# NL-complete

In computational complexity theory, **NL-complete** is a complexity class containing the languages that are complete for **NL**, the class of decision problems that can be solved by a nondeterministic Turing machine using a logarithmic amount of memory space. The NL-complete languages are the most "difficult" or "expressive" problems in **NL**. If a method exists for solving any one of the NL-complete problems in logarithmic memory space, then **NL** = **L**.

## Definitions

**NL** consists of the decision problems that can be solved by a nondeterministic Turing machine with a read-only input tape and a separate read-write tape whose size is limited to be proportional to the logarithm of the input length. Similarly, **L** consists of the languages that can be solved by a deterministic Turing machine with the same assumptions about tape length. Because there are only a polynomial number of distinct configurations of these machines, both **L** and **NL** are subsets of the class **P** of deterministic polynomial-time decision problems.

Formally, a decision problem is NL-complete when it belongs to **NL**, and has the additional property that every other decision problem in **NL** can be reduced to it. Unless otherwise specified, the reductions in this definition are assumed to be many-one reductions by a deterministic logarithmic-space algorithm.

## Properties

If an NL-complete language *X* could belong to **L**, then so would every other language *Y* in **NL**. For, suppose (by NL-completeness) that there existed a deterministic logspace reduction *r* that maps an instance *y* of problem *Y* to an instance *x* of problem *X*, and also (by the assumption that *X* is in **L**) that there exists a deterministic logspace algorithm *A* for solving problem *X*. With these assumptions, a problem *y* in language Y could be solved in logarithmic space by an algorithm that simulates the behavior of algorithm *A* on input *r*(*y*), using the reduction algorithm to simulate each access to the read-only tape for *r*(*y*).

It follows from the Immerman–Szelepcsényi theorem that, if a language is co-NL-complete (that is, if its complement is NL-complete) then the language is also NL-complete itself.

## Examples

One important NL-complete problem is ST-connectivity (or "Reachability") (Papadimitriou 1994 Thrm. 16.2), the problem of determining whether, given a directed graph *G* and two nodes *s* and *t* on that graph, there is a path from *s* to *t*. ST-connectivity can be seen to be in **NL**, because we start at the node *s* and nondeterministically walk to every other reachable node. ST-connectivity can be seen to be NL-hard by considering the computation state graph of any other **NL** algorithm, and considering that the other algorithm will accept if and only if there is a (nondetermistic) path from the starting state to an accepting state.

Another important NL-complete problem is 2-satisfiability (Papadimitriou 1994 Thrm. 16.3), the problem of determining whether a boolean formula in conjunctive normal form with two variables per clause is satisfiable.

The problem of unique decipherability of a given variable-length code was shown to be co-NL-complete by Rytter (1986); Rytter used a variant of the Sardinas–Patterson algorithm to show that the complementary problem, finding a string that has multiple ambiguous decodings, belongs to **NL**. Because of the Immerman–Szelepcsényi theorem, it follows that unique decipherability is also NL-complete.

Additional NL-complete problems on Propositional Logic, Algebra, Linear System, Graph, Finite Automata, Context-free Grammar are listed in Jones (1976).

## References

- Papadimitriou, Christos H. (1994).
*Computational Complexity*. Reading, Massachusetts: Addison-Wesley. ISBN 0-201-53082-1. - Rytter, Wojciech (1986), "The space complexity of the unique decipherability problem",
*Information Processing Letters*,**23**(1): 1–3, doi:10.1016/0020-0190(86)90121-3, MR 0853618. - Jones, Neil D.; Lien, Y. Edmund; Laaser, William T (1976). "New problems complete for nondeterministic log space".
*Mathematical systems theory*.**10**(1): 1–17. doi:10.1007/BF01683259.