[Date Prev][Date Next] [Chronological] [Thread] [Top]

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

On 2011-10-09 11:56:29 +0000, Josef Urban said:


On 10/9/11, Jesse Alama <jesse.alama@gmail.com> wrote:

On 2011-10-01 09:26:45 +0000, Josef Urban said:


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%.


Nonetheless, the situation is not entirely satisfactory.  What we have
is a kind of hybrid proof: part is in the mizar proof formalism, and
the other part is in the proof formalism of whatever ATP is used.  From
a proof-theoretic perspective from a mizar proof we find a natural
deduction proof with lots of axioms (corresponding to its "by" steps).


The whole (completed) proofs are in one formalism: TPTP format with
assumptions (see the paper).
The TPTP notation for derivations is a strong start, certainly. Looking at the paper, I see that they have a richer structure than I thought they had. 




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.


I agree that it probably would be a rather complex project to have bona
fide proof objects for mizar.  In addition to the problem of exposing
"by"/"from" steps, there remains the problem of exporting a mizar proof
-- even taking for granted that its "by" and "from" steps can be filled
in some acceptable way, treating them as black boxes (or as axioms, in
a natural deduction/sequent calculus context) -- into a vanilla natural
deduction format.  I think there are lots of design problems here.  For
example, consider the text fragment:

set X = the set;
theorem X = X;

Do we translate this as the universal formula:

for X : set (X = the set --> X = X)

Or as the equation:

the set = the set


There are indeed various choice points when translating from Mizar to
pure first-order logic, but their default solutions have been provided
in MPTP since 2005. See
http://www.springerlink.com/content/kw9075426q26pu32/ . "set foo =
bar" is just a syntactic macro in Mizar, so already Mizar semantic
layer is "the set = the set", there is not really much of a choice
here. The choice comes when translating "the" (Hilbert's epsilon) to
first-order logic. This is not described in the paper above, because
"the" came later in Mizar (2009?). But its default translation is very
similar to translation of Faenkel terms (de-anonymization).


Well, semantically the fragment

set X = the set;
X = X
may simply be the equation "the set = the set", but it isn't the case that one can simply eliminate all 'set' statements from the MML by substituting the right-hand sides for free occurrences of the left-hand side. If we really were to have natural deductions for mizar proof objects, I'd prefer the approach of introducing a universal formula to deal with 'set' statements, because then one could have a label for the equation 

 X = the set
and obtain find-grained information about where this equation is needed. Simply uniformly plugging in the right-hand side for the left-hand side would, in general, produce unreadable proofs. I'd like ideal proof objects which don't yet exist to record this information. Working with "semantic mizar proofs" (for lack of a better term) -- ones where macro substitution is absent because it has already been carried out -- is (at least) one step too far, to my taste, since we've lost some information in the mizar text. 



To sum up: there is a default TPTP translation both of the Mizar
natural deduction proof format and of the Mizar extensions over
first-order logic, and both have been around and productively used for
a couple of years now.
From my point of view, this is just the first step. To my knowledge,
we still lack even one bona fide natural deduction proof -- consisting exclusively of axioms of set theory, equality, and introduction and elimination of connectives -- of any (non-trivial) mizar theorem. Would you agree, or am I missing something? 



But I'd be happy to see a continuation of the 2008 cross-verification
paper that would attempt ATP-based cross-verification of the full MML
(not just large random parts of it). Based on that (on the full proof
objects), it would be probably fairly easy to import MML into HOL
Light, Isabelle, Coq, ..., and that could in turn trigger further work
on communication between the libraries of these systems.
Portability of the library is certainly one ripe fruit of such an export. My own interest is in foundations. There are all sorts of experiments that I'd like to carry out on a library of natural deduction proofs of genuine mathematical results, which, as far as I can see, I can't carry out with the TPTP derivation notation. (I'm happy to concede that some of this is simply due to my ignorance of the details of TPTP derivations.) 

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