Page 1 Lecture 27 207 Page 2 Lecture 27 207 Beyond NP Many complexity classes are worse, e.g. time 2 2 n , 2 2 2 n , … Others seem to be “worse” in a different sense, e.g., not in NP, but still exponential time. E.g., let Lp = “assignment y satis?es formula x”, ? P Then : SAT = { x | ?y ?x,y??L P } UNSAT = { x | ?y ?x,y??L P } QBF k = { x | ?y 1 ?y 2 ?y 3 … k y k ?x,y 1 …y k ??L P } QBF 8 = { x | ?y 1 ?y 2 ?y 3 … ?x,y 1 … ??L P } 208 Q Page 3 Lecture 27 207 Beyond NP Many complexity classes are worse, e.g. time 2 2 n , 2 2 2 n , … Others seem to be “worse” in a different sense, e.g., not in NP, but still exponential time. E.g., let Lp = “assignment y satis?es formula x”, ? P Then : SAT = { x | ?y ?x,y??L P } UNSAT = { x | ?y ?x,y??L P } QBF k = { x | ?y 1 ?y 2 ?y 3 … k y k ?x,y 1 …y k ??L P } QBF 8 = { x | ?y 1 ?y 2 ?y 3 … ?x,y 1 … ??L P } 208 Q S P 2 : { x | ?y?z ?x,y,z??L P } ? ? P 0 : P The “Polynomial Hierarchy” ? P 1 : P time given SAT S P 1 (NP): { x | ?y ?x,y??L P } SAT, Clique, VC, HC, Knap,… ? P 1 (co-NP): { x | ?y ?x,y??L P } UNSAT,… ? P 2 : { x | ?y?z ?x,y,z??L P } Potential Utility: It is often easy to give such a quantifier-based characterization of a language; doing so suggests (but doesn’t prove) whether it is in P, NP, etc. and suggests candidates for reducing to it. Page 4 Lecture 27 207 Beyond NP Many complexity classes are worse, e.g. time 2 2 n , 2 2 2 n , … Others seem to be “worse” in a different sense, e.g., not in NP, but still exponential time. E.g., let Lp = “assignment y satis?es formula x”, ? P Then : SAT = { x | ?y ?x,y??L P } UNSAT = { x | ?y ?x,y??L P } QBF k = { x | ?y 1 ?y 2 ?y 3 … k y k ?x,y 1 …y k ??L P } QBF 8 = { x | ?y 1 ?y 2 ?y 3 … ?x,y 1 … ??L P } 208 Q S P 2 : { x | ?y?z ?x,y,z??L P } ? ? P 0 : P The “Polynomial Hierarchy” ? P 1 : P time given SAT S P 1 (NP): { x | ?y ?x,y??L P } SAT, Clique, VC, HC, Knap,… ? P 1 (co-NP): { x | ?y ?x,y??L P } UNSAT,… ? P 2 : { x | ?y?z ?x,y,z??L P } Potential Utility: It is often easy to give such a quantifier-based characterization of a language; doing so suggests (but doesn’t prove) whether it is in P, NP, etc. and suggests candidates for reducing to it. Examples QBF k in S P k Given graph G, integers j & k, is there a set U of = j vertices in G such that every k-clique contains a vertex in U? Given graph G, integers j & k, is there a set U of = j vertices in G such removal of any k edges leaves a Hamilton path in U? 210 Page 5 Lecture 27 207 Beyond NP Many complexity classes are worse, e.g. time 2 2 n , 2 2 2 n , … Others seem to be “worse” in a different sense, e.g., not in NP, but still exponential time. E.g., let Lp = “assignment y satis?es formula x”, ? P Then : SAT = { x | ?y ?x,y??L P } UNSAT = { x | ?y ?x,y??L P } QBF k = { x | ?y 1 ?y 2 ?y 3 … k y k ?x,y 1 …y k ??L P } QBF 8 = { x | ?y 1 ?y 2 ?y 3 … ?x,y 1 … ??L P } 208 Q S P 2 : { x | ?y?z ?x,y,z??L P } ? ? P 0 : P The “Polynomial Hierarchy” ? P 1 : P time given SAT S P 1 (NP): { x | ?y ?x,y??L P } SAT, Clique, VC, HC, Knap,… ? P 1 (co-NP): { x | ?y ?x,y??L P } UNSAT,… ? P 2 : { x | ?y?z ?x,y,z??L P } Potential Utility: It is often easy to give such a quantifier-based characterization of a language; doing so suggests (but doesn’t prove) whether it is in P, NP, etc. and suggests candidates for reducing to it. Examples QBF k in S P k Given graph G, integers j & k, is there a set U of = j vertices in G such that every k-clique contains a vertex in U? Given graph G, integers j & k, is there a set U of = j vertices in G such removal of any k edges leaves a Hamilton path in U? 210 Space Complexity DTM M has space complexity S(n) if it halts on all inputs, and never visits more than S(n) tape cells on any input of length n. NTM …on any input of length n on any computation path. DSPACE(S(n)) = { L | L acc by some DTM in space O(S(n)) } NSPACE(S(n)) = { L | L acc by some NTM in space O(S(n)) } 211Read More

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