4 th Week. Relativizations and Hierarchies

Transcription

1 4 th Week Relativizations and Hierarchies Synosis. Oracle Turing Machines and Relativization The Polynomial Hierarchy Generic Oracles Collasing Recursive Oracles The Hierarchy Aril 30, :59

2 Course Schedule: 16 Weeks Subject to Change Week 1: Basic Comutation Models Week 2: NP-Comleteness, Probabilistic and Counting Comlexity Classes Week 3: Sace Comlexity and the Linear Sace Hyothesis Week 4: Relativizations and Hierarchies Week 5: Structural Proerties by Finite Automata Week 6: Stye-2 Comutability, Multi-Valued Functions, and State Comlexity Week 7: Crytograhic Concets for Finite Automata Week 8: Constraint Satisfaction Problems Week 9: Combinatorial Otimization Problems Week 10: Average-Case Comlexity Week 11: Basics of Quantum Information Week 12: BQP, NQP, Quantum NP, and Quantum Finite Automata Week 13: Quantum State Comlexity and Advice Week 14: Quantum Crytograhic Systems Week 15: Quantum Interactive Proofs Week 16: Final Evaluation Day (no lecture)

3 YouTube Videos This lecture series is based on numerous aers of T. Yamakami. He gave conference talks (in English) and invited talks (in English), some of which were videorecorded and uloaded to YouTube. Use the following keywords to find a laylist of those videos. YouTube search keywords: Tomoyuki Yamakami conference invited talk laylist Conference talk video

4 Main References by T. Yamakami L. Fortnow and T. Yamakami. Generic searation. Journal of Comuter and System Sciences, Vol. 52, (1996) S. A. Cook, R. Imagliazzo, and T. Yamakami. A tight relationshi between generic oracles and tye-2 comlexity Theory. Inf. Comut. 137(2): (1997) T. Yamakami. Collasing Recursive Oracles for Relativized Polynomial Hierarchies. In the Proc. of FCT 2005, Lecture Notes in Comuter Science, vol. 3623, (2005) T. Yamakami and Y. Kato. The dissecting ower of regular languages. Inf. Process. Lett. 113(4): (2013) T. Yamakami. Oracle ushdown automata, nondeterministic reducibilities, and the hierarchy over the family of context-free languages. In Proc. of SOFSEM 2014, Lecture Notes in Comuter Science, vol. 8327, (2014). Comlete version is available at arxiv:

5 I. Oracle Turing Machines 1. Oracle Turing Machines 2. Oracle Comutation 3. Turing Reductions 4. Turing Comlete Problems 5. Relativizations and Relativized Worlds

6 Oracle Turing Machines I (revisited) An oracle is an external information source, which can rovide an underlying machine with necessary information via a rocess of query and answer. An oracle Turing machine (OTM) is equied with an extra one-way write-only tae, called a query tae, by which the machine make a query to an oracle. query y oracle one-way: write-only query tae y query word y

7 Oracle Turing Machines II (revisited) Inner state answer oracle two way Inut/work tae query one way query tae (write-only)

8 Oracle Comutation I Let M be an oracle Turing machine (OTM). Let x be any string in Σ*. Let B be an oracle (which is now a language). 1. M starts with inut x. 2. Whenever M writes a query word y on its query tae and enters a query state q query, y is automatically sent to oracle B. 3. The oracle B returns its answer (YES/NO) by changing M s inner state from q query to either q yes or q no, deending on whether y B or y B, resectively. 4. M resumes its comutation, starting with q yes or q no. 5. If M halts, outut M(x). Otherwise, go to Ste 2.

9 Oracle Comutation II (revisited) M: OTM, B: oracle inut x query y 1 Σ* 1. M starts with inut x. 2. Whenever M writes a query word y on its query tae and enters a query state q query, y is automatically sent to B. 3. The oracle B returns its answer (YES/NO) by changing M s inner state from q query, to either q yes or q no. 4. M resumes its comutation, starting with q yes or q no. 5. If M halts, outut M(x). Otherwise, go to Ste 2. outut M(x) answer q NO query y 2 answer q YES y 1 y 2 B

10 Languages Recognized by OTMs Let M be an OTM. Let B be an oracle (which is a language). We define a language recognized by M relative to B. L(M,B) = { x Σ* M B accets x with oracle B }. Note that L(M,B) is deending on the choice of oracle B. If we choose a different oracle, say, C, then L(M,C) may be different from L(M,B).

11 Turing Reductions We have already defined olynomial-time many-one reductions. Here, we define olynomial-time Turing reductions. For two languages A and B, we say that A is olynomialtime Turing reducible (-T-reducible or T -reducible) to B (written as A T B) if there is an OTM M such that A = L(M,B); that is, for every inut x, x A M B accets x via making queries to the oracle B, M runs in olynomial time. In other words, A T B M: -time OTM [ A = L(M,B) ]

12 Turing Comlete Problems I Similar to m -educibility, T -reducibility gives rise to a notion of comleteness for a comlexity class. To emhasize the use of Turing reductions, we often call this comleteness notion by Turing comleteness. A decision roblem (or a language) L is said to be Turing comlete for NP (-T-comlete for NP or T -comlete for NP) if 1) L NP, and 2) A T L holds for every language A NP.

13 Turing Comlete Problems II Note that all T -comlete roblems are also m - comlete, because olynomial-time many-one reductions are also olynomial-time Turing reductions. However, the converse does not hold; namely, X,Y s.t. X m Y and X T Y. (Claim) SAT is Turing comlete for NP. [Cook (1971)] Oen Problem: are all -T-comlete roblems for NP also -m-comlete for NP?

14 Relativizations and Relativized Worlds Providing an oracle to an underlying OTM M (res., a comlexity class C defined by OTMs) is called a relativization of M (res., C). An informal term relativized world means a situation that is caused by a certain oracle. P B = collection of all languages L(M,B) for olynomialtime deterministic oracle Turing machines M NP B = collection of all languages L(M,B) for olynomialtime nondeterministic oracle Turing machines M We will see later a relativized world where P = NP and another relativized word where P NP.

15 II. Relativizations and the Hierarchy 1. Relativizations of P and NP 2. Relativized Worlds for P, NP, and co-np 3. The Polynomial Hierarchy 4. Proerties of the Polynomial Hierarchy 5. Comlexity Class Θ k 6. Advised Comlexity Classes

16 Relativizations of P and NP I Recall that relativization means that we relace the original machines by oracle machines of the same tyes. Recall that P B and NP B are relativizations of P and NP relative to oracle B, resectively. Baker, Gill, and Solvay (1975) roved the following conflicting results. 1) (Claim) There exists an oracle A s.t. P A = NP A. 2) (Claim) There exists an oracle B s.t. P B NP B. Next, we will exlain why these claims are true. However, the above claims do not imly P = NP or P NP.

17 Relativizations of P and NP II (Claim) There exists an oracle A s.t. P A = NP A. Proof Sketch: Choose a PSPACE-comlete set A. Note that P A = P PSPACE = PSPACE because PSPACE is closed under olynomial-time Turing reductions. Similarly, NP A = NP PSPACE = PSPACE by the same reasoning and NP PSPACE. Therefore, P A = NP A follows. QED

18 Relativizations of P and NP III (Claim) There exists an oracle B s.t. P B NP B. Proof Sketch 1. Enumerate all olynomial-time DTMs as M 1,M 2, 2. Let i be a olynomial that bounds the runtime of M i. 3. Define an examle language L A as L A = { x y [ y = x y A ] } for any oracle A. 4. Note that L A NP A for any A (by guessing y and querying it to A). 5. We want to construct the desired set B in the following way.

19 Relativizations of P and NP VI 6. Let B 0 = and let t(1) = 1 (1) For each n 1, run M n on inut 1 t(n) with oracle B n M n makes only n (t(n)) queries to B n-1. Let Q be the set of all queried words. Note that Q Σ t(n) n (t(n)) < 2 t(n). 9. There is a y 0 such that y 0 {0,1} t(n) but y 0 Q. 10.Define B n = B n-1 { y 0 } if M n rejects 1 t(n) with B n-1, and B n = B n-1 otherwise. 11.Thus, x L(M i,b n ) x L Bn. 12.Define t(n+1) s.t. t(n+1) > t(n) and 2 t(n+1) > n+1 (t(n+1)). 13.This means that M i on inut 1 t(n) cannot query strings in { y y {0,1} t(n+1) }. 14.Finally, we set B = Bn n 1 QED

20 Relativized Worlds for P, NP, and co-np Three oracles A,B,C below rovide quite different relativized worlds. (Claims) [Baker-Gill-Solvay (1975)] 1) There is an oracle A s.t. P A NP A conp A NP A. 2) There is an oracle B s.t. P B NP B = conp B. 3) There is an oracle C s.t. P C = NP C conp C NP A. co-np NP NP = co-np co-np NP P P 1) 2) 3) P = NP co-np

21 The Polynomial Hierarchy I Meyer and Stockmeyer (1972,1973) introduced a notion of the olynomial-time hierarchy over NP. Customarily nowadays, we dro the word -time and call this hierarchy the olynomial hierarchy. The olynomial hierarchy consists of the following comlexity classes: for every index k 1, = P; Σ = NP; Π = co-np = P ; Σ = NP ; Π = co-σ Σk Σk k+ 1 k+ 1 k k relativizations comlementation

22 The Polynomial Hierarchy II.. Level 3 Σ 3 Π 3 3 Σ 2 Π 2 Level 2 2 Σ 1 = NP Π 1 = conp Level 1 1 = P inclusion

23 Proerties of the Polynomial Hierarchy We define the comlexity class PH as follows: ( ) PH = Σk Πk k The comlexity class PH includes all classes in the olynomial hierarchy. (Claim) NP PH PSPACE (Claim) If P = NP, then P = PH. (Claim) P PH = NP PH = PH. Oen Problems Is k = Σ k for each k 1? Is k = Σ k Π k for each k 1? PH NP PSPACE

24 Comlexity Class Θ k Wagner (1987) introduced the comlexity class Θ 2. A language A is in Θ 2 there exist a olynomial-time deterministic OTM M, and a language B NP such that 1. A = L(M,B), and 2. on each inut x, M makes queries O(log( x )) times. We can naturally extend Θ 2 to Θ k for any k 2 by setting Θ 1 = P and by relacing NP (= Σ 1 ) in the above definition with Σ k-1. (Claim) For any k 2, Σ k-1 Π k-1 Θ k k. (Claim) Θ 2 PP. [Beigel-Hemachandra-Wechsung (1991)]

25 Advised Comlexity Classes We have seen the notion of advice in Week 3. Let us suly advice to the olynomial-time hierarchy. For any index k 1 and for any language L, L k /oly there exists a language A k and an advice function h such that 1) h(n) = n O(1) for any n N, and 2) L = { x x,h( x ) A }. Similarly, we can define Σ k /oly and Θ k /oly. (Claim) k /oly Θ k /oly Σ k /oly for every k 1.

26 III. Collasing Recursive Oracles 1. Relativized Polynomial Hierarchies 2. Basic Oracle Searations and Collases 3. Collasing Recursive Oracles 4. Searations and Collases 5. Sarse Sets

27 Relativized Polynomial Hierarchies We can relativize the olynomial hierarchy by taking an oracle A. We write k (A), Σ k (A), and Π k (A) as relativizations of k, Σ k, and Π k, resectively, with resect to oracle A. More formally, for every index k 1, we define: ( A) = P ; Σ ( A) = NP ; Π ( A) = co-np A A A ( A) = P ; Σ ( A) = NP ; Π ( A) = co- Σ ( A) Σk ( A) Σk ( A) k+ 1 k+ 1 k k A relativized olynomial hierarchy relative to A is { k (A), Σ k (A), Π k (A) k 1 }. We can also relativize Θ k to oracle A and obtain Θ k (A).

28 Basic Oracle Searations and Collases Yao (1985) constructed a recursive oracle A s.t. (Claim) k (A) Σ k (A) Π k (A) for all k 1. Ko (1989) (Claim) Σ k (B k ) Σ k+1 (B k ) = Σ k+2 (B k ) for each k 1. Heller (1984) (Claim) 2 (C) Σ 2 (C) = Π 2 (C). (Claim) Σ 1 (D) 2 (D) = Σ 2 (D). Bruschi (1992) (Claim) k (E) Σ k (E) = Π k (E) for all k 3. (Claim) Σ k-1 (F) k (F) = Σ k (F) for all k 3. Sheu and Long (1994) (Claim) Θ k (H) k (H) Σ k (H) for all k 2. (Claim) Σ k-1 (J) Θ k (J) = Σ k (J) for all k 2.

29 Collasing Recursive Oracles As we have seen earlier, Ko (1989) constructed recursive oracles (i.e., comutable oracle) that force a relativized olynomial-time hierarchy to collase to any fixed level. We call such oracles collasing oracles for simlicity. Yamakami (2005) defined the collasing recursive oracle olynomial (CROP) hierarchy as: CROΣ k ={ A REC Σ k (A) = Σ k+1 (A) } CRO k ={ A REC k (A) = k+1 (A) } CROΘ k ={ A REC Θ k (A) = Θ k+1 (A) } CROPH = i 1 CRO i REC = class of recursive languages

30 Basic Proerties (Claim) For any k 1, it follows that CRO k CROΣ k CROΘ k CROΠ k. (Claim) All PSPACE-comlete roblems are in CRO 1. A set is said to be coinfinite if its comlement (with resect to a fixed universe) is infinite. Lemma: [Yamakami (2005)] Let k 1 and assume that k Σ k. If A CROΣ k, then A is infinite and coinfinite.

31 Searations and Collases Yamakami (2005) showed the following roerties of the CROP hierarchy. Theorem: [Yamakami (2005)] Let k 1. CRO k CROΣ k CROΘ k CROΠ k. Proosition: [Yamakami (2005)] Let k 1. The following equivalences hold. NP CRO k k = Σ k. NP CROΣ k Σ k = Π k. NP CROΘ k Θ k = Σ k.

32 Sarse Sets A set S over alhabet Σ is called olynomially sarse (or simly sarse) if there is a olynomial such that S Σ n ( x ) for all n N, where Σ n = { x Σ* x n }. Let rsparse = { A A is recursive and sarse }. Proosition: [Yamakami (2005)] Let k rsparse CRO k Σ k k /oly 2. rsparse CROΣ k Π k Σ k /oly 3. rsparse CROΘ k Σ k Θ k /oly

33 IV. Generic Oracles 1. Conditions and Extensions 2. Generic Oracles 3. Comlexity Class UP 4. Searations by Generic Oracles 5. Extension of UP to U k 6. Generic Searations 7. Proof Ideas

34 Conditions and Extensions I Blum and Imagliazzo (1987) considered a class of oracles, which are called generic oracles. Recall that χ A denotes the characteristic function of language A (i.e., x [(x A χ A (x)=1) [(x A χ A (x)=0)]). A condition σ is a artial function from { 0,1 }* to { 0,1 } with a finite domain dom(σ). Hence, a condition is often identified with a string. A condition τ extends σ (denoted by σ τ) if dom(σ) dom(τ) and σ(x) = τ(x) for every x dom(σ). A set A extends σ (denoted by σ A) if χ A (x) = σ(x) for every x dom(σ).

35 Conditions and Extensions II Consider the following examle for (σ,τ,a). x: λ σ: * * * 1 0 * * 1 *... τ: 0 * * * χ A : dom(σ) = { 00, 01, 000,... } A = { 0, 1, 00, 10, 000,... } dom(τ) = { λ, 00, 01, 10, 000, 001,... } (Claim) σ τ A, because: 1. dom(σ) dom(τ) blue & red circles 2. σ(x) = τ(x) for all x dom(σ) red circles 3. τ(x) = χ A (x) for all x dom(τ) blue & red circles

36 Generic Oracles I For an oracle Turing machine M, the machine M σ behaves like M with access to the oracle { x σ(x)=1 } and all queries may be made within dom(σ). When some queries are made outsides of dom(σ), we treat M σ as being undefined. oracle { x σ(x)=1 } queries dom(σ) Σ* M: OTM answers

37 Generic Oracles II A set S of conditions is dense if, for every condition σ, there is another condition τ S such that τ extends σ. A set A Σ* meets a set S of conditions if there is a condition σ S such that A extends σ. A set S is arithmetical if S is exactly definable in firstorder arithmetic. A set G is (Cohen) generic if G meets every dense arithmetic set of conditions. S: dense σ S τ s.t. σ τ meaning: easy to define in an arithmetic way

38 Comlexity Class UP We introduce a comlexity class called UP. For any language L, L UP there exists an NTM M such that, for any x, 1. x L M accets x, 2. x L M rejects x, and 3. M has at most one acceting ath on inut x. We can relativize UP to UP A, using oracle A. x L x L a unique acceting ath accet reject reject

39 Searations by Generic Oracles With generic oracles, we do not have conflicting results for most comlexity classes. For examle: If a single generic oracle G satisfies P G = NP G or P G NP G, then, for all generic oracles H, we obtain P H = NP H or P H NP H, resectively (Claim) [Blum-Imagliazzo (1987)] For any generic oracle G, P G NP G. If P = NP, then P G = UP G = NP G co-np G for any generic oracle G. (*) Can we extend this result to any higher level of the olynomial hierarchy?

40 The Polynomial Hierarchy (revisited).. Level 3 Σ 3 Π 3 3 Σ 2 Π 2 Level 2 2 Σ 1 = NP Π 1 = conp Level 1 1 = P inclusion

41 Extension of UP to U k We have already defined the comlexity class UP. Here, we extend it to fit into the k-th level of the olynomial hierarchy. Fortnow and Yamakami (1996) defined the following extension of UP. Σk U = P; U = UP for any k 1. 1 k+ 1 (Claim) k U k Σ k Π k for each index k 1.

42 Generic Searations Fortnow and Yamakami (1996) roved that the olynomial hierarchy is infinite regarding generic oracles. Theorem: [Fortnow-Yamakami (1996)] Let k 2. U k (G) Π k (G) k (G) for any generic oracle G. As a result, we obtain Σ k (G) Π k (G) k (G). Proof Idea: Let A be any oracle. We define a function f na : {0,1} n {0,1} as follows.

43 Proof Idea I This a new function f n A is defined as: f x = x x x x A n n 1 n 2 2 n 1 n ( ) χa( 0 ) χa( 0 1) χa( 0 1 ) χa( 01 ) Here, we define a ermutation as a bijection (i.e., a oneone and onto function) on all binary strings of length n. Let PERM A = { 1 n f A 2n is a ermutation } Let S 2n = 1 n (0+1) n +(0+1) n-1 0 n+1 (regular exression) Define L(A) = { 1 n PERM A y {0,1} n z {0,1} n [ f 2nA (yz) S 2n ] }

44 Proof Idea II It suffices to rove that, for any k 2, 1) L(A) U k (A) Π k (A) for any oracle A, and 2) L(G) k (G) for any generic oracle G. Item 1) can be shown directly. For the case of k = 2, item 2) can be directly roven. When k 3, the roof of 2) requires the notion of random restrictions and Håstad s switching lemma for circuits. QED

45 Oen Problems Concerning generic oracles, there is still a wide room to obtain interesting results. 1. Is it true that k (G) U k (G) co-u k (G) for any k 2 and for any generic oracle G? 2. Prove more searations and collases of comlexity classes relative to generic oracles. (*) The notion of generic oracle turns out to be closely related to tye-2 comutability. This subject will be discussed in Week 6.

46 V. Many-One Reductions by 1nda s 1. m-reduction 1nda s as Oracle 1nda s 2. Nondeterministic Pushdown Automata with Query Taes 3. Many-One -Reducibility 4. Examles 5. Characterization by Dyck Languages 6. K-Fold Alication of Reductions

47 m-reduction 1nda s as Oracle 1nda s Yamakami (2014) considered nondeterministic many-one reducibility based on 1nda s. An m-reduction 1nda or an oracle 1nda M M = (Q,Σ,{,$},Θ,Γ,δ,q 0,Z 0,Q acc,q rej ) is a standard 1nda lus a write-only query tae and a secial transition function δ: This is because all context-free languages * δ : ( Q Q ) ( Σ { λ}) Γ PQ ( Γ ( Θ { λ})) halt Termination condition of M: are recognized by O(n)-time 1nda s. All comutation aths (both acceting and rejecting) should terminate (reaching halting states) within O(n) stes. ACC M (x) = set of acceting comutation aths of M on x

48 One-Way Nondeterministic Pushdown Automata with Query Taes alhabet Γ Inner state answer oracle one way alhabet Σ one way inut tae (read-only) query stack (write/read) alhabet Θ query tae (write-only)

49 Many-One -Reducibility Let L be a language over Σ and A be a language over Θ. L is many-one -reducible to A M (m-reduction 1nda) s.t. x Σ* 1) along any acceting ath, M on x roduces a valid string y Θ* (called a query word) on a query tae, and 2) x L ACC M (x) [ y A ]. In this case, we write L A m or L m (A). The language A is customarily called an oracle. Given a language family F, F m = m (F) = union of A m for all A F.

50 Examle: Du 2 I Consider the following language Du 2. Du 2 = { xx x Σ* } (dulication) over alhabet Σ Is not context-free Is many-one -reducible to A = { x R #x x Σ* }. Belongs to m A m. A reduction is made by the following oracle 1nda M. 1. On inut w, nondeterministically slit it into xy. 2. Using a stack, roduce x R #y on a query tae, where # is a secial symbol not in Σ. 3. Make a query to oracle. 4. If oracle answers yes, accet w; otherwise, reject w. (*) See the next slide for illustration.

51 Examle: Du 2 II The oracle 1nda M works as follows with oracle A. (again) 1. On inut w, nondeterministically slit it into xy. 2. Using a stack, roduce x R #y on a query tae, where # is a secial symbol not in Σ. 3. Make a query to oracle. 4. If oracle A answers yes, accet w; otherwise, reject w. w = 1010 λ#1010 1#010 01#10 A A A no no yes reject reject accet inut query answer slit & reverse A = { x R #x x Σ* }

52 Examle: Du 2 III The oracle nda M for Du 2 (again) 1. On inut w, nondeterministically slit it into xy. 2. Using a stack, roduce x R #y on a query tae, where # is a secial symbol. 3. Make a query to oracle. 4. If oracle answers yes, accet w; otherwise, reject w. More formally, we define M as follows: δ( q0, cz, 0) = {( q0, Z0, λ)} δ ( q0,$, Z0) = {( qacc, Z0,#)} δ( q0, σ, Z0) = {( q1, σz0, λ)} δ ( q, σ, τ ) = {( q, στ, λ),( q, στ, λ)} where στ Σ, δ( q2, λτ, ) = {( q2, λτ, )} δ( q2, λ, Z0) = {( q3, Z0,#)} δ( q3, σ, Z0) = {( q3, Z0, σ)} δ( q,$, Z ) = {( q, Z, λ)} 3 0 acc 0 δ λ λ * : ( Q Qhalt ) ( Σ { }) Γ PQ ( Γ ( Θ { }))

53 Similar Examles Let us consider other but similar languages. Du 3 = { xxx x Σ* } (3 coies) over Σ Is not context-free Is many-one -reducible to B = { x R #x#x#x R x Σ* }. Belongs to B m m. Match = { x#w u,v Σ* [ w = uxv ] } (matching) over Σ Is not context-free Is many-one -reducible to C = { x R #x x Σ* } Belongs to C m m. Consequence: m. Comare this with NP = NP NP m.

54 Examle: Sq I Consider a slightly more comlex language. 2 n n Sq = {0 1 n 1} (squared length) over alhabet {0,1} Is not context-free Is -m-reducible to D = = i j1 j2 j { w 0 #1 #1 # #1 k (i) and (ii) }, where (i) j = j, j = j, (ii) i = k D Belongs to m D m. The second item shown above can be roven by the following oracle 1nda M using D as an oracle.

55 Examle: Sq II D = = i j1 j2 j { w 0 #1 #1 # #1 k (i) and (ii) }, where (i) j = j, j = j, (ii) i = k The desired oracle 1nda M works as follows. 1. On inut w, check if w is of the form 0 i 1 j for some i,j Simultaneously, guess (j 1,j 2,,j k ) to satisfy: (1) j = j1+ j2 + + j k (2) j = j, j = j, Produce w (shown in the above box) on a query tae. 4. Make a query to oracle D. 5. If yes, then accet w; otherwise, reject w.

56 Characterization by Dyck Languages Dyck languages over Σ = { σ 1,σ 2,,σ d } { σ 1,σ 2,,σ d } A Dyck language L is generated by a context-free grammar with the following roduction set: { S λ SS σ Sσ : i = 1,2,..., d } E.g., When d = 2, L is a set of all balanced arentheses. DYCK = set of all Dyck languages Proosition: [Yamakami (2014)] i m = m D = m DYCK = NFA m DYCK = m (NFA m DYCK ). This lemma suggest that DYCK may be considered as the most difficult language in under many-one reductions. i

57 K-Fold Alication of Reductions Many-one -reducibility lacks the transitivity roerty. K-fold alication of reductions m[1] A = m A m[k+1] A = m ( m[k]a ) m[k] F = union of m[k] A for all A F. K-conjunctive closure of (k) = { A B 1,B 2,,B k [ A = B 1 B 2 B k ] } (ω) = (1) (2) Namely, m. { (k) k N + } is an infinite hierarchy. [Liu-Weiner (1973)] Lemma: [Yamakami (2014)] m[k] = m (k) and U k 1 m[k] = m (ω).

58 Inclusion Relations among Language Families TC 1 CSL DSPACE(O(n)) H roer inclusion inclusion no inclusion NC 2 Π 3 Σ 3 AC 0 () = LOG = SAC 1 NL P Π 2 BH (ω) Σ 2 m (ω) m (2) = m[2] L AC 0 (REG) = NC 1 BP 3 2 co- = Π 1 (3) (2) Σ 1 = m (1) = m[1] /n REG/n REG

59 VI. Turing Reductions by 1nda s 1. T-Reduction 1nda s as Oracle 1nda s 2. Adative Queries in Nondeterministic Comutation 3. Turing -Reducibility 4. Examle 5. The Boolean Hierarchy over

60 T-Reduction 1nda s as Oracle 1nda s Similarly to m-reductions, Yamakami (2014) considered Turing reductions. A T-reduction 1nda or an oracle 1nda M M = (Q,Σ,{,$},Θ,Γ,δ,q 0,Z 0, Q oracle,q acc,q rej ) is a standard 1nda with a write-only query tae, a secial state set Q oracle, and δ such that Q = { q, q, q } oracle query yes no * : ( Q Qhalt { qquery}) ( Σ { }) Γ P(( Q { qyes, qno}) Γ ( Θ { })) δ λ λ Termination condition of M: All comutation aths (both acceting and rejecting aths) should terminate (reaching halting states) within O(n) time, no matter what oracle is rovided.

61 Adative Queries in Nondeterministic Comutation Let M be an oracle 1nda. Inut x is given. M queries y 1 along a comutation ath. The query tae is automatically reset. Deending on an oracle answer, M makes another query on y 2 along ath. M continues making queries. At the time when ath terminates, M must enter a halting state. A oracle A answer query answer ath accet query x y 1 y 1 y 2 y 2 reject

62 Turing -Reducibility Let L be a language over Σ and A be a language over Θ. L is Turing -reducible to A M (T-reduction 1nda) s.t. x Σ* x L M accets x using oracle A termination condition of M In the above case, we write L T A or L T (A). Given a language family F, let All comutation aths (both acceting and rejecting) should terminate (reaching halting states) within O(n) time, no matter what oracle is rovided. T F = T (F) = union of T A for all A F.

63 The Boolean Hierarchy over There is a nice relationshi between T A and m A (as well as NFA m A ). Proosition [Yamakami (2014)] T = m ( 2 ) = NFA m ( 2 ). We define the notation of the Boolean hierarchy (over ). Boolean hierarchy over [Yamakami-Kato (2013)] 1 = 2k = { A B A 2k-1 B co- } 2k+1 = { A B A 2k B } BH = E.g., 2 = { A B A, B co- } Since co- 2, we obtain 2.

64 Inclusion Relations among Language Families TC 1 CSL DSPACE(O(n)) H roer inclusion inclusion no inclusion NC 2 Π 3 Σ 3 AC 0 () = LOG = SAC 1 NL P Π 2 BH (ω) Σ 2 m (ω) m (2) = m[2] L AC 0 (REG) = NC 1 BP 3 2 co- = Π 1 (3) (2) Σ 1 = m (1) = m[1] /n REG/n REG

65 VII. The Hierarchy 1. The Hierarchy (H) 2. Closure Proerties of H 3. Examles 4. Structural Proerties of H 5. Relationshis to the Polynomial Hierarchy

66 The Hierarchy (H) I Yamakami (2014) defined a hierarchy over using oracle 1nda s. The hierarchy is { k, Σ k, Π k k N } whose elements are defined as follows. 1 = D Σ 1 = k+1 = D T (Σ k ) for any k 1 Σ k+1 = T (Σ k ) for any k 1 Π k = co- Σ k for any k 1 We further define H as H = Σ 1 Σ 2 Σ 3.

67 The Hierarchy (H) II The hierarch satisfies the following basic roerties. Proosition: [Yamakami (2014)] Let k 1. 1) T (Σ k ) = T (Π k ) 2) D T (Σ k ) = D T (Π k ) 3) Σ k Π k k+1 Σ k+1 Π k+1 4) H DSPACE(O(n)) NOTE: item 4) comes from the termination condition of oracle 1nda s.

68 Closure Proerties of H Each Σ k (k 1) is closed under the following oerations. Length-nondecreasing substitution Concatenation Union Reversal Kleene closure λ-free homomorhism Inverse homomorhism A substitution s:σ P(Θ*) is length nondecreasing s(σ) and λ s(σ) for all σ Σ. A homomorhism h:σ Θ* is λ- free h(σ) λ for all σ Σ. Σ 1 is not closed under intersection nor comlementation, because Σ 1 =.

69 Examles of Languages Some languages are nicely laced in the hierarchy. Du 2 = { xx x {0,1}* } (dulication) Du 3 = { xxx x {0,1}* } (3 coies) Sq = { 0 n 1 k k = n 2, n N } (squared length) Prim = { 0 n n is a rime } (rime length) Σ 2 Π 2 Σ 1 Π 1 Σ 1 Π 1

70 Inclusion Relations among Language Families TC 1 CSL DSPACE(O(n)) H roer inclusion inclusion no inclusion NC 2 Π 3 Σ 3 AC 0 () = LOG = SAC 1 NL P Π 2 BH (ω) Σ 2 m (ω) m (2) = m[2] L AC 0 (REG) = NC 1 BP 3 2 co- = Π 1 (3) (2) Σ 1 = m (1) = m[1] /n REG/n REG

71 Structural Proerties of H The hierarchy has the following roerties. Theorem (uward collase roerties): [Yamakami (2014)] Let k 2 1. Σ k = Σ k+1 H = Σ k. 2. Σ k = Π k BHΣ k = Σ k. 3. Σ k = Π k Σ k = Σ k+1. Similarly to e and BH, Yamakami (2014) defined: Boolean Hierarchy over Σ k Σ k,e = e-th level of the Boolean hierarchy over Σ k. BHΣ k = union of Σ k,e for all e N.

72 Relationshis to the Polynomial Hierarchy The hierarchy has a close connection to the olynomial hierarchy. Theorem: [Yamakami (2014)] Let k 1. If Σ k+1 = Σ k+2, then Σ k = Σ k+1. In other words, if the olynomial hierarchy is truly an infinite hierarchy, then so is the hierarchy. Proof Idea: It suffices to show that each Σ k+1 contains a language that is -T-comlete for Σ k.

73 Oen roblems Prove that Σ k+1 Σ k+2 for all k 1. Note that roving that Σ k+1 = Σ k+2 is much more difficult because this imlies Σ k P = Σ k+1p. Find interesting relativized worlds regarding Σ k. E.g., BP A Σ 2 (A) Π 2 (A) for some A. Prove the REG-dissectability of Σ k+1. The dissectability will be discussed in Week 5.

74

75 Q & A I m hay to take your question!

76 END