Divmeq

Divmeq (derived from 'DIVide and branch, if Modulo EQuals 0') is a one-instruction Turing-complete esoteric programming language created by User:TheCanon2.

Divmeq was designed with the goal of being the simplest Turing-complete language. Divmeq's unusual properties help accomplish this.

• One instruction, with only two operands
• A single accumulator
• No self-modifying code
• No output

Thus, Divmeq should be able to be easily implemented in any reasonable language.

Commands

Divmeq has a single unbounded accumulator, x. The Divmeq command accepts two operands, A and B.

If x is divisible by A, divide x by A and jump to line B. Else, move 1 line forward.

Line numbers start at 0 and increment by 1 for each line in the program. The program halts if the line number is greater than the length of the program.

Divmeq accepts a numeric input at the beginning of a program, which is written to the accumulator. The accumulator is set to 1 if no input is given.

Examples

Hello, World!

0: 1/72 1
1: 72/101 2
2: 101/108 3
3: 1 4
4: 108/111 5
5: 111/44 6
6: 1.375 7
7: 32/87 8
8: 29/37 9
9: 37/38 10
10: 19/18 11
11: 1.08 12
12: 100/33 13

The ASCII values in 'Hello, World!' are in the pseudo-output.

Truth machine

0: 2 2
1: 1 1

This halts if the input is 0 and never halts if the input is 1.

XKCD Random Number

0: 0.25 1

Sets the accumulator to 4.

Digital root

0: 2 1
1: 512 1
2: 0.5 3

When given input $ 2^{a} $, returns $ 2^{dr_{10}(a)} $

A+B problem

0: 1 2
1: 0.5 2
2: 3 1

When given input $ 2^{a}3^{b} $, returns $ 2^{a+b} $. Prime factorisation is essential for storing more than one value in Divmeq's accumulator.

This program can be shortened to:

0: 1.5 0

A-B problem

0: 1 2
1: 2 2
2: 3 1

When given input $ 2^{a}3^{b} $, returns $ 2^{a-b} $.

This program can be shortened to:

0: 6 0

A*B problem

0: 3 2
1: 1 7
2: 7 2
3: 2/77 3
4: 11/2 4
5: 7/5 5
6: 1 0
7: 2 7
8: 5/2 8

When given input $ 2^{a}3^{b} $, returns $ 2^{ab} $. Does not work for larger values of $ a $ due to errors caused by floating point precision.

Squaring

0: 2/15 0
1: 5/2 1
2: 3 4
3: 1 9
4: 7 4
5: 2/77 5
6: 11/2 6
7: 7/5 7
8: 1 2
9: 2 9
10: 5/2 10

When given input $ 2^{a} $, returns $ 2^{a^{2}} $. Does not work for larger values of $ a $ due to errors caused by floating point precision.

Computational class

Divmeq is Turing-complete for it can simulate any section 14.2 single-register Minsky machine.

0: 1/n 1 Multiply r by n

0: n A Check if r is divisible by n, and if so, divide by n and jump to A

However, the nature of quines in Divmeq is uncertain, thus the Turing-completeness of Divmeq may be disproven.

Implementations

The following Python script is an interpreter.

prgm = [[]]
prgm.pop(0)
while True:
    line = input('%s: ' % len(prgm)) # Displays line numbers automatically
    if line != "eof":
        line = list(line.split(" "))
        prgm.append([eval(line[0]), int(line[1])])
    else:
        break
init = input('>i ') # Grabs user input
if init == "":
    x = 1
else: x = float(init)
# Executes the program
def divmeq(A, B):
    global x
    global inst
    if x % A < 1e-16 or x % A > (A - 1e-16): # 1e-16 may need to be adjusted
        x /= A
        x = round(x, 0) # Fixes errors caused by float division
        inst = B
    else: inst += 1
inst = 0
while inst < len(prgm):
    divmeq(prgm[inst][0], prgm[inst][1])
    print(x, end=" ")

This interpreter uses the symbol eof to mark the end of a program.

Any subsequent parameters after A and B are ignored, allowing for comments. This interpreter also prints the value of the accumulator after every instruction for ease of programming.