lacaranian / surreal Goto Github PK

There should be robust conversion functions between surreal numbers and the other numerical types that lie within the field. These include:

• Integers
• Rationals
• Reals

The code is currently sitting in a monolithic file without any modularity. This should be changed to a module which does not export utility functions.

Find a way to represent infinite sets of surreals in the left or right set of a surreal, allowing for representation of infinite ordinals and infinitesimals, and numbers such as 1/3.

Division of surreal numbers is a monster. The left set and right set of the result of division are defined in terms of one another, which has the potential to require an infinite sequence of surreals to provide a valid result. The formula for the division of two surreal numbers (restricting z_L to positive options) is:

x = y / z = y * {0 , (1+(z_R-z)_x_L)/z_R , (1+(z_L-z)_x_R)/z_L | (1+(z_L-z)_x_L)/z_L , (1+(z_R-z)_x_R)/z_R }

An example: 1/3 is equal to {0|} / {2|}, which produces {0 , (1-x_R)/2 | (1-x_L)/2} (null left or right sets render their "blueprint" unusable, so it won't contribute new elements). This produces {0 , (1-x_R)/2 | 1/2 , (1-x_L)/2}, which then allows {0 , 1/4 , (1-x_R)/2 | 1/2 , (1-x_L)/2}, and so on, until the infinite sequences become {0 , 1/4 , 5/16 ... | 1/2 , 3/8 , 11/32 ...}. This is a correct representation of 1/3, but takes an infinite amount of memory to represent in the current paradigm.

The evaluation speed for simple arithmetic operations is dismal. This is due primarily to the recursive structure of a surreal, which causes any recursively defined operation to explode as the number of operations increases. I need to ensure that the canonical form of a surreal is preferred and the evaluation of operations that would typically involve large thunks when evaluated lazily is made strict.