# Space complexity - PowerPoint PPT Presentation

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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

[AB 4]

### 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

• 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

### Configurations

For a given input string x  {0,1}n

||s

|Q|

n

s

### 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.

### 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

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