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The Self-Reducibility Technique The Self-Reducibility Technique Amal Fahad, Chetan Bhole, Jonathan Gordon, Mehdi Manshadi Fall 2007 Amal Fahad, Chetan Bhole, Jonathan Gordon, Mehdi Manshadi Department of Computer Science University of

  1. The Algorithm Self-Reducibility Technique Amal Fahad, Chetan Stage i Bhole, Jonathan Gordon, Input to stage i : C = { F 1 , . . . , F l } (the output from the previous Mehdi Manshadi stage) Step 1: Let C = { F 1 [ v i = True ] , . . . F l [ v i = True ] , Overview F 1 [ v i = False ] , . . . F l [ v i = False ] } Definitions Sparse sets Step 2: Set C ′ = ∅ Self-reducibility Hardness Step 3: For each formula f in C : NP -hard Tally Sets Statement 1 Compute g ( f ) Proof Correctness 2 If for no formula h ∈ C ′ does g ( f ) = g ( h ) , add f to C ′ Runtime Proof coNP -hard Sparse Step 4: If C ′ contains at least p d ( p k ( | F | )) + 1 elements, stop and Sets Statement immediately declare that F ∈ SAT . Proof Correctness Runtime Proof Stage m + 1 Summary References If some member of C evaluates to true, F ∈ SAT . Otherwise, F �∈ SAT . 1.15

  2. The Correctness Self-Reducibility Technique Amal Fahad, Chetan Bhole, Jonathan Gordon, Mehdi Manshadi 1 If we reach stage m + 1 then trivially we have a collection of formulae in which all variables are assigned and we can check if F is in SAT by checking if any one of the formulae in our collection is satisfied. 2 If we stop abruptly at step 4 of any stage then we use the Overview following argument to show that F ∈ SAT Definitions Sparse sets We know, Self-reducibility Hardness F i ∈ SAT ⇐ ⇒ g ( F i ) ∈ S NP -hard Tally Sets Statement We had p d ( p k ( | F | ))+ 1 or more unique elements generated by Proof Correctness g . But our sparse set can only be as big as p d ( p k ( | F | )) . That Runtime Proof means at least one element computed by g was not in S . coNP -hard Sparse Sets But, Statement Proof g ( F i ) �∈ S ⇒ F i �∈ SAT Correctness Runtime Proof Therefore, F i ∈ SAT meaning F ∈ SAT Summary References 1.16

  3. The Proof of polynomial time Self-Reducibility Technique Amal Fahad, Chetan Bhole, Jonathan Gordon, Mehdi Manshadi Overview Definitions Sparse sets Self-reducibility Hardness NP -hard Tally Sets Statement Proof Correctness • Number of formulae at the output of each stage is ≤ Runtime Proof p d ( p k ( | F | )) coNP -hard Sparse Sets • At the most m + 1 levels where m is the number of variables Statement Proof in F Correctness Runtime Proof • At each node we call g ( F i ) which is also bounded by | F | k + k Summary The algorithm clearly runs in polynomial time on the size of the References input F and we can show whether F is in SAT or not thus proving P = NP 1.17

  4. The Summary Self-Reducibility Technique Amal Fahad, Chetan Bhole, Jonathan Gordon, Mehdi Manshadi We showed that we can show P = NP using the pruning method Overview on self-reducibility trees in the following cases: Definitions Sparse sets 1 If there is a tally set that is ≤ p m -hard for NP , then P = NP Self-reducibility Hardness 2 If there is a sparse set that is ≤ p m -hard for coNP , then NP -hard Tally Sets Statement P = NP Proof Correctness Runtime Proof coNP -hard Sparse Sets Statement Proof Correctness Runtime Proof Summary References 1.18

  5. The References Self-Reducibility Technique Amal Fahad, Chetan Bhole, Jonathan Gordon, Mehdi Manshadi Overview Definitions • Lane Hemaspaandra and Mitsunori Ogihara. The Complexity Sparse sets Theorem Companion . Springer: Springer, 2002. Self-reducibility Hardness NP -hard Tally Sets Statement Proof Correctness Runtime Proof coNP -hard Sparse Sets Statement Proof Correctness Runtime Proof Summary References 1.19

  6. The Self-Reducibility Technique Amal Fahad, Chetan Bhole, Jonathan Gordon, Mehdi Manshadi Part II Overview Introduction Mahaney’s Theorem Proof of Theorem 1.4 Breadth-first Search Method Depth-first Search Method Algorithm Correctness Running Time Mahaney’s Theorem Complement of Sparse Set Pseudocomplement of Sparse Set Algorithm and Correctness Summary References 1.20

  7. The Overview Self-Reducibility Technique Amal Fahad, Chetan Bhole, Jonathan Gordon, 4 Introduction Mehdi Manshadi 5 Proof of Theorem 1.4 Breadth-first Search Method Depth-first Search Method Overview Algorithm Correctness Introduction Running Time Proof of Theorem 1.4 Breadth-first Search Method 6 Mahaney’s Theorem Depth-first Search Method Complement of Sparse Set Algorithm Correctness Pseudocomplement of Sparse Set Running Time Algorithm and Correctness Mahaney’s Theorem Complement of Sparse Set 7 Summary Pseudocomplement of Sparse Set Algorithm and Correctness 8 References Summary References 1.21

  8. The Introduction Self-Reducibility Technique Amal Fahad, Chetan Bhole, Jonathan Gordon, Mehdi Manshadi Last Lecture We proved: • If there is an NP -complete tally set, then P = NP . Overview Introduction • If there is a coNP -complete sparse set, then P = NP . Proof of Theorem 1.4 Breadth-first Search Method Depth-first Search Method Algorithm Correctness Running Time Mahaney’s Theorem Complement of Sparse Set Pseudocomplement of Sparse Set Algorithm and Correctness Summary References 1.22

  9. The Introduction Self-Reducibility Technique Amal Fahad, Chetan Bhole, Jonathan Gordon, Mehdi Manshadi Last Lecture We proved: • If there is an NP -complete tally set, then P = NP . Overview Introduction • If there is a coNP -complete sparse set, then P = NP . Proof of Theorem 1.4 Breadth-first Search Method Depth-first Search This Lecture Method Algorithm Correctness We will prove Mahaney’s Theorem: Running Time • If there is an NP -complete sparse set, then P = NP . Mahaney’s Theorem Complement of Sparse Set Pseudocomplement of Sparse Set Algorithm and Correctness Summary References 1.22

  10. The Proof of Theorem 1.4 Self-Reducibility Technique Theorem 1.4 Amal Fahad, Chetan Bhole, If there is a sparse set that is ≤ p Jonathan Gordon, m -hard for coNP , then P = NP Mehdi Manshadi Suppose S is a sparse set that is ≤ p m -hard for coNP , then SAT ≤ p m S . There is a polynomial-time function g : g : SAT → S Overview Introduction Proof of Theorem ∀ x | g ( x ) | ≤ P g ( | x | ) 1.4 Breadth-first Search Method Depth-first Search ∀ n C s ( n ) ≤ P s ( n ) Method Algorithm Correctness for monotonically increasing polynomials P g and P s and where Running Time C s ( n ) = || S ≤ n || is called the census function. Mahaney’s Theorem Complement of Sparse Set Pseudocomplement of Sparse Set Algorithm and Correctness Summary References 1.23

  11. The Breadth-first Search Method Self-Reducibility Technique g : SAT → S Amal Fahad, Chetan Bhole, ∀ x | g ( x ) | ≤ P g ( | x | ) Jonathan Gordon, Mehdi Manshadi ∀ n C s ( n ) ≤ P s ( n ) Overview Introduction Proof of Theorem 1.4 Breadth-first Search Method Depth-first Search Method Algorithm Correctness Running Time Mahaney’s Theorem Complement of Sparse Set Pseudocomplement of Sparse Set Algorithm and Correctness Summary References 1.24

  12. The Breadth-first Search Method Self-Reducibility Technique g : SAT → S Amal Fahad, Chetan Bhole, ∀ x | g ( x ) | ≤ P g ( | x | ) Jonathan Gordon, Mehdi Manshadi ∀ n C s ( n ) ≤ P s ( n ) Overview Introduction Proof of Theorem 1.4 Breadth-first Search Method Depth-first Search Method Algorithm Correctness Running Time Mahaney’s Theorem Complement of Sparse Set Pseudocomplement of Sparse Set Algorithm and Correctness Summary References This doesn’t work if S is NP -complete. Why? SAT ≤ p m S 1.24

  13. The Depth-first Search Method Self-Reducibility Technique Amal Fahad, Chetan Another proof for Theorem 1.4 (a depth-first search method): Bhole, Jonathan Gordon, Mehdi Manshadi Overview Introduction Proof of Theorem 1.4 Breadth-first Search Method Depth-first Search Method Algorithm Correctness Running Time Mahaney’s Theorem Complement of Sparse Set Pseudocomplement of Sparse Set Algorithm and Correctness Summary References 1.25

  14. The Depth-first Search Method Self-Reducibility Technique Amal Fahad, Chetan Another proof for Theorem 1.4 (a depth-first search method): Bhole, Jonathan Gordon, Mehdi Manshadi Overview Introduction Proof of Theorem 1.4 Breadth-first Search Method Depth-first Search Method Algorithm Correctness Running Time Mahaney’s Theorem Complement of Sparse Set Pseudocomplement of Sparse Set Algorithm and Correctness Summary References 1.25

  15. The Depth-first Search Method Self-Reducibility Technique Amal Fahad, Chetan Another proof for Theorem 1.4 (a depth-first search method): Bhole, Jonathan Gordon, Mehdi Manshadi Overview Introduction Proof of Theorem 1.4 Breadth-first Search Method Depth-first Search Method Algorithm Correctness Running Time Mahaney’s Theorem Complement of Sparse Set Pseudocomplement of Sparse Set Algorithm and Correctness Summary References 1.26

  16. The Depth-first Search Method Self-Reducibility Technique Amal Fahad, Chetan Another proof for Theorem 1.4 (a depth-first search method): Bhole, Jonathan Gordon, Mehdi Manshadi Overview Introduction Proof of Theorem 1.4 Breadth-first Search Method Depth-first Search Method Algorithm Correctness Running Time Mahaney’s Theorem Complement of Sparse Set Pseudocomplement of Sparse Set Algorithm and Correctness Summary References 1.27

  17. The Depth-first Search Method Self-Reducibility Technique Amal Fahad, Chetan Another proof for Theorem 1.4 (a depth-first search method): Bhole, Jonathan Gordon, Mehdi Manshadi Overview Introduction Proof of Theorem 1.4 Breadth-first Search Method Depth-first Search Method Algorithm Correctness Running Time Mahaney’s Theorem Complement of Sparse Set Pseudocomplement of Sparse Set Algorithm and Correctness Summary References 1.28

  18. The Depth-first Search Method Self-Reducibility Technique Amal Fahad, Chetan Another proof for Theorem 1.4 (a depth-first search method): Bhole, Jonathan Gordon, Mehdi Manshadi Overview Introduction Proof of Theorem 1.4 Breadth-first Search Method Depth-first Search Method Algorithm Correctness Running Time Mahaney’s Theorem Complement of Sparse Set Pseudocomplement of Sparse Set Algorithm and Correctness Summary References 1.29

  19. The Depth-first Search Method Self-Reducibility Technique Amal Fahad, Chetan Another proof for Theorem 1.4 (a depth-first search method): Bhole, Jonathan Gordon, Mehdi Manshadi Overview Introduction Proof of Theorem 1.4 Breadth-first Search Method Depth-first Search Method Algorithm Correctness Running Time Mahaney’s Theorem Complement of Sparse Set Pseudocomplement of Sparse Set Algorithm and Correctness Summary References 1.30

  20. The Depth-first Search Method Self-Reducibility Technique Amal Fahad, Chetan Another proof for Theorem 1.4 (a depth-first search method): Bhole, Jonathan Gordon, Mehdi Manshadi Overview Introduction Proof of Theorem 1.4 Breadth-first Search Method Depth-first Search Method Algorithm Correctness Running Time Mahaney’s Theorem Complement of Sparse Set Pseudocomplement of Sparse Set Algorithm and Correctness Summary References 1.31

  21. The Depth-first Search Method Self-Reducibility Technique Amal Fahad, Chetan Another proof for Theorem 1.4 (a depth-first search method): Bhole, Jonathan Gordon, Mehdi Manshadi Overview Introduction Proof of Theorem 1.4 Breadth-first Search Method Depth-first Search Method Algorithm Correctness Running Time Mahaney’s Theorem Complement of Sparse Set Pseudocomplement of Sparse Set Algorithm and Correctness Summary References 1.32

  22. The Depth-first Search Method Self-Reducibility Technique Amal Fahad, Chetan Another proof for Theorem 1.4 (a depth-first search method): Bhole, Jonathan Gordon, Mehdi Manshadi Overview Introduction Proof of Theorem 1.4 Breadth-first Search Method Depth-first Search Method Algorithm Correctness Running Time Mahaney’s Theorem Complement of Sparse Set Pseudocomplement of Sparse Set Algorithm and Correctness Summary References 1.33

  23. The Depth-first Search Method Self-Reducibility Technique Amal Fahad, Chetan Another proof for Theorem 1.4 (a depth-first search method): Bhole, Jonathan Gordon, Mehdi Manshadi Overview Introduction Proof of Theorem 1.4 Breadth-first Search Method Depth-first Search Method Algorithm Correctness Running Time Mahaney’s Theorem Complement of Sparse Set Pseudocomplement of Sparse Set Algorithm and Correctness Summary References 1.34

  24. The Depth-first Search Method Self-Reducibility Technique Amal Fahad, Chetan Another proof for Theorem 1.4 (a depth-first search method): Bhole, Jonathan Gordon, Mehdi Manshadi Overview Introduction Proof of Theorem 1.4 Breadth-first Search Method Depth-first Search Method Algorithm Correctness Running Time Mahaney’s Theorem Complement of Sparse Set Pseudocomplement of Sparse Set Algorithm and Correctness Summary References 1.35

  25. The Depth-first Search Method Self-Reducibility Technique Amal Fahad, Chetan Another proof for Theorem 1.4 (a depth-first search method): Bhole, Jonathan Gordon, Mehdi Manshadi Overview Introduction Proof of Theorem 1.4 Breadth-first Search Method Depth-first Search Method Algorithm Correctness Running Time Mahaney’s Theorem Complement of Sparse Set Pseudocomplement of Sparse Set Algorithm and Correctness Summary References 1.36

  26. The Depth-first Search Method Self-Reducibility Technique Amal Fahad, Chetan Another proof for Theorem 1.4 (a depth-first search method): Bhole, Jonathan Gordon, Mehdi Manshadi Overview Introduction Proof of Theorem 1.4 Breadth-first Search Method Depth-first Search Method Algorithm Correctness Running Time Mahaney’s Theorem Complement of Sparse Set Pseudocomplement of Sparse Set Algorithm and Correctness Summary References 1.37

  27. The Depth-first Algorithm Self-Reducibility Technique Amal Fahad, Chetan Decide(F) Bhole, Jonathan Gordon, Mehdi Manshadi SL = { g(False) }; Search(F); Declare unsatisfiable and halt. End. Overview Introduction Proof of Theorem 1.4 Breadth-first Search Method Depth-first Search Method Algorithm Correctness Running Time Mahaney’s Theorem Complement of Sparse Set Pseudocomplement of Sparse Set Algorithm and Correctness Summary References 1.38

  28. The Depth-first Algorithm Self-Reducibility Technique Amal Fahad, Chetan Decide(F) Bhole, Jonathan Gordon, Mehdi Manshadi SL = { g(False) }; Search(F); Declare unsatisfiable and halt. End. Overview Search(G) Introduction Proof of Theorem if G = True 1.4 Breadth-first Declare satisfiable and halt. Search Method Depth-first Search else if g(G) is in SL Method Algorithm return; Correctness Running Time else Mahaney’s For v , the first variable in G : Theorem Complement of Search(G{v = False}); Sparse Set Pseudocomplement Search(G{v = True}); of Sparse Set Algorithm and Add g(G) to SL; Correctness return; Summary End. References 1.38

  29. The Correctness Self-Reducibility Technique Amal Fahad, Chetan Bhole, Jonathan Gordon, Mehdi Manshadi Overview Introduction Proof of Theorem 1.4 Breadth-first Search Method Depth-first Search Method Algorithm Correctness Running Time Mahaney’s Theorem Complement of Sparse Set Pseudocomplement of Sparse Set Algorithm and Correctness Summary References 1.39

  30. The Correctness Self-Reducibility Technique Amal Fahad, Chetan Bhole, Jonathan Gordon, Mehdi Manshadi Argument • The only way that Search declares a formula satisfiable is by Overview finding a satisfying assignment. Introduction Proof of Theorem 1.4 Breadth-first Search Method Depth-first Search Method Algorithm Correctness Running Time Mahaney’s Theorem Complement of Sparse Set Pseudocomplement of Sparse Set Algorithm and Correctness Summary References 1.39

  31. The Correctness Self-Reducibility Technique Amal Fahad, Chetan Bhole, Jonathan Gordon, Mehdi Manshadi Argument • The only way that Search declares a formula satisfiable is by Overview finding a satisfying assignment. Introduction • That is: If Search declares G is satisfiable, then G is definitely Proof of Theorem 1.4 satisfiable. Breadth-first Search Method Depth-first Search Method Algorithm Correctness Running Time Mahaney’s Theorem Complement of Sparse Set Pseudocomplement of Sparse Set Algorithm and Correctness Summary References 1.39

  32. The Correctness Self-Reducibility Technique Amal Fahad, Chetan Bhole, Jonathan Gordon, Mehdi Manshadi Argument • The only way that Search declares a formula satisfiable is by Overview finding a satisfying assignment. Introduction • That is: If Search declares G is satisfiable, then G is definitely Proof of Theorem 1.4 satisfiable. Breadth-first Search Method • The only way that we don’t expand a node is that we already Depth-first Search Method know that the corresponding formula is unsatisfiable. Algorithm Correctness Running Time Mahaney’s Theorem Complement of Sparse Set Pseudocomplement of Sparse Set Algorithm and Correctness Summary References 1.39

  33. The Correctness Self-Reducibility Technique Amal Fahad, Chetan Bhole, Jonathan Gordon, Mehdi Manshadi Argument • The only way that Search declares a formula satisfiable is by Overview finding a satisfying assignment. Introduction • That is: If Search declares G is satisfiable, then G is definitely Proof of Theorem 1.4 satisfiable. Breadth-first Search Method • The only way that we don’t expand a node is that we already Depth-first Search Method know that the corresponding formula is unsatisfiable. Algorithm Correctness • That is: If G is satisfiable, then finally Search finds a Running Time Mahaney’s satisfying assignment. Theorem Complement of Sparse Set Pseudocomplement of Sparse Set Algorithm and Correctness Summary References 1.39

  34. The Running Time Self-Reducibility Technique How many nodes does the Search method visit? Amal Fahad, Chetan Bhole, Jonathan Gordon, Mehdi Manshadi Overview Introduction Proof of Theorem 1.4 Breadth-first Search Method Depth-first Search Method Algorithm Correctness Running Time Mahaney’s Theorem Complement of Sparse Set Pseudocomplement of Sparse Set Algorithm and Correctness Summary References 1.40

  35. The Running Time Self-Reducibility Technique How many nodes does the Search method visit? Amal Fahad, Chetan Bhole, Jonathan Gordon, Unsatisfiable Nodes Mehdi Manshadi | g ( F i ) | ≤ P g ( | F i | ) ≤ P g ( | F | ) || SL || ≤ P s ( P g ( | F | )) Overview Introduction Proof of Theorem 1.4 Breadth-first Search Method Depth-first Search Method Algorithm Correctness Running Time Mahaney’s Theorem Complement of Sparse Set Pseudocomplement of Sparse Set Algorithm and Correctness Summary References 1.40

  36. The Running Time Self-Reducibility Technique How many nodes does the Search method visit? Amal Fahad, Chetan Bhole, Jonathan Gordon, Unsatisfiable Nodes Mehdi Manshadi | g ( F i ) | ≤ P g ( | F i | ) ≤ P g ( | F | ) || SL || ≤ P s ( P g ( | F | )) Claim : No two interior nodes can be labelled with the same value in SL unless they are on the same path from the root. Overview Introduction Proof of Theorem 1.4 Breadth-first Search Method Depth-first Search Method Algorithm Correctness Running Time Mahaney’s Theorem Complement of Sparse Set Pseudocomplement of Sparse Set Algorithm and Correctness Summary References 1.40

  37. The Running Time Self-Reducibility Technique How many nodes does the Search method visit? Amal Fahad, Chetan Bhole, Jonathan Gordon, Unsatisfiable Nodes Mehdi Manshadi | g ( F i ) | ≤ P g ( | F i | ) ≤ P g ( | F | ) || SL || ≤ P s ( P g ( | F | )) Claim : No two interior nodes can be labelled with the same value in SL unless they are on the same path from the root. Overview Introduction Proof of Theorem 1.4 Breadth-first Search Method Depth-first Search Method Algorithm Correctness Running Time Mahaney’s Theorem Complement of Sparse Set Pseudocomplement of Sparse Set Algorithm and Correctness Summary References Maximum number of unsatisfiable interior nodes: m · P s ( P g ( | F | )) where m is the number of variables in F . 1.40

  38. The Running Time Self-Reducibility Technique How many nodes does the Search method visit? Amal Fahad, Chetan Bhole, Jonathan Gordon, Satisfiable Interior Nodes Mehdi Manshadi Maximum: m Overview Introduction Proof of Theorem 1.4 Breadth-first Search Method Depth-first Search Method Algorithm Correctness Running Time Mahaney’s Theorem Complement of Sparse Set Pseudocomplement of Sparse Set Algorithm and Correctness Summary References 1.41

  39. The Running Time Self-Reducibility Technique How many nodes does the Search method visit? Amal Fahad, Chetan Bhole, Jonathan Gordon, Satisfiable Interior Nodes Mehdi Manshadi Maximum: m Overview Introduction Proof of Theorem 1.4 Breadth-first Search Method Depth-first Search Method Algorithm Correctness Running Time Mahaney’s Theorem Complement of Sparse Set Pseudocomplement of Sparse Set Algorithm and Correctness Summary References 1.41

  40. The Running Time Self-Reducibility Technique How many nodes does the Search method visit? Amal Fahad, Chetan Bhole, Jonathan Gordon, Satisfiable Interior Nodes Mehdi Manshadi Maximum: m Overview Introduction Proof of Theorem 1.4 Breadth-first Search Method Depth-first Search Method Algorithm Correctness Running Time Mahaney’s Theorem Complement of Sparse Set Pseudocomplement of Sparse Set Algorithm and Correctness Summary Maximum number of interior nodes that the Search method visits: References m + m · P s ( P g ( | F | )) 1.41

  41. The Mahaney’s Theorem Self-Reducibility Technique Amal Fahad, Chetan Theorem Bhole, Jonathan Gordon, If there is an NP-complete sparse set, then P = NP. Mehdi Manshadi Overview Introduction Proof of Theorem 1.4 Breadth-first Search Method Depth-first Search Method Algorithm Correctness Running Time Mahaney’s Theorem Complement of Sparse Set Pseudocomplement of Sparse Set Algorithm and Correctness Summary References 1.42

  42. The Mahaney’s Theorem Self-Reducibility Technique Amal Fahad, Chetan Theorem Bhole, Jonathan Gordon, If there is an NP-complete sparse set, then P = NP. Mehdi Manshadi Suppose S is an NP-complete sparse set, then there is a polynomial-time function f : Overview Introduction f : SAT → S Proof of Theorem 1.4 Breadth-first Search Method f : SAT → S Depth-first Search Method Algorithm Correctness Running Time Mahaney’s Theorem Complement of Sparse Set Pseudocomplement of Sparse Set Algorithm and Correctness Summary References 1.42

  43. The Mahaney’s Theorem Self-Reducibility Technique Amal Fahad, Chetan Theorem Bhole, Jonathan Gordon, If there is an NP-complete sparse set, then P = NP. Mehdi Manshadi Suppose S is an NP-complete sparse set, then there is a polynomial-time function f : Overview Introduction f : SAT → S Proof of Theorem 1.4 Breadth-first Search Method f : SAT → S Depth-first Search Method Algorithm Correctness Running Time Mahaney’s Is S in NP ? If yes, Theorem Complement of Sparse Set Pseudocomplement h : S → S of Sparse Set Algorithm and Correctness Summary g = h ◦ f : SAT → S References 1.42

  44. The Is S in NP ? Self-Reducibility Technique Amal Fahad, Chetan Bhole, Jonathan Gordon, Mehdi Manshadi A nondeterministic, polynomial-time TM that accepts S : Overview On input x , Introduction Let n = | x | Proof of Theorem Let k = C s ( n ) 1.4 Breadth-first In a nondeterministic fashion, guess distinct strings Search Method Depth-first Search s 1 , s 2 , . . . , s k (such that | s i | ≤ n ) Method Algorithm If all of these strings are in S , Correctness Running Time then accept x if it is not among the s i ’s. Mahaney’s Theorem Complement of Sparse Set Pseudocomplement of Sparse Set Algorithm and Correctness Summary References 1.43

  45. The Is S in NP ? Self-Reducibility Technique Amal Fahad, Chetan Bhole, Jonathan Gordon, Mehdi Manshadi A nondeterministic, polynomial-time TM that accepts S : On input x , Overview Let n = | x | Introduction Let k = C s ( n ) ? Proof of Theorem In a nondeterministic fashion, guess distinct strings 1.4 Breadth-first s 1 , s 2 , . . . , s k (such that | s i | ≤ n ) Search Method Depth-first Search If all of these strings are in S , Method Algorithm then accept x if it is not among the s i ’s. Correctness Running Time Mahaney’s But: This algorithm is wrong. C s ( n ) may not be P-time Theorem Complement of computable. Sparse Set Pseudocomplement of Sparse Set Algorithm and Correctness Summary References 1.44

  46. The Pseudocomplement of S Self-Reducibility Technique Amal Fahad, Chetan Bhole, p (the Jonathan Gordon, A nondeterministic, polynomial-time TM that accepts S Mehdi Manshadi pseudo-complement of S ): On input � x , k , 0 n � • If | x | > n , reject. Overview Introduction • If k > P s ( n ) , reject. Proof of Theorem • Guess distinct strings s 1 , s 2 , . . . , s k in a nondeterministic 1.4 Breadth-first fashion (such that | s i | ≤ n ) and guess proofs that each Search Method Depth-first Search belongs to S . Method Algorithm Correctness • If this guess succeeded, then accept x if x is not among the Running Time s i ’s. Mahaney’s Theorem Complement of Sparse Set Pseudocomplement of Sparse Set Algorithm and Correctness Summary References 1.45

  47. The Pseudocomplement of S Self-Reducibility Technique Amal Fahad, Chetan Bhole, p (the Jonathan Gordon, A nondeterministic, polynomial-time TM that accepts S Mehdi Manshadi pseudo-complement of S ): On input � x , k , 0 n � • If | x | > n , reject. Overview Introduction • If k > P s ( n ) , reject. Proof of Theorem • Guess distinct strings s 1 , s 2 , . . . , s k in a nondeterministic 1.4 Breadth-first fashion (such that | s i | ≤ n ) and guess proofs that each Search Method Depth-first Search belongs to S . Method Algorithm Correctness • If this guess succeeded, then accept x if x is not among the Running Time s i ’s. Mahaney’s Theorem p ⇔ x ∈ S (for | x | ≤ n ) Complement of Sparse Set k = C s ( n ) : � x , k , 0 n � ∈ S Pseudocomplement of Sparse Set Algorithm and Correctness Summary References 1.45

  48. The Pseudocomplement of S Self-Reducibility Technique Amal Fahad, Chetan Bhole, p (the Jonathan Gordon, A nondeterministic, polynomial-time TM that accepts S Mehdi Manshadi pseudo-complement of S ): On input � x , k , 0 n � • If | x | > n , reject. Overview Introduction • If k > P s ( n ) , reject. Proof of Theorem • Guess distinct strings s 1 , s 2 , . . . , s k in a nondeterministic 1.4 Breadth-first fashion (such that | s i | ≤ n ) and guess proofs that each Search Method Depth-first Search belongs to S . Method Algorithm Correctness • If this guess succeeded, then accept x if x is not among the Running Time s i ’s. Mahaney’s Theorem p ⇔ x ∈ S (for | x | ≤ n ) Complement of Sparse Set k = C s ( n ) : � x , k , 0 n � ∈ S Pseudocomplement of Sparse Set Algorithm and Correctness (What happens if k > C s ( n ) or k < C s ( n ) ?) Summary References 1.45

  49. The Pruning Functions Self-Reducibility Technique Amal Fahad, Chetan Bhole, f : SAT → S ∀ x | f ( x ) | ≤ P f ( | x | ) Jonathan Gordon, Mehdi Manshadi Overview Introduction Proof of Theorem 1.4 Breadth-first Search Method Depth-first Search Method Algorithm Correctness Running Time Mahaney’s Theorem Complement of Sparse Set Pseudocomplement of Sparse Set Algorithm and Correctness Summary References 1.46

  50. The Pruning Functions Self-Reducibility Technique Amal Fahad, Chetan Bhole, f : SAT → S ∀ x | f ( x ) | ≤ P f ( | x | ) Jonathan Gordon, p → S ∀ x | h ( x ) | ≤ P h ( | x | ) Mehdi Manshadi h : S where P f and P h are monotonically increasing polynomials. Overview Introduction Proof of Theorem 1.4 Breadth-first Search Method Depth-first Search Method Algorithm Correctness Running Time Mahaney’s Theorem Complement of Sparse Set Pseudocomplement of Sparse Set Algorithm and Correctness Summary References 1.46

  51. The Pruning Functions Self-Reducibility Technique Amal Fahad, Chetan Bhole, f : SAT → S ∀ x | f ( x ) | ≤ P f ( | x | ) Jonathan Gordon, p → S ∀ x | h ( x ) | ≤ P h ( | x | ) Mehdi Manshadi h : S where P f and P h are monotonically increasing polynomials. Let n = P f ( | F | ) Overview Pruning functions: g k ( F i ) = h ( � f ( F i ) , k , 0 n � ) (for k ≤ P s ( n ) ) Introduction Proof of Theorem 1.4 Breadth-first Search Method P h ( |� f ( F i ) , k , 0 n �| ) | g k ( F i ) | ≤ Depth-first Search Method ≤ P g ( | F | ) Algorithm Correctness Running Time for some polynomial P g . Mahaney’s Theorem Complement of Sparse Set Pseudocomplement of Sparse Set Algorithm and Correctness Summary References 1.46

  52. The Pruning Functions Self-Reducibility Technique Amal Fahad, Chetan Bhole, f : SAT → S ∀ x | f ( x ) | ≤ P f ( | x | ) Jonathan Gordon, p → S ∀ x | h ( x ) | ≤ P h ( | x | ) Mehdi Manshadi h : S where P f and P h are monotonically increasing polynomials. Let n = P f ( | F | ) Overview Pruning functions: g k ( F i ) = h ( � f ( F i ) , k , 0 n � ) (for k ≤ P s ( n ) ) Introduction Proof of Theorem 1.4 Breadth-first Search Method P h ( |� f ( F i ) , k , 0 n �| ) | g k ( F i ) | ≤ Depth-first Search Method ≤ P g ( | F | ) Algorithm Correctness Running Time for some polynomial P g . Mahaney’s Theorem Complement of Sparse Set Pseudocomplement of Sparse Set When k = C s ( n ) Algorithm and Correctness Summary References 1.46

  53. The Pruning Functions Self-Reducibility Technique Amal Fahad, Chetan Bhole, f : SAT → S ∀ x | f ( x ) | ≤ P f ( | x | ) Jonathan Gordon, p → S ∀ x | h ( x ) | ≤ P h ( | x | ) Mehdi Manshadi h : S where P f and P h are monotonically increasing polynomials. Let n = P f ( | F | ) Overview Pruning functions: g k ( F i ) = h ( � f ( F i ) , k , 0 n � ) (for k ≤ P s ( n ) ) Introduction Proof of Theorem 1.4 Breadth-first Search Method P h ( |� f ( F i ) , k , 0 n �| ) | g k ( F i ) | ≤ Depth-first Search Method ≤ P g ( | F | ) Algorithm Correctness Running Time for some polynomial P g . Mahaney’s Theorem Complement of Sparse Set Pseudocomplement of Sparse Set When k = C s ( n ) Algorithm and Correctness F i ∈ SAT ⇔ g k ( F i ) ∈ S Summary References 1.46

  54. The Pruning Functions Self-Reducibility Technique Amal Fahad, Chetan Bhole, f : SAT → S ∀ x | f ( x ) | ≤ P f ( | x | ) Jonathan Gordon, p → S ∀ x | h ( x ) | ≤ P h ( | x | ) Mehdi Manshadi h : S where P f and P h are monotonically increasing polynomials. Let n = P f ( | F | ) Overview Pruning functions: g k ( F i ) = h ( � f ( F i ) , k , 0 n � ) (for k ≤ P s ( n ) ) Introduction Proof of Theorem 1.4 Breadth-first Search Method P h ( |� f ( F i ) , k , 0 n �| ) | g k ( F i ) | ≤ Depth-first Search Method ≤ P g ( | F | ) Algorithm Correctness Running Time for some polynomial P g . Mahaney’s Theorem Complement of Sparse Set Pseudocomplement of Sparse Set When k = C s ( n ) Algorithm and Correctness F i ∈ SAT ⇔ g k ( F i ) ∈ S Summary g k : SAT → S References 1.46

  55. The Pruning Functions Self-Reducibility Technique Amal Fahad, Chetan Bhole, f : SAT → S ∀ x | f ( x ) | ≤ P f ( | x | ) Jonathan Gordon, p → S ∀ x | h ( x ) | ≤ P h ( | x | ) Mehdi Manshadi h : S where P f and P h are monotonically increasing polynomials. Let n = P f ( | F | ) Overview Pruning functions: g k ( F i ) = h ( � f ( F i ) , k , 0 n � ) (for k ≤ P s ( n ) ) Introduction Proof of Theorem 1.4 Breadth-first Search Method P h ( |� f ( F i ) , k , 0 n �| ) | g k ( F i ) | ≤ Depth-first Search Method ≤ P g ( | F | ) Algorithm Correctness Running Time for some polynomial P g . Mahaney’s Theorem Complement of Sparse Set Pseudocomplement of Sparse Set When k = C s ( n ) Algorithm and Correctness F i ∈ SAT ⇔ g k ( F i ) ∈ S Summary g k : SAT → S References || S L || ≤ P s ( P g ( | F | )) 1.46

  56. The Polynomial Algorithm for SAT Self-Reducibility Technique Decide(F) Amal Fahad, Chetan Bhole, Jonathan Gordon, for k = 0 to P s ( n ) { Mehdi Manshadi SL = { g k ( False ) } Run the Search method on F with pruning function g k . If the number of interior nodes visited by the Search method exceeds m + m · P s ( P g ( | F | )) , halt the search for Overview this k . Introduction } Proof of Theorem Declare unsatisfiable and halt. 1.4 Breadth-first End. Search Method Depth-first Search Method Algorithm Correctness Running Time Mahaney’s Theorem Complement of Sparse Set Pseudocomplement of Sparse Set Algorithm and Correctness Summary References 1.47

  57. The Polynomial Algorithm for SAT Self-Reducibility Technique Decide(F) Amal Fahad, Chetan Bhole, Jonathan Gordon, for k = 0 to P s ( n ) { Mehdi Manshadi SL = { g k ( False ) } Run the Search method on F with pruning function g k . If the number of interior nodes visited by the Search method exceeds m + m · P s ( P g ( | F | )) , halt the search for Overview this k . Introduction } Proof of Theorem Declare unsatisfiable and halt. 1.4 Breadth-first End. Search Method Depth-first Search Method Running time: polynomial in | F | . Algorithm Correctness Running Time Mahaney’s Theorem Complement of Sparse Set Pseudocomplement of Sparse Set Algorithm and Correctness Summary References 1.47

  58. The Polynomial Algorithm for SAT Self-Reducibility Technique Decide(F) Amal Fahad, Chetan Bhole, Jonathan Gordon, for k = 0 to P s ( n ) { Mehdi Manshadi SL = { g k ( False ) } Run the Search method on F with pruning function g k . If the number of interior nodes visited by the Search method exceeds m + m · P s ( P g ( | F | )) , halt the search for Overview this k . Introduction } Proof of Theorem Declare unsatisfiable and halt. 1.4 Breadth-first End. Search Method Depth-first Search Method Running time: polynomial in | F | . Algorithm Correctness Running Time Correctness Mahaney’s Theorem Complement of Sparse Set Pseudocomplement of Sparse Set Algorithm and Correctness Summary References 1.47

  59. The Polynomial Algorithm for SAT Self-Reducibility Technique Decide(F) Amal Fahad, Chetan Bhole, Jonathan Gordon, for k = 0 to P s ( n ) { Mehdi Manshadi SL = { g k ( False ) } Run the Search method on F with pruning function g k . If the number of interior nodes visited by the Search method exceeds m + m · P s ( P g ( | F | )) , halt the search for Overview this k . Introduction } Proof of Theorem Declare unsatisfiable and halt. 1.4 Breadth-first End. Search Method Depth-first Search Method Running time: polynomial in | F | . Algorithm Correctness Running Time Correctness Mahaney’s Theorem • Decide declares F as satisfiable only if it finds a satisfying Complement of Sparse Set assignment. Pseudocomplement of Sparse Set Algorithm and Correctness Summary References 1.47

  60. The Polynomial Algorithm for SAT Self-Reducibility Technique Decide(F) Amal Fahad, Chetan Bhole, Jonathan Gordon, for k = 0 to P s ( n ) { Mehdi Manshadi SL = { g k ( False ) } Run the Search method on F with pruning function g k . If the number of interior nodes visited by the Search method exceeds m + m · P s ( P g ( | F | )) , halt the search for Overview this k . Introduction } Proof of Theorem Declare unsatisfiable and halt. 1.4 Breadth-first End. Search Method Depth-first Search Method Running time: polynomial in | F | . Algorithm Correctness Running Time Correctness Mahaney’s Theorem • Decide declares F as satisfiable only if it finds a satisfying Complement of Sparse Set assignment. Pseudocomplement of Sparse Set • If F is satisfiable, when k = C s ( n ) , the Search method finally Algorithm and Correctness will find a satisfying assignment. Summary References 1.47

  61. The Polynomial Algorithm for SAT Self-Reducibility Technique Decide(F) Amal Fahad, Chetan Bhole, Jonathan Gordon, for k = 0 to P s ( n ) { Mehdi Manshadi SL = { g k ( False ) } Run the Search method on F with pruning function g k . If the number of interior nodes visited by the Search method exceeds m + m · P s ( P g ( | F | )) , halt the search for Overview this k . Introduction } Proof of Theorem Declare unsatisfiable and halt. 1.4 Breadth-first End. Search Method Depth-first Search Method Running time: polynomial in | F | . Algorithm Correctness Running Time Correctness Mahaney’s Theorem • Decide declares F as satisfiable only if it finds a satisfying Complement of Sparse Set assignment. Pseudocomplement of Sparse Set • If F is satisfiable, when k = C s ( n ) , the Search method finally Algorithm and Correctness will find a satisfying assignment. Summary References • F is declared as satisfiable if and only if F is satisfiable. 1.47

  62. The Summary Self-Reducibility Technique Amal Fahad, Chetan Bhole, Jonathan Gordon, Mehdi Manshadi Mahaney’s Theorem If there is an NP-complete sparse set, then P = NP . Overview Introduction What about an NP -hard sparse set? Proof of Theorem 1.4 Breadth-first Search Method Depth-first Search Method Algorithm Correctness Running Time Mahaney’s Theorem Complement of Sparse Set Pseudocomplement of Sparse Set Algorithm and Correctness Summary References 1.48

  63. The Summary Self-Reducibility Technique Amal Fahad, Chetan Bhole, Jonathan Gordon, Mehdi Manshadi Mahaney’s Theorem If there is an NP-complete sparse set, then P = NP . Overview Introduction What about an NP -hard sparse set? Proof of Theorem 1.4 Breadth-first Theorem 1.9 (see Hemaspaandra-Ogihara) Search Method Depth-first Search NP has sparse ≤ p m -hard sets if and only if NP has sparse Method Algorithm ≤ p m -complete sets. Correctness Running Time Mahaney’s Theorem Complement of Any questions? Sparse Set Pseudocomplement of Sparse Set Algorithm and Correctness Summary References 1.48

  64. The References Self-Reducibility Technique Amal Fahad, Chetan Bhole, Jonathan Gordon, Mehdi Manshadi • S. Mahaney. “Sparse Sets and Reducibilities”. Studies in Overview Complexity Theory , pages 63-118. John Wiley and Sons, Introduction 1986. Proof of Theorem 1.4 • Lane Hemaspaandra and Mitsunori Ogihara. The Complexity Breadth-first Search Method Depth-first Search Theorem Companion . Springer: Springer, 2002. Method Algorithm Correctness Running Time Mahaney’s Theorem Complement of Sparse Set Pseudocomplement of Sparse Set Algorithm and Correctness Summary References 1.49

  65. The Self-Reducibility Technique Amal Fahad, Chetan Bhole, Jonathan Gordon, Mehdi Manshadi Part III Overview The Hartmanis–Immerman–Sewelson Introduction E and NE Encoding Theorem Lemma 1.19 Warm-up Proof Lemma 1.21 Summary References 1.50

  66. The Overview Self-Reducibility Technique Amal Fahad, Chetan Bhole, Jonathan Gordon, Mehdi Manshadi 9 Introduction 10 E and NE Overview Introduction 11 Theorem E and NE Lemma 1.19 Theorem Warm-up Proof Lemma 1.19 Warm-up Proof Lemma 1.21 Lemma 1.21 Summary References 12 Summary 13 References 1.51

  67. The Introduction Self-Reducibility Technique Amal Fahad, Chetan Bhole, Jonathan Gordon, Mehdi Manshadi The Question Is there any sparse set in NP − P ? Overview That is, is there a sparse set in NP that’s so hard it has no Introduction polynomial-time algorithm? E and NE Theorem Lemma 1.19 Warm-up Proof Lemma 1.21 Summary References 1.52

  68. The Introduction Self-Reducibility Technique Amal Fahad, Chetan Bhole, Jonathan Gordon, Mehdi Manshadi The Question Is there any sparse set in NP − P ? Overview That is, is there a sparse set in NP that’s so hard it has no Introduction polynomial-time algorithm? E and NE Theorem Lemma 1.19 Such a set would, necessarily, not be NP -complete. Why? Warm-up Proof Lemma 1.21 Summary References 1.52

  69. The Introduction Self-Reducibility Technique Amal Fahad, Chetan Bhole, Jonathan Gordon, Mehdi Manshadi The Question Is there any sparse set in NP − P ? Overview That is, is there a sparse set in NP that’s so hard it has no Introduction polynomial-time algorithm? E and NE Theorem Lemma 1.19 Such a set would, necessarily, not be NP -complete. Why? Warm-up Proof Lemma 1.21 Remember Mahaney’s Theorem from last lecture: We proved that Summary if any sparse set is NP -complete, then P = NP . Well, if P = NP , References then NP − P is empty, so our set can’t exist. 1.52

  70. The E and NE Self-Reducibility Technique Amal Fahad, Chetan Bhole, Jonathan Gordon, Mehdi Manshadi E and NE are exponential-time analogs of P and NP . Recall: � DTIME [ n k ] = P k Overview � NTIME [ n k ] NP = Introduction E and NE k Theorem Lemma 1.19 Warm-up Proof Lemma 1.21 Summary References 1.53

  71. The E and NE Self-Reducibility Technique Amal Fahad, Chetan Bhole, Jonathan Gordon, Mehdi Manshadi E and NE are exponential-time analogs of P and NP . Recall: � DTIME [ n k ] = P k Overview � NTIME [ n k ] NP = Introduction E and NE k Theorem Lemma 1.19 Warm-up Proof Lemma 1.21 � DTIME [ 2 cn ] E = Summary c > 0 References � NTIME [ 2 cn ] = NE c > 0 1.53

  72. The E and NE Self-Reducibility Technique Amal Fahad, Chetan Bhole, Jonathan Gordon, Mehdi Manshadi E and NE are exponential-time analogs of P and NP . Recall: � DTIME [ n k ] = P k Overview � NTIME [ n k ] NP = Introduction E and NE k Theorem Lemma 1.19 Warm-up Proof Lemma 1.21 � DTIME [ 2 cn ] E = Summary c > 0 References � NTIME [ 2 cn ] = NE c > 0 And just as P ⊆ NP , E ⊆ NE . 1.53

  73. The Theorem Self-Reducibility Technique Amal Fahad, Chetan Bhole, Jonathan Gordon, Mehdi Manshadi Theorem Overview The following are equivalent: Introduction 1 E = NE E and NE Theorem 2 NP − P contains no sparse sets Lemma 1.19 Warm-up Proof 3 NP − P contains no tally sets Lemma 1.21 Summary References 1.54

  74. The Theorem Proof: Part One Self-Reducibility Technique Amal Fahad, Chetan Bhole, Jonathan Gordon, Mehdi Manshadi Lemma 1.19 If NP − P contains no tally sets, then E = NE . Overview Proof: Since we know that E ⊆ NE , we can prove this by Introduction showing that NE ⊆ E . E and NE Theorem Set up Lemma 1.19 Warm-up Proof Lemma 1.21 L ∈ NE , so there must exist a nondeterministic, exponential-time Summary TM N s.t. L ( N ) = L . References Define a tally set L ′ = { 1 k | ( ∃ x ∈ L )[ k = ( 1 x ) 2 ] } 1.55

  75. Proof L ′ ∈ NP The Self-Reducibility Technique Amal Fahad, Chetan Bhole, Does L ′ ∈ NP ? We give an NPTM for L ′ : Jonathan Gordon, Mehdi Manshadi Algorithm for L ′ On input y , • Reject if y is not of the form 1 k (for some k > 0). Overview • Otherwise, simulate nondeterministically N ( w ) where w is all Introduction E and NE digits of k except the leftmost 1. Theorem Lemma 1.19 Warm-up Proof Lemma 1.21 Run-time: | w | is logarithmic in the length of y , so the run-time is Summary at most O ( 2 c log | y | ) = O ( | y | c ) for some c . So, the algorithm runs References nondeterministically (because N is nondeterministic) and runs in polynomial-time. Since L ′ is a tally set in NP and our assumption is that NP − P contains no tally sets, this means that L ′ ∈ P . 1.56

  76. The Proof L ∈ E Self-Reducibility Technique Amal Fahad, Chetan Bhole, There is a deterministic, polynomial-time TM M s.t. L ( M ) = L ′ . Jonathan Gordon, Mehdi Manshadi We use this to construct a TM that accepts L : Algorithm for L On input y , • Compute b = 1 ( 1 y ) 2 Overview Introduction • Simulate M ( b ) . E and NE • Accept iff M accepts. Theorem Lemma 1.19 Warm-up Proof Lemma 1.21 Run-time: Since M is polynomial-time and | b | ≤ 2 | y | + 1 , the Summary number of steps M ( b ) requires is exponential-time: References ( 2 | y | + 1 ) c = 2 c | y | + c . Thus L ∈ E . Since our proof holds for any L ∈ NE , NE ⊆ E and thus our lemma is proved: So, if NP − P contains no tally sets, then E = NE . 1.57

  77. The Warm-up Proof Self-Reducibility Technique Amal Fahad, Chetan Bhole, Jonathan Gordon, Mehdi Manshadi We want to prove: Lemma 1.21 Overview If E = NE then NP − P contains no sparse sets Introduction E and NE Theorem First, though, we prove the simpler claim: Lemma 1.19 Warm-up Proof Lemma 1.21 Prove Summary If E = NE then NP − P contains no tally sets. References 1.58

  78. The E = NE ⇒ NP − P contains no tally sets Self-Reducibility Technique Amal Fahad, Chetan Bhole, Let L be a tally set in NP . Jonathan Gordon, Mehdi Manshadi Define L ′ = { x | ( x is 0 or x is a binary string of nonzero length with no leading zeros ) and 1 ( x ) 2 ∈ L } Overview Introduction Using the NPTM N that accepts L , we can give an algorithm to E and NE accept L ′ in nondeterministic exponential time: Theorem Lemma 1.19 Algorithm for L ′ Warm-up Proof Lemma 1.21 On input y , Summary References • Reject if ( y � = 0 and has leading zeros) or y = ǫ . • Otherwise, simulate N ( x ) nondeterministically where x = 1 ( y ) 2 . L ′ ∈ NE , so (by our assumption) L ′ ∈ E . 1.59

  79. The L ∈ P Self-Reducibility Technique Amal Fahad, Chetan Since L ′ ∈ E , there is a deterministic, exponential-time TM that Bhole, Jonathan Gordon, accepts L ′ . We’ll call this TM ME . We can use it to construct a Mehdi Manshadi polynomial-time algorithm for L . Algorithm for L On input y , Overview • Reject if y �∈ 1 k for some k Introduction • Otherwise, write k as 0 if k = 0 or as ( k ) 2 with no leading E and NE Theorem zeros. Lemma 1.19 • Then simulate ME for L ′ on ( k ) 2 . Warm-up Proof Lemma 1.21 Summary References Why is this polynomial-time? Why doesn’t this work for sparse sets? 1.60

  80. The L ∈ P Self-Reducibility Technique Amal Fahad, Chetan Since L ′ ∈ E , there is a deterministic, exponential-time TM that Bhole, Jonathan Gordon, accepts L ′ . We’ll call this TM ME . We can use it to construct a Mehdi Manshadi polynomial-time algorithm for L . Algorithm for L On input y , Overview • Reject if y �∈ 1 k for some k Introduction • Otherwise, write k as 0 if k = 0 or as ( k ) 2 with no leading E and NE Theorem zeros. Lemma 1.19 • Then simulate ME for L ′ on ( k ) 2 . Warm-up Proof Lemma 1.21 Summary References Why is this polynomial-time? Why doesn’t this work for sparse sets? This proof uses the fact that the length of a string in a tally set determines the string (a string of length n must be 1 n ). This is not true for all sparse sets. We need a new encoding. 1.60

  81. The Theorem Proof: Part Two Self-Reducibility Technique Amal Fahad, Chetan Bhole, Jonathan Gordon, Lemma 1.21 Mehdi Manshadi If E = NE then NP − P contains no sparse sets. Let L be a sparse set in NP . We need to show it’s in P . There is some polynomial q such that ( ∀ n )[ || L = n || ≤ q ( n )] Overview Introduction E and NE Hartmanis–Immerman–Sewelson Encoding Theorem Lemma 1.19 Warm-up Proof Lemma 1.21 Summary L ′ { 0 # n # k | || L = n || ≥ k } ∪ = References { 1 # n # c # i # j | ( ∃ z 1 , z 2 , . . . , z c ∈ L = n ) [ z 1 < lex z 2 < lex . . . < lex z c ∧ the j th bit of z i is 1 ] } Since L ∈ NP , we can show L ′ ∈ NE . 1.61

  82. L ′ ∈ NE The Self-Reducibility Technique Amal Fahad, Chetan Algorithm for L ′ Bhole, Jonathan Gordon, Mehdi Manshadi On input y , • If first bit is 0 • Get binary values of n and k • Guess k distinct strings that are n bits long. • If there is such a set of strings that each string is accepted by Overview N L then accept. Introduction • Otherwise, reject. E and NE Theorem • If first bit is 1 Lemma 1.19 • Get binary values of n , c , i , and j . Warm-up Proof Lemma 1.21 • Guess c distinct strings that are n bits long. Summary • Put them in lexical order. References • If there is a set such that each string is accepted by N L and the j th bit of the i th string is 1, accept. • Otherwise, reject. Algorithm is nondeterministic, exponential-time. L ′ ∈ NE , so L ′ ∈ E (by our assumption). This means there is a deterministic, exponential-time TM M ′ which accepts L ′ . 1.62

  83. The L ∈ P Self-Reducibility Technique We give an algorithm to prove that L ∈ P : Amal Fahad, Chetan Bhole, Jonathan Gordon, Mehdi Manshadi On input x , • Let n = | x | • Simulate M ′ on 0 # n # 0, 0 # n # 1, . . . , 0 # n # q ( n ) to see which of them belong to L ′ Overview • Let c = max { k | 0 ≤ k ≤ q ( n ) ∧ 0 # n # k ∈ L ′ = || L = n ||} Introduction E and NE • Now simulate M ′ on Theorem 1 # n # c # 1 # 1 , 1 # n # c # 1 # 2 , . . . , 1 # n # c # 1 # n , Lemma 1.19 Warm-up Proof 1 # n # c # 2 # 1 , 1 # n # c # 2 # 2 , . . . , 1 # n # c # 2 # n , Lemma 1.21 Summary . . . References 1 # n # c # c # 1 , 1 # n # c # c # 2 , . . . , 1 # n # c # c # n • If x is in the set of the n -length strings given by M ′ ’s answers, then accept. Otherwise, reject. This is polynomially many queries. L ′ ∈ E , but each of the polynomially-many queries to the TM M ′ is of length O ( log n ) . Thus the algorithm is polynomial-time. 1.63

  84. The Summary Self-Reducibility Technique Amal Fahad, Chetan Bhole, Jonathan Gordon, Mehdi Manshadi Theorem The following are equivalent: Overview 1 E = NE Introduction 2 NP − P contains no sparse sets E and NE Theorem 3 NP − P contains no tally sets Lemma 1.19 Warm-up Proof Lemma 1.21 We have shown that if NP − P has no tally sets, E = NE and if Summary References E = NE , NP − P has no sparse sets, thus we have proved our theorem. 1.64

  85. The References Self-Reducibility Technique Amal Fahad, Chetan Bhole, Jonathan Gordon, Mehdi Manshadi Overview Introduction • Lane Hemaspaandra and Mitsunori Ogihara. The Complexity E and NE Theorem Companion . Springer: Springer, 2002. Theorem Lemma 1.19 Warm-up Proof Lemma 1.21 Summary References 1.65

  86. The Self-Reducibility Technique Amal Fahad, Chetan Bhole, Jonathan Gordon, Mehdi Manshadi Part IV Overview Sparse NP One More Thing Hardness and Completeness Theorem Proof Algorithm End of Proof Summary References 1.66

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