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Problems Solvable in Polynomial

Space

Alice Lewis

May 6, 2002

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Polynomial Space Problems

Allow Turing machine M to use polynomial

amount o space in input, regar!less o

amount o time nee!e!

"onsi!er !eterministic as well as non#

!eterministic

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Polynomial#space#boun!e!

Turing Machine

$inite

"ontrol

### input w#####

n cells

######cells ever use!#########

n cells

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Polynomial Space Problems

polynomial p%n& given input w o length

n, the TM never visits more than p%n& cells

o its tape

'y Theorem ()*2, stu!ie! previously, we

may assume that the tape is semi#ininite

an! the TM never moves let o the irst cell

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PS an! +PS

PS class o polynomial space languages

-.nclu!es L%M& or some polynomial#space#

boun!e!, !eterministic TM M

+PS class o non#!eterministic

polynomial space languages

-.nclu!es L%M& or some non#!eterministicpolynomial#space#boun!e! TM M)

"learly, PS is containe! in +PS

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To Prove/ PS+PS

Theorem **)/

-. M is a polynomial#space#boun!e! TM %1TM

or +TM&, an! p%n& is its polynomial spaceboun!, then there is a constant c such that i M

accepts its input w o length n, it !oes so within

c*p%n&moves

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To Prove/ PS+PS

Theorem **)3

-. L is a language in PS %or +PS&, then L is

accepte! by a polynomial#space#boun!e!!eterministic %non!eterministic& TM that halts

ater ma4ing at most c5%n&moves, or some

polynomial 5%n& an! constant c *

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To Prove/ PS+PS

The proo o P +P is !iicult to imagine7

PS +PS is easy

Proo involves

-Simulation o +TM that has a polynomial space

boun! p%n& with a 1TM with polynomial space

boun! 8%p2%n&&-9ses recursive unction reach!escribe! on

page 3:6 o te;t

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To Prove/ PS+PS

Theorem **)

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>ecursive

PS+PS

co#+P

+P

P