1. Eric Allender Rutgers University New Surprises from Self- Reducibility CiE 2010, Ponta Delgada, Azores

2. New Surprises from Self-Reducibility, CiE 2010, Ponta Delgada, Azores < 2 >< 2 > Why a “Fantastic Voyage”?  It’s apt.  It’s a bad pun on “self-reduction”.  It is contemporary with the birth of self- reducibility.

3. New Surprises from Self-Reducibility, CiE 2010, Ponta Delgada, Azores < 3 >< 3 > 40 Years of Self-Reducibility  Boris A. Trakhtenbrot, On Autoreducibility, Dokl. Akad. Nauk. SSSR 11, 1970.

4. New Surprises from Self-Reducibility, CiE 2010, Ponta Delgada, Azores < 4 >< 4 > Self-Reducibility  A set B is said to be “self-reducible” if B ≤ r B

5. New Surprises from Self-Reducibility, CiE 2010, Ponta Delgada, Azores < 5 >< 5 > Self-Reducibility  A set B is said to be “self-reducible” if B ≤ r B via a reduction that, on input x, does not ask about whether x is in B.  Very well-studied notion.  For example, φ is in SAT if and only if (φ 0 is in SAT) or (φ 1 is in SAT).

6. New Surprises from Self-Reducibility, CiE 2010, Ponta Delgada, Azores < 6 >< 6 > Self-Reducibility  A set B is said to be “self-reducible” if B ≤ r B via a reduction that, on input x, does not ask about whether x is in B.  Very well-studied notion.  In fact, this is such a simple notion, the really surprising thing is that, for four decades, slight variations on this theme have yielded surprising and powerful insights.  We will not survey all 40 years of work on this topic! (See [Selke].)

7. New Surprises from Self-Reducibility, CiE 2010, Ponta Delgada, Azores < 7 >< 7 >

8. < 8 >< 8 >

9. < 9 >< 9 > Plan for Today  Give a brief review of some (historical) settings where self-reducibility has been useful in complexity theory.  Present a few recent examples of work at the intersection of complexity theory and computability theory, where self-reducibility plays a central role.  But first, let’s recall some of the grand challenges in complexity theory that motivate these investigations.

10. New Surprises from Self-Reducibility, CiE 2010, Ponta Delgada, Azores What Crypto Needs from Complexity  Factoring (or some other suitable trap-door function) is hard for some fixed input size (corresponding to the size of a public key).  That is: we need to talk about hardness of finite functions.  Complexity theory can do this:  Theorem: Any circuit that takes as input a logical formula (in WS1S) of length 616 and produces as output a correct answer, saying if the formula is valid or not, has at least 10 123 gates. (Stockmeyer, 1974)

11. New Surprises from Self-Reducibility, CiE 2010, Ponta Delgada, Azores Circuits vs Turing Machines  2 Basic models of computation – Programs (one program – works for every input length) – Circuits (different circuit for each input length)  One crucial difference: circuit lower bounds can be used to prove intractability results for fixed input sizes.  Program run-time lower bounds can’t.

12. New Surprises from Self-Reducibility, CiE 2010, Ponta Delgada, Azores An example: the Game of Checkers  Computing strategies for Checkers requires exponential time. – More precisely, given an n -by- n Checkers board with checkers on it, no program can compute an optimal next move in fewer than c 2 n – d steps, for some constants c and d. – n -by- n Checkers is complete for EXP.

13. New Surprises from Self-Reducibility, CiE 2010, Ponta Delgada, Azores An example: the Game of Checkers  Computing strategies for Checkers requires exponential time. – More precisely, given an n -by- n Checkers board with checkers on it, no program can compute an optimal next move in fewer than c 2 n – d steps, for some constants c and d. – Thus any program solving this problem must run very slowly on large inputs. This is the essence of asymptotic analysis.

14. New Surprises from Self-Reducibility, CiE 2010, Ponta Delgada, Azores An example: the Game of Checkers  Computing strategies for Checkers requires exponential time. – More precisely, given an n -by- n Checkers board with checkers on it, no program can compute an optimal next move in fewer than c 2 n – d steps, for some constants c and d. – but…Conceivably, there is a hand-held device that computes optimal moves, even for Checker boards of size 1000-by-1000! – …because we don’t know if EXP is in P/poly (the class of problems with small circuits).

15. New Surprises from Self-Reducibility, CiE 2010, Ponta Delgada, Azores Two Fundamental Questions: SAT є PSAT є P/poly coNP NP = NP NP [Karp-Lipton, 1980]

16. New Surprises from Self-Reducibility, CiE 2010, Ponta Delgada, Azores Two Fundamental Questions: SAT є PSAT є P/poly coNP NP = NP NP [Karp-Lipton, 1980] Guess a circuit, and use the NP oracle to see if it computes SAT.

17. New Surprises from Self-Reducibility, CiE 2010, Ponta Delgada, Azores Autoreducibility of Complete Sets  Here are a few longstanding open questions in complexity theory: – EXP = NP – EXP = PH (= NP U NP NP U NP NP NP …) – PSPACE = NP – PSPACE = PH (= NP U NP NP U NP NP NP …)  [Buhrman, Fortnow, van Melkebeek, Torenvliet] showed that resolving some innocent-sounding questions about auto- reducibility would solve these questions!

18. New Surprises from Self-Reducibility, CiE 2010, Ponta Delgada, Azores Autoreducibility of Complete Sets  [BFvMT]: All ≤ P -Complete sets for EXP are autoreducible.  There is an oracle A, relative to which not all ≤ P -Complete sets for EXP are autoreducible. – Thus the proof of the preceding theorem does not “relativize”. (That’s a good thing!)  Not all ≤ P -Complete sets for EEXPSPACE (doubly-exponential space) are autoreducible.  How about classes between EXP and EEXPSPACE? (E.g., EXPSPACE & EEXP.)

19. New Surprises from Self-Reducibility, CiE 2010, Ponta Delgada, Azores Autoreducibility of Complete Sets  Are all ≤ P -Complete sets for EEXP autoreducible? – If YES, then PH ≠ EXP. – If NO, then P ≠ PSPACE.  Are all ≤ P -Complete sets for EXPSPACE autoreducible?  Usually questions about “big” classes like EXPSPACE and EEXP are not too hard to answer. Diagonalization techniques work there, that don’t work for “smaller” classes.

20. New Surprises from Self-Reducibility, CiE 2010, Ponta Delgada, Azores Autoreducibility of Complete Sets  Are all ≤ P -Complete sets for EEXP autoreducible? – If YES, then PH ≠ EXP. – If NO, then P ≠ PSPACE.  Are all ≤ P -Complete sets for EXPSPACE autoreducible? – If YES then PH ≠ PSPACE.

21. New Surprises from Self-Reducibility, CiE 2010, Ponta Delgada, Azores Autoreducibility of Complete Sets  Are all ≤ P -Complete sets for EEXP autoreducible? – If YES, then PH ≠ EXP. – If NO, then P ≠ PSPACE & NL ≠ NP.  Are all ≤ P -Complete sets for EXPSPACE autoreducible? – If YES then PH ≠ PSPACE. – If NO, then NL ≠ NP.

22. New Surprises from Self-Reducibility, CiE 2010, Ponta Delgada, Azores Big Complexity Classes  NP PP .. ..  NC  NL (Nondeterministic Logspace)  L (Deterministic Logspace)

23. New Surprises from Self-Reducibility, CiE 2010, Ponta Delgada, Azores  TC 0 O(1)-Depth Circuits of MAJ gates  AC 0 [6]  NC 1 Log-Depth Circuits  AC 0 can’t compute Mod 2 [FSS,A]  AC 0 O(1)-Depth Circuits of AND/OR gates Objects of Interest: Small Complexity Classes

24. New Surprises from Self-Reducibility, CiE 2010, Ponta Delgada, Azores  TC 0 O(1)-Depth Circuits of MAJ gates  AC 0 [6]  NC 1 Log-Depth Circuits  AC 0 can’t compute Mod 2 [FSS,A]  AC 0 O(1)-Depth Circuits of AND/OR gates Objects of Interest: Small Complexity Classes

25. New Surprises from Self-Reducibility, CiE 2010, Ponta Delgada, Azores  TC 0 O(1)-Depth Circuits of MAJ gates  NC 1 Log-Depth Circuits  AC 0 [2] can’t compute Mod 3 [R,S]  AC 0 [2]  AC 0 O(1)-Depth Circuits of AND/OR gates Objects of Interest: Small Complexity Classes

26. New Surprises from Self-Reducibility, CiE 2010, Ponta Delgada, Azores  NC 1 Log-Depth Circuits  TC 0 O(1)-Depth Circuits of MAJ gates  AC 0 [6]  AC 0 [2]  AC 0 O(1)-Depth Circuits of AND/OR gates Objects of Interest: Small Complexity Classes

27. New Surprises from Self-Reducibility, CiE 2010, Ponta Delgada, Azores  NC 1 poly-size formulae  TC 0 O(1)-Depth Circuits of MAJ gates  AC 0 [6]  AC 0 [2]  AC 0 O(1)-Depth Circuits of AND/OR gates Objects of Interest: Small Complexity Classes

28. New Surprises from Self-Reducibility, CiE 2010, Ponta Delgada, Azores  NP has complete sets (under polynomial time reducibility ≤ P )  These small classes have complete sets, too (under ≤ AC° )  Amazingly, even with restricted reductions, the classes of complete sets for “big” complexity classes (EXP, NP, …) are essentially unchanged. Complete Problems

29. New Surprises from Self-Reducibility, CiE 2010, Ponta Delgada, Azores Reductions  A ≤ AC° B means that there is a constant-depth circuit computing A that has the usual AND and OR gates, and also has ‘oracle gates’ for B. B

30. New Surprises from Self-Reducibility, CiE 2010, Ponta Delgada, Azores  NC 1  TC 0  AC 0 [6]  AC 0 [2]  AC 0 Complete Problems  sorting, multiplication, division  [Naor,Reingold] Pseudorandom Generator

31. New Surprises from Self-Reducibility, CiE 2010, Ponta Delgada, Azores  NC 1  TC 0  AC 0 [6]  AC 0 [2]  AC 0 Complete Problems  BFE: Balanced Boolean Formula Evaluation (AND,OR,XOR)  Word problem over S 5

32. New Surprises from Self-Reducibility, CiE 2010, Ponta Delgada, Azores The Word Problem Over S 5  A regular set complete for NC 1 =

33. New Surprises from Self-Reducibility, CiE 2010, Ponta Delgada, Azores Complexity Classes are not Invented – They’re Discovered  NP (SAT, Clique, TSP,…)  P (Linear Programming, CVP, …)  NL (Connectivity, Shortest Paths, 2SAT, …)  L (Undirected Connectivity, Acyclicity, …)  NC 1 (BFE, Regular Sets)  TC 0 (Sorting, Multiplication, Division) We’re interested in NC 1 (for instance) not because we want to build formulae for these functions…

34. New Surprises from Self-Reducibility, CiE 2010, Ponta Delgada, Azores Complexity Classes are not Invented – They’re Discovered  NP (SAT, Clique, TSP,…)  P (Linear Programming, CVP, …)  NL (Connectivity, Shortest Paths, 2SAT, …)  L (Undirected Connectivity, Acyclicity, …)  NC 1 (BFE, Regular Sets)  TC 0 (Sorting, Multiplication, Division) … but because we want to know if the blocks of this partition are distinct.

35. New Surprises from Self-Reducibility, CiE 2010, Ponta Delgada, Azores Complexity Classes are not Invented – They’re Discovered  NP (SAT, Clique, TSP,…)  P (Linear Programming, CVP, …)  NL (Connectivity, Shortest Paths, 2SAT, …)  L (Undirected Connectivity, Acyclicity, …)  NC 1 (BFE, Regular Sets)  TC 0 (Sorting, Multiplication, Division) These classes are real. They’re important.

36. New Surprises from Self-Reducibility, CiE 2010, Ponta Delgada, Azores Other Longstanding Open Problems  Is P = NP?  Is AC 0 [6] = NP?  Is depth 3 AC 0 [6] = NP? We’ll focus on questions such as : Is BFE in TC 0 ? Is BFE in AC 0 [6]?

37. New Surprises from Self-Reducibility, CiE 2010, Ponta Delgada, Azores How Close Are We to Proving Circuit Lower Bounds?  Conventional Wisdom: Not Close At All!  No new superpolynomial size lower bounds in over two decades.  Razborov and Rudich: Any “natural” argument proving a lower bound against a circuit class C yields a proof that C can’t compute a pseudorandom function generator.  Since the [Naor, Reingold] generator is computable in TC 0, this is bad news.

38. New Surprises from Self-Reducibility, CiE 2010, Ponta Delgada, Azores More Modest Goals  Problems requiring formulae of size n 3 [Håstad]  Problems requiring branching programs of size nearly n loglog n [Beame, Saks, Sun, Vee]  Problems requiring depth d TC 0 circuits of size n 1+ c [Impagliazzo, Paturi, Saks]  Time-Space Tradeoffs [Fortnow, Lipton, Van Melkebeek, Viglas]  There is little feeling that these results bring us any closer to separating complexity classes.

39. New Surprises from Self-Reducibility, CiE 2010, Ponta Delgada, Azores  How close are the following two statements?  TC 0 Circuits for BFE must be of size n 1+Ω(1)  For some c >0, TC 0 Circuits for BFE must be of size n 1+ c. How Close Are We to Proving Circuit Lower Bounds?

40. New Surprises from Self-Reducibility, CiE 2010, Ponta Delgada, Azores  How close are the following two statements?  TC 0 Circuits for BFE must be of size n 1+Ω(1)  For some c >0, TC 0 Circuits for BFE must be of size n 1+ c How Close Are We to Proving Circuit Lower Bounds? This is known [IPS’97] This implies TC 0 ≠ NC 1 [A, Koucky]

41. New Surprises from Self-Reducibility, CiE 2010, Ponta Delgada, Azores

42. New Surprises from Self-Reducibility, CiE 2010, Ponta Delgada, Azores Self-Reducibility  [Goldwasser et al]: Many of the important problems in (or near) NC 1 have a special self- reducibility property:

43. New Surprises from Self-Reducibility, CiE 2010, Ponta Delgada, Azores Self-Reducibility  [Goldwasser et al]: Many of the important problems in (or near) NC 1 have a special self- reducibility property: Instances of length n are AC 0 -Turing reducible to instances of length n ½ via reductions of linear size.  Examples: – BFE – the word problem over S 5 – MAJORITY

44. New Surprises from Self-Reducibility, CiE 2010, Ponta Delgada, Azores Self Reducibility  BFE A subformula near the root Subformulae near inputs

45. New Surprises from Self-Reducibility, CiE 2010, Ponta Delgada, Azores Self Reducibility S5S5

46. New Surprises from Self-Reducibility, CiE 2010, Ponta Delgada, Azores Self Reducibility  The self-reduction of S 5, on inputs of size n, uses ( n ½ + 1) oracle gates of size n ½.  Thus if S 5 has TC 0 circuits of size n k, it also has circuits of size ( n ½ + 1) n k/ 2 = O(n (k+ 1)/2 ).  Similar arguments hold for other classes (such as AC 0 [6] and NC 1 ).  More complicated self-reductions can be presented for MAJORITY and other problems.

47. New Surprises from Self-Reducibility, CiE 2010, Ponta Delgada, Azores A Corollary  If BFE has TC 0 or AC 0 [6] circuits, then it has such circuits of nearly linear size.  If S 5 has TC 0 or AC 0 [6] circuits, then it has such circuits of nearly linear size.  If MAJ has AC 0 [6] circuits, then it has such circuits of nearly linear size. (Etc.)  Thus, e.g., to separate NC 1 from TC 0, it suffices to show that BFE requires TC 0 circuits of size n 1.0000001.

48. New Surprises from Self-Reducibility, CiE 2010, Ponta Delgada, Azores Prospects for Progress  The [Razborov & Rudich] framework of natural proofs assumes that a “natural” proof of a lower bound will make use of a combinatorial property that (among other things) is shared by a large fraction of the functions on n bits.  In contrast, we are making use of a self- reducibility property that allows us to boost a n 1+ ε lower bound to a superpolynomial lower bound. This self-reducibility property holds for only a vanishingly small fraction of all functions.

49. New Surprises from Self-Reducibility, CiE 2010, Ponta Delgada, Azores Prospects for Progress  The [Razborov & Rudich] framework of natural proofs assumes that a “natural” proof of a lower bound will make use of a combinatorial property that (among other things) is shared by a large fraction of the functions on n bits.  Thus, it’s conceivable that a “natural” proof can be given of a modest lower bound of the form: BFE requires TC 0 circuits of size n 1.0000001. This would yield an “unnatural” proof separating NC 1 from TC 0.

50. New Surprises from Self-Reducibility, CiE 2010, Ponta Delgada, Azores Recall…  If BFE has TC 0 or AC 0 [6] circuits, then it has such circuits of nearly linear size.  If S 5 has TC 0 or AC 0 [6] circuits, then it has such circuits of nearly linear size.  If MAJ has AC 0 [6] circuits, then it has such circuits of nearly linear size. (Etc.)  How widespread is this phenomenon? Is it true for SAT? (I.e., if SAT is in TC 0, does it have TC 0 circuits of size n 1.0000001 ?)

51. New Surprises from Self-Reducibility, CiE 2010, Ponta Delgada, Azores Different Flavors of Self-Reducibility  If A is “word-decreasing self-reducible” (the self-reduction queries only lexicographically smaller strings) then A is in EXP. Some EXP- complete sets have this property.  If A is “downward self-reducible” (the self- reduction queries only shorter strings) then A is in PSPACE. Some PSPACE-complete sets have this property.  If A is “strongly downward self-reducible” (the self-reduction queries only very short strings) then A is in NC. Some NC 1 -complete sets have this property. (This is not tight!)

52. New Surprises from Self-Reducibility, CiE 2010, Ponta Delgada, Azores Different Flavors of Self-Reducibility  If A is “strongly downward self-reducible” (the self-reduction queries only very short strings) then A is in NC. Some NC 1 -complete sets have this property. (This is not tight!)  There are lots of classes between NC 1 and NC (such as L and NL, among others).  Are there sets that are complete for L and NL that are strongly downward self-reducible? Would this imply something unlikely?

53. New Surprises from Self-Reducibility, CiE 2010, Ponta Delgada, Azores How Powerful is Randomness?  Recall the basic definitions of Kolmogorov Complexity: – C(x) = min {|d| : U(d) = x}. – C(x) ≤ |x| + O(1). – x is random if |x| ≤ C(x). – R C is the set of Kolmogorov-random strings.  [ABKMR]: PSPACE is poly-time Turing reducible to R C.

54. New Surprises from Self-Reducibility, CiE 2010, Ponta Delgada, Azores How Useful is this Theorem?  PSPACE is poly-time Turing reducible to R C.  Is it trivial? After all, R C isn’t even computable!  Note that R C is not hard for NP under poly- time many-one reductions, unless P=NP. (This follows, since R C has no infinite enumerable subset.)  No simple direct reduction from PSPACE to R C is known; the known proofs rely on techniques from derandomization, interactive proof systems, and …

55. New Surprises from Self-Reducibility, CiE 2010, Ponta Delgada, Azores How Useful is this Theorem?  PSPACE is poly-time Turing reducible to R C.  Is it trivial? After all, R C isn’t even computable!  Note that R C is not hard for NP under poly- time many-one reductions, unless P=NP. (This follows, since R C has no infinite enumerable subset.)  No simple direct reduction from PSPACE to R C is known; the known proofs rely on techniques from derandomization, interactive proof systems, and … self-reducibility.

56. New Surprises from Self-Reducibility, CiE 2010, Ponta Delgada, Azores How Useful is this Theorem?  PSPACE is poly-time Turing reducible to R C.  Is this inclusion optimal in some sense? Is there some larger complexity class that is reducible to R C ?  An intriguing possibility: can PSPACE be characterized in some sense, in terms of efficient reductions to R C ?  …or is the Halting Problem poly-time reducible to R C ?

57. New Surprises from Self-Reducibility, CiE 2010, Ponta Delgada, Azores A Strange Characterization of P  Here is an illustration of what such a characterization might look like. Instead of poly-time truth-table reductions, consider poly- time dtt reductions. (I.e., in poly-time, output a list of queries, and accept if at least one of them is in R C.)  Fact: For every computable time bound t, there is a decidable set D that is not in Dtime(t) that is poly-time dtt-reducible to R C.  This would seem to kill any possibility of characterizing complexity classes.

58. New Surprises from Self-Reducibility, CiE 2010, Ponta Delgada, Azores A Strange Characterization of P  Here is an illustration of what such a characterization might look like. Instead of poly-time truth-table reductions, consider poly- time dtt reductions. (I.e., in poly-time, output a list of queries, and accept if at least one of them is in R C.)  Fact: For every computable time bound t, there is a decidable set D that is not in Dtime(t) that is poly-time dtt-reducible to R C.  …but the set D crucially depends on the universal Turing machine that defines C(x)!

59. New Surprises from Self-Reducibility, CiE 2010, Ponta Delgada, Azores A Strange Characterization of P  [A,Buhrman,Koucky]: P consists precisely of the decidable sets that are poly-time dtt- reducible to R C no matter which universal Turing machine is used in the definition of the Kolmogorov complexity function C(x).  It would be very interesting if a similar characterization of PSPACE could be obtained.  Conjecture: There is a decidable set that is not poly-time reducible to R C. (Self-reducibility may be necessary, to make use of R C.)

60. New Surprises from Self-Reducibility, CiE 2010, Ponta Delgada, Azores Closing Remarks  Self-Reducibility is a simple idea that has been surprisingly useful over a span of four decades.  Self-Reducibility points to promising avenues to separate complexity classes. – Autoreducibility of EEXP-complete sets. – Non-natural proofs in circuit complexity.  …and it may help us to forge a new connection between complexity theory and computability, by clarifying the power of efficient reducibility to R C.

61. New Surprises from Self-Reducibility, CiE 2010, Ponta Delgada, Azores Closing Remarks Obrigado!