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lecture5 - P= CSci 5403 COMPLEXITY THEORY LECTURE V SPACE...

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1 COMPLEXITY THEORY CSci 5403 LECTURE V: SPACE COMPLEXITY AND MORE CENTRAL QUESTIONS P = DTIME(n c ) c N NP = NTIME(n c ) c N EXP = DTIME(2 n c ) c N NEXP = NTIME(2 n c ) c N coNP = { L | L ̄ 2 NP } Definition: A language B is C-complete if: 1. B C 2. Every A in C is poly-time reducible to B (i.e. B is C-hard) HARDEST PROBLEMS IN C Definition: Let M be a deterministic TM that halts on all inputs. The space complexity of M is the function ƒ : N N , where ƒ(n) is the rightmost work tape position that M reaches on any input of length n. Definition: Let M be a non-deterministic TM that halts on all inputs. The space complexity of M is the function ƒ : N N , where ƒ(n) is the rightmost work tape position that M reaches on any branch of its computation on any input of length n.

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2 { L | L is a language decided by a O(t(n)) space non-deterministic Turing Machine } Definition: SPACE(t(n)) = Definition: NSPACE(t(n)) = { L | L is a language decided by a O(t(n)) space deterministic Turing Machine } Theorem. 3SAT ˥ SPACE(n) SAVITCH’S THEOREM Theorem: For any function ƒ where ƒ(n) n NSPACE(ƒ(n)) SPACE(ƒ(n) 2 ) Proof: Let N be a non-deterministic TM with space complexity ƒ(n) Construct a deterministic machine M that simply tries each branch of N Since each branch of N uses space at most ƒ(n), then M uses space at most ƒ(n)
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lecture5 - P= CSci 5403 COMPLEXITY THEORY LECTURE V SPACE...

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