On the Computational Power of Radio Channels

Radio networks can be a challenging platform for which to develop distributed algorithms, because the network nodes must contend for a shared channel. In some cases, though, the shared medium is an advantage rather than a disadvantage: for example, many radio network algorithms cleverly use the shared channel to approximate the degree of a node, or estimate the contention. In this paper we ask how far the inherent power of a shared radio channel goes, and whether it can efficiently compute "classicaly hard" functions such as Majority, Approximate Sum, and Parity. Using techniques from circuit complexity, we show that in many cases, the answer is "no". We show that simple radio channels, such as the beeping model or the channel with collision-detection, can be approximated by a low-degree polynomial, which makes them subject to known lower bounds on functions such as Parity and Majority; we obtain round lower bounds of the form Omega(n^{delta}) on these functions, for delta in (0,1). Next, we use the technique of random restrictions, used to prove AC^0 lower bounds, to prove a tight lower bound of Omega(1/epsilon^2) on computing a (1 +/- epsilon)-approximation to the sum of the nodes\u27 inputs. Our techniques are general, and apply to many types of radio channels studied in the literature

Two Sides of the Coin Problem

In the coin problem, one is given n independent flips of a coin that has bias b > 0 towards either Head or Tail. The goal is to decide which side the coin is biased towards, with high confidence. An optimal strategy for solving the coin problem is to apply the majority function on the n samples. This simple strategy works as long as b > c(1/sqrt n) for some constant c. However, computing majority is an impossible task for several natural computational models, such as bounded width read once branching programs and AC^0 circuits. Brody and Verbin proved that a length n, width w read once branching program cannot solve the coin problem for b < O(1/(log n)^w). This result was tightened by Steinberger to O(1/(log n)^(w-2)). The coin problem in the model of AC^0 circuits was first studied by Shaltiel and Viola, and later by Aaronson who proved that a depth d size s Boolean circuit cannot solve the coin problem for b < O(1/(log s)^(d+2)). This work has two contributions: 1. We strengthen Steinberger\u27s result and show that any Santha-Vazirani source with bias b < O(1/(log n)^(w-2)) fools length n, width w read once branching programs. In other words, the strong independence assumption in the coin problem is completely redundant in the model of read once branching programs, assuming the bias remains small. That is, the exact same result holds for a much more general class of sources. 2. We tighten Aaronson\u27s result and show that a depth d, size s Boolean circuit cannot solve the coin problem for b < O(1/(log s)^(d-1)). Moreover, our proof technique is different and we believe that it is simpler and more natural

Smaller ACC0 Circuits for Symmetric Functions

What is the power of constant-depth circuits with $MOD_m$ gates, that can count modulo $m$? Can they efficiently compute MAJORITY and other symmetric functions? When $m$ is a constant prime power, the answer is well understood: Razborov and Smolensky proved in the 1980s that MAJORITY and $MOD_m$ require super-polynomial-size $MOD_q$ circuits, where $q$ is any prime power not dividing $m$. However, relatively little is known about the power of $MOD_m$ circuits for non-prime-power $m$. For example, it is still open whether every problem in $EXP$ can be computed by depth-$3$ circuits of polynomial size and only $MOD_6$ gates. We shed some light on the difficulty of proving lower bounds for $MOD_m$ circuits, by giving new upper bounds. We construct $MOD_m$ circuits computing symmetric functions with non-prime power $m$, with size-depth tradeoffs that beat the longstanding lower bounds for $AC^0[m]$ circuits for prime power $m$. Our size-depth tradeoff circuits have essentially optimal dependence on $m$ and $d$ in the exponent, under a natural circuit complexity hypothesis. For example, we show for every $\varepsilon > 0$ that every symmetric function can be computed with depth-3 $MOD_m$ circuits of $\exp(O(n^{\varepsilon}))$ size, for a constant $m$ depending only on $\varepsilon > 0$. That is, depth-$3$ $CC^0$ circuits can compute any symmetric function in \emph{subexponential} size. This demonstrates a significant difference in the power of depth-$3$ $CC^0$ circuits, compared to other models: for certain symmetric functions, depth-$3$ $AC^0$ circuits require $2^{\Omega(\sqrt{n})}$ size [H{\aa}stad 1986], and depth-$3$ $AC^0[p^k]$ circuits (for fixed prime power $p^k$) require $2^{\Omega(n^{1/6})}$ size [Smolensky 1987]. Even for depth-two $MOD_p \circ MOD_m$ circuits, $2^{\Omega(n)}$ lower bounds were known [Barrington Straubing Th\'erien 1990].Comment: 15 pages; abstract edited to fit arXiv requirement

Advice coins for classical and quantum computation

We study the power of classical and quantum algorithms equipped with nonuniform advice, in the form of a coin whose bias encodes useful information. This question takes on particular importance in the quantum case, due to a surprising result that we prove: a quantum finite automaton with just two states can be sensitive to arbitrarily small changes in a coin’s bias. This contrasts with classical probabilistic finite automata, whose sensitivity to changes in a coin’s bias is bounded by a classic 1970 result of Hellman and Cover. Despite this finding, we are able to bound the power of advice coins for space-bounded classical and quantum computation. We define the classes BPPSPACE/coin and BQPSPACE/coin, of languages decidable by classical and quantum polynomial-space machines with advice coins. Our main theorem is that both classes coincide with PSPACE/poly. Proving this result turns out to require substantial machinery. We use an algorithm due to Neff for finding roots of polynomials in NC; a result from algebraic geometry that lower-bounds the separation of a polynomial’s roots; and a result on fixed-points of superoperators due to Aaronson and Watrous, originally proved in the context of quantum computing with closed timelike curves