The Greek philosopher Empedocles famously postulated that the four elements comprising all matter are Fire (F), Water (W), Earth (E), and Air (A). More recently, the video game “Doodle God” has turned this theory into a fun way to waste time: by repeatedly combining these $4$ elements, the player can construct novel entities. For example:
Dust = Earth + Air = EA
Energy = Air + Fire = AF
Storm = Energy + Air = (AF)A
Swamp = Earth + Water = EW
Life = Energy + Swamp = (AF)(EW)
Here are some of the more complex advanced entities that can be created:
Quicksilver = W(F(A(FE)))
Glass = F(A(A(FE)))
Lizard = ((A(FE))((AF)(EW)))(EW)
Note that parentheses must be used to disambiguate these combinations, because “combination” is a non-associative operation, i.e. A(BC) is not necessarily the same as (AB)C.
QUESTION: Can we come up with a Doodle-God-esque theory of matter that starts with fewer starting elements? Can we come up with one that uses only one starting element to generate all other entities, but still uses roughly intuitive/guessable rules of combination? How would you generate Fire, Water, Earth, and Air from your proposed “first element” and what would the intermediate steps be?
(Here’s an analogous concept in SKI combinator calculus, for you computer science nerds.)