Understanding Problems Outside of NP: NP-hard vs. Non-NP-hard

It is a common misconception that every problem outside of NP (Nondeterministic Polynomial time) is NP-hard. However, this is not the case. Not all problems that are outside of NP are NP-hard, as there are distinct categories of problems that lie outside these complexity classes. Let's delve into the classification of problems outside of NP and explore why not all of them are NP-hard.

Definitions: Classes of Problems in Computational Theory

Before we proceed, it's crucial to define the classes of problems discussed here:

NP: Nondeterministic Polynomial time - The class of decision problems where a given solution can be verified in polynomial time. NP-hard - A class of problems that are at least as hard as the hardest problems in NP. If an NP-hard problem can be solved in polynomial time, then all problems in NP can also be solved in polynomial time. PSPACE - The class of decision problems that can be solved using a polynomial amount of space but not necessarily in polynomial time. EXPTIME-complete - Problems that can be solved in exponential time but are not necessarily verifiable in polynomial time. NEXP-complete - Non-deterministic Exponential time complete problems, which are a subset of NP problems that require exponential time to solve and are not in NP but are in NEXP (Non-deterministic Exponential Time).

Problems Outside of NP that are not NP-hard

Not every problem outside of NP is NP-hard. Some problems fall into categories where they are neither NP nor NP-hard. Here, we will explore two examples of such problems:

EXPTIME-complete Problems

EXPTIME-complete problems are a class of decision problems that can be solved in exponential time. These problems are not necessarily verifiable in polynomial time. For instance, certain games and decision problems that require exponential time to solve fall into this category. EXPTIME-complete problems are a subset of problems that are harder than those in NP but cannot be categorized as NP-hard due to their inability to be reduced to NP problems.

The Halting Problem

The Halting Problem is a classic example of a problem that is outside of NP and not NP-hard. It is undecidable, meaning no algorithm can deterministically solve it for all possible program-input pairs. While it does not belong to NP because solutions cannot be verified in polynomial time, it also doesn't fall into the category of NP-hard problems because it cannot be reduced to an NP problem. This makes it a unique case that exists outside of both NP and NP-hard categories.

Some Problems in PSPACE

Problems in PSPACE are those that can be solved using a polynomial amount of space but not in polynomial time. While PSPACE is a superset of NP, not all problems in PSPACE are in NP. Some PSPACE problems, by virtue of their space complexity, do not belong to NP and thus are not NP-hard. This highlights the distinction between NP and PSPACE, where PSPACE encompasses a broader range of problems including those with high space requirements, which are often more complex than NP problems.

NP-hard but not in NP

For problems that are NP-hard but not in NP, one can look at NEXP-complete problems. These problems are a subset of NP that extend beyond the polynomial time solvability of NP problems into the realm of exponential time. An example of an NEXP-complete problem is the SUCCINCT-SAT:

SUCCINCT-SAT: Given a circuit C over n variables of polynomial size, decide whether the truth table of the circuit, interpreted as a 3-CNF formula, is satisfiable. The formula is of exponential size, hence the succinct representation of the problem.

This problem is NP-hard but not in NP, as the truth table interpretation results in an exponential-size formula that cannot be verified in polynomial time. Therefore, SUCCINCT-SAT is an example of a problem that is NP-hard but not in NP.

Complexity of Deriving a Non-NP Problem in co-NP

As of December 2017, no one has found a problem in co-NP that is definitely not in NP. If such a problem could be found, it would not only provide a problem as requested but also prove that NP is not equal to co-NP. The inequality of these classes remains unproven, contributing to the challenge and intrigue in complexity theory. Experts often describe complexity theory as "really friggin difficult," reflecting the deep and ongoing exploration in understanding computational problems.

For further reading on this topic, see the following resources:

Wikipedia - Nondeterministic computational complexity Wikipedia - co-NP

Understanding the nuances of these complex classes of problems is crucial for advancing our knowledge in computational theory and practical applications in computer science.