hwsol1.4

# hwsol1.4 - Example solution 1.4 Avery Musbach February 9...

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Example solution: 1.4 Avery Musbach February 9, 2010 (a) We claim that a language L { 0 , 1 } * is in roNTIME ( f ( n )) i ff there exists a polynomial p : N N , a natural number c N and a f ( n c )-time TM M (which I shall call the verifier for L ) such that for every x { 0 , 1 } * , x L ⇐⇒ ∃ u { 0 , 1 } f ( p ( | x | )) M ( x, u ). If x L and u { 0 , 1 } f ( p ( | x | )) satisfy M ( x, u ), then we shall call u a certificate for x (with respect to the language L and machine M ). Suppose L is decided by a NDTM N that runs in time f ( n c ) (for some c N ). For every x L , there is a sequence of nondeterministic choices that makes N reach q accept on input x . We can use this sequence as a certificate for x . This certificate has length f ( | x | c ) and can be verified in O ( f ( n c )) time by a deterministic Turing machine, which simulates the action of N using these nondeterministic choices. Conversely, if L meets the criterion in question, then we describe a f ( n c )- time NDTM N that decides L . On input x , it uses the ability to make

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• Spring '08
• Sturtivant,C
• Computational complexity theory, Quantification, Universal quantification, Existential quantification

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