Noise threshold for universality of 2-input gates

During the last decades, computers have become faster and faster, mainly due to advances in hardware miniaturization. However, there are physical limits to the possible extent of this miniaturization, and the closer one gets to these limits, the less robust and more error-prone the components become [3], [2]. It is estimated that the time when processor architects face these limitations is within the next decade [4].

Gates, the smallest components of any processor, can fail in (at least) two ways. The first is that they do not work at all. The second is that they work most of the time correctly, and fail sometimes. This type of errors is called “soft errors” by hardware engineers. We deal with faults of the second type.

In particular, we consider the computational model of noisy formulas. Formulas are a special kind of circuits in which each gate has exactly one output wire 1 . We ask how much noise on the gates is tolerable, such that any function can still be computed by some formula with bounded-error. We will assume throughout that gates fail independently of each other.

This question has been studied earlier. Already in 1956 von Neumann discovered that reliable computation is possible with noisy 3-majority gates if each F. Unger works at the Centrum voor Wiskunde en Informatica in Amsterdam, The Netherlands. (Falk.Unger@cwi.nl) 1 Precise definitions for all terms used can be found in Section II.

gate fails independently with probability less than 0.0073 [12]. The first to prove an upper bound on the tolerable noise was Pippenger [11]. He proved that formulas with gates of fan-in at most k, where each gate fails independently with probability at least ǫ ≥ 1 2 -1 2k , are not sufficient for universal computation (i.e. not all functions can be computed with bounded error). Feder proved that this bound also applies to circuits [8]. Later, Feder’s bound was improved to 1 2 -1 2 √ k by Evans and Schulman [6]. For formulas with gates of fan-in k and k odd, Evans and Schulman [7] proved the tight bound

. Tight here means that if all gates fail independently with the same fixed probability ǫ < β k , then any function can be bounded-error computed, and if each gate fails with some probability at least β k (which does not need to be the same for all gates), universal computation is not possible. For k = 3 the threshold was first established by Hajek and Weller [9]. However, so far it has not been possible to establish thresholds for gates with even fan-in (or even prove their existence), as pointed out in [7]. In particular, the most basic case of fan-in 2, which is most commonly used, had been elusive. An intuitive argument why even fan-in is different is that for even fan-in threshold gates (and in particular majority gates) can never be “balanced”, in the sense that the number of inputs on which they are 1 cannot be the same as the number of inputs on which they are 0.

Evans and Pippenger [5] made some progress in this direction. First, they show that all functions can be computed by formulas with noisy NANDgates with fan-in 2, if each NAND-gate fails with probability exactly ǫ, for any 0 ≤ ǫ < β 2 = 3- Theorem 1: Assume ∆ > 0. Functions that are computable with bias ∆ by a formula in which all gates have fan-in at most 2 and fail independently with probability at least β 2 = (3 -√ 7)/4, depend on at most a constant number of input bits.

Together with the first mentioned result from [5] this gives the exact threshold for formulas with gates of fan-in 2. It extends the second result from [5] in the following ways: (1) We allow all gates of fan-in 2, instead of only NAND-gates. (2) We prove that if the noise is exactly β 2 , then no universal boundederror computation is possible. (3) In contrast to our result, the upper bound in [5] only applies to “soft” inputs. They show that gates with noise more than β 2 cannot increase the bias. More precisely, if the inputs to the formula are noisy themselves and have bias at most ∆ > 0, then the output of the formula cannot have larger bias than ∆. This left open the case where the input bits are not noisy and either 0 or 1, which is the case we care about most. Our argument shows that even with perfect inputs faulttolerant computation is not possible for noise at least β 2 .

To prove Theorem 1 we introduce a new technique, which is also applicable in the case of fan-in 2. We expect that it can be extended to other fan-in cases.

We conjecture that our bound also holds for circuits.

For any function f : {0, 1} n → {0, 1} we will choose an input bit x i which f depends on, and fix all other bits such that f still depends on x i . Assume that f is computed by a formula F with noisy gates that fail independently with probability at least β 2 . Then, for each gate in the formula F with input wires A and B and output wire C we can define a = 1

and analogously for B and C. The variable a can be seen as the average probability of A being 0. We call δ a the bias of A wi

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