
>        community, but not having deductions is a serious shortcoming.
>        We want proof objects for all mizar proofs.

The "ATP proof object completion" used for ATP cross-verification of
Mizar in http://www.springerlink.com/content/b7383x43l55p1183/ does
not rely on re-proving whole Mizar theorems.

It is based on re-proving the Mizar "by" and "from" steps, where the
ATP success rate is (so far) over 99% , and on ATP verification of the
Mizar structural natural deduction steps, where the success rate is
(so far) 100%.

>
>      * The calculi of the proofs that are found this way depend on the
>        theorem prover employed.  One finds proofs in the superposition
>        calculi, resolution calculi... The plurality of proof calculi is
>        welcome -- we want to view mizar proof objects through various
>        formal lenses -- but none of these is satisfactory because none
>        is a natural deduction proof in the style of mizar.  (To some
>        extent this problem can be overcome: an ATP could either do
>        search in a natural deduction setting, or emit natural
>        deductions by translating whatever calculus it uses into natural
>        deduction.  But proof search in natural deduction is generally
>        not as efficient as search in other calculi.  And the
>        translation from other calculi to a natural deduction is not
>        always clear.)  What is wanted from proof objects for mizar is a
>        sort of natural deduction proof that adheres more or less to the
>        mizar format, but which brings out all logical details.  It is
>        unpleasant and awkward to switch from natural deduction-style
>        mizar proofs to unnatural resolution deductions.

It is trivial to translate Prover9 proof objects to Mizar proofs, It
was done in 2003: https://github.com/JUrban/ott2miz . The translated
detailed proofs can be further post-processed by running the Mizar
native proof-improving ("irrelevant") utilities in a fixpoint loop
(this was added to Emacs mode on Freek's request in 2005).

>
>      * There is no assurance that the proofs discovered by an ATP
>        thanks to Josef's translations are the same as the mizar proofs
>        with which one started.  Genuinely new proofs can be (and are)
>        discovered.  An ATP might exploit a premise or combination of
>        premises in an unusual way that diverges from the input mizar
>        proof.  An even when the ATP-discovered proof is more or less
>        congruent to the mizar proof from which it came, because it is
>        expressed in a different formal calculus there might be some
>        uncertainty about whether we are looking at the same proof.

This again is more about using ATPs to find new big proofs than about
reproducing the (typically quite limited) "by" steps. But technically
you are right.

>
>      * By diverging from mizar's natural deduction format, one loses
>        the ability to carry out experiments and investigations that
>        require that one works with natural deductions.  Thus, one might
>        wish to investigate the notion of obviousness.  One might ask,
>        for example, what instances of which universal formulas were
>        used to carry out a particular by step.  One might wish to carry
>        out certain transformations of the the deduction (e.g., rewrite
>        a natural deduction by represent applications of definitions as
>        rules of inference, rather than as applications of the rule of
>        modus ponens).

I totally agree that having detailed proof objects for all of MML
would be a great resource for study and data-mining of mathematical
proofs. The ATP path to this turned out to be very cheap once I had
practically complete ATP export and strong ATP methods.

Doing it the hard way inside Mizar is certainly possible (I have
partially done it in 2000), but it will be a serious amount of work on
Mizar, and even if you do it, it will be in a constant danger of
becoming obsolete by later re-implementations of parts of Mizar. Bill
McCune told me in 2004 that the detailed Otter proof objects were so
much added code that it lead him to reimplement the whole thing as
Prover9. My experience from 2000 with the Mizar kernel was similar: it
was a large blow-up of the code, and a cleaner complete rewrite would
be needed.

Josef

>
> --
> Jesse Alama
> http://centria.di.fct.unl.pt/~alama/
>
>