CS 3240 – Chapter 9
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Turing Machines
 From Alan Turing, 1936
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Turns out to be the LAST machine
 The only machine you'll ever need
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The Church-Turing Thesis
 All algorithms have a Turing Machine equivalent
▪ Some TMs are not algorithms, however
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We have shown that anbncn is not context free
 By the Pumping Lemma
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What if we add an extra stack to a PDA?
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input, pop1/pop2, push1/push2
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an is accepted by an FA
anbn is accepted by a PDA
anbncn is accepted by a 2PDA
What about anbncndn?
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An FA is a machine with no auxiliary memory
 A PDA is like an FA but with unlimited, restricted
memory
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 LIFO access
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A Turing Machine is like an FA but with unlimited,
unrestricted memory
 We can move to any memory cell
 We can both read and write there
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Language
Machine
Grammar
Regular
Finite Automaton
Regular Expression,
Regular Grammar
Context-Free
Pushdown Automaton
Context-Free
Grammar
Recursively
Enumerable
Turing Machine
Unrestricted PhraseStructure Grammar
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Control Unit (Deterministic)
 states (1 start state; 1 or more or halt states)
 transitions: input,output,move (L or R)
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I/O Tape (2-way, infinite)
 pre-populated with (finite) input
 output left on tape after processing
 special blank symbol ☐
▪ not part of input alphabet
▪ used as a separator and/or a boundary
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Contents left on tape is ignored
Typically has only one accepting state
 with no out-edges
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Any string that causes the TM to halt in an
accepting state is in the TM’s language
 otherwise it is not in the language
 (an explicit reject state may also be used)
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Pseudocode:
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Replace the first a with an X
Move right to first b, replace with Y
Move to right first c, replace with Z
Move left to right-most X
Repeat
When no more a’s, skip over all Y’s and Z’s
Halt and accept or reject
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aabbcc
Xabbcc
Xabbcc
XaYbcc
XaYbcc
XaYbZc
XaYbZc
XaYbZc
XaYbZc
XaYbZc
XXYbZc
XXYbZc
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XXYYZc
XXYYZc
XXYYZZ
XXYYZZ
XXYYZZ
XXYYZZ
XXYYZZ
XXYYZZ
XXYYZZ
XXYYZZ
XXYYZZ☐
XXYYZZ☐☐ (accept)
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q0aabbcc
Xq1abbcc
Xaq1bbcc
XaYq2bcc
XaYbq2cc
XaYq3bZc
Xaq3YbZc
Xq3aYbZc
q3XaYbZc
Xq0aYbZc
XXq1YbZc
XXYq1bZc
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XXYYq2Zc
XXYYZq2c
XXYYq3ZZ
XXYq3YZZ
XXq3YYZZ
Xq3XYYZZ
XXq0YYZZ
XXYq4YZZ
XXYYq4ZZ
XXYYZq4Z
XXYYZZq4☐
XXYYZZ☐q5☐ (accept)
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A sequence of configurations in a TM leading
to a halt state
 i.e., it doesn’t crash
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If the TM is an acceptor, it explicitly accepts or
rejects
If the TM represents some function, f, it
leaves f(x) on the tape, for each valid input x
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What would be a reasonable strategy?
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The initial configuration of the tape is the
input
What’s left on the tape is the output
A model of a function
Exercising such a TM constitutes a
computation
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Suppose f: D→R is a function from domain D
to range R
If there is a TM that transforms every d in D to
f(d)
 That is, it reads d from the tape and leaves f(d) on
the tape…
 Then f is Turing-computable
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This is the formal definition of an algorithm
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Must first decide how to encode numbers on a
TM tape
Then we must separate the numbers, so we
can distinguish between them
Plan:
 represent the numbers in unary notation
 then move the 1’s from one number to the other!
▪ i.e., just remove the separator and move the 1’s
together
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11101111
11101111
11101111
11101111
11111111
11111111
11111111
11111111
11111111_
1111111_
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Using unary notation, as usual
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Trace ababb
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TMs typically have one start and one or two
halt states (but can have many):
 one if the TM is a function
 two if the TM is an acceptor
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You can “call” a TM as a routine by setting up
appropriate “linkage”
 return to the desired state in the calling machine
 make sure the tape workspace is in an acceptable
configuration
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Tape configuration:
 start: [x]0[y]0
 finish: [x]0[y]0[xy]
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First, modify copy to accommodate a 0delimiters:
 0[y]0… → 0[y]0…[y]
 and finish by positioning at left of y
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Then, mark a 1 in x with a, copy y to end, and
repeat; then restore x’s a’s to 1’s
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See Example 9.11 for an explanation of Comparer
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T = (Q, ∑, Γ, δ, q0, H)
Q = set of states
∑ = input alphabet
Γ = tape alphabet
 includes ∑ and blank symbol (☐)
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q0 = start state
H = one or more halt states
 each typically represents a different meaning
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δ(q0,1) = (q0,1,R)
δ(q0,0) = (q1,1,R)
δ(q1,1) = (q1,1,R)
δ(q1,☐) = (q2,☐,L)
δ(q2,1) = (qh,☐,R)
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δ
q0
q1
q2
☐
0
1
(q1,1,R)
(q0,1,R)
(q1,1,R)
(q2,☐,L)
(qh,☐,R)
δ(q0,1) = (q0,1,R)
δ(q0,0) = (q1,1,R)
δ(q1,1) = (q1,1,R)
δ(q1,☐) = (q2,☐,L)
δ(q2,1) = (qh,☐,R)
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q0,1,q0,1,R
q0,0,q1,1,R
q1,1,q1,1,R
q1,B,q2,B,L
q2,1,qh,B,R
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(B = ☐; Use in Program 4)
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Input:
 A TM
▪ quintuple format: state1,input,state2,output,direction
 An input string (initial tape contents)
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Output:
 A trace of the actions of the machine
▪ with the current state positioned as the read-write head
 Final state and tape contents
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TMs can make multiple passes over the data
 in multiple directions
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And can process an arbitrary amount of
auxiliary data in addition to the original input
They can give more meaningful output
 vs. just yes/no
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Interesting fact:
 TMs came first!
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TMs suffer from a vulnerability that FAs and
PDAs do not:
They are not guaranteed to halt
 They may hang (loop forever)!
 Depends on the nature of the computation
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More on this later
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“Any computation that can be carried out by
mechanical means can be performed by some TM”
1. Anything that can be done on any digital computer
can be done by a TM
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 it’s the “lowest-level” programming language
2.
No one has yet found a solvable problem for which a
TM cannot be written
3.
No alternative computation model invented is more
powerful then a TM
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
The Turing Machine is the model for most
programming as we know it
 Imperative Programming
 Defines a machine architecture
 Characterized by reading and writing memory
 Instructions and data share the same memory
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There are other models of computation
 Lambda Calculus (Functional paradigm; CS 4450)
 Post Systems, Markov Systems, Structured Prog…
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