Definition of empty alternations

[DivergentClouds]

README.md says

Finally, since there is no way to represent them syntactically, we don't define the behavior of empty alternations (although it seems logical to define an empty alternation as a function that fails on all input, since this is the identity element of function alternation).

however, it is possible to represent empty alternations syntactically as { | ... } where ... is any further code.

[crb233]

By empty alternation, I mean an alternation expression with no arguments. So your example wouldn't count since it has two arguments ` and.... I think the only reason there's syntax for an empty composition is because I defined the empty expression to represent a composition with no arguments (extending the idea thata bcomposes two arguments anda` composes just one).

Semantically, the empty alternation should be something that behaves as the identity element of the alternation operator (analogously to an empty sum being 0 or an empty product being 1). This is another reason your example wouldn't work. The empty alternation must be a function F that fails on all inputs, since F | x = x and x | F = x for all x. It's possible construct F like this:

0 { - 0 | }
F { 0 - }

So F is semantically equivalent to an empty alternation, but I wouldn't call it an empty alternation because it's syntactically very different.

Thinking about this issue when I first wrote this part of the README led me to design an alternative syntax for Unarian which has an elegant syntax for empty alternations. In this alternative syntax, [ a b c ] represents the composition of a, b, c, and { a b c } represents the alternation of a, b, c. Then [ ] is the empty composition and { } is the empty alternation. Using this syntax, we can write the code above like this:

0 { [ - 0 ] [ ] }
F [ 0 - ]