Space Complexity Costas Busch - LSU

for deterministic Turing machine: Space complexity for deterministic Turing machine: Maximum number of cells that the machine scans on any input string of size …… …… …… …… input string total space used Costas Busch - LSU

for non-deterministic Turing machine: Space complexity for non-deterministic Turing machine: Maximum number of cells that the machine scans on any input string of size in any computation path …… …… …… …… input string total space used Costas Busch - LSU

all deterministic decidable languages that use space all non-deterministic decidable languages that use space Costas Busch - LSU

Example linear space problem SAT: given a formula check in linear space each possible assignment to variables Costas Busch - LSU

An interesting problem: it is not known if it is in NP However, the problem is in Costas Busch - LSU

Savitch’s Theorem Costas Busch - LSU

Given NTM that decides in space Build DTM that decides in space Costas Busch - LSU

Assume NTM clears the tape when it aceepts and enters a special configuration Costas Busch - LSU

Does configuration yield in steps? Key idea: Does configuration yield in steps? Costas Busch - LSU

Recursion Costas Busch - LSU

recursion Possibilities for : Costas Busch - LSU

Info stored at each tree level: Run: Depth of search tree: Info stored at each tree level: Total space: Costas Busch - LSU

Checks also if space used by NTM is more than f(n) Difficulty: is not known Guess it: Checks also if space used by NTM is more than f(n) Costas Busch - LSU

Trivially: From Savitch theorem: Costas Busch - LSU

Costas Busch - LSU