Automating Mathematics

Working journal

Differentiable Function Combinators

Basic functions and Combinators (all implemented)

We build differentiable functions for our basic learning system using some basic ones and combinators:

  • compositions (in class)
  • sums and scalar products of differentiable functions. (done)
  • product of a real valued and a vector valued function - the subtlest case (done).
  • identity function (done).
  • projections on (V, W) (done)
  • inclusions to (V, W) (dome)
  • evaluation of a finite distribution at a point (done).
  • atomic distribution as a function of weight (done).
  • point-wise multiplication of a finite distribution by a given function (done).
  • sum of a set of functions, with even the set depending on argument (done).
    • this can be interpreted as the sum of a fixed set of functions, but with all but finitely many zero.
  • repeated squaring $k$ times, with $k=0$ and $k<0$ cases (done).
  • recursive definitions for families indexed by integers - generic given zero. Done (for vector spaces).

Derived from these

  • (optional) moves
  • (optional) pairings
  • linear combinations.

Convenience code

  • Have implicit conversion from
    • a type T implicitly depending on a linear structure on T to have methods ++ and *:
    • DiffbleFunction[V, V] to a class with a multiplication method **:

To Do:

  • Construct differentiable functions for Finite Distributions - atom, evaluate and point-wise product.