Re: Axiom musings ... runtime computation of first-class dependent types

Tim

In this series of videos Robert Harper gives an provocative take on 
these subjects:
https://www.youtube.com/watch?v=LE0SSLizYUI

At about 37 min into this video  he says: "what is true has unbounded 
quantifier complexity whereas any formalism is always relentlessly 
recursively innumerable" - Gödels theorem.

"there is a distinction between proof and truth"
(by formalism here I think he means formal logic?)

It looks to be like the programming style he is putting forward is 
dynamic typing? I cant find it now, in these videos, but I think he 
might have suggested that this style of programming is required to model 
all of mathematics (be foundational)? Even cubical type theory might not 
be foundational?

Do you think that dynamic types and/or the sort of self modifying 
programs you are suggesting are necessary to model all of mathematics?

These videos are 4 years old, do you know if this research has come to 
any conclusions on these issues?

MJB

On 29/05/2022 08:50, Tim Daly wrote:
> Well, SANE is certainly turning into a journey.
> 
> When we look at the idea of first-class dependent types
> as implemented in languages like Agda and Idris we find
> what I call "peano types". For example, Fin is a type
> of "finite numbers", usually defined as
> 
> data Fin : Nat -> Type where
>    FZ : Fin (S n)             -- the base case
>    FS : Fin n -> Fin (S n)    -- the recursive case
> 
> Fin 7 describes 7 numbers 0..6
> 
> This construction is popular because the compiler can
> compute that the type construct is total. Thus the
> type construct is its own proof.
> 
> A dependent type allows computation to occur during
> its construction. The canonical example is defining
> a list of length m, which depends on the value of m.
> Another list of length n is a separate type. An
> append function of these lists computes a new typed
> list of length (m + n).
> 
> The current approach, Agda, Idris, etc., is "bottom up"
> where restrictions are placed which will allow proofs
> in all contexts.
> 
> There are excellent reasons for such carefully
> styled dependent types.
> 
> Reading the literature you come across the famous
> Dan Friedman admonition "Recursion is not allowed",
> usually in reference to his Pie language [0]. While
> Pie does not allow it, he admits 2 kinds of recursion
> (pp 358-359).
> 
> The general case of first-class dependent types I'm
> considering is capable of much broader and more
> ill-founded behavior. This is deliberate because some
> of the mathematics in Axiom can only be proven correct
> under limited cases. These cases have to be constructed
> "on the fly" from more general dependent type definitions.
> 
> For example, a type might not be total in general but
> it may be possible to compute a bounded version of the
> type when used in context. This can only be decided at
> run time with the given parameters in the given context.
> In general, this means computing that "in this context
> with these bindings the result can be proven".
> 
> This general case occurs because a SANE type G is able to
> invoke all of lisp with an environment that contains the
> type G under construction as well as its environment
> which contains the program. This allows the type G to self
> reference and self modify based on context.
> 
> The general type G:List(G,REPL) in a context of a statement
> M:G := (append '(1 2 3) '(4 5 6 7)) as effectively typed
> M:List(7).
> 
> The SANE type G is neither sound nor complete but List(7) is.
> The game is to construct a contextually dependent type and
> the associated proof "on the fly".
> 
> An intuitive Gedankenexperiment is defining an arm for a
> general robot. One could have a 2 arm robot defined as
> Robot : TwoArm(Robot,Repl).
> 
> The problem, based on context, might generate each of the
> two arms separately, one with 7 degress of freedom to
> reach the whole of the workspace (in context) and a second
> 4 degree robot that can position and hold a part in the
> reach of the first arm. The same TwoArm type might resolve
> to a different configuration in a different context.
> 
> Self-reference is already a metamathematical problem
> (witness Godel's proof). Self-modification is, as Monty
> Python famously says, "right out". One might argue with
> Godel but contradicting Python is heresy.
> 
> The SANE game is to create the general case of first-class
> dependent types with such abilities and then find certain
> restrictions as necessary to try to construct the proof
> in a given context. This SOUNDs COMPLETEly crazy, of course.
> 
> One has to construct a lisp program "on the fly" in the
> dependent type context, prove it correct, and return the
> new type with the type-program and proof. Values, during the
> run time computation may have to be dynamically substituted
> in the type-program, re-running the proof with those values.
> 
> At best this ranges from the nearly impossible to horribly
> inefficient. We all know this can never work.
> 
> This seems necessary as Axiom's computer algebra algorithms
> were never developed with proof in mind. It seems necessary
> to start "from the top" and work downward to the instance
> of existing code, adding restrictive assumptions as needed.
> 
> The moral of the story is "Never give a lisper a REPL".
> 
> There are some who call me ... Tim [1]
> 
> 
> 
> 
> Amusing historical note: Code linters were discovered during an
> omphaloskepsis session.
> 
> [0] Friedman, Daniel P. and Christiansen, David Thrane
> "The Little Typer", MIT Press (2018) ISBN 978-0-262-53643-1
> 
> [1] https://www.youtube.com/watch?v=co3ygE6H7PU 
> <https://www.youtube.com/watch?v=co3ygE6H7PU>