I got the tongue-in-cheek aspect but Axiom's formal logic basis is
important.
In the 1980s Expert Systems were going to take over the world. We tried to
prove our version
correct using the Boyer-Moore theorem proving technology. We failed
miserably, partly
because the technology was not up to the task but mostly because we were
not up to the
task. Proving programs correct, at the time, was pretty much based on
reasoning about
pre- and -post-conditions. That's the "assembly language" of proof
techniques these days.
There was a complete mismatch with the rule-based expert systems technology.
Because Axiom is so close to the math and Coq is so close to Axiom it seems
there
is hope that this effort will make a lot more progress.
Coq is astonishingly good at proving things. It has a very small trusted
kernel based on
logic. But it has things like a 'tactics' language that enables automation
of proof steps
to the point where a proof might be just a single tactic, a 1-line
statement. This really
simplifies what is usually a hair-tearing struggle. Lean (a Microsoft/CMU)
system is
the basis of a course I'm taking at CMU and it seems even more
user-affectionate.
Even better, Coq has dependent types so it is possible to state, for
instance, the size
of a matrix. Since types can be user-defined we can create Axiom-specific
types in Coq
that mirror the Category structure.Even sweeter is that the syntax is very
close to Spad.
For example, Spad
gcd(x:NNI,y:NNI):NNI ==
zero? x => y
gcd(y rem x, x)
and Coq
Fixpoint gcd x y :=
match x with
| O => y
| S x' => gcd (y mod (S x')) (S 'x)
where S is the Peano Successor function. Coq can prove this directly and
automatically infer that the type signature is
gcd(x:nat,y:nat):nat
Coq proofs are now integrated into the build. The next commit will likely
include complete
automation of the proof, calling Coq during the build process with GCD as
the first case.
Once that happens the game seems to be creating the Axiom type tower in Coq.
Axiom's NNI becomes Coq's 'nat' type. This enables writing Axiom types
directly
in Coq proofs.
The ideal case would be to modify Axiom's compiler to accept the Coq/ML
syntax
but that's a large effort.
Another, more likely effort, would be to implement Coq's trusted kernel in
Axiom.
That would permit the compiler and interpreter to prove running code.
It is time to formalize computational mathematics and put computer algebra
on
a firm mathematical basis. The technolgy is here (although the funding is
not).
On Wed, Jan 25, 2017 at 3:18 AM, Gabriel Dos Reis <
g...@integrable-solutions.net> wrote:
> I was largely poking, tongue in cheek, at the earlier syllogism:
>
> >> The point is that Lisp has a formal logic basis and, as Spad is really
> >> just a domain specific language implemented in Lisp then Spad shares
> >> the formal logic basis.
>
> -- Gaby
>
> On Thu, Jan 12, 2017 at 10:35 PM, Tim Daly <axiom...@gmail.com> wrote:
>
>> > There were implementations of C in Lisp. So C shares that formal logic
>> basis, or that it was discovered?
>>
>> According to Wadler, C is an "invented" language, not a "discovered"
>> language.
>>
>> Wadler (https://www.youtube.com/watch?v=aeRVdYN6fE8) at around 40
>> minutes
>> shows the logician's view versus the computer science view of various
>> theories.
>> He makes a distinction betwen invented and discovered langauges.
>>
>> (begin quote of Wadler)
>>
>> Every interesting logic has a corresponding language feature:
>>
>> Natural Deduction (Gentzen) <==> Typed Lambda Calculus (Church)
>> Type Schemes (Hindley) <==> ML Type System (Milner)
>> System F (Girard) <==> Polymorphic Lambda Calculus (Reynolds)
>> Modal Logic (Lewis) <==> Monads (state, exceptions) (Kleisli, Moggi)
>> Classical-Intuitionistic Embedding (Godel) <==> Continuation Passing
>> (Reynolds)
>> Linear Logic (Girard) <==> Session Types (Honda)
>>
>> Languages which were "discovered" (based on theory):
>>
>> Lisp (McCarthy)
>> Iswim (Landin)
>> Scheme (Steele and Sussman)
>> ML (Milner, Gordon, Wadsworth)
>> Haskell (Hudak, Hughes, Peyton Jones, Wadler)
>> O'Caml (Leroy)
>> Erlang (Armstrong, Virding, Williams)
>> Scala (Odersky)
>> F# (Syme)
>> Clojure (Hickey)
>> Elm (Czaplicki)
>>
>> At about minute 43 we hear:
>>
>> "Not all of these languages are based on lambda calculus ...
>> but there is a core that is "discovered". Now most of you work
>> in languages like Java, C++, or Python and those languages I will
>> claim are not discovered; they are "invented". Looking at them
>> you can tell they are invented. So this is my invitation to you
>> to work in languages that are "discovered".
>>
>> Somebody asked before about "dependent types". It turns out that
>> when you extend into dependent types you now get languages that
>> have "for all" and "there exists" that corresponds to a feature
>> called "dependent types". So this means that you can encode very
>> sophisticated proofs, pretty much everything you can name as a
>> program. And so the standard way to get a computer to check a
>> proof for you and to help you with that proof is to represent that
>> as a program in typed lambda calculus.
>>
>> All of these systems are based on a greater or lesser extent on
>> that idea:
>>
>> Automath (de Bruijn)
>> Type Theory (Martin Lof)
>> Mizar (Trybulec)
>> ML/LCF (Milner, Gordon, Wadsworth)
>> NuPrl (COnstable)
>> HOL (Gordon, Melham)
>> Coq (Huet, Coquand)
>> Isabelle (Paulson)
>> Epigram (McBride, McKinna)
>> Agda (Norell)
>>
>> (end quote)
>>
>> Axiom is using Coq for its proof platform because Axiom needs
>> dependent types (e.g. specifying matrix sizes by parameters).
>>
>> In addition, Coq is capable of generating a program from a
>> proof and the plan is to reshape the Spad solution to more
>> closely mirror the proof-generated code. Also, of course, we need
>> to remove any errors in the Spad code found during proofs.
>>
>> It seems there must be an isomorphism between Coq and Spad.
>>
>> At the moment it seems that Coq's 'nat' matches Axiom's
>> NonNegativeInteger. Coq also has a 'Group' type which needs
>> to be matched with the Axiom type. The idea is to find many
>> isomorphisms of primitive types which will generate lemmas
>> that can be used to prove more complex code.
>>
>> Given that Axiom has an abstract algebra scaffold it seems likely
>> that a reasonable subset can be proven (modulo the fact that there
>> are arguable differences from the abstract algebra type hierarchy).
>>
>> Currently Coq is run during the Axiom build to check any proofs
>> of Spad code that are included. ACL2 is also run during the build
>> to check any proofs of Lisp code that are included.
>>
>> I'm taking a course at Carnegie-Mellon this semester using Lean
>> (a Coq offshoot) in order to try to make some working examples
>> of proving Spad code correct.
>>
>>
>> On Thu, Jan 12, 2017 at 10:14 PM, Gabriel Dos Reis <
>> g...@integrable-solutions.net> wrote:
>>
>>> There were implementations of C in Lisp. So C shares that formal logic
>>> basis, or that it was discovered?
>>>
>>> On Wed, Jan 11, 2017 at 8:17 PM, Tim Daly <axiom...@gmail.com> wrote:
>>>
>>>> I'm making progress on proving Axiom correct both at the Spad level and
>>>> the Lisp level. One interesting talk by Phillip Wadler on "Propositions
>>>> as
>>>> Types", a very entertaining talk, is here:
>>>> https://www.youtube.com/watch?v=IOiZatlZtGU
>>>>
>>>> He makes the interesting point late in the talk that some languages are
>>>> "discovered" based on fundamental logic principles (e.g.Lisp) and others
>>>> are "invented" with no formal basis (e.g. C). As he says, "you can tell
>>>> whether your language is discovered or invented".
>>>>
>>>> The point is that Lisp has a formal logic basis and, as Spad is really
>>>> just a domain specific language implemented in Lisp then Spad shares
>>>> the formal logic basis.
>>>>
>>>>
>>>>
>>>> _______________________________________________
>>>> Axiom-developer mailing list
>>>> Axiom-developer@nongnu.org
>>>> https://lists.nongnu.org/mailman/listinfo/axiom-developer
>>>>
>>>>
>>>
>>
>
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