Probabilistic Autoreductions

Transcription

1 Probabilistic Autoreductions Liyu Zhang University of Texas Rio Grande Valley Joint Work with Chen Yuan and Haibin Kan SOFSEM

2 Overview Introduction to Autoreducibility Previous Results Main Result 1: Complete sets for NP are autoreducible for the probabilistic many-one (Karp) reduction. Main Result 2: Complete sets for classes in the truth-table polynomial hierarchy are autoreducible for the probabilistic truth-table reduction. Q & A 2

3 Autoreducibility: Definition A set A is r-autoreducible set if A reduces to itself via a r-reduction that does not query on the input element itself. Being autoreducible means redundancy: Information about x A? can be retrieved from the information about whether y 1, y 2,..., y m A, where each y i x. 3

4 Autoreducibility: Background The research on autoreducibilities of (complete) sets started as early as 1970 s (Trakhtenbrot) mainly in the recursive setting. Ambos-Spies (1984) translated the notion of autoreducibility to the polynomial-time setting. It attracted more attention in the past two decades (see survey by Glaßer et al., 2009) because it was considered a promising property to apply the Post s program (1944) in order to attacking major unsolved separation problems in complexity theory (Buhrman and Torenvliet, 2005 and Buhrman et al. 2000). 4

5 Current Status: Autoreducibility of Complete Sets p m p 1-tt p tt p T NP yes yes open yes Σ P i yes yes open yes Π P i yes yes open yes P i yes yes yes yes PSPACE yes yes open yes EXP yes yes open yes NEXP yes yes open open Solving highlighted problems will lead to major breakthrough in complexity theory (Buhrman et al, ) 5

6 Autoreducibility: This Work This work: we investigate autoreducibilities under probabilistic reductions and attempt to extend some of the previous results on deterministic reductions to their probabilistic counterparts. 6

7 Notations We use standard notations: Complexity classes: P, NP, Σ P k, PH, RP, BPP,etc. reductions (all polynomial time): p m, p tt, p T, etc. 7

8 Many-one Autoreducible Sets For the common many-one ( p m, Karp) reduction, a set A is autoreducible if there exists a polynomial-time computable function f such that for every x, x A f(x) A, and f(x) x 8

9 Many-one Autoreducible Sets: An Illustration x f(x) A f(f(x))... Figure 1: Many-one Autoreducible Sets. x A f(x) A f(f(x)) A x f(x) f(f(x)) (but f(f(x)) might equal x) 9

10 Autoreducibility of Known NP-complete Sets Most known NP-complete (for the many-one reduction) sets are autoreducible simply because they are p-cylinders. Example: f(x) = x TRUE is an autoreduction of SAT, the set of satisfiable propositional formulas: x, x TRUE x. x, x SAT x TRUE SAT. Question: Are all NP-complete sets are autoreducible? Note that if the Isomorphism Conjecture by Hartmanis and Berman is true, then trivially all NP-complete sets are many-one autoreducible. 10

11 p m -Autoreducibility of Complete Sets Glaßer, Ogihara, Pavan, Selman and Zhang, 2007: For the p m reduction, all Nontrivial NP-complete sets are autoreducible, and so are complete sets of many other classes: P, EXP, PSPACE, etc. Note: Here we define a set A to be nontrivial if both A and A are no less than 2. We assume that all sets in our discussion are nontrivial. 11

12 Autoreducibility Proof of NPC Sets: Basic Setup We show the proof idea of Glaßer et al. s result using an NP-complete Set. Basic Setup: Let A be an NPC set accepted by an NTM N in time p(n) for a polynomial p. Assume each computation path of N is a string over {0, 1} of length p(n). Define the following left set (Ogiwara and Watanabe, 1991) B of A: B = { x, y N accepts x along some path w such that y w }, where is based on the common lexicographical ordering. Clearly B NP and therefore reduces to A via some reduction f. Let {a, b} A and {c, d} A. 12

13 Autoreducibility Proof for NPC Sets: Constructing Autoreduction Constructing the autoreduction g ; Case 1: z = f( x, 0 m ) x. Output z. Case 2: z = f( x, 1 m ) = x. Subcase 2.1: N accepts x along the path 1 m. Output {a, b} {x} Subcase 2.2: N does not accept x along the path 1 m. Output {c, d} x. Case 3: Not in cases 1 or 2. Find w such that f( x, w ) = x and z = f( x, w + 1 ) x. Subcase 3.1: N accepts x along the path w. Output {a, b} {x}. Subcase 3.2: N does not accept x along the path w + 1. Output z. 13

14 Autoreducibility Proof for NPC Sets: An Illustration x 0^m w w+1 1^m f(x,0^m)=x f(x,w) = x f(x,w+1)!= x f(x,1^m)!= x Figure 2: Computation Tree of N on input x. Each leave if labeled with f( x, y ), where y is the computation path between the leave and the root. 14

15 Autoreducibility Proof for NPC Sets: Correctness The function g defined in the last slide is an p m -autoreduction for A due to the following observations: g is polynomial-time computable. ( u {0, 1} p( x ) ) f( x, u ) A. f( x, 0 m ) A iff x A. f( x, 1 m ) A iff N accepts x along the path 1 m. For w {0, 1} m {0 m, 1 m }, if N does not accept x along the path w, then f( x, w ) A iff f( x, w + 1 ) A. 15

16 Autoreducibility of Other Complete Sets Extending the autoreducibility result to complete sets of other classes, for example, PSPACE, is not straightforward. Additional properties of those complete sets must be used in a nontrivial way in order to prove their autoreducibility. 16

17 RP Many-one Autoreducible Sets Now we consider the following RP-type probabilistic many-one reduction due to Valiant and Vazirani (1986): A set A is RP many-one reducible ( rp m ) to another set B if there exists a probabilistic polynomial-time computable function f and a polynomial q such that for every input x, x A Pr[f(x) A] 1/q( x ), and x A Pr[f(x) A] = 0. Furthermore, A is rp m -autoreducible if A rp m A via a function f and f(x) x for every input x. 17

18 rp m -Autoreducibility Proof for NPC Sets To prove rp m -autoreducibility of rp m -complete sets for NP, use the following function f in place of the function f in the proof for the p m -autoreducibility of NPC sets: Function f : 0 Input x, y, where x = n and y = p(n) 1 Run f on x, y for r(n) times for some polynomial r and let S be the set of the r(n) outputs from Step 2. 2 If x S, output x 3 Else output z, where z is a randomly selected from S. Note: Here r(n) is a polynomial to be determined later. 18

19 rp m -Autoreducibility Proof for NPC Sets, cont d The rp m -autoreduction g for a rp m -complete set A for NP is similar to the one for the NPC set A with two changes: f is replaced by f. The output in Subcase 3.2 will be now determined in the following way: Run f on x, w + 1 again and yield output z Output z ifz x, or {b1, b2} {x} otherwise. Note: The 2nd change above is important and to ensure the autoreduction outputs an appropriate element with sufficient probability. 19

20 rp m -Autoreducibility Proof for NPC Set, cont d rp m -autoreduction for rp m -complete sets for NP: Case 1: z = f ( x, 0 m ) x. Output z. Case 2: z = f ( x, 1 m ) = x. Subcase 2.1: N accepts x along the path 1 m. Output {a, b} {x} Subcase 2.2: N does not accept x along the path 1 m. Output {c, d} x. Case 3: Not in cases 1 or 2. Find w such that f ( x, w ) = x and z = f ( x, w + 1 ) x. Subcase 3.1: N accepts x along the path w. Output {a, b} {x}. Subcase 3.2: Run f on x, w + 1 again and yield output z. Output z ifz x, or {b1, b2} {x} otherwise. 20

21 Proof of Correctness for rp m -autoreduction To prove the correctness of g, we first make the following straightforward observations: g is polynomial-time computable. For every input x, Pr[g (x) x] = 1. For every input x A, Pr[g (x) A] = 0. 21

22 Proof of Correctness for rp m -autoreduction, cont d Now it only remains to show that ( polynomial q ) ( x A) Pr[g(x) A] 1/q ( x ). We consider the following types of strings u {0, 1} p(n) for a fixed x with x = n: good: if Pr[f( x, u ) = x] 1/2q(n). (f( x, u ) output a non-x element with sufficient likelihood.) bad: otherwise. 22

23 Proof of Correctness for rp m -autoreduction, cont d We arrived at three cases: Case 1: 0 m is good. Easy since f( x, 0 m ) x with sufficiently large probability. Case 2: 1 m is bad. Easy since f( x, 1 m ) = x with sufficiently large probability. Case 3: 0 m is bad and 1 m is good. Need more work. 23

24 Proof of Correctness for rp m -autoreduction, cont d Case 3 for rp m -autoreduction: 0 m is bad and 1 m is good. Still need to find w, where f ( x, w ) = x and z = f ( x, w + 1 ) x. but need to ensure that an appropriate w is found with sufficiently large probability. We observe that it suffices that all the events S i defined below occur in order to warrant an appropriate w : S i : u i is bad f ( x, u i ) = x, where u i denotes the i-th string for which f( x, u i ) is computed during the search for w. 24

25 Proof of Correctness for rp m -autoreduction: The Last Step Finally we were able to prove that the probability for i S i is at least 1/t(n) for some polynomial t with the following observation where appropriate r(n) is used in f : For every u Σ m and s = f ( x, u ), i. if u is good, then (a) Pr[s x] 1 2p(n), and (b) x, u B Pr[s L s x] 1 r(n) for some polynomial r(n). ii. if u is bad, then Pr[s = x] > 1 2 p(n). 25

26 BPP Truth-table Reduction: Definition BPP Truth-table Reduction: A bpp tt B if there exist a probabilistic polynomial-time oracle algorithm A, and a (deterministically) polynomial-time computable function g, where for every input x: g(x) outputs all queries A will make to B. If x A, then Pr[A B accepts x] 2 3. If x A, then Pr[A B accepts x] 1 3. Using probability amplification: x, x. 26

27 bpp tt -Autoreducibility of NPC Sets Let L be a bpp tt -complete set for NP. Fix the following: r L : a length-increasing p m -reduction from L to SAT. r vv : The rp m -reduction from SAT to USAT with one-sided error 1/4 by Valiant and Vazirani (1986). T = φ, 0i φ SAT has a satisfying assignment where the i-th variable is true. 27

28 bpp tt -Autoreducibility of NPC Sets: an Algorithm Buhrman et al. (2000) used the following algorithm M to show that p tt -complete sets for NP are bpp tt -autoreducible: 0 Input x 1 Compute φ = r L (x) and ψ = r vv (φ). 2 Determine the memberships of ψ, 0 i in T by making truth-table queries to L {x} and L {x}, respectively. 3 Let a 0 and a 1 be the two assignments induced by the two sets of memberships of ψ, 0 i in T, as determined in Line 2 4 Evaluating ψ(a 0 ) and ψ(a 1 ). 5 If the above ψ(a 0 ) or ψ(a 1 ) evaluates to true, ACCEPT. 6 REJECT. 28

29 bpp tt -Autoreducibility of NPC Sets: Correctness The correctness of the algorithm M is as follows: x L φ = r L (x) SAT Pr[ψ = r vv (φ) SAT] = 0 (By Valiant and Vazirani (1986)) Pr[ M accepts φ] = 0 (Due to verification of the satisfying assignment at line 5) x L φ SAT Pr[ψ = r vv (φ) SAT] 1 4 (By Valiant and Vazirani (1986)) Pr[ M accepts φ] 1 4 (since a 0 or a 1 is a satisfying assignment) 29

30 Generalizing Algorithm M We attempt to generalize algorithm M to a bpp tt -autoreduction for an arbitrary complete set for a higher class in PH: Define SAT (k) to serve as the role of SAT for Σ P k : SAT (1) = SAT and SAT (k) = SAT SAT(k 1) for every k 2. SAT A is the same as SAT except literals can be of the form A(w), where w consists of variables and T/F s and A(w) = 1 iff w A. It s well known that SAT (k) is p m -complete for Σ P k. 30

31 Generalizing Algorithm M, cont d Generalizing Algorithm M, cont d: Consider a bpp tt -complete set L k for Σ P k Use a p m -reduction r k from L k to SAT (k)., where k 2. The other components of M can be generalized in a similar fashion: r vv, T, etc. However, there is a critical problem... 31

32 Generalizing Algorithm M: A Critical Problem Problem: Line 4 of the algorithm M now requires p T -access to a p T -complete set for ΣP k 1 since now they might contain literals of the form SAT (k 1) (w). 4 Evaluating ψ(a 0 ) and ψ(a 1 ). 32

33 Generalizing Algorithm M: A Critical Problem, cont d To address the problem in the last slide: limit L k to be a bpp tt Σ P k in the PH defined using bpp Now the desired bpp tt a bpp tt -complete set for Σ P,tt k, the counterpart of tt instead of p T. -autoreduction A for Σ P,tt k -reduction (not autoreduction) from SAT (k 1) to L k in needs to utilize order to resolve queries to SAT (k 1), which recursively requires the same reduction from SAT (k 2) to L k, and so on. Some extra care is also needed in order for the bpp tt -autoreduction A to decide L k with sufficiently small two-sided error probability. 33

34 Putting It Together: An bpp tt -autoreduction for L k A k : An probabilistic algorithm with truth-table oracle access to L k that on input φ, y, 0 n decides whether φ SAT (k) with high probability ( 1/2 n ), without querying on y. To construct A k, we also need to construct A k 1, A k 2,..., A 1. A: the bpp tt -autoreduction for L k that on input x computes φ = r k (x), runs A k on φ, x, 0 n and accepts iff A k accepts. 34

35 Corollaries of Main Result 2 Corollary: All bpp tt -autoreducible: bpp tt -complete sets in the following classes are PH tt : the counterpart of PH for p tt reductions. Θ P -levels (Wagner, 1990): Θ P 0 = P, and Θ P k+1 = LΣP k, where L denotes a log-space oracle Turing machine. 35

36 Probabilistic Autoreductions Questions? Thank You! 36