## A Turing Machine Simulator

### March 27, 2009

A turing machine is a hypothetical computer, invented in 1936 by the British mathematician Alan Turing, that can evaluate any computable function. Though hypothetical, turing machines play a large role in computer science. Many variants of turing machines have been devised; the purpose of this exercise is to build a simulator for a particular type of turing machine.

The hardware of our turing machine consists of a read/write head and an infinite tape divided into cells that passes across the read/write head. Each cell on the tape contains a symbol, which we take as an ascii character; there is a special *blank* symbol that is present in all cells which are not yet written. The read/write head can see one cell at a time, read its contents, and write a new symbol (possibly the *blank* symbol) to the cell.

The software of our turing machine consists of a table encoding a transition function and a state register that maintains the current state. In our machine, states are numbered with non-negative integers; the machine always starts in state zero, and halts if it enters a negative-numbered state. The program being executed by the turing machine is encoded in the transition function, which takes two inputs, the current state and the symbol in the current cell, and then writes a new symbol (or the same symbol, or the special symbol *blank*) in the current cell, then either moves the read/write head one cell to the left or one cell to the right or remains in the same place, then resets the state to a new state. The transition function is thus encoded in a table of 5-tuples *q1* *s1* s*2* *d* *q2* where *q1* is the current state, *s1* is the symbol currently under the read/write head, *s2* is the symbol to be written, *d* is the direction to move the tape, and *q2* is the next state to enter.

As an example, consider the tape `_ _ _ [1] 1 1 + 1 1 1 1 1 _ _ _`

. This is our notation for a tape that contains, in the middle of an infinite number of blanks, the nine symbols consisting of three one’s, a plus sign, and five one’s, with the read/write head positioned over the first one; the tape can be considered to model the problem of adding the two numbers three and five. The program

` 0 1 1 R 0`

0 + 1 R 1

1 1 1 R 1

1 _ _ L 2

2 1 _ L -1

adds the two numbers and writes `_ _ _ 1 1 1 1 1 1 1 [1] _ _ _`

as its output.

Write a function that takes a turing-machine program and a tape and simulates the action of a turing machine, writing the final tape as output.

Here is my try

There should be a preview option.

Here’s the second try,http://codepad.org/pb3pJtxl

I decided to interpret the Turing Machine as an object, so I made a class for it which has methods that simulate actually operating with a physical machine.

I’ve included the entire class, and some test code that runs the function you posted.

Cheers!

class Turing_Machine():

def __init__(self, input_tape=[], function_matrix=[]):

“””Constructs a functional Turing Machine with the optional inputs.”””

self.set_input_tape(input_tape)

self.set_transition_matrix(function_matrix)

print

print “Turing Machine created as: ”

print

print self

direction = ”

# hardware

head_position = 0

input_tape = []

# software

state_register = 0

“””a list of lists with indices as states demarking a list with form

[symbol(being read 1 or 0 or _), symbol(to write; 1 or 0 or _), d(direction to move tape; R or L),

state(to enter; 0 or integer, -1 to end)]”””

transition_function = []

# output

output_tape = []

def run(self):

“””Runs the Turing Machine. (Doesn’t print)”””

self._place_head()

self._actual_compute()

self.output_tape.insert(0,’_’) # for looks

self.reset()

def set_transition_matrix(self, matrix):

“””Takes in a list of 5-length lists, puts them into the format that we need,

into self.transition_function. Could make this move intuitive to input?”””

for function in matrix:

self._add_transition_line(function)

def set_input_tape(self, list):

“””Takes in a list (starting with an arbitrary number of _) to be the input to the program.

Could make this take an easier input (a string? space seperatd, null terminated?)”””

self.input_tape = list

def clear_matrix(self):

“””Resets the transition function.”””

self.transition_function = []

def clear_input(self):

“””Resets the input_tape”””

self.input_tape = []

def reset(self):

“””Sets the machine to a naive state (but keeps function matrix and input tape)”””

self.state_register = 0

self.head_position = 0

def _place_head(self):

“””Places the head on the first non-blank index of the input_tape”””

for i in range(len(self.input_tape)):

if self.input_tape[i] != ‘_’:

self.head_position = i

self.input_offset = i #how many blanks we had to skip (for output formatting)

break

def _add_transition_line(self, line):

“””Adds a transition statement to the function matrix self.transition_function. One line at a time. Takes in an array of length 5.”””

if line[0] >= len(self.transition_function): #checks to see if state number is already in the function marrix

self.transition_function.append([]) #if not, add a state array

self.transition_function[line[0]].append(line[1:]) #append the rest of the function to the matrix (minus the state)

def _actual_compute(self):

“””Recursively computes the output of the transition table applied to the input, writes that to the output array.”””

for function in self.transition_function[self.state_register]:

if self.input_tape[self.head_position] == function[0]:

if self.direction == ‘L’: #if we’re moving backwards, the output tape already has that position, so don’t append anything to the output tape array

self.output_tape[self.head_position-self.input_offset] = function[1]

else:

self.output_tape.append(function[1]) #if we’re moving right, we need to append to the output tape array

self.state_register = function[3] #sets the next state

self.direction = function[2] # sets the next direction

if self.direction == ‘R’:

self.head_position += 1 #moves the head to the right if dir = ‘R’

else: #otherwise, it’s left, so move left

self.head_position -+ 1

if self.state_register >= 0: #if the state isn’t negative, compute again

self._actual_compute()

def __str__(self):

“””Basic printing of the Turing Machine. Could be improved.”””

return “program: ” + str(self.transition_function) + “\ninput: ” + str(self.input_tape) + “\noutput: ” + str(self.output_tape) + “\n”

def _test_machine():

_ = ‘_’

plus = ‘+’

R = ‘R’

L = ‘L’

matrix = [ [0, 1, 1, R, 0], [0, plus, 1, R, 1], [1, 1, 1, R, 1], [1, _, _, L, 2], [2,1, _, L, -1] ]

input_tape = [_,_,_,1,1,1,’+’,1,1,1,1,1,_,_,_]

tm = Turing_Machine(input_tape, matrix)

tm.run()

print tm

https://github.com/ftt/programming-praxis/blob/master/20090327-a-turing-machine-simulator/turing.py