Randomness Vs. Memory: Prospects and Barriers

With insights courtesy of MoniNaor, Ran Raz, Luca Trevisan, SalilVadhan, AviWigderson, many more … Randomness Vs. Memory:Prospects and Barriers Omer Reingold, Microsoft Research and Weizmann

Randomness In Computation (1) • Distributed computing (breaking symmetry) • Cryptography: Secrets, Semantic Security, … • Sampling, Simulations, … Can’t live without you

Randomness In Computation (2) • Communication Complexity (e.g., equality) • Routing (on the cube [Valiant]) - drastically reduces congestion You change my world

Randomness In Computation (3) • In algorithms – useful design tool, but many times can derandomize (e.g., PRIMES in P). Is it always the case? • RL=L means that every randomized algorithm can be derandomized with only a constant factor increase in memory Do I really need you?

Talk’s Premise: Many Frontiers of RL=L Barriers of previous proofs  wealth of excellent research problems. • RL in L3/2 RL=L • And Beyond

RL  (NL ) L2[Savitch 70] • Configuration graph (per RL algorithm forP & inputx): 0 s = start config 1 0 poly(|x|) configs 1 0 1 1 0 t = accept config 1 0 • x P  random walk from s ends at tw.p. ≥ ½ • xPt unreachable from s • Enumerating all possible paths – too expensive. Main idea: 1st half of computation only transmits log n bits to 2nd half transitions oncurrent random bit duplicate (running time T) ≤ poly(|x|) times

Oblivious Derandomiztion of RL • Pseudorandom generators that fool space-bounded algorithms [AKS87, BNS89, Nisan90, NZ93, INW94] • Nisan’s generator has seed length log2n • Proof that RL in L2 via oblivious derandomization • Major tool in the study of RL vs. L • Applications beyond [Ind00, Siv02, KNO05,…] • Open problem: PRGs with reduced seed length

G x,y x, Ext(x,y) Randomness Extractors@Your Service • Basic idea [NZ93] (related to Nisan’s generator): • Let log-space A read a random100lognbit string x. • Since A remembers at most lognbits, xstill contains (roughly) 99lognbits of entropy (independent of A’s state). • Can recycle x:

Randomness Extractors@Your Service • NZ generator: • Possible setting of parameters: x is O(log n) long. Each yi is O(log½n) long and have log½n yi’s. • Expand O(log n) bits to O(log3/2n) (get any poly) • Error >> 1/n ([AKS87] gets almost log2n bits w. error 1/n) G x,y1,y2, … x, Ext(x,y1), Ext(x,y2),

Randomness Extractors@Your Service • NZ generator: • Error >> 1/n ([AKS87] gets almost log2n bits w. error 1/n) • Open: get any polynomial expansion w. error 1/n • Open: super polynomial expansion with logarithmic seed and constant error (partial result [RR99]). G x,y1,y2, … x, Ext(x,y1), Ext(x,y2),

G x,y x, Ext(x,y) Nisan,INW Generators via Extractors • Recall basic generator: • Lets flip it …

Nisan,INW Generators via Extractors Altogether: seed length = log 2 n Given state of machine in the middle, Ext(x,y) still -random Loss at each level: log n (possible entropy in state). + log 1/έfor extractor seed, where έ = /n x,y x log n Ext(x,y)

Nisan,INW + NZ RL=L • Let M be an RL machine • Using [Nisan] get M’ that uses only log2n random bits • Fully derandomizeM’ using [NZ] • Or does it? • M’ is not an RL machine (access to seed of [Nisan, INW] not read once) • Still, natural approach – derandomize seed of [Nisan] Can we build PRGs from read once ingredients? Not too promising …

RL  L3/2[SZ95] - “derandomized” [Nis] • Nisan’s generator has following properties: • Seed divided into h (length log2n) and x (length logn). • Given h in input tape, generator runs in L. • M, w.h.p over h, fixing h and ranging over x implies a good generator for M. • h is shorter if we generate less than n bits

[SZ95] - basic idea • Fix h, divide run of M to segments: • Enumerate over x, estimate all transition probs. • Replace each segment with a single transition • Recurse using the same h • Now M’ depends on h M’ close to some t-power of M. [SZ] perturb M’ to eliminate dependency on h

[SZ95] –further progress • Open: Translate [SZ] to a better generator against space bounded algorithms! • Potentially, can then recursively apply [SZ] and get better derandomization of RL (after constant number of iterations may get RL in L1+) • Armoni showed an interesting extrapolation between [NZ] and [INW] and as a result got a slight improvement (RL in L3/2/(log L)1/2)

Thoughts on Improving INW Loss at each level: log n (possible entropy in state). + log 1/έfor extractor seed, where έ = /n Even for combinatorial rectangles we do not know “optimal” PRGs x,y x Ext(x,y) • Avoiding loss due to entropy in state: • [RR99] Recycle the entropy of the states. • Challenge: how to do it when do not know state probabilities? Open: better PRGs against constant width branching programs

Thoughts on Improving INW Loss at each level: log n (possible entropy in state). + log 1/έfor extractor seed, where έ = /n x x,y x Ext(x,y(x)) Ext(x,y) • Avoiding loss due to extractor seeds: • Can we recycle y from previous computation? • Challenge: contain dependencies … Do we need a seed at all? Use seedless extractors instead?

Thoughts on Improving INW Loss at each level: log n (possible entropy in state). + log 1/έfor extractor seed, where έ = /n x,y x Ext(x,y) • Extractor seed is long because we need to work with small error έ = /n • Error reduction for PRGs? If use error έ = /(log n) sequence still has some unpredictability property, is it usable? (Yes for SL [R04,RozVad05]!)

Final Comment on Improving INW • Perhaps instead on reducing the loss per level we should reduce the number of levels? • This means that at each level the number of pseudorandom strings we have should increase more rapidly (e.g., quadraticaly). • Specific approach based on ideas from Cryptography (constructions of PRFs based on PR Synthesizers [NR]), more complicated to apply here.

Its all About Graph Connectivity • Directed Connectivity captures NL • Undirected Connectivity is in L [R04]. • Oblivious derandomization: pseudo-converging walks for consistently labelled regular digraphs [R04,RTV05] • Where is RL on this scale? • Connectivityfor digraphs w/polynomial mixing time [RTV05] • Outgoing edges have labels. • Consistent labelling means that each label forms a permutation on vertices • A walk on consistently labelled graph cannot lose entropy

in L Suffice toprove RL=L Towards RL vs. L? Connectivity for undirected graphs [R04] Connectivityfor regular digraphs [RTV05] • It is not about reversibility but about regularity • In fact it is about having estimates on stationary probabilities [CRV07] Pseudo-converging walks for consistently-labelled, regular digraphs [R04, RTV05] Pseudo-converging walksfor regular digraphs [RTV05] Connectivityfor digraphs w/polynomial mixing time [RTV05] RL

Some More Open Problems • Pseudo-converging walks on an (inconsistently labelled) clique. (Similarly, universal traversal sequence). • Undirected Dirichlet Problem: • Input: undirected graph G, a vertex s, a set B of vertices, a function f: B → [0, 1]. • Output: estimation of f(b) where b is the entry point of the random walk into B.

Conclusions • Richness of research directions and open problems towards RL=L and beyond: • PRGs against space bounded computations • Directed connectivity. Even if you think that NL=L is plain crazy, many interesting questions and some beautiful research …

Widescreen Test Pattern (16:9) Aspect Ratio Test (Should appear circular) 4x3 16x9

Randomness Vs. Memory: Prospects and Barriers