Space complexity
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Space complexity. [AB 4]. Input/Work/Output TM. Configurations (First try). The recorded state of a Turing machine at a specific time. How many distinct configurations may a Turing machine that uses s cells have?. |  | T. |  | n. . |  | s. | Q |. n. . s. . T. . . .

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Space complexity

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Space complexity

Space complexity

[AB 4]


Input work output tm

Input/Work/Output TM


Configurations first try

Configurations (First try)

The recorded state of a Turing machine at a specific time

  • How many distinctconfigurationsmay a Turing machine that uses s cells have?

||T

||n

||s

|Q|

n

s

T


Configurations

Configurations

  • The input stays fixed (Read only tape)

  • The output (or the output tape head) does not affect

  • the next transitions (Right only, write only tape).

||T

||n

||s

|Q|

n

s

T


Configurations1

Configurations

For a given input string x  {0,1}n

||s

|Q|

n

s


S pace complexity

Space complexity

Def: The space complexity of a Turing Machine T on input x is the maximal number of tape cells used throughout the computation.


Space complexity1

Space-Complexity


Space vs time

Space vs. Time


The configuration graph for a given input x

The Configuration graph for a given input x.

Gx=(V,E)

V – All possible configurations for the given input x.

E – An edge between configurations c and c’,

if there exists a legal  step from c to c’.


Pspace exp

PSPACE  EXP

Proof: A deterministic run that halts must avoid repeating a configuration 

its running time is bounded from above by the number of configurations the machine has,

which, for a PSPACE machine, is at most exponential


Name the class

Name the Class


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