NC (complexity)

Unsolved problems in computer science

Is NC = P ?

In complexity theory, the class NC (for "Nick's Class") is the set of decision problems decidable in polylogarithmic time on a parallel computer with a polynomial number of processors. In other words, a problem is in NC if there exist constants c and k such that it can be solved in time O(log^c n) using O(n^k) parallel processors. Stephen Cook coined the name "Nick's class" after Nick Pippenger, who had done extensive research on circuits with polylogarithmic depth and polynomial size.

Just as the class P can be thought of as the tractable problems, so NC can be thought of as the problems that can be efficiently solved on a parallel computer. NC is a subset of P because polylogarithmic parallel computations can be simulated by polynomial-time sequential ones. It is unknown whether NC = P, but most researchers suspect this to be false, meaning that there are probably some tractable problems which are "inherently sequential" and cannot significantly be sped up by using parallelism. Just as the class NP-Complete can be thought of as "probably intractable", so the class P-Complete, when using NC reductions, can be thought of as "probably not parallelizable" or "probably inherently sequential".

The parallel computer in the definition can be assumed to be a parallel, random-access machine (PRAM). That is a parallel computer with a central pool of memory, and any processor can access any bit of memory in constant time. The definition of NC is not affected by the choice of how the PRAM handles simultaneous access to a single bit by more than one processor. It can be CRCW, CREW, or EREW. See PRAM for descriptions of those models.

Equivalently, NC can be defined as those decision problems decidable by a uniform Boolean circuit (which can be calculated from the length of the input) with polylogarithmic depth and a polynomial number of gates.

The NC Hierarchy

NCⁱ is the class of decision problems decidable by uniform boolean circuits with a polynomial number of gates and depth O(logⁱ n), or the class of decision problems solvable in time O(logⁱ n) on a parallel computer with a polynomial number of processors. Clearly, we have

$\textbf{NC}^1 \subseteq \textbf{NC}^2 \subseteq \cdots \subseteq \textbf{NC}^i \subseteq \cdots \textbf{NC}$

which forms the NC-hierarchy.

We can relate the NC classes to the space classes L and NL. From Papadimitriou 1994, Theorem 16.1:

$\mathbf{NC}^1 \subseteq \mathbf{L} \subseteq \mathbf{NL} \subseteq \mathbf{NC}^2 \subseteq \mathbf{P}.$

Similarly, we have that NC is equivalent to the problems solvable on an alternating Turing machine with O(log n) space and (log n)^O(1) alternations.^[1]

Open problem: Is NC proper?

One major open question in complexity theory is whether or not every containment in the NC hierarchy is proper. It was observed by Papadimitriou that, if NCⁱ = NCⁱ⁺¹ for some i, then NCⁱ = NC^j for all j ≥ i, and as a result, NCⁱ = NC. This observation is known as NC-hierarchy collapse because even a single equality in the chain of containments

$\textbf{NC}^1 \subseteq \textbf{NC}^2 \subseteq \cdots$

implies that the entire NC hierarchy "collapses" down to some level i. Thus, there are 2 possibilities:

1. $\textbf{NC}^1 \subset \cdots \subset \textbf{NC}^i \subset ... \subset \textbf{NC}^{i+j} \subset \cdots \textbf{NC}$
2. $\textbf{NC}^1 \subset \cdots \subset \textbf{NC}^i = ... = \textbf{NC}^{i+j} = \cdots \textbf{NC}$

It is widely believed that (1) is the case, although no proof as to the truth of either statement has yet been discovered.

Barrington's theorem

A branching program with n variables of width k and length m consists of a sequence of m instructions. Each of the instructions is a tuple (i, p, q) where i is the index of variable to check (1 ≤ i ≤ n), and p and q are functions from {1,2,...,k} to {1,2,...,k}. The program initially starts in state 1, and each instruction (i, p, q) changes the state from x to p(x) or q(x), depending on whether i-th variable is 0 or 1.

A family of branching programs consists of a branching program with n variables for each n.

It's easy to show that every language L on {0,1} can be decided using a family of branching programs of width 4 and exponential length, or using a family of exponential width and linear length.

Every regular language on {0,1} can be recognized with a family of branching programs of constant width and linear number of instructions (since a DFA can be converted to a branching program).

Barrington's theorem^[2] says that the class of languages recognized with a family of branching programs of width 5 and polynomial length is exactly nonuniform NC¹. The proof uses nonsolvability of symmetric group S₅.

The theorem is rather surprising. It implies that majority function can be computed with a family of branching programs of constant width and polynomial size, while intuition might suggest that to achieve polynomial size, one needs linear number of states.

Proof of Barrington's theorem

A branching program of constant width and polynomial size can be easily converted (via divide-and-conquer) to a circuit in NC¹.

Conversely, suppose a circuit in NC¹ is given. Without loss of generality, assume it uses only AND and NOT gates.

Lemma 1: If there exists a branching program that sometimes works as permutation P and sometimes as Q, by right-multiplying permutations in first instruction by α, and in last instruction left-multiplying by β, we can make a circuit of the same length that behaves as βPα or βQα respectively.

Call a branching program α-computing a circuit C if it works as identity when C's output is 0, and as α when C's output is 1.

As a consequence of lemma 1 and the fact that all cycles of length 5 are conjugate, for any two 5-cycles α,β if there exists a branching program α-computing a circuit C, then there exists a branching program β-computing the circuit C, of the same length.

Lemma 2: There exists 5-cycles γ,δ such that their commutator $\gamma \delta \gamma^{-1} \delta^{-1} = \epsilon$ is a 5-cycle. For example, γ = (12345), δ = (13542).

We will now prove Barrington's theorem by induction.

Assume that for all subcircuits D of C and 5-cycles α, there exists a branching program α-computing C. We will show that for all 5-cycles α, there exists a branching program α-computing C.

• If the circuit outputs x_i, the branching program has one instruction checking x_i and outputting identity or α respectively.
• If the circuit outputs $\neg C$, where C is a different circuit. Create a branching program α ^{− 1}-computing C, and multiply output of the program (using lemma 1) by α to get a branching program outputting id or α, i.e. α-computing $\neg C$.
• If the circuit outputs $C \wedge D$, join the branching programs that δ ^{− 1}-compute D, γ ^{− 1}-compute C,δ-compute D, γ-compute C. If one of the circuits outputs 0, the resulting program will be identity; if both circuits output 1, the resulting program will work as $\epsilon$. In other words, we get a program $\epsilon$-computing $C \wedge D$. Because $\epsilon$ and α are two 5-cycles, they are conjugate, and there exists a program α-computing $C \wedge D$.

The size of the branching program is at most 4^d, where d is the depth of the circuit. If the circuit has logarithmic depth, the branching program has polynomial length.

References

1. ^ S. Bellantoni and I. Oitavem (2004). "Separating NC along the delta axis". Theoretical Computer Science 318: 57–78. 
2. ^ D. A. Barrington. Bounded-Width Polynomial-Size Branching Programs Recognize Exactly Those Languages in NC¹

• Greenlaw, Raymond, James Hoover, and Walter Ruzzo. Limits To Parallel computation; P-Completeness Theory. ISBN 0-19-508591-4
• Heribert Vollmer. Introduction to Circuit Complexity -- A Uniform Approach. ISBN 3-540-64310-9
• Christos Papadimitriou (1993). Computational Complexity (1st edition ed.). Addison Wesley. ISBN 0-201-53082-1.  Section 15.3: The class NC, pp.375–381.
• Dexter Kozen (2006). Theory of Computation. Springer. ISBN 1-84628-297-7.  Lecture 12: Relation of NC to Time-Space Classes