FRACTRAN is a Turing-complete esoteric programming language invented by the mathematician John Conway. A FRACTRAN program is an ordered list of positive fractions together with an initial positive integer input n. The program is run by updating the integer n as follows:
Contents
- Understanding a FRACTRAN program
- Addition
- Multiplication
- Subtraction and division
- Conways prime algorithm
- Other examples
- References
- for the first fraction f in the list for which nf is an integer, replace n by nf
- repeat this rule until no fraction in the list produces an integer when multiplied by n, then halt.
In The Book of Numbers, John Conway and Richard Guy gave a formula for primes in FRACTRAN:
Starting with n=2, this FRACTRAN program generates the following sequence of integers:
2, 15, 825, 725, 1925, 2275, 425, 390, 330, 290, 770, ... (sequence A007542 in the OEIS)After 2, this sequence contains the following powers of 2:
which are the prime powers of 2.
Understanding a FRACTRAN program
A FRACTRAN program can be seen as a type of register machine where the registers are stored in prime exponents in the argument n.
Using Gödel numbering, a positive integer n can encode an arbitrary number of arbitrarily large positive integer variables. The value of each variable is encoded as the exponent of a prime number in the prime factorization of the integer. For example, the integer
represents a register state in which one variable (which we will call v2) holds the value 2 and two other variables (v3 and v5) hold the value 1. All other variables hold the value 0.
A FRACTRAN program is an ordered list of positive fractions. Each fraction represents an instruction that tests one or more variables, represented by the prime factors of its denominator. For example:
tests v2 and v5. If
Since the FRACTRAN program is just a list of fractions, these test-decrement-increment instructions are the only allowed instructions in the FRACTRAN language. In addition the following restrictions apply:
Addition
The simplest FRACTRAN program is a single instruction such as
This program can be represented as a (very simple) algorithm as follows:
Given an initial input of the form
Multiplication
We can create a "multiplier" by "looping" through the "adder". In order to do this we need to introduce states into our algorithm. This algorithm will take a number
State B is a loop that adds v3 to v5 and also moves v3 to v7, and state A is an outer control loop that repeats the loop in state B v2 times. State A also restores the value of v3 from v7 after the loop in state B has completed.
We can implement states using new variables as state indicators. The state indicators for state B will be v11 and v13. Note that we require two state control indicators for one loop; a primary flag (v11) and a secondary flag (v13). Because each indicator is consumed whenever it is tested, we need a secondary indicator to say "continue in the current state"; this secondary indicator is swapped back to the primary indicator in the next instruction, and the loop continues.
Adding FRACTRAN state indicators and instructions to the multiplication algorithm table, we have:
When we write out the FRACTRAN instructions, we must put the state A instructions last, because state A has no state indicators - it is the default state if no state indicators are set. So as a FRACTRAN program, the multiplier becomes:
With input 2a3b this program produces output 5ab.
Subtraction and division
In a similar way, we can create a FRACTRAN "subtracter", and repeated subtractions allow us to create a "quotient and remainder" algorithm as follows:
Writing out the FRACTRAN program, we have:
and input 2n3d11 produces output 5q7r where n = qd + r and 0 ≤ r < d.
Conway's prime algorithm
Conway's prime generating algorithm above is essentially a quotient and remainder algorithm within two loops. Given input of the form
Other examples
The following FRACTRAN program:
calculates the Hamming weight H(a) of the binary expansion of a i.e. the number of 1s in the binary expansion of a. Given input 2a, its output is 13H(a). The program can be analysed as follows: