Gravity

Gravity was invented by Safalra (Stephen Morley) on 4th August 2005, based on a idea from Isaac Newton. Gravity is very different from any other existing programming language. Although its behavior is well-defined and deterministic, the evolution of its space is in general non-computable, due to the nature of the differential equations that govern it.

Introduction
The execution of a Gravity program consists of the simulation of the effects of gravity on a countably infinite number of point particles. The initial space configuration is an infinite two-dimensional grid of point particles positioned with one at every point (x,y), where x and y are integers, with all but a finite number of unit mass. When particles collide they are replaced according to a set of collision rules specified by the program. Input, output, and halting can happen only when collisions occur. It can be shown that a Turing machine cannot compute, in the general case, whether even a single collision will ever happen.

Examples
The language specification states that "comments are not permitted, but each program should ideally be accompanied by a mathematical paper offering a proof that the program behaves in the desired way." The examples below do not include proofs, but these can be found under "External resources".

Hello World program
The following code is a Gravity Hello, world! program. It outputs ASCII values corresponding to the phrase "Hello world", and then, assuming no input was provided, it halts.

(0,0) : 2 (1,1,1,1, 2) -> 3 :  72 (1,1,1,1, 3) ->  4 : 101 (1,1,1,1, 4) ->  5 : 108 (1,1,1,1, 5) ->  6 : 108 (1,1,1,1, 6) ->  7 : 111 (1,1,1,1, 7) ->  8 :  32 (1,1,1,1, 8) ->  9 : 119 (1,1,1,1, 9) -> 10 : 111 (1,1,1,1,10) -> 11 : 114 (1,1,1,1,11) -> 12 : 108 (1,1,1,1,12) ->  # : 100

Cat program
The following code is a cat program. The encoding used can be any encoding that maps characters to a series of 3s and 4s (for example, mapping characters to their Unicode positions, and then turning 0 and 1 bits into 3s and 4s respectively).

(0,0) : 2 (1,1,1,1,2) -> # : (1,1,1,1,3) -> # : 3 (1,1,1,1,4) -> # : 4

Digital root program
The following code is the first few lines of a digital root program. The program works with the standard ASCII encoding of the digits 0 to 9. The full program is 113 lines long.

(0,0) : 2 (1,1,1,1, 2)   ->  # : (1,1,1,1,48)    ->  # : 48 (1,1,1,1,49)    ->  # : 49 (1,1,1,1,50)    ->  # : 50 (1,1,1,1,51)    ->  # : 51 (1,1,1,1,52)    ->  # : 52 (1,1,1,1,53)    ->  # : 53 (1,1,1,1,54)    ->  # : 54 (1,1,1,1,55)    ->  # : 55 (1,1,1,1,56)    ->  # : 56 (1,1,1,1,57)    ->  # : 57 (1,1,1,1,48) 48 -> 48 : (1,1,1,1,48) 49 -> 49 :

External resources

 * Gravity (at Safalra's Website), with examples and proofs