Technical Perspective: Patterns Hidden from Simple Algorithms

Is the number 9021960864034418159813 random? Educated opinions might vary from "No! No single string can be random," to the more contemptuous "Come on! Those are just the 714th to 733rd digits of π." Yet, to my limited mind, the string did appear random. Is there a way to use some formal mathematics to justify my naïveté? The modern theory of pseudorandomness^2,5 indeed manages to explain such phenomena, where strings appear random to simple minds. The key, this theory argues, is that randomness is really in the "eyes of the beholder," or rather in the computing power of the tester of randomness. More things appear random to simpler, or resource-limited, algorithms than to complex, powerful, algorithms.

Why should we care? Because randomness is a key resource in computation. Monte Carlo simulations abound in the use of computation for prediction. On the theoretical side too, algorithms get simplified or speeded up incredibly if they use randomness. Fundamental tasks in distributed computing (such as synchronization) can be solved only with randomness. And there is no hope for maintaining privacy and secrecy without randomness. At the same time much of the randomness we deal with in reality is not "pure." They don’t come as a collection of independent random bits, but are generated by other processes. Knowing how an algorithm, or a computational process, works in the presence of somewhat random strings becomes the essence of a recurring question. (For example, should you really trust nuclear power safety calculations made by a Monte-Carlo algorithm using randomness from the C++ rand program?)

Such questions get formalized in the theory of pseudo-randomness as follows: It considers some source of randomness X (formally given by some probability distribution over n-bit strings), and a class of Boolean algorithms A (algorithms that decide Boolean questions) and asks if some algorithm in A behaves very differently given access to a random string generated by X, than it does on pure random strings? If every algorithm in A behaves roughly the same on X as on pure randomness, we say X is pseudo-random to A. In the example here, X = X_π may be the source that picks a random integer i between 1 and 10000 and outputs π [i + 1];…;π [i + 20] where π [j] denotes the jth digit of π. Given some class of algorithms A one could now ask, does X_π look pseudo-random to A?

Unfortunately answering such questions leads to a fundamental challenge in the theory of computing. Proving, say, that X_π looks random to A involves showing that no algorithm in A can detect patterns shown by the digits of π. And proving that some class of algorithms can’t do some task is a major challenge to theoretical computer science (the most notorious question being "Is P = NP?").

Given such major obstacles to analyzing the seeming randomness in strings, it is no surprise that the study of pseudo-randomness remains in its infancy. The following paper by Mark Braverman sheds new light on this field. It illustrates that a broad class of sources, as long as they have sufficient local randomness, fool a broad, but relatively simple, class of algorithms—those that compute "AC⁰ functions." AC⁰ functions are those whose input-output behavior can be described by a Boolean logic circuit consisting n input wires and polynomially many NOT gates and AND and OR gates of arbitrary fan-in. AC⁰ functions—that correspond to tasks computable in constant time on highly parallel machines—are at the forefront of the class of computational tasks that we do seem to understand. For instance, AC⁰ functions can compute the sum of two n/2-bit integers, but not their product (and so we do know things they can’t do). In fact, we even knew a explicit source of relatively small entropy that appeared pseudo-random to this class.⁴ Yet our understanding of the essence of what sources fooled this class of algorithms was unknown, leading Linial and Nisan³ to conjecture that a certain form of "local randomness" was sufficient to seem random to this class of algorithms. The local randomness they pointed to is widely termed "limited independence." A source X is said to be k-wise independent if any particular k bits of the random source are totally random and independent. Linial and Nisan conjectured that for every constant depth circuit, some (log n)^O(1) -wise independence would look like pure n bits of randomness. For over two decades this conjecture remained unresolved. Only recently, Bazzi¹ resolved a special case of the conjecture (for the case of AC⁰ functions corresponding to depth 2 circuits, or "DNF formulae").

Now Braverman’s work resolves the conjecture affirmatively. In the process he reveals novel, elegant ways in which AC⁰ functions can be described by low-degree multivariate polynomials, showing some of the interplay between Boolean computation and more classical mathematics. We need to see many more such connections before we can hope to address the wide challenges of computational complexity (and P vs. NP). Indeed even to address the question implied by the opening paragraph "Are the digits of π pseudo-random to AC⁰ functions?," one needs to understand much more. Fortunately, Braverman’s work may have reduced this question to a purely number-theoretic one: Are the digits of π locally random?

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