Complete Problems In Semantic Complexity Classes: Do They Exist?

The Concept of Completeness

A central concept in computational complexity theory is that of problem completeness. A problem is said to be complete for a particular complexity class if it is among the hardest or most expressive problems within that class. More formally, a problem P is complete for a complexity class C if:

• P is in C
• Any problem in C can be reduced to P in polynomial time

Thus, complete problems capture the full difficulty or expressiveness of a complexity class. They are the problems to which all others in the class can be mapped. Studying complete problems is key to understanding the nature of complexity classes.

The most famous examples are NP-complete problems like Boolean satisfiability. If any NP-complete problem can be solved quickly, then all problems in NP can be as well. However, despite decades of research, no efficient solutions are known for NP-complete problems, strongly suggesting that P ≠ NP.

Complete Problems in P

The class P consists of decision problems that can be solved in polynomial time on a deterministic Turing machine. Intuitively, P represents the set of “efficiently solvable” or “tractable” problems.

Some canonical complete problems for P include:

• Circuit Value Problem: Given a Boolean circuit comprised of AND, OR, and NOT gates and an input assignment, determine if the circuit outputs 1.
• Linear Programming: Given a set of linear inequalities over real variables and an objective function, determine if there is a variable assignment that satisfies the inequalities and optimizes the objective.
• Shortest Path: Given a weighted graph and two nodes, determine the shortest path between the nodes.

All problems in P can be reduced to any of these complete problems in polynomial time. There are many other natural P-complete problems capturing tasks like network flow optimization, string pattern matching, and more. The existence of these reductions formally proves that all problems in P have efficient solutions.

Complete Problems in NP

The defining complete problems for NP are the NP-complete problems like 3SAT, vertex cover, and thousands more. To be NP-complete, a problem X must satisfy:

• X is in NP (solutions for X can be verified in polynomial time)
• Any problem in NP can be polynomial-time reduced to X

It is widely believed (though not proved) that no NP-complete problem has a polynomial time algorithm. Resolving this question – determining whether P = NP or P ≠ NP – stands as one of the great open challenges in computer science and mathematics.

The NP-complete problems have natural definitions but seem inherently inefficient to solve exactly. For example, given a Boolean formula like (x AND y) OR (NOT z), determining whether there exists an assignment to the variables that makes the formula evaluate to TRUE appears to require laborious search through all possibilities.

Open Questions Around NP-Completeness

Despite the definition of NP-completeness being established for decades, open questions remain around the concept. Some key open areas of research include:

• Are there many intermediate levels of complexity between P and the NP-complete problems? For instance, are there natural problems demonstrably harder than Linear Programming but easier than 3SAT? Some problems like Graph Isomorphism seem to occupy this gray area, but there are relatively few known examples.
• Is P = NP? Or failing that, can any NP-complete problem be solved in slightly super-polynomial time like O(n^{log n})?
• Can interactive or quantum proofs demonstrate P = NP? These models allow a hypothetical prover additional power to convince a verifier.
• Are there useful alternate characterizations of complexity besides NP? Concepts like fixed parameter tractability, approximation, and space-bounded complexity provide additional ways of quantifying problem difficulty.

So while the NP-complete problems have a robust theoretical characterization, much work remains exploring their limitations and connections to practical computation.

Complete Problems in PSPACE

The complexity class PSPACE contains decision problems that can be solved with a polynomial amount of memory (but perhaps exponential time). Complete problems for PSPACE have the remarkable property that any problem solvable with a sane amount of memory reduces to them.

Some canonical PSPACE-complete problems are:

• Generalized Geography: Given an initial configuration of a generalized geography-style game (players in groups occupying marked territories on a graph), determine whether the players can reach a position that fulfills a certain end condition using the allowed moves.
• Boolean Formula Games: Two players take turns assigning TRUE/FALSE values to the variables of a Boolean formula to make the overall formula evaluate to TRUE or FALSE. Determine which player (if any) has a winning strategy from a given configuration.

These PSPACE-complete problems formally express the incredible difficulty of problems involving strategy, knowledge, and perfect information. Solving them optimally may sometimes require considering an entire game tree or logical system. Just as NP-complete emphasizes the probable difficulty of search, PSPACE-complete emphasizes the inherent difficulty of planning.

The Exponential Hierarchy and Complete Problems

Beyond polynomial time, complexity classes can have complete problems that express differing types and quantities of computational resources. An important example is the exponential hierarchy EH.

For each nonnegative integer k, the complexity class Σ_kP in the exponential hierarchy contains decision problems solvable by a nondeterministic Turing machines in O(n^k) time. Π_kP is similar but refers to co-nondeterministic machines (accepting inputs with no computational path leading to a reject state).

Some complete problems for classes in EH include:

• Σ₁P-complete: Succinct Set Cover – Given a small explicit universe U and a collection of subsets of U described implicitly by a Boolean circuit, determine if there exists k subsets that exactly cover U.
• Π₂P-complete: Polynomially Bounded Tiling – Given a set of tile types and rules for allowed neighboring tile types, determine if a width n grid can be fully tiled without any tile type appearing more than O(n^k) times for some constant k.

These give precise characterizations of completeness for higher complexity classes using quantified Boolean formulas and intricate tiling rules.

Complete Problems in EXPTIME

At an even more rarefied level of complexity lie the EXPTIME-complete problems. EXPTIME consists of problems solvable by a deterministic Turing machines in O(2ⁿ) time or exponential space. Canonical EXPTIME-complete problems include:

• Deterministic Generalized Geography: This two-player game version requires determining which player has a winning strategy given perfect information.
• Succinct Chess: Determine which player can guarantee a win in an n x n chess game encoded on a small circuit board.

These stunningly difficult problems essentially require computing optimal solutions over game trees or combinatorial search spaces with an exponential number of configurations. Some researchers believe that the class EXPTIME may contain fundamental barriers to computational power—that this level of complexity transcends even nondeterminism as a practical resource. Understanding EXPTIME-complete problems is thus crucial to exploring the limits of efficient computation itself.

The Relationship Between Completeness and Complexity

The existence of complete problems gets at the heart of the complexity class definitions. Completeness notions formally prove that entire classes of problems are fundamentally equivalent with respect to efficiency. Hence, by showing that one natural problem essentially encodes what a whole class can do, complete problems demonstrate robust lower bounds on the fastest algorithms possible within a class.

Complete problems also illuminate how adding computational resources like nondeterminism (going from P to NP) or memory (going from NP to PSPACE) increases expressiveness. The complete problems for the higher classes harness those resources in a maximally hard way. Determining whether those same complete problems can be solved with fewer resources relates directly to major open questions like P vs NP.

In summary, complete problems elucidate both the inherent complexity and relationships between classes. Their study is thus integral, not peripheral, to the field of computational complexity theory itself.

Practical Implications of Intractable Completeness

The theory of NP-, PSPACE-, and EXPTIME-completeness has direct implications for the algorithms and applications that can or cannot exist in practice. Knowing that thousands of natural problems are likely inherently inefficient to solve exactly pinpoints where researchers should concentrate efforts.

Some examples include:

• Not attempting vainly to find polynomial-time solutions for NP-complete problems.
• Instead, focusing on approximation algorithms or heuristics that find near-optimal solutions.
• Restricting to specialized cases of problem instances that avoid worst-case completeness.
• Using complete problems themselves as cryptographic primitives based on their conjectured hardness.

Thus, while completeness results dash hopes for universally efficient algorithms, they properly direct practical efforts into feasible channels. The unyielding barriers exposed by complete problems liberate researchers to explore realistic, tangent directions rather than wasting time on likely impossibilities.

Future Research Directions

Some promising avenues for further research into complete semantic complexity problems include:

• Continuing the classification program for natural problems across different complexity classes.
• Generalizing completeness definitions to finer-grained, restricted models like fixed-parameter tractability and approximation.
• Exploring completeness under alternate computational models like quantum computing.
• Matching canonical complete problems to real-world applications in domains like scheduling, genetics, and economics.
• Harnessing complete problems with strong cryptographic properties for secure protocols.
• Connecting semantic classes like NL and L to traditional complexity classes via complete problems.

While longstanding completions notions have crystallized difficulty across broad classes, much potential exists to sharpen boundaries further for both theory and application. Pursuing complete problems in semantic complexity therefore remains an exciting frontier after over 50 years of foundational progress. The barriers illuminated by completions help calibrate both solvability expectations and research directions moving forward across computing.