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1. Turing Machines
(At last!)

2. Designing Universal Computational
Devices Was Not The Only
Contribution from Alan Turing…
Enter the year 1940:
•The world is at war
•Nazi Germany has succeeded in conquering most of west
Europe
•Britain is under siege
•British supply lines are threaten by German
•Germany used the Enigma Code, considered unbreakable
•Alan Turing led a group of scientist that broke the
enigma code
Enter the year 2006: The National Security Agency (NSA)
needs people! (www.nsa.gov)

3. Background
•Automata accepts regular languages only
For example, {an
bn
: n = 0, 1, …} is not regular
•Pushdown Automata accepts context-free languages only
For example, {an
bn
cn
: w  *} is not context-free
•We can easily write an algorithm (in C) recognizing if a
sequence of characters have the form an
bn
cn
or not

4. The Key Question
•Is it possible to design a formal model of a computational
device that capture the capabilities of any algorithm?
Alan Turing, 1940’s: Yes!

5. Basic Idea
Turing decomposed operations in the device as follows:
•Computational steps erase a symbol observed
by the pencil and write a new one in its place
•The decision about which symbol should be
written and which will be observed next depend on:
•A pencil will be “observing” symbols from a paper
1. The current symbol being observed
2. The “state of mind” of the device
This device is called a Turing Machine
•The pencil can move left and right

6. Turing Machine Looks Similar to
Finite Automata
Common:
•A “state of mind” sounds like the collection of states
•A “pencil” sounds like the pointer for the next word
Main differences:
•Turing machines can write on the “paper”
•Turing machines can backup and read symbols again
This turns out to be a major difference!

7. In Turing’s Own Words
... which can be made to do the work of any special-purpose
machine, that is to say to carry out any piece of computing,
if a tape bearing suitable "instructions" is inserted into it.

8. Expressivity of Turing Machines (TMs)
•TMs can simulate any data structure
•TMs can simulate major components of imperative
languages: sequence, branching and loop
•TMs can control branching and loops
•Several extensions have been attempted: adding multiple
heads (pencils), tapes, non-determinism, etc. But we can
show that TMs can simulate each of these extensions

9. Church-Turing Thesis
Every computer algorithm can be
implemented as a Turing machine
Therefore, C, C++, Prolog, Lisp, Small talk, and Java programs
can be simulated in Turing machines
Definition: a programming language is Turing-complete if it is
equivalent to a Turing machine.

10. Informal Description
…
a1 a2
Control
Tape
cell
head
The head:
•Reads the symbol from the cell it is
pointing to, and
•Either:
•Writes a new symbol in the cell, or
•Moves one cell to the left or right

11. Informal Description (II)
•New cells can be added to the right of the tape as needed
(similar to RAM memory)
•These new cells contain the blank symbol, 
•The tape is bounded to the left

12. Preliminaries to Formal Description
 denotes the alphabet of characters in language as usual.
 denotes the set of symbols that can be written on tape
•It contains  and all symbols in 
• Transitions can be described by (Case I):
((s,a),(q,b,)) If the machine is in state s and the
current cell has an a then jump to state q
and write b in the current cell and moves
head to right

13. Preliminaries to Formal Description (II)
• Transitions can be described by (Case II):
((s,a),(q,b,)) If the machine is in state s and the
current cell has an a then jump to state q
and write b in the current cell and moves
head to left

14. Formal Definition
Definition. A Turing machine is a 7-tuple (Q, , , , q0, qaccept, qreject),
where:
•Q is a set of states
 is a set of symbols (the alphabet)
 is a set of symbols that can be written in tape,    and   
•q0  Q is the initial state
• qaccept is the accepting state
•qreject is the rejecting state, qreject  qaccept

15. Formal Definition (II)
  is a collection of transitions defined by the function:
: (Q  {qaccept, qreject })    Q    {, }

16. My First Turing Machine
Construct a Turing machine such that:
• input: a substring over the alphabet  ={a,b}, the
substring contains at least 1 character b
• the header is pointing to the first cell in tape
• When the Turing machine halts, the header must be
pointing to the first occurrence of b in the input
a b a  a b a 

17. Discussion: Are you sure that such a simple model can simulate
my C++ program for computing the Minimum Spanning Tree?

18. The “Eraser” TM
Construct a Turing machine that receives as input a
substring of a’s and replace each a for a blank space, 
a a a     

19. Configuration for Turing Machines
Configuration:
•The current state, q
•The symbol pointed by the head, h
•The string left of the head, LS
•The string right of the head, RS
(q,h,LS,RS)

20. Configuration for Turing Machines (2)
Instead of writing: (q,h,LS,RS), we write:LSqhRS
Examples:
- Initial configuration with 3 a’s for the eraser TM: eq0aaa
- Accepting configuration starting with eq0aaa: qaccept
If q = qaccept, then LSqhRS is called an accepting configuration
If q = qreject, then LSqhRS is called a rejecting configuration

21. Computation in Turing Machines
•A configuration C1 = LS1q1h1RS1 yields a configuration C2
= LS2h2q2RS2 in one step, written C1 C2, if:
There is a transition: (q1, h1) = (q2,h2,)

C yields C’, written C * C’:

There is a sequence of configurations in one step:
C C0 C1 … C’




• The other case is analogous for (q1, h1) = (q2,h2,)
• Example: eq0aaa yields eq0

22. Two Forms of Computability
• Definition. A word w is accepted by a Turing machine M if
M yields an accepting configuration starting from the
configuration eq0w
• Definition. The language L(M) recognized by a Turing
machine M:
L(M) = {w | w accepted by M}
•Definition. A language L is Turing-recognizable if there is a
Turing machine that recognizes L
• Definition. A language L is decidable if there is a Turing
machine M such that:
L(M) = L
For any word w L, M yields a rejecting configuration
starting from the configuration eq0w

23. Constructing Complex Turing Machines
Idea:
Use a modular approach similar to the one use to
construct complex programs
Example (programming):
Given two nn matrixes A and B to compute ABT
we:
1. Construct a procedure to compute BT
,
transverse(B,C)
2. Construct a procedure to multiply two matrices,
multiply(B,C,D)
3. Call transverse(B,C) and then multiply(A,C,D). D
contains the answer

24. Graphical Conventions
We will use a convention similar to a flow chart to indicate how
to put together Turing machines (TMs).
Convention. Let A and B be 2 TMs, and a be a character in 
then:
A
a
B
Means that:
1. If A halts in a configuration of the form w1qacceptaw2
where w1or w2 are any words
2. Then B starts with the configuration w1q0aw2
A;
if a then B

25. Graphical Conventions (2)
Means that:
• If M halts in a configuration of the form w1qacceptaw2,
then A starts with the configuration w1q0aw2
• If M halts in a configuration of the form w1 qacceptw2
with   a, then B starts with the configuration
w1q0w2
A
a
B
M
a M;
If a then A
else B

26. Graphical Conventions (3)
Means that:
• If the initial configuration has the form w1q0w2,
then A starts with this configuration if   a, where
s is the initial state in A
• If A halts in a configuration of the form w1qacceptw2
with   a, then A starts with the configuration
A
a while (not a) do
A
a

27. Basic Turing Machines
a Denotes “write a on the current cell” (a is a
character in )
 Denotes “move to the right of the current
cell” (the book uses “R” instead of “”)
 Denotes “move to the left of the current
cell” (the book uses “L” instead of “”)

28. Other Simple Turing Machines
b
a
Machine What it does
“if current cell has an a, then
replace it with a b”
a
 “if current cell has a
character a different than ,
then move once to the right
and write that a”

29. The  Turing Machine



•What it does?
–
–
Move to the right until if finds 

30. The  TuringMachine



–
–
Draw diagram for 

31.  and 
– –


–


–
Move to the left until the first non-blank space cell
is found
Move to the right until the first non-blank space
cell is found

32. The Conventions
• 2 simple rules:
• Rule # 1:
• Rule # 2:
R
“move to the right until you find “
Note: first check. Then move (think of a “while”)
“Never forget Rule # 1”
(Mr Miogi, The Karate Kid)
R
“move to the right until you find a symbol other than “

33. Example: Deciding the language (ab)*
a b
 a b
a b
 a a
“yes”
“no”