Cornucopia

Cornucopia

Paradigm(s)	Functional
Designed by	User:Hakerh400
Appeared in	2026
Computational class	Unknown
Major implementations	Implemented
File extension(s)	.txt

Cornucopia is an esolang invented by User:Hakerh400 in 2026.

Overview

Functions operate on natural numbers. [1] There are only two built-in functions [2] [3] :

• succ(n) - returns the successor of n
• sub(n, m) - returns the difference n - m, or 0 if n < m

User can define any number of functions, but there must be a unary function named main that represents the entry point. [4] Functions that take no arguments are called constants.

If there exists a unique [5] assignment of functions that satisfy all the definitions (the program has a unique model), then the program is executed. Otherwise (the definitions are contradictory, or have multiple models) throw an error.

Examples

All examples are formally verified and each claim about programs is supported by a proof. If you want to submit a new program or function, please discuss it on the talk page before updating the article.

Identity function

main(a) := a

This program represents the identity function. [6] It has a unique model. [7]

Loop

main(a) := main(a)

This program has infinitely many models and therefore is not a valid program. [8] [9]

Loop with successor

main(a) := succ(main(a))

This program does not have a model. [10]

Another loop with successor

main(a) := main(succ(a))

This program has infinitely many models (constant function for each natural number). [11] [12]

Constant one

main(a) := 1
1 := sub(succ(1), 1)

This program defines constant 1 that is equal to number 1. [13] The main function ignores the argument and outputs 1. [14] This program has a unique model. [15] This definition of constant 1 is submitted by User:Blashyrkh.

Arithmetic

main(a) := a
0 := sub(0, 0)
1 := succ(0)
not(a) := sub(1, a)
bool(a) := not(not(a))
add(a, b) := sub(sub(sub(succ(succ(add(sub(a, 1), sub(b, 1)))), not(a)), not(b)), add(0, 0))
ite(a, b, c) := sub(sub(sub(succ(ite(a, sub(b, 1), sub(c, 1))), sub(bool(a), b)), sub(not(a), c)), ite(a, 0, 0))

This program has a unique model. [16] The main function is just the identity function. [17] The intention is to show how to implement arithmetical functions.

• Constant 0 has value 0. [18]
• Constant 1 has value 1. [19]
• Unary function not takes a number; if the number is 0, returns 1, otherwise returns 0. [20]
• Unary function bool takes a number; if the number is 0, returns 0, otherwise returns 1. [21]
• Binary function add represents addition. [22].
• Ternary function ite takes numbers a, b and c; if a is 0, returns c, otherwise returns b. [23] This function is submitted by User:PkmnQ.

Computational class

The requirement that there exists a unique assignment of functions satisfying the definitions is probably undecidable, so the language is maybe hypercomputable. It is also unknown whether the language is Turing complete.

External resources

• Implementation in Lean.