Re: [Axiom-developer] Axiom Sane musings (SEL4)

[Veer Singh]

If you have looked at ATS lang and rejected it for your purpose
then just ignore rest of mail.

ATS programming language (http://www.ats-lang.org/)
For more information (https://github.com/githwxi/ATS-Postiats/wiki)

I think ATS is worth a look at least once if you are looking for
lang/system that produces efficient code, can encode and prove theorems,
can do programming with theorem proving, have dependent types and
many more rich types for example viewtypes.

ATS compiles to C.
It also compile to clojure, _javascript_ and may be
some more, with some restriction.

In ATS one can return a proof along with the computation result.
Proof is consumed/checked only during compilation, and actual result is
produced when code is executed.

For example:
 val x : int = 2                 // normal binding
 val (pf | x)  =  afun (...)    //

First case is example of normal binding found in most langs.

In second case "pf" is bound to proof return by "afun" and
"x" is bound to the result of computation, where "|" separtes them.
(During runtime there is no proof object , it is used only during compilation)

To see the power of ATS,small part of factorial example
taken from ATS book is described below .

Factorial of natural number is first "encoded" as relation
in ATS using "dataprop":

//---------------------------------------------------------------

dataprop FACT (int,int) =
  // Base case
  | FACT_bas (0,1) of ()

  // Inductive case
  | {n:nat}{r1,r:int}
    FACT_ind (n,r) of (FACT (n-1,r1), MUL (n,r1,r))

//-----------------------------------------------------------------


-----------------------------------------------------------------
SOME COMMENTS:
-----------------------------------------------------------------
Two proof constructor are FACT_bas and FACT_ind

{n:nat} means n is a natural number

{r1,r: int} means r1 and r are integers

Type of FACT is (int,int) -> prop  

MUL (int,int,int) is also defined via dataprop and
  MUL (a,b,c) encodes a * b = c

----------------------------------------------------------------------


Function to compute factorial is defined as follows:

//----------------------------------------------------------------------
fun fact {n: nat} .<n>.
(m: int (n)):<> [r: int] (FACT (n,r) | int (r) ) =
  if m = 0
  then (FACT_bas () | 1 ) // base case
  else let
    val (pf1 | r1) = fact (m - 1)           // pf1: FACT(m-1,r1)
    val (pfmul | r ) = imul2 (m,r1)         // pf2: MUL(m,r1,r)
    in (FACT_ind (pf1,pfmul) | r) end       // inductive case

//----------------------------------------------------------------------



----------------------------------------------------------------------
SOME COMMENTS:
----------------------------------------------------------------------
.<n>. is a termination metric

{n: nat} means for all natural numbers n

(m: int (n)) means m is value of type int (n)

imul2 is function which also returns a proof of mul and its result

[r: int] (FACT (n,r) | int (r) ) means there exist a value
   of type integer r such that proof FACT (n,r) holds.

Type sig of "fact" is :
{n:nat} (int (n)) -> [r: int] (FACT (n,r) | int (r))

Finally I can read above as:
 for any natural number n , the fact (n) produces
 the factorial of n , say r such that prop FACT (n,r) holds .

--------------------------------------------------------------------------

On Sun, Sep 22, 2019 at 11:30 PM Tim Daly <address@hidden> wrote:

Of particular interest is clarity.

I've been working with LEAN. The code is in C++ and is very
clever. For instance, there is a beautiful macro embedded in
data structures to perform reference counting.

Unfortunately, I can't reverse-engineer the logic rules that are
embedded in the C++ code.

HOL, on the other hand, seems to have a very clear connection
betwen the code and the logic rules.

In a proof system it is vital that the logic rules and their
implementation is "obviously correct" and transparent.

I have not yet looked at Idris so I can't comment on that.

Tim


On 9/21/19, Tim Daly <address@hidden> wrote:
> Hmm. The problem to be solved involves several parts.
> Idris is of interest in PART 6, 7, and 8 below.
>
> PART 1: We have the domain
>
> We have GCD in NAT (axiom: NonNegativeInteger or NNI)
>
> NonNegativeInteger is what Axiom calls a "Domain", which means
> that it contains signatures, such as
>
>   quo : (%,%) -> %
>   rem : (%,%) -> %
>   gcd : (%,%) -> %
>
> which says that gcd takes 2 NonNegativeIntegers (NATs) and
> returns a NonNegativeInteger (NAT).
>
> The NonNegativeInteger domain also includes information about
> how its elements are represented.
>
> PART 2: We have an implementation of gcd in the domain
>
> The NNI domain contains an implementation of gcd:
>
> gcd(x,y) ==
>   zero? x => y
>   gcd(y rem x,x)
>
> PART 3: We have a way to inherit things for the domain
>
> The NNI domain inherits properties from what Axiom
> (unfortunately) calls Categories. Categories provide
> additional signatures and default implementations.
>
> PART 4: We have the FUNDAMENTAL PROBLEM
>
> The PROBLEM to be solved is that we want to prove
> that the above code for gcd is correct.
>
> Of course, the first question is "correct with respect to..."
>
> PART 5: We need a specification language
>
> There needs to be a specification of the gcd function.
> What are the properties it should fulfill?
> What are the invariants?
> What are the preconditions?
> What are the postconditions?
>
> Some parts of the specification will be inherited.
>
> Which means we need a language for specification.
>
> PART 6: We need a theorem language
>
> Given a specification, what theorems are available?
> Some theorems are inherited from the categories,
> usually as axioms.
>
> Some theorems and axioms are directly stated in
> the NNI domain.
>
> Some lemmas need to be added to the domain to help
> the proof process.
>
> Which means we need a language for theorems.
>
> PART 7: We need a proof engine
>
> Now that we have an implementation, a specification,
> a collection of theorems and pre- and post-conditions,
> lemmas, and invariants we need a proof.
>
> Which engine will we use for the proof?
> What syntax does it require?
> Does it provide a verifier to re-check proofs?
>
> PART 8: We need to prove many GCD algorithms
>
> Axiom contains 22 signatures for gcd. For example,
> it contains a gcd for polynomials. The above machinery
> needs to support proofs in those domains also.
>
> PART 9: LOOP
>
> GOTO part 4 above, pick a new function, and repeat.
>
>
> PART 10: ISSUES
>
> PART 10a: "Down to the metal"
>
> THere are a pile of "side issues". I'm re-implementing Axiom
> using Common Lisp CLOS. THe defclass macro in CLOS
> creates new Common Lisp types. This allows using the types
> for type-checking (currently looking at bi-directional checking
> algorithms)
>
> Axiom sits on Common Lisp. There is a question of using a
> "trusted core". I'm looking into Milawa
> https://www.cl.cam.ac.uk/~mom22/soundness.pdf
> with a deeply layered design.
>
> I'm also looking at SEL4 on ARM
> https://ts.data61.csiro.au/publications/nicta_full_text/3783.pdf
> which is a trustworthy operating system.
>
> I wrote a paper on the semantics of the Intel instruction set:
> Daly, Timothy Intel Instruction Semantics Generator SEI/CERT Research
> Report, March 2012
> http://daly.axiom-developer.org/TimothyDaly_files/publications/sei/intel/intel.pdf
> so SEL4 on Intel is interesting.
>
>
> PART 10b: Dependent type theory
>
> Dependent types are undecidable. Axiom contains several
> heuristics to resolve types at runtime. The heuristic type
> algorithm needs to be explicit and declarative.
>
> PART 10c: Size
>
> Axiom contains about 10,000 algorithms in 1100 categories,
> domains, and packages. This is going to take a while.
>
> PART 10d: Mathematics
>
> Many of the algorithms are partial. Many are PhD thesis
> work (and hard to understand). Many are ad hoc and have
> no mathematical specification.
>
> PART 10e: Time
>
> The target delivery date is April, 2023.
> There is much to do.
>
> Tim
>
>
> On 9/21/19, Henri Tuhola <address@hidden> wrote:
>> Idris has a way to present equalities like this:
>>
>> addition_unit : (a:Nat) -> (a + 0) = a
>> addition_s : (a,b:Nat) -> (a + S b) = S (a + b)
>> add_commutative : (a,b:Nat) -> (a + b = b + a)
>>
>> They can be used to prove more things:
>>
>> try_out : (x,y:Nat) -> ((x + 0) + y) = y + x
>> try_out x y = rewrite addition_unit x in add_commutative x y
>>
>> It's rewriting the left _expression_ to right _expression_, though you can
>> easily flip the direction. For clarity I show these few dissections:
>>
>> try_out x y = ?a1
>> a1 : plus (plus x 0) y = plus y x
>>
>> try_out x y = rewrite addition_unit x in ?a2
>> a2 : plus x y = plus y x
>>
>> Idris has this feature called "Elaborator reflection". It allows you
>> to describe automated tactics for writing proofs/programs.
>> The "getGoal" and "getEnv" allow you to examine types in the context:
>>
>> getGoal : Elab (TTName, TT)
>> getEnv : Elab (List (TTName, Binder TT))
>>
>> The elaborator reflection also allows accessing the term rewriting. I
>> suppose that's all you need in order to write a program that
>> simplifies equations inside the type context?
>>
>> -- Henri Tuhola
>>
>> On Sat, 21 Sep 2019 at 11:50, Martin Baker <address@hidden> wrote:
>>>
>>> I'm a fan of both Axiom and Idris. I think my ideal would be Axiom
>>> mathematics build on top of the Idris type system.
>>>
>>> The Axiom type system was incredibly advanced for its time but I suspect
>>> the Idris type system has finally caught up and overtaken it? Correct me
>>> if I'm wrong but I think the Axiom type system does not have the
>>> following capabilities that Idris does:
>>>
>>> * Enforcement of pure functions.
>>> * Ability to flag a function as total as opposed to partial (automatic
>>> in some cases).
>>> * Universes (types of types hierarchy).
>>>
>>> I'm no expert but I would have guessed these things would be almost
>>> indispensable for proving Axiom correct?
>>>
>>> Also Idris makes it far more practical to use these things, I don't
>>> think Axiom can implement category theory constructs like monads. Also,
>>> although both have dependent types, Axiom does not use them for say,
>>> preventing the addition of a 2D vector to a 3D vector. In Idris this is
>>> more likely to be compile time error than a runtime error, I know there
>>> are theoretical limits to this but I think Idris has capabilities to
>>> make this practical in more cases.
>>>
>>> I don't pretend I know how an Idris type system could be used with Axiom
>>> in practice. For instance I think the proofs Henri is talking about are
>>> equalities in the type system (propositions as types). So how would
>>> these equations relate to equations acted on by equation solvers (which
>>> might be an element of some equation type). Could there be some way to
>>> lift equations into the type system and back?
>>>
>>> Sorry if I'm confusing things here but I just have an intuition that
>>> there is something even more powerful here if all this could be put
>>> together.
>>>
>>> Martin
>>>
>>> On 21/09/2019 04:28, Tim Daly wrote:
>>> > Axiom has type information everywhere. It is strongly
>>> > dependently typed. So give a Polynomial type, which
>>> > Axiom has, over a Ring or Field, such as
>>> > Polynomial(Integer) or Polynomial(Fraction(Integer))
>>> > we can use theorems from the Ring while proving
>>> > properties of Polynomials.
>>>
>>> _______________________________________________
>>> Axiom-developer mailing list
>>> address@hidden
>>> https://lists.nongnu.org/mailman/listinfo/axiom-developer
>>
>

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