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 >
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< 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 PP .. .. 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 S5S5
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!