Axiom

An axiomatic number system in python

Synopsis

We are given four "axiom" functions,

1. zero() returns a zero object
2. is_zero(obj) returns True if the given object is a zero
3. next(obj) returns the next object from the given object
4. prev(obj) returns the previous object from the given object

This is similar to how a stack works,

>>> from axiom import zero, is_zero, next, prev
>>> is_zero(zero())
True
>>> is_zero(next(zero()))
False
>>> is_zero(prev(next(zero())))
True

Nothing is less than than a zero object.

Objects are not necessarily immutable (e.g., a list). This constraint might be added later.

Arbitrary loops (e.g., while, recursion) are forbidden. Iterate over generators instead.

We have one general-purpose generator,

• compose(fn, arg) yields arg, fn(arg), fn(fn(arg)), etc.

Derived functions are grouped by depth from the axioms. For example, we group dist and add because they are derived from the axioms. We group eq and multiples because they are derived from dist and add (respectively).

There are exceptions, such as dividing by zero.

By convention, 0⁰ equals 1 and does not throw an exception.

Sequences

Some useful sequences may be generated,

• counting() yields 0 1 2 3 4...
• fib() yields 0 1 1 2 3 5 8 13...
• multiples(n) yields n 2n 3n 4n 5n...
• primes() yields 2 3 5 7 11 13 17 19...
• catalan() yields 1 1 2 5 14 42 132...
• fact() yields 1 1 2 6 24 120...
• powers(n) yields 1 n n² n³ n⁴ n⁵...
• pascal_column(k) yields kth column of Pascal's triangle
• pascal_row(n) yields nth row of Pascal's triangle