# NL (complexity)

Open problem in computer science: |

In computational complexity theory, **NL** (Nondeterministic Logarithmic-space) is the complexity class containing decision problems which can be solved by a nondeterministic Turing machine using a logarithmic amount of memory space.

**NL** is a generalization of **L**, the class for logspace problems on a deterministic Turing machine. Since any deterministic Turing machine is also a nondeterministic Turing machine, we have that **L** is contained in **NL**.

**NL** can be formally defined in terms of the computational resource nondeterministic space (or NSPACE) as **NL** = **NSPACE**(log *n*).

Important results in complexity theory allow us to relate this complexity class with other classes, telling us about the relative power of the resources involved. Results in the field of algorithms, on the other hand, tell us which problems can be solved with this resource. Like much of complexity theory, many important questions about **NL** are still open (see Unsolved problems in computer science).

Occasionally **NL** is referred to as **RL** due to its probabilistic definition below; however, this name is more frequently used to refer to randomized logarithmic space, which is not known to equal **NL**.

## Contents

## NL-complete problems

Several problems are known to be **NL-complete** under log-space reductions, including ST-connectivity and 2-satisfiability. ST-connectivity asks for nodes *S* and *T* in a directed graph whether *T* is reachable from *S*. 2-satisfiability asks, given a formula of which each clause is the disjunction of two literals, if there is a variable assignment that makes the formula true. An example instance, where indicates *not*, might be:

## Containments

It is known that **NL** is contained in **P**, since there is a polynomial-time algorithm for 2-satisfiability, but it is not known whether **NL** = **P** or whether **L** = **NL**. It is known that **NL** = **co-NL**, where **co-NL** is the class of languages whose complements are in **NL**. This result was independently discovered by Neil Immerman and Róbert Szelepcsényi in 1987 (Immerman-Szelepcsényi Theorem), who received the 1995 Gödel Prize for this work.

In circuit complexity, **NL** can be placed within the **NC** hierarchy. In Papadimitriou 1994, Theorem 16.1, we have:

More precisely, **NL** is contained in AC^{1}. It is known that **NL** is equal to **ZPL**, the class of problems solvable by randomized algorithms in logarithmic space and unbounded time, with no error. It is not, however, known or believed to be equal to **RLP** or **ZPLP**, the polynomial-time restrictions of **RL** and **ZPL** which some authors refer to as **RL** and **ZPL**.

We can relate **NL** to deterministic space using Savitch's theorem, which tells us that any nondeterministic algorithm can be simulated by a deterministic machine in at most quadratically more space. From Savitch's theorem, we have directly that:

This was the strongest deterministic-space inclusion known as of 1994^{[update]} (Papadimitriou 1994 Problem 16.4.10, "Symmetric space"). Since larger space classes are not affected by quadratic increases, the nondeterministic and deterministic classes are known to be equal, so that for example we have **PSPACE** = **NPSPACE**.

## Probabilistic definition

Suppose *C* is the complexity class of decision problems solvable in logarithmithic space with probabilistic Turing machines that never accept incorrectly but are allowed to reject incorrectly less than 1/3 of the time; this is called *one-sided error*. The constant 1/3 is arbitrary; any *x* with 0 ≤ *x* < 1/2 would suffice.

It turns out that *C* = **NL**. Notice that *C*, unlike its deterministic counterpart **L**, is not limited to polynomial time, because although it has a polynomial number of configurations it can use randomness to escape an infinite loop. If we do limit it to polynomial time, we get the class **RL**, which is contained in but not known or believed to equal **NL**.

There is a simple algorithm which establishes that *C* = **NL**. Clearly *C* is contained in **NL**, since:

- If the string is not in the language, both reject along all computation paths.
- If the string is in the language, an
**NL**algorithm accepts along at least one computation path and a*C*algorithm accepts along at least two-thirds of its computation paths.

To show that **NL** is contained in *C*, we simply take an **NL** algorithm and choose a random computation path of length *n*, and do this 2^{n} times. Because no computation path exceeds length *n*, and because there are 2^{n} computation paths in all, we have a good chance of hitting the accepting one (bounded below by a constant).

The only problem is that we don't have room in log space for a binary counter that goes up to 2^{n}. To get around this we replace it with a *randomized* counter, which simply flips *n* coins and stops and rejects if they all land on heads. Since this event has probability 2^{-n}, we expect to take 2^{n} steps on average before stopping. It only needs to keep a running total of the number of heads in a row it sees, which it can count in log space.

Because of the Immerman–Szelepcsényi theorem, according to which NL is closed under complements, the one-sided error in these probabilistic computations can be replaced by zero-sided error. That is, these problems can be solved by probabilistic Turing machines that use logarithmic space and never make errors. The corresponding complexity class that also requires the machine to use only polynomial time is called ZPLP.

Thus, when we only look at space alone, it seems that randomization and nondeterminism are equally powerful.

## Descriptive complexity

There is a simple logical characterization of **NL**: it contains precisely those languages expressible in first-order logic with an added transitive closure operator.

## References

*Complexity Zoo*: NL- Papadimitriou, C. (1994). "Chapter 16: Logarithmic Space".
*Computational Complexity*. Addison-Wesley. ISBN 0-201-53082-1. - Michael Sipser (1997). "Sections 8.4–8.6: The Classes L and NL, NL-completeness, NL equals coNL".
*Introduction to the Theory of Computation*. PWS Publishing. pp. 294–302. ISBN 0-534-94728-X. - Introduction to Complexity Theory: Lecture 7. Oded Goldreich. Proposition 6.1. Our
*C*is what Goldreich calls badRSPACE(log n).