1. Eric Allender Rutgers University Chipping Away at P vs NP: How Far Are We from Proving Circuit Size Lower Bounds? Joint work with Michal Koucky ʹ Czech Academy of Sciences

2. Eric Allender: How Close Are We to Proving Circuit Lower Bounds? < 2 >< 2 > Introduction  How far are we from proving circuit lower bounds?  I have no idea!  There is a lot of pessimism, based on – The lack of any good circuit lower bounds – The [Razborov,Rudich] “natural proofs” obstacle  Today, we’ll make some observations that may cause some of you to be less pessimistic.

3. Eric Allender: How Close Are We to Proving Circuit Lower Bounds? < 3 >< 3 > But First…Why Circuits?  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.

4. Eric Allender: How Close Are We to Proving Circuit Lower Bounds? < 4 >< 4 > 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.

5. Eric Allender: How Close Are We to Proving Circuit Lower Bounds? < 5 >< 5 > 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.

6. Eric Allender: How Close Are We to Proving Circuit Lower Bounds? < 6 >< 6 > 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. – This is a much stronger statement about complexity than we are able to prove for most problems (such as NP-complete problems).

7. Eric Allender: How Close Are We to Proving Circuit Lower Bounds? < 7 >< 7 > 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).

8. Eric Allender: How Close Are We to Proving Circuit Lower Bounds? < 8 >< 8 > An Example of what can be done, given a circuit size lower bound  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)  (Proof sketch): There is a problem requiring exponential circuit size that is efficiently reducible to WS1S.

9. Eric Allender: How Close Are We to Proving Circuit Lower Bounds? < 9 >< 9 > An Example of what can be done, given a circuit size lower bound  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)  What we need: Similar lower bounds, but for problems in NP such as SAT or FACTORING.  We would even be glad to get lower bounds for restricted classes of circuits.

10. Eric Allender: How Close Are We to Proving Circuit Lower Bounds? Big Complexity Classes  NP PP .. ..  NC  L (Deterministic Logspace)

11. Eric Allender: How Close Are We to Proving Circuit Lower Bounds?  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 The Main Objects of Interest: Small Complexity Classes

12. Eric Allender: How Close Are We to Proving Circuit Lower Bounds?  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 The Main Objects of Interest: Small Complexity Classes

13. Eric Allender: How Close Are We to Proving Circuit Lower Bounds?  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 The Main Objects of Interest: Small Complexity Classes

14. Eric Allender: How Close Are We to Proving Circuit Lower Bounds?  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 The Main Objects of Interest: Small Complexity Classes

15. Eric Allender: How Close Are We to Proving Circuit Lower Bounds?  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 The Main Objects of Interest: Small Complexity Classes

16. Eric Allender: How Close Are We to Proving Circuit Lower Bounds?  NP has complete sets (under polynomial time reducibility ≤ P )  These small classes have complete sets, too (under ≤ AC° ) Complete Problems

17. Eric Allender: How Close Are We to Proving Circuit Lower Bounds? 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

18. Eric Allender: How Close Are We to Proving Circuit Lower Bounds?  NC 1  TC 0  AC 0 [6]  AC 0 [2]  AC 0 Complete Problems  sorting, multiplication, division  [Naor,Reingold] Pseudorandom Generator

19. Eric Allender: How Close Are We to Proving Circuit Lower Bounds?  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

20. Eric Allender: How Close Are We to Proving Circuit Lower Bounds? The Word Problem Over S 5  A regular set complete for NC 1 =

21. Eric Allender: How Close Are We to Proving Circuit Lower Bounds? 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…

22. Eric Allender: How Close Are We to Proving Circuit Lower Bounds? 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.

23. Eric Allender: How Close Are We to Proving Circuit Lower Bounds? 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.

24. Eric Allender: How Close Are We to Proving Circuit Lower Bounds? How far are we in this talk?  We’ve explained why circuit lower bounds are important.  …even for restricted classes of circuits.  What is currently known about these classes?

25. Eric Allender: How Close Are We to Proving Circuit Lower Bounds? 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]?

26. Eric Allender: How Close Are We to Proving Circuit Lower Bounds? 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.

27. Eric Allender: How Close Are We to Proving Circuit Lower Bounds? 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.

28. Eric Allender: How Close Are We to Proving Circuit Lower Bounds?  How close are the following two statements?  TC 0 Circuits for BFE must be of size n 1+Ω(1)  TC 0 Circuits for BFE must be of size n 1+ c for some c >0 How Close Are We to Proving Circuit Lower Bounds?

29. Eric Allender: How Close Are We to Proving Circuit Lower Bounds?  How close are the following two statements?  TC 0 Circuits for BFE must be of size n 1+Ω(1)  TC 0 Circuits for BFE must be of size n 1+ c for some c >0 How Close Are We to Proving Circuit Lower Bounds? This is known [IPS’97] This implies TC 0 ≠ NC 1

30. Eric Allender: How Close Are We to Proving Circuit Lower Bounds? Self-Reducibility  A set B is said to be “self-reducible” if B ≤ P B

31. Eric Allender: How Close Are We to Proving Circuit Lower Bounds? Self-Reducibility  A set B is said to be “self-reducible” if B ≤ P 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)

32. Eric Allender: How Close Are We to Proving Circuit Lower Bounds? Self-Reducibility  Many of the important problems in (or near) NC 1 have a special self-reducibility property:

33. Eric Allender: How Close Are We to Proving Circuit Lower Bounds? Self-Reducibility  Many of the important problems in (or near) NC 1 have a special self-reducibility property: Instances of length n are AC 0 -Turing (or TC 0 - Turing) reducible to instances of length n ½ via reductions of linear size.  Examples: – BFE – the word problem over S 5 – MAJORITY – Iterated Product of 3-by-3 Integer Matrices

34. Eric Allender: How Close Are We to Proving Circuit Lower Bounds? Self Reducibility  BFE A subformula near the root Subformulae near inputs

35. Eric Allender: How Close Are We to Proving Circuit Lower Bounds? Self Reducibility S5S5

36. Eric Allender: How Close Are We to Proving Circuit Lower Bounds? 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 Iterated Product of 3-by-3 matrices.

37. Eric Allender: How Close Are We to Proving Circuit Lower Bounds? 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.

38. Eric Allender: How Close Are We to Proving Circuit Lower Bounds? 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.)  How widespread is this phenomenon? Is it true for SAT? (I.e., can we show NP ≠ TC 0 by proving that SAT requires TC 0 circuits of size n 1.0000001 ?)

39. Eric Allender: How Close Are We to Proving Circuit Lower Bounds? Limitations of Self-Reducibility  Any problem for which instances of length n are TC 0 -Turing reducible to instances of length n ½ via poly-size reductions lies in NC.  Thus there is no obvious way to apply these techniques to SAT or to problems complete for P.  …but perhaps, rather than showing directly that SAT has this strong form of self- reducibility, one can argue that if SAT is in TC 0 then it has TC 0 circuits of nearly-linear size.

40. Eric Allender: How Close Are We to Proving Circuit Lower Bounds? Limitations of Self-Reducibility  Any problem for which instances of length n are TC 0 -Turing reducible to instances of length n ½ via poly-size reductions lies in NC.

41. Eric Allender: How Close Are We to Proving Circuit Lower Bounds? Limitations of Self-Reducibility  Any problem for which instances of length n are TC 0 -Turing reducible to instances of length n ½ via poly-size reductions lies in NC. d levels of oracle gates

42. Eric Allender: How Close Are We to Proving Circuit Lower Bounds? Limitations of Self-Reducibility  Any problem for which instances of length n are TC 0 -Turing reducible to instances of length n ½ via poly-size reductions lies in NC. d 2 levels of oracle gates

43. Eric Allender: How Close Are We to Proving Circuit Lower Bounds? Limitations of Self-Reducibility  Any problem for which instances of length n are TC 0 -Turing reducible to instances of length n ½ via poly-size reductions lies in NC. d 3 levels of oracle gates After log log rounds, the depth is log O(1) n

44. Eric Allender: How Close Are We to Proving Circuit Lower Bounds? Some Lower Bounds  Recall that [IPS] showed: – TC 0 Circuits for BFE must be of size n 1+Ω(1)  Thus SAT also requires TC 0 circuits of this size.  The [IPS] bound actually shows that PARITY requires circuits of this size.  We do NOT know similar bounds for AC 0 [6].

45. Eric Allender: How Close Are We to Proving Circuit Lower Bounds? A Lower Bound for AC 0 [6]  For every d there is an ε >0 such that SAT requires depth d AC 0 [6] circuits of size n 1+ ε  The same proof shows that the same bound holds for TC 0.

46. Eric Allender: How Close Are We to Proving Circuit Lower Bounds? A Lower Bound for AC 0 [6]  For every d there is an ε >0 such that SAT requires depth d AC 0 [6] circuits of size n 1+ ε  If no such ε exists, then for all δ >0, SAT is in TimeSpace( n 1+ δ,n 1- δ ).  This violates the time-space tradeoff results of [Van Melkebeek].

47. Eric Allender: How Close Are We to Proving Circuit Lower Bounds? A Lower Bound for AC 0 [6]  For every d there is an ε >0 such that SAT requires depth d AC 0 [6] circuits of size n 1+ ε  If no such ε exists, then for all δ >0, SAT is in TimeSpace( n+n dδ,n 1- δ ).  Hint of proof: Do a depth-first evaluation of the circuit, using the space bound to store the value of all gates having large fan-in (and re- computing all other values as needed).

48. Eric Allender: How Close Are We to Proving Circuit Lower Bounds? Prospects for Progress  We have seen that existing techniques prove bounds that are “nearly” good enough to separate NC 1 and TC 0. Some of these proofs are “natural”.  Don’t the results of [Razborov & Rudich] indicate that further progress will require very different approaches?  Not necessarily!

49. Eric Allender: How Close Are We to Proving Circuit Lower Bounds? 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.

50. Eric Allender: How Close Are We to Proving Circuit Lower Bounds? Prospects for Progress  These observations are simple, but …  they have forever changed the way that we look at quadratic (and smaller) lower bounds.  We are not claiming to have found a way around the obstacles identified by [Razborov & Rudich]. (Such a claim will have to wait until someone proves that NC 1 ≠ TC 0.) But we do believe that this avenue deserves further exploration.

51. Eric Allender: How Close Are We to Proving Circuit Lower Bounds? Conclusions  Circuit lower bounds are necessary. – Program run-time lower bounds do not yield bounds for fixed input sizes.  We even need circuit lower bounds for small circuit classes.  Seemingly-modest improvements to existing lower bounds would yield exciting separations of complexity classes.  There may be cause for renewed optimism.