Rrr

Turing-complete programming language

Overview

Rrr is a programming language to describe a Turing machine using a binary alphabet consisting of symbols, R and r.

A program in Rrr is a sequence of instructions for a universal machine. Each command represents a transition, which includes a starting state, a conditional read symbol, a write symbol, a move action, and a next state.

How it works

The universal Turing machine reads a description of a Turing machine encoded in Rrr and simulates it.

In each state q_i with the current symbol s on the tape, the universal machine searches for a command with the starting state q_i and the read symbol s. If found, the symbol on the tape is replaced by the write symbol from the command, the reading head moves left or right according to the move action in the command, and the current state is updated to the next state q_i+1 from the command.

Syntax

The syntax of Rrr consists of quintuplets of elements encoded as unary numbers of symbols (digits) r. The elements are separated by the symbol R:

The quintuplets are as well separated by the symbol R:

Semantic

The semantic of Rrr corresponds to the Turing machine quintuples encoded via unary numbers:

Encoded	Value	State	Read/write symbol	Move action
r	0	q0	alphabet[0]	left
rr	1	q1	alphabet[1]	right
rrr	2	q2	alphabet[2]	invalid
r × N	N - 1	qN-1	alphabet[N - 1]	invalid for N > 2

A concrete alphabet is provided to the simulating machine as a parameter.

Interpreter

There is a simple JavaScript interpreter for Rrr:

cd interpreter
npm install
npm test

const tm = new TM(code, input, alphabet)
const out = tm.compute()

Examples

All source code can be found in examples.

Generating a sequence of Rrr words

Read empty cells continuously, write R followed by two rs and repeat.

The program can be described by the flowchart:

It is represented by the following quintuplets:

(q₀, ε, R, >, q₁)
(q₁, ε, r, >, q₂)
(q₂, ε, r, >, q₀)

The corresponding Rrr code reads:

rRrRrrRrrRrrRrrRrRrrrRrrRrrrRrrr
RrRrrrRrrRr

The program alphabet is { ε, R, r }

Adding two unary numbers

Encode numbers using r digits and use R as separators. The number 1 is encoded as r, 2 as rr, 3 as rrr, and so on.

For instance, the tape for add 2 and 3 is initialized as follows:

rrRrrr

To carry out the calculation, shift the second number one cell to the left.

The program can be described by the flowchart:

It is represented by the following quintuplets:

(q₀, r, r, >, q₀)
(q₀, R, r, >, q₁)
(q₁, r, r, >, q₁)
(q₁, ε, ε, <, q₂)
(q₂, r, ε, >, q₃)

The corresponding Rrr code reads:

rRrrrRrrrRrrRrRrRrrRrrrRrrRrrRrr
RrrrRrrrRrrRrrRrrRrRrRrRrrrRrrrR
rrrRrRrrRrrrr

The program alphabet is { ε, R, r }

Copier

The copier program effectively duplicates the given input, appending it to the end of the existing tape. This doubling process repeats the input string.

For example, if the input is r the output will be rr. If the input is rrr, the output will be rrrrrr, and so on.

1. Start at the beginning of the input string.
2. Mark the first symbol.
3. Move to the end of the input string and print it after an empty cell, which acts as a separator.
4. Move back to the marked symbol, unmark it, and then mark the next symbol in the input.
5. Repeat this process until all symbols have been copied.
6. Finally, clear the last cell of the new string and rewrite the separator with a symbol.

The program can be described by the flowchart:

It is represented by the following quintuplets:

(q₀,r,R,>,q₁)
(q₁,r,r,>,q₁)
(q₁,ε,ε,>,q₂)
(q₂,r,r,>,q₂)
(q₂,ε,r,<,q₃)
(q₃,r,r,<,q₃)
(q₃,ε,ε,<,q₄)
(q₄,r,r,<,q₄)
(q₄,R,r,>,q₀)
(q₀,ε,r,>,q₅)
(q₅,r,r,>,q₅)
(q₅,ε,ε,<,q₆)
(q₆,r,ε,<,q₇)

The corresponding Rrr code reads:

rRrrrRrrRrrRrrRrrRrrrRrrrRrrRrrR
rrRrRrRrrRrrrRrrrRrrrRrrrRrrRrrr
RrrrRrRrrrRrRrrrrRrrrrRrrrRrrrRr
RrrrrRrrrrRrRrRrRrrrrrRrrrrrRrrr
RrrrRrRrrrrrRrrrrrRrrRrrrRrrRrRr
RrRrrrRrrRrrrrrrRrrrrrrRrrrRrrrR
rrRrrrrrrRrrrrrrRrRrRrRrrrrrrrRr
rrrrrrRrrrRrRrRrrrrrrrr

The program alphabet is { ε, R, r }

Palindromes

A palindrome is a sequence of symbols that reads the same backward as forward, such as rRr or RRrRR.

1. Read a symbol from the input.
2. Move to the last symbol of the input.
3. If the last symbol matches the one you read in Step 1, clear the cell at the end, move back to the beginning, and start over.
4. If the last symbol is different from the one you read in Step 1, clear the entire input and reject by printing R.
5. If the input is empty (all symbols have been matched and cleared), accept by printing r.

It is represented by the following quintuplets:

(q₀,r,ε,>,q₁)
(q₁,R,R,>,q₁)
(q₁,r,r,>,q₁)
(q₁,ε,ε,<,q₂)
(q₂,ε,ε,<,q₀)
(q₂,r,ε,<,q₃)
(q₃,R,R,<,q₃)
(q₃,r,r,<,q₃)
(q₃,ε,ε,>,q₀)
(q₀,R,ε,>,q₄)
(q₄,R,R,>,q₄)
(q₄,r,r,>,q₄)
(q₄,ε,ε,<,q₅)
(q₅,ε,ε,<,q₀)
(q₅,R,ε,<,q₃)
(q₀,ε,r,>,q₇)
(q₂,R,ε,<,q₆)
(q₅,r,ε,<,q₆)
(q₆,R,ε,<,q₆)
(q₆,r,ε,<,q₆)
(q₆,ε,R,>,q₇)

The corresponding Rrr code reads:

rRrrrRrRrrRrrRrrRrrRrrRrrRrrRrrR
rrrRrrrRrrRrrRrrRrRrRrRrrrRrrrRr
RrRrRrRrrrRrrrRrRrRrrrrRrrrrRrrR
rrRrRrrrrRrrrrRrrrRrrrRrRrrrrRrr
rrRrRrRrrRrRrRrrRrRrrRrrrrrRrrrr
rRrrRrrRrrRrrrrrRrrrrrRrrrRrrrRr
rRrrrrrRrrrrrRrRrRrRrrrrrrRrrrrr
rRrRrRrRrRrrrrrrRrrRrRrRrrrrRrRr
RrrrRrrRrrrrrrrrRrrrRrrRrRrRrrrr
rrrRrrrrrrRrrrRrRrRrrrrrrrRrrrrr
rrRrrRrRrRrrrrrrrRrrrrrrrRrrrRrR
rRrrrrrrrRrrrrrrrRrRrrRrrRrrrrrr
rr

The program alphabet is { ε, R, r }

Hello World

It simply prints Hello World.

The program can be described by the flowchart:

It is represented by the following quintuplets:

(q₀,ε,H,>,q₁)
(q₁,ε,e,>,q₂)
(q₂,ε,l,>,q₃)
(q₃,ε,l,>,q₄)
(q₄,ε,o,>,q₅)
(q₅,ε,ε,>,q₆)
(q₆,ε,W,>,q₇)
(q₇,ε,o,>,q₈)
(q₈,ε,r,>,q₉)
(q₉,ε,l,>,q₁₀)
(q₁₀,ε,d,>,q₁₁)

The corresponding Rrr code reads:

rRrRrrRrrRrrRrrRrRrrrRrrRrrrRrrr
RrRrrrrRrrRrrrrRrrrrRrRrrrrRrrRr
rrrrRrrrrrRrRrrrrrRrrRrrrrrrRrrr
rrrRrRrRrrRrrrrrrrRrrrrrrrRrRrrr
rrrRrrRrrrrrrrrRrrrrrrrrRrRrrrrr
RrrRrrrrrrrrrRrrrrrrrrrRrRrrrrrr
rRrrRrrrrrrrrrrRrrrrrrrrrrRrRrrr
rRrrRrrrrrrrrrrrRrrrrrrrrrrrRrRr
rrrrrrrRrrRrrrrrrrrrrrr

The program alphabet is { ε, H, e, l, o, W, r, d }

Universal Turing Machine in Rrr

Rrr is obviously Turing-complete as it corresponds one-to-one to the description of how a Turing machine works.

A universal machine for Rrr, that is, a program written in Rrr that can interpret any programs in Rrr.

The alphabet of the universal machine is { ε, R, r, +, #, * }

The programs for the machine are restricted to the alphabet { R, r } which requires some adaptions of the algorithms.

For instance, a single R must be added to the end of the input to denote the end of the input when adding two unary numbers. So instead of rrRrrr we must provide rrRrrrR.

The adapted program can be described by the flowchart:

The corresponding code reads:

rRrrRrrRrrRrRrRrRrrRrrRrrRrrRrrR
rrRrrRrrRrrRrRrRrRrrrRrrrRrrRrRr
rRrrrr

The given program is provided on the input of the universal machine together with the input of the program joined by a separator RR:

<program>RR<input>

The whole universal machine in Rrr reads as follows:

rRrrrRrrrRrrRrRrRrrRrrRrrRrrRrrR
rrrRrrrRrrRrRrrRrrRrrrrrRrRrrrRr
rrRrrRrrrrrRrrRrrrrRrrrrRrrrrrRr
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rrrrRrrrrrRrrRrrRrrRrrrrrRrrrrrR
rRrrrrrRrrRrrrrrrRrrrrrrRrRrrrrr
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rRrrRrrrrrrrrrrRrrrrrrrrrrRrRrrr
RrRrrrrrrrrRrrrrrrrrRrrrrrRrrrrr
RrrRrrrrrrrrrrrRrrrrrrrrrrrRrRrr
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