Axiomatic Approach to Barriers in Complexity

1. Axiomatic Approach to Barriers in Complexity Russell Impagliazzo (IAS & UCSD) Valentine Kabanets (IAS & SFU) AntoninaKolokolova (MUN)

2. P ? NP BPP =? P Algebrization ridge IP=PSPACE Relativization gorge P  EXP

3. Let’s compute everything! Hilbert You can’t. Gödel, Turing Let’s prove we can’t in efficient realm, too! Cook, Karp,... The way you are trying, you can’t. BGS, RR,AW...

4. Proving unprovability • 1900: Hilbert’s program • Let’s axiomatize all of mathematics! • 1930: Godel’s incompleteness theorem • There are true mathematical statements that cannot be proven in a (reasonable) theory. • 1937: Turing’s uncomputability • Uses Cantor’s diagonalization as a major tool. • If both L and its complement are semi-decidable (verifiable), then L is decidable. Halt ____ Halt

5. Diagonalization • Idea: list all possible Turing machines • For each of them, look at the inputs it accepts (its “language”). • Show that there is a TM that is not on the list by constructing one different from any TM in the list on some input. • Therefore, there is a language D on which any Turing machine will make an error.

6. Scaling down to complexity • 1964-66 Cobham, Edmonds, Rabin • Polynomial time = efficient. • 1965: Hartmanis/Stearns • There are problems solvable in exponential time, but not polynomial time (use diagonalization). • Same kind of hierarchy theorems can be proven for space and non-deterministic time and space. • Can this be extended to solve P vs. NP?

7. P vs. NP • 1956: Yablonky (in Russia) • Thought he solvedthe “perebor” (brute-force search) problem. • Showed that for some class of algorithms it is impossible to eliminate brute-force search. • 1956: Godel’s lost letter • Do tautologies have proofs of size “not too much larger” than the size of formulas? • 1971: Cook,Levin, then Karp • Concept of NP-completeness, showing many problems are NP-complete.

8. The Plot Summary • c. 1971: Quest for “P  NP” • 1975:Relativization • 1990s:IP=PSPACE, NEXP=MIP, NP=PCP[log n, 1], … • 2008:Algebrization ~ ~ PBNPB PA = NPA PBNPB PA = NPA

9. “Natural Proof” Monster … Razborov 95, bounded arithmetic framework P/poly NP [RR’97]

10. Oracles • 1939: Turing, already, considered allowing his machines access to a “source of intuition” • Such a machine could “ask queries” to some source of knowledge (call it an “oracle”) by writing a query on a tape and immediately getting an answer • But the diagonalization argument for undecidability works for such machines!

11. Relativization • 1975: Baker/Gill/Solovay • There is an oracle A such that some polynomial-time machine with access to it is as powerful as any NP machine with access to that oracle. • Take A to be any PSPACE-complete language. • There is another oracle B such that NP with access to it is provably stronger than P with access to it. • Construct B using diagonalization, or take a random oracle. • Diagonalization alone cannot be used to resolve P vs NP. • Same is true for many other complexity questions. PBNPB PA = NPA

12. Logically speaking … What does it mean to have contradictory relativizations ? Intuitively, “P vs. NP” should be independent of the “Relativizing Complexity Theory”. Oracle worlds ¼ models of a theory [BGS’75] PBNPB PA = NPA

13. Oracle worlds and models

14. AIV92 approach • Take a theory of arithmetic (unbounded) and add function symbols. • Give limited amount of information about properties of these functions. • What can be proven about these functions with the full power of arithmetic, but knowing only a few facts about them?

15. Relativizing Complexity Theory (RCT) [Arora, Impagliazzo, Vazirani ’92] Axiomatization of PolyTime Computation s.t. { PA | any oracle A } = { stand. models of RCT }. • Interpretation: Complexity statements provable in RCT are relativizing statements. • Consequence:Non-relativizing statements (such as P  NP ) are independent of RCT.

16. Cobham’s characterization of P [Cob64]: P is the minimal class of functions that • Contain basic functions • +, |x|,x*y, bit(x,i),projection, 2|x|*|x| • Closed under function composition • If f and g are in P, then so is fo g. • And under limited recursion on notation • f(x,0)=g(x), f(x,k)=h(x,f(x,k/2)), • |f(x,k)|<poly(x,k)

17. RCT : more details 2-sorted FO extension of Number Theory. f,g,h,… functions from NtoN; x,y,z,…numbers. Cobham’s axioms for P’ (without minimality) • basic functions: e.g., +, -, *, # … 2 P’ • closures: e.g., f, g 2 P’ ) f o g 2 P’ Add: • Induction (unrestricted), • Length axiom (outputs of f2P’ polybounded), • Universality ( P’ has a universal function)

18. RCT : Defining other classes Define all complexity classes in terms ofP’. • f 2 NP’ iff9 c2 N 9 g2 P’ 8 x 2 N f(x) = 1 ,9 y |y| < |x|c & g(x,y) = 1 • f 2 EXP’ iff9 c 2 N 9 g2 P’ 8 x 2 N f(x) = 1 ,g( x, 2|x|c ) = 1 • PSPACE’ = Reachability Problems on digraphs with Adjacency(x,y)2 P’

19. Bounded arithmetic vs. AIV92 • Bounded arithmetic • Reasoning power of a system (induction) is restricted • If polynomial-time functions are introduced, they are exactly the polytime functions • Question: what can be proved about polytime functions with restricted power of reasoning? RCT • Reasoning power of a system is not restricted • Function symbols have some properties of polytime functions • Question: what can be proved about functions from limited known properties?

20. RCT : Computation is a “black box” [AIV’92]:Using RCT’s definition of PolyTime, can only prove relativizing complexity statements.

21. Opening up the “black box” What other properties does PolyTime have ?

22. Local Checkability of Computation tape symbols time 2x3 window tableu of computation Turing machine computation is correct iff all 2x3 “local windows” are legal. (Used by Cook & Levin to show 3-SAT is NP-complete)

23. LCT: Local Checkability Theorem [AIV’92]: Extend RCT with an axiom stating that computation is locally checkable. LCT Axiom: L  NP  P-uniform 3-cnf family {n} s.t. x1…xn L iffy1…ym , n+m (x1,…,xn, y1,…,ym), for m = poly(n). LCT holds in “real world” (by Cook-Levin).

24. LCT Theory = RCT + LCT axiom [AIV’92]: • LCT proves PSPACE=IP, NP=PCP[log n, 1], … • If oracle O satisfies LCT, then O 2 NP/poly coNP/poly. LCT is “too strong”: Provability in LCT is almost the same as provability in “real world”. Versions of LCT with more uniformity are even more restrictive.

25. RCT vs. LCT Theory capturing “arithmetization” ??? “Non-black-box”/ essentially all techniques “Black-box”/ relativizing techniques RCT is strictly weaker than LCT. e.g., IP=PSPACE is provable in LCT, but not in RCT.

26. Arithmetization • Technique used to prove non-relativizing results such as IP=PSPACE • Treat Boolean formulas as polynomials over a larger field. Over Boolean values, the two agree. • (x Æ y) turns into x*y, (x Ç y) into 1-(1-x)*(1-y). • Now, the values of the polynomial can be computed for any integer, giving extra information.

27. IP=PSPACE [LFKN, Shamir, 1990] • PSPACE: True Quantified Boolean Formulae problem • Interactive Proofs: L 2 IP[k] if there is a probabilistic polytime verifier V such that • x 2 L then 9prover P so that Pr(V accepts)¸ 2/3 • : (x 2 L) then 8provers P, Pr(V accepts) · 1/3 Á2 TQBF? ArithmetizeÁ as p() p(x1) V P if p(0)+p(1) is ok, send random a1 p(a1, x2) ...

28. Two Approaches to capturing arithmetization: [For’94 ] & [AW’08]

29. Fortnow’s self-algebrizing oracles ~ Def: Language A is self-algebrizing if its multilinear polynomial extension A is in PA. ( PA-computable family of multilinear poly’s {pn} over Zs.t., for all Boolean x1, …, xn , A(x1, …, xn ) = pn (x1, …, xn). ) [Fortnow’94]:For every language L, there is a self-algebrizing language A such that • L 2PA , and • A 2PSPACEL .

30. Self-algebrizing oracles [Fortnow’94]:For every self-algebrizing language A, IPA = PSPACEA. Say that a complexity statement Fortnow-algebrizes if it holds relative to every self-algebrizing oracle. But what complexity statements fail to Fortnow-algebrize ???

31. Aaronson & Wigderson’s Approach ~ Def [algebrizing inclusion]:For complexity classes C and D, the inclusion Cµ DAW-algebrizes if for any A, CADA, where A is a low-degree polynomial extension of A. Def [algebrizing separation]: For C and D, C  D AW-algebrizes if for any A, CADA. ~ ~

32. AW-Algebrization [AW’08]: Many known results algebrize (e.g., PSPACE=IP, OWF  NP ZKIP, MAEXP not in P/poly, ...) [AW’08]: Many open questions fail to algebrize (e.g., P vs NP, NP vs BPP, ...). Closure under logical deduction ? Statements other than inclusions / separations ? Is all hope really lost ?

33. Algebrization II:Enter the Logic [AW’08] Al Jabra ~ ~ PBNPB Logic PA = NPA PA = NPA PBNPB Black Box [BGS’75]

34. AW-Algebrization vs. This Work [AW’08] pointed out an important new barrier to progress in complexity theory. [Impagliazzo,Kabanets,K’09]: • Axiomatizing this barrier leads to fine-tuning. • Fine-tuned placement of the barrier is less pessimistic about the possibility of progress with current techniques.

35. Arithmetic Checkability Axiom • For every language in NP, there exists a low-degree polynomial verifier. L  NP  polynomial family f={fn}, fn : Zn Z, • deg(fn) = poly(n) & f  P, so that  Boolean x1 …xn, x1 …xn L iff  Boolean y1,…,yms.t. fn+m (x1,…, xn, y1,…, ym) 0.

36. Arithmetic Checkability: Example “ G has an independent set of size t ” • input variables: xi,j= indicator for edge (i,j) • witness variables: yi= indicator for i being in Independent Set f(x,y) = i,j (xi,j + yi + yj -3) * k<t (yi - k) = 0 if not Independent Set = 0 if size < t

37. ACT = RCT + Arithmetic Checkability • LCT implies Arithmetic Checkability Axiom. • ACT does not imply LCT. (Arithmetic Checkability holds relative to any self-algebrizing, arbitrarily powerful oracle; but LCT doesn’t.)

38. ACT proves the following: • PSPACE = IP, • OWF  NP  ZKIP, • MAEXP not in P/poly, … (similar to [AW’08])

39. Independence from ACT • NP vs P, • EXP vsio-P/poly, • BPP vs P, … (construct models using communication complexity lower bounds, similar to [AW’08])

40. NEXP = MIP … is not provable in ACT either !

41. NEXP vs MIP Theorem: There is an oracle A satisfying Arithmetic Checkability, but EXPAMIPA. Proof. Construct L such that EXPLMIPPSPACEL(using that MIP and PSPACE make only poly-size oracle queries). Let A be the self-algebrizing (Fortnow-) encoding of L, and so A satisfies Arithmetic Checkability. We get EXPL EXPA, and MIPA MIPPSPACEL. Hence, EXPAMIPA. QED

42. Local vs Arithmetic Checkability as in [AW’08]: For every A satisfying Arithmetic Checkability, NEXPA[poly] = MIPA( with only poly-length queries allowed ). But, we can construct an Arithmetically Checkable oracle A so that NEXPA[poly] =PA= MIPA ( A = { (N, x, 1L) | NTM NA accepts x on some path with oracle queries of length  L } ) So ACT cannot prove MIP  P …

43. Local vs Arithmetic Checkability [Meir’09]: Combinatorial proof of NEXP = MIP (without using “arithmetization”). Meir’s proof relies on strongly-uniform local checkability of NEXP computation.

44. ACT and PCP Theorems We also show that ACT does not prove the PCP Theorem that NP=PCP[log n, 1]. Cf. [Dinur’07]: Combinatorial proof of PCP Thm. Also, ( ACT + PCP Theorem ) do not prove NEXP=MIP.

45. RCT vs ACT vs LCT < < PSPACE = IP NP = PCP

46. Interpretation • (strong) Local Checkabilityis enough to prove any provable complexity statement. • Algebraic Checkabilityprovides a useful way to exploit Local Checkability(and show, e.g., PSPACE = IP) • However, NP = PCP and NEXP = MIP require more than just Algebraic Checkability. Moreover, [Dinur’07] and [Meir’09] show that Algebraic Checkability is not necessary for these results.

47. Oracle worlds and models ACT

48. Conclusion • AIV’92 framework provides a way to formalize (axiomatize) barriers by stating which properties of (polynomial-time) computation existing techniques are using. • For example, ACT is a useful characterization of provability by using arithmetization-based techniques. • Arithmetizationalone cannot resolve most open complexity questions. Can Local Checkabilitybe used more?

49. Open Questions • New ways to use Local Checkability ? • Combinatorial proof of PSPACE = IP ? • Other barriers formalized in the same framework?

50. P ? NP BPP =? P Algebrization ridge IP=PSPACE Thank you! Relativization gorge P  EXP