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

Slides:



Advertisements
Similar presentations
COMPLEXITY THEORY CSci 5403 LECTURE VII: DIAGONALIZATION.
Advertisements

Complexity Theory Lecture 6
Are lower bounds hard to prove? Michal Koucký Institute of Mathematics, Prague.
Lecture 9. Resource bounded KC K-, and C- complexities depend on unlimited computational resources. Kolmogorov himself first observed that we can put resource.
Gillat Kol joint work with Ran Raz Competing Provers Protocols for Circuit Evaluation.
Lecture 16: Relativization Umans Complexity Theory Lecturess.
Amplifying lower bounds by means of self- reducibility Eric Allender Michal Koucký Rutgers University Academy of Sciences Czech Republic Czech Republic.
Great Theoretical Ideas in Computer Science for Some.
Complexity 12-1 Complexity Andrei Bulatov Non-Deterministic Space.
Complexity 15-1 Complexity Andrei Bulatov Hierarchy Theorem.
Computability and Complexity 13-1 Computability and Complexity Andrei Bulatov The Class NP.
CPSC 411, Fall 2008: Set 12 1 CPSC 411 Design and Analysis of Algorithms Set 12: Undecidability Prof. Jennifer Welch Fall 2008.
Hardness Results for Problems P: Class of “easy to solve” problems Absolute hardness results Relative hardness results –Reduction technique.
1 Undecidability Andreas Klappenecker [based on slides by Prof. Welch]
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.
Arithmetic Hardness vs. Randomness Valentine Kabanets SFU.
Eric Allender Rutgers University The Audacity of Computational Complexity Theory.
CS151 Complexity Theory Lecture 12 May 6, CS151 Lecture 122 Outline The Polynomial-Time Hierarachy (PH) Complete problems for classes in PH, PSPACE.
Computability and Complexity 32-1 Computability and Complexity Andrei Bulatov Boolean Circuits.
RELATIVIZATION CSE860 Vaishali Athale. Overview Introduction Idea behind “Relativization” Concept of “Oracle” Review of Diagonalization Proof Limits of.
Analysis of Algorithms CS 477/677
Eric Allender Rutgers University Circuit Complexity, Kolmogorov Complexity, and Prospects for Lower Bounds DCFS 2008.
Chapter 11: Limitations of Algorithmic Power
Eric Allender Rutgers University Cracks in the Defenses: Scouting Out Approaches on Circuit Lower Bounds CSR 2008.
Chapter 11 Limitations of Algorithm Power Copyright © 2007 Pearson Addison-Wesley. All rights reserved.
1.1 Chapter 1: Introduction What is the course all about? Problems, instances and algorithms Running time v.s. computational complexity General description.
Definition: Let M be a deterministic Turing Machine that halts on all inputs. Space Complexity of M is the function f:N  N, where f(n) is the maximum.
Complexity Classes Kang Yu 1. NP NP : nondeterministic polynomial time NP-complete : 1.In NP (can be verified in polynomial time) 2.Every problem in NP.
Chapter 11 Limitations of Algorithm Power. Lower Bounds Lower bound: an estimate on a minimum amount of work needed to solve a given problem Examples:
CSCE350 Algorithms and Data Structure
Computational Complexity Polynomial time O(n k ) input size n, k constant Tractable problems solvable in polynomial time(Opposite Intractable) Ex: sorting,
CSCI 4325 / 6339 Theory of Computation Zhixiang Chen.
Theory of Computing Lecture 15 MAS 714 Hartmut Klauck.
Optimal Proof Systems and Sparse Sets Harry Buhrman, CWI Steve Fenner, South Carolina Lance Fortnow, NEC/Chicago Dieter van Melkebeek, DIMACS/Chicago.
The Complexity of Optimization Problems. Summary -Complexity of algorithms and problems -Complexity classes: P and NP -Reducibility -Karp reducibility.
Theory of Computing Lecture 17 MAS 714 Hartmut Klauck.
Eric Allender Rutgers University The Strange Link between Incompressibility and Complexity China Theory Week, Aarhus August 13, 2012.
Eric Allender Rutgers University Dual VP Classes Joint work with Anna Gál (U. Texas) and Ian Mertz (Rutgers) MFCS, Milan, August 27, 2015.
Week 10Complexity of Algorithms1 Hard Computational Problems Some computational problems are hard Despite a numerous attempts we do not know any efficient.
1 Lower Bounds Lower bound: an estimate on a minimum amount of work needed to solve a given problem Examples: b number of comparisons needed to find the.
CSCI 3160 Design and Analysis of Algorithms Tutorial 10 Chengyu Lin.
Theory of Computing Lecture 21 MAS 714 Hartmut Klauck.
1 The Theory of NP-Completeness 2 Cook ’ s Theorem (1971) Prof. Cook Toronto U. Receiving Turing Award (1982) Discussing difficult problems: worst case.
Umans Complexity Theory Lectures Lecture 1a: Problems and Languages.
1 Chapter 34: NP-Completeness. 2 About this Tutorial What is NP ? How to check if a problem is in NP ? Cook-Levin Theorem Showing one of the most difficult.
NP-COMPLETE PROBLEMS. Admin  Two more assignments…  No office hours on tomorrow.
Amplifying lower bounds by means of self- reducibility Eric Allender Michal Koucký Rutgers University Academy of Sciences Czech Republic Czech Republic.
Eric Allender Rutgers University Circuit Complexity meets the Theory of Randomness SUNY Buffalo, November 11, 2010.
NP-Complete problems.
NP-complete Problems Prof. Sin-Min Lee Department of Computer Science.
My Favorite Ten Complexity Theorems of the Past Decade II Lance Fortnow University of Chicago.
Umans Complexity Theory Lectures Lecture 17: Natural Proofs.
1 How to establish NP-hardness Lemma: If L 1 is NP-hard and L 1 ≤ L 2 then L 2 is NP-hard.
Donghyun (David) Kim Department of Mathematics and Computer Science North Carolina Central University 1 Chapter 7 Time Complexity Some slides are in courtesy.
CS151 Complexity Theory Lecture 16 May 20, The outer verifier Theorem: NP  PCP[log n, polylog n] Proof (first steps): –define: Polynomial Constraint.
Comparing Notions of Full Derandomization Lance Fortnow NEC Research Institute With thanks to Dieter van Melkebeek.
Eric Allender Rutgers University Curiouser and Curiouser: The Link between Incompressibility and Complexity CiE Special Session, June 19, 2012.
Lecture. Today Problem set 9 out (due next Thursday) Topics: –Complexity Theory –Optimization versus Decision Problems –P and NP –Efficient Verification.
Homework 8 Solutions Problem 1. Draw a diagram showing the various classes of languages that we have discussed and alluded to in terms of which class.
1 Finite Model Theory Lecture 5 Turing Machines and Finite Models.
The NP class. NP-completeness Lecture2. The NP-class The NP class is a class that contains all the problems that can be decided by a Non-Deterministic.
Complexity Theory and Explicit Constructions of Ramsey Graphs Rahul Santhanam University of Edinburgh.
From Classical Proof Theory to P vs. NP
NP-Completeness Yin Tat Lee
Perspective on Lower Bounds: Diagonalization
Intractable Problems Time-Bounded Turing Machines Classes P and NP
Chapter 11 Limitations of Algorithm Power
Introduction to Oracles in Complexity Theory
The Polynomial Hierarchy Enumeration Problems 7.3.3
Switching Lemmas and Proof Complexity
Presentation transcript:

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

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.

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.

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

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).

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].)

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

< 8 >< 8 >

< 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.

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 gates. (Stockmeyer, 1974)

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.

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.

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.

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).

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]

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.

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!

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.)

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.

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.

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.

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

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

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

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

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

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

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

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

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

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

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

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…

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.

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.

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]?

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.

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.

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?

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]

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

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:

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

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

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

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.

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

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.

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 This would yield an “unnatural” proof separating NC 1 from TC 0.

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 ?)

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!)

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?

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.

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 …

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.

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 ?

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.

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)!

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.)

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.

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