Computability and Complexity

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Computability and Complexity
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Computability and Complexity Andrei Bulatov
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What is a computation?
Alan Turing considered this question in 1936 He
argued that any computation can be done using
the following steps

• Concentrate on one part of the problem
  (for
  example, one piece of paper, one arithmetic
  operation, etc.)
• Depending on this, and present state of the
  computer
• - Modify this part of problem
• - Move to another part of problem
• - Change into a new state.
• Repeat this until finished.

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Implications

• If the computation can be described in a finite
  space, then there can
• can be only a finite number of different
  situations to deal with
• Therefore, there can be only a finite number of
  states and a
• finite number of symbols.
• But note that the computation might continue
  forever and never
• finish, and it might use (potentially)
  infinite storage space

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The Turing Machine
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Turing Machine Configuration
Tape contains a sequence (infinite to the
right) of symbols from an alphabet
? almost all of the members of this sequence
(except for a finite number) are the
blank symbol ? At the first step it
contains the input, i.e. the non-blank
symbols on the tape constitutes the input
Head reads one cell (one symbol on the tape)
at the first step it reads the leftmost
symbol on the tape
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Turing Machine Operation

• At each step of operation machine is in one of
  the states, say q (initially state is )
• The machine is reading one symbol on tape, say s
• To execute one step, the machine looks up the
  value of a transition function ?(q,s), which is,
  say, (q,s,m)
• The machine
• changes to state q
• overwrites the symbol s with
  s
• moves the head along the tape
  according to m
• (RRight, LLeft,
  SStatic).
• Execution stops when a final state is reached

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Turing Machine Definition
Definition A Turing Machine is a 6-tuple
where

• Q is a finite set of states
• ? is the input alphabet not containing the
  blank symbol ?
• ? is the tape alphabet, where ?? ? and ? ?
  ?
• ? is a transition function
• is the initial state
• F ? Q is a set of final states

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Question
What do the following machines do?
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TM Example
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TM Example
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TM Diagram
e
??S
0?01LS
c
a
10L
0?R
1?R
?0S
?1S
11R
01R
b
d
00R
10R
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Configurations
Computation a (possibly infinite) sequence
of configurations entered by a TM
beginning with an initial configuration
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Acceptors and Transducers
Acceptor machines have 2 final states

• Accept ( )
• Reject ( )

These machines can be used to solve decision
problems input x is accepted by T if T(x)
is finite and ends in the Accept state.
Language accepted set of inputs accepted by
T, denoted by L(T)
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A?
R?
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Transducer T computes (partial) function
f if computation T(x) halts in
a final state with f(x) on the output tape
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O
U
T
P
U
T
H?
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Turings Thesis
(Also called Churchs Hypothesis, Sequential
Computing Hypothesis)
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k-tape TM
Transition function
Initial configuration

• the first tape contains an input
• the other tapes contain only blank symbols

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Nondeterministic TM
We can relax the definition of a TM to allow more
than one possible transition for any pair ?q,s?
of state and symbol
This gives a nondeterministic TM (i.e. it has a
choice of actions)
Nondeterministic computation a
tree of possible configurations resulting from a
start configuration For machine NT and
input x this is denoted NT(x)
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Nondeterministic Acceptor
Computation path is any path (finite or
infinite) from the root of NT(x)
Accepting computation is a finite computation
path ending in the Accept state
Language accepted is the set of inputs x
having some accepting computation in
NT(x), denoted L(NT).
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Time
R
R
R
R
R
A
R
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Random Access Machines
Another model of computation is the Random
Access Machine, which has

• an infinite series of memory locations,
• where each can contain an arbitrary
  integer
• a processor which can perform basic
  operations, including
• addition, subtraction, integer division by 2,
  indirect addressing,
• conditional goto, and halt
• a finite program specifying a sequence of
  basic operations
• involving the contents of memory locations

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• READ 1
• STORE 1
• STORE 5
• READ 2
• STORE 2
• HALF
• STORE 3
• ADD 3
• SUB 2
• JZERO 14
• LOAD 4
• ADD 5
• STORE 4
• LOAD 5
• ADD 5
• STORE 5
• LOAD 3
• JZERO 20
• JUMP 5

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4
5
6
7
8
9
H?
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Simulations
The following machines can all be simulated by a
TM with one tape and alphabet 0,1

• TMs with tape infinite in both directions
• TMs with more than one tape
• TMs with 2-dimensional tape (or more!)
• TMs with several read/write heads on each tape
• Nondeterministic TMs
• RAM Machines

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Other Models
The following people also proposed models of
general computation

• Post (1936) Posts normal systems
• Kleene (1936) recursive functions
• Church (1941) lambda calculus
• Markov (1954) Markovs rewriting systems

All of these have been shown to be equivalent to
Turing Machines
see Machine Models and Simulations in
the Handbook of Theoretical Computer Science,
MIT Press, 1990